WILLIAM N. GILL ROBERT COLE E. JAMES DAVIS JOSEPH ESTRIN RICHARD J. NUNGE HOWARD LITTMAN
Fluid uynamics
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I
Emphasis is on development of basic principles his one-year review covers papers published during the period T from January to December 1968. I n constructing any review which deals with as wide-ranging and difficult a subject as fluid dynamics, a t the outset, one has to determine whether to concentrate o n breadth or depth of coverage. In the present review, we have attempted to strike a balance in this regard. However, since we found it necessary to report approximately 800 references, some preference has been given to the breadth of coverage. I t is difficult to choose those papers to be included in a review. I n general we have tried to restrict coverage to papers which emphasize the development and delineation of basic principles rather than practical applications; such work is more likely to have permanent value. No doubt some very significant papers reIevant to fluid dynamics have not been reported. Single-Phase Laminar Flow of Newtonian Fluids in Channels
Stability. T h e failure of linear stability theory to predict a critical Reynolds number for fully developed tube flow which may, in fact, be unstable experimentally has motivated further investigations of the reasons for this discrepancy. Examined in recent studies are the alternatives that the instability may be due to disturbances which are not axially symmetric or to asymmetries in the fully developed velocity distribution. An azimuthly periodic disturbance in fully developed tube flow was considered experimentally by Fox ed al. ( 7 A ) and theoretically
by Lessen et al. (27A). Unfortunately no linear instabilities were found theoretically for the mode sets with a n azimuthal periodicity of unity, while the experiments indicated instability for this case. An experimentally measured asymmetrical velocity profile for flow between parallel plates was used by Potter and Smith ( 3 0 A ) to investigate the effect of asymmetries on the flow stability. Such a flow was found to have a higher critical Reynolds number than the symmetrical case so that it is unlikely that asymmetries i n the velocity profile can be the cause of the discrepancy between linear stability theory and experiment for tube flow. This same effect of asymmetry in the velocity distribution on the stability was noted by Mott and Joseph (26A)who considered flow in annuli. For deliberately skewed velocity profiles in a n annulus with a fixed radius ratio, the flow was found to become more stable as the maximum velocity shifted closer to either solid boundary. A second interesting result of both papers (26A, 304) dealing with asymmetrical velocity profiles is finding the existence of two minima on the neutral stability curve, either of which may yield the lowest value of the critical Reynolds number depending on the degree of asymmetry. As shown by Mott and Joseph (26A),these two minima have very different Reynolds stress distributions. T h e limiting case of the Reynolds number approaching infinity in plane Couette flow has been examined (29A) and it was concluded that instability is possible only for finite values of the Reynolds number. A contribution to the theory of stability of compressible flows has been reported in (20A). VOL. 6 2
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Rosenblat (35A) has begun a n investigation of the linear stability to axisymmetric disturbances of time-dependent flows by examining the limiting situation of inviscid periodic azimuthal flow in annuli. The meaning of instability in unsteady flows was discussed, and treatments were made for periodic flows with both zero and nonzero means. Flows with a zero mean are generally unstable but for flows with a nonzero mean, the periodic component stabilizes unstable mean flow and destabilizes stable mean flow. A review of existing correlations, taken largely from experimental data, for friction factors and critical Reynolds number in coiled tubes has been presented (47A), and new equations for the critical Reynolds number have been suggested. Isothermal Flow. T h e inlet region of a pipe for laminar flow a t moderate Reynolds numbers where the boundary layer approximation is not valid has been investigated by Friedmann et al. (8A) using a numerical solution of the Navier-Stokes equations. For a uniform inlet velocity the dimensionless momentum entrance xla 2auo length --, where Re = -, a is the tube radius, and u o the inlet Re Y velocity, was found to vary between 0.176 for a Reynolds number of 10 to0.112forRe = 500 (or m ) . The change in the momentum entrance length in a tube caused by positioning several devices with different geometries a t the tube entrance has been investigated experimentally ( 5 A ) with the goal of shortening the entrance length. However, reductions were achieved in only a limited number of cases and deleterious effects were noted in others, so that it was concluded that the approach was inappropriate. Experiments on water flowing through eccentric annuli with an eccentricity ratio of unity have yielded information on the friction factor and transition zone between laminar and turbulent flow in such geometries (3A). Additions to the theory of steady, fully developed laminar flow of incompressible fluids have been made for tubes which are lenticular in cross-section ( 4 A ) ,curved circular tubes (23A),polygonal shaped ducts (32A), porous annuli (42A), and porous flat plate systems (38A). Applications of the results of ( 4 A ) are primarily to open channel flow. McConalogue and Srivastava ( 2 3 A ) have extended Dean's original work on flow in curved tubes to higher values of the Reynolds number i n the laminar regime. However, the assumption that the ratio of the tube radius to the radius of curvature is small is retained in the simplification of the equations of motion. T h e key step in the mathematical analysis of flow in the annular space between regular polygonal shaped ducts and a circular central core by Ratkowsky and Epstein (324) is the fitting of a linear combination of a finite number of harmonic functions to the boundary conditions. Cases ranging from no core to a core touching the outer walls were considered. Results for the inverted geometry are reported in ( 7 4 4 ) . Terrill ( 4 2 A ) has treated flow in a permeable annulus with arbitrary injection or suction a t either wall including the inertial terms, thus clarifying and generalizing a previous analysis of this problem in which these terms were ignored. T h e results presented in (38A) for a porous wall, flat plate system deal with the case of large blowing rates a t both walls. I n a n experimental investigation, the excess pressure drop, the pressure losses above those associated with developed flow, has been measured in a sharp edged contraction connecting two circular tubes of different diameter ( 7 A ) . The results indicate that usual correlations tend to significantly underestimate the excess losses in such systems. Steady incompressible flow in a variable area duct with mass transfer a t the walls including the injection of a foreign gas has been studied (&A) by using a linearization of the Navier-Stokes equations similar to that in the classical work of Oseen. A discussion of the range of applicability of the approximate solutions is given. Experimental determinations of the flow characteristics, such as the friction factor and lower critical Reynolds number as a function of the diameter ratio in an annulus of unit eccentricity, have been reported by Bourne et al. ( 3 A ) . Over a range of Reynolds numbers between 200 and 20,000, the product of the friction factor and the Reynolds number showed a minimum a t a diameter ratio of 0.75. Phosphorescent powder, excited to emit visible radiation, has been used (75A) to observe velocity distributions in a rectangular duct for various aspect ratios in laminar flow. Further experiments on a rectangular duct with a large aspect ratio equipped with a movable partition have been reported in (444). Measurements 50
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
of the volumetric flow rate us. the pressure drop in creeping flow for various positions of the partition were ,made. A good deal of interest in unsteady flows continues to be indicated in the literature, in keeping with the widespread occurrence of such flows i n natural and industrial processes. The development with axial position of periodic laminar flow in a tube has been investigated experimentally ( 6 4 ) with the conclusion that the total velocity may be considered as a fully developed pulsating flow superimposed on a developing mean flow. Experimental data on the value of the critical Reynolds number and the frictional losses in the transition region for oscillating pipe flow have been given (36A). Further treatment of developed periodic flow in tubes has been made and used in the interpretation of the water-hammer phenomena i n viscous fluids (47A). Some experiments on laminar pipe flow in which the pipe vibrates in a plane perpendicular to the flow direction have been described (25A). An interesting analogy to flow in bends is drawn and a correlation in terms of a modified Dean number presented. T h e speed of pressure pulses in expansible tubes has been investigated both experimentally and analytically (37A). I n the analysis, the assumption that the rate of change of cross-sectional flow area with respect to time is proportional to the rate of change of velocity with time was made, and led to predictions in good agreement with experiment. The flow induced by sinusoidal traveling wave motion of the walls of a parallel plate system has been treated analytically by Fung and Yih ( I O A ) for moderate amplitudes. This study includes the effects of inertia and wavelength on the flow. For motion opposed to the longitudinal pressure gradient, a backward flow will occur in the central region of the stream if a critical pressure gradient is exceeded. A general method of solving linear unsteady fluid flow problems in closed conduits has been developed ( 4 5 A ) and is applicable to problems both with and without initial conditions when the pressure gradient is a n arbitrary function of time. Flow developing from rest between parallel plates subject to the condition that the volumetric flow rate remains constant ( Q A )has been compared analytically to the same situation subject to a coustant pressure gradient. For a constant volumetric flow rate, the velocity distribution approaches the fully developed state much more rapidly. Coupled Flows. Velocity distributions for nonisothermal flows in which the temperature dependence of the viscosity cannot be ignored have been obtained in ( I I A , 43A). An investigation of the effect of viscous heating on the velocity and temperature distributions in plane and circular Couette flow with the viscosity exhibiting an exponential dependence on the temperature has yielded the result that solutions of the governing equations are double-valued i n the applied shear stress ( I I A ) . This occurrence has not been wholly recognized i.1 previous analyses. A physical interpretation of the double-valued behavior of the solutions is given. T h e effects of a temperature dependent viscosity in laminar forced convection heat transfer in a pipe were investigated by Test (43A) for a n oil. Local velocity and temperature profiles were measured in experiments designed to eliminate free convection. One important feature of the analytical work in this paper is a n estimation of the magnitude of the terms appearing in the full momentum and energy equations which are normally dropped on the basis that they are negligible. Some of these are shown to have significant effects on the results. T h e frictional and heat transfer characteristics of laminar flow in porous tubes with uniform wall mass transfer assuming constant fluid properties have been obtained ( I S A ) . T h e results are presented in the form of friction factor and Nusselt number curves over a wide range of injection and suction wall Reynolds numbers. T h e dispersion of a solute material, which is described by the convective-diffusion equation, is a n important example of fluid dynamical effects on the transport processes. Natural convection has been demonstrated to influence markedly the length of the mixed zone between two miscible phases in vertical upward displacements ( 3 4 A ) . T h e dispersion is enhanced when the lighter phase displaces a heavier phase and inhibited when the situation is reversed compared to the case when buoyancy effects are negligible. Dispersion taking place i n laminar flows developing from rest and in the velocity entrance region of tubes has been studied ( 7 2 A ) . The dispersion coefficient is, i n general, smaller in developing fields than i n fully developed situations because the variation in the residence time across the flow is largest i n the fully developed case. Hence, the length of the mixed zone is shorter i n a developing velocity field.
An empirical correlation of experimental results for longitudinal dispersion occurring in the bends of a circular pipe over a range of Reynolds numbers characteristic of turbulent flow has been reported by Aunicky ( 2 A ) . For the parameters investigated, the dispersion coefficient was found to increase by curvature over that which would occur in a straight pipe. Experimental measurements for the liquid phase velocity profiles in vertical cocurrent air-water flow have been demonstrated also to yield sufficient data for computing liquid phase dispersion coefficients in ( 7 3 A ) . Rarefied Gas Flows. Localized measurements of the molecular velocity distribution function in rarefied helium flows based on the observation of the Doppler profile of a helium emission line excited by a n electron beam have been described in (27A). An investigation of the limitations of the technique is made and the applicability discussed. A study of the behavior of a rarefied gas flow with an initial discontinuity in the tangential velocity has been reported in ( 3 9 A ) . The case studied has an initial condition of equal magnitude but oppositely directed velocities in the two semi-infinite regions normal to the flow direction. Developing flow in the momentum entrance region of tubes and flat plates with a slip flow boundary condition has been solved numerically by Quarmby ( 3 7 A ) for Knudsen numbers between 0 and 0.1. The assumptions that the velocity is uniform a t the entrance and that the momentum component perpendicular to the flow can be ignored are made. Extensive comparisons with previous approximate solutions and experimental results are presented. The slip coefficient a t a plane surface in rarefied flows has been computed from the BGK model of the Boltzman equation by the discrete ordinate method (16A, 3 3 A ) . Reddy ( 3 3 A ) has also presented details of the velocity field in the Knudsen layer. Sone and Yamamoto (4OA) have investigated analytically the asymptotic behavior a t small Knudsen numbers of rarefied flow in a tube. For this case, the gas flow may be expressed by a continuum model except in the Knudsen layer adjacent to the wall, and matched inner and outer solutions were obtained simultaneously. Some discussion of the second-order slip condition and limitations of previous work on the same problem is given. The molecular flow rate in rarefied gas flow through annuli (244) has been predicted by using a continuum model with slip a t the walls to which is added a term due to self diffusion. The theoretical results are in excellent agreement with experiments on argon and previous theoretical results obtained with the BGK model. Plane Poiseuille flow in the transition regime between free molecular and continuum flow has been studied (22A) using the moment method. The primary result is the prediction of a minimum volumetric flow rate a t an inverse Knudsen number of 3 which is higher than that obtained previously. A physical explanation of the occurrence of a minimum in the volumetric flow rate based on the interactions of flow due to self-diffusion, laminar convection, and slip flow is offered. The discrete ordinate technique has also been used to study transient plane Couette flow ( 7 7 A ) and nonlinear Couette flow with heat transfer (78A). The latter problem was treated as a test of the technique for applications in nonlinear rarefield gas dynamics since exact solutions were available. The agreement obtained indicates the usefulness of the discrete ordinate method. Plane Couette flow and rarefied gas flow between parallel plates with heat transfer have also been studied (28A)using a model sampling procedure. Two-Phase Flow
The literature on two-phase flow continues to be heavily weighted toward predictions of pressure drop, void fraction distribution, and velocity distribution. Additional topics of interest include the use of dispersion models to treat the two-phase flow and consideration of two-phase liquid metal flows. An International Symposium on Research in Cocurrent Gas-Liquid Flow was held a t the University of Waterloo, Waterloo, Ontario, Canada, September 18 and 19,1968. Kriegel (27B)claims to have derived and formulated expressions for the pressure drop and phase contributions in liquid films in such a general fashion that they are valid for any two-phase flow. After simplification, they are shown to be in agreement with the empirical Lockhart-Martinelli relations. Two-phase non-Newtonian flows have been studied by Oliver and Hoon (33B, 3 4 B ) . T h e non-Newtonian mixtures behave differently in the various flow regimes (e.g., slug flow) and, consequently, marked discrepancies
appear when using the Lockhart-Martinelli relations. Forced convection boiling inside helically coiled tubes and annular airwater flow in a heated horizontal tube have been investigated by Owhadi et d. ( 3 5 B ) and Pletcher and McManus ( 3 6 B ) , respectively. I n both situations, except for low vapor flow rates in the former, the Lockhart-Martinelli parameter was successful in correlating the two-phase pressure drop. Rogers ( 3 8 B ) , using parahydrogen between one atmosphere and its critical pressure, has developed a means for determining the two-phase pressure drop for single component systems over the entire range of gas-liquid coexistence, provided that a good estimate can be made of the Martinelli term. For para-hydrogen, this term varied greatly with pressure. A simple procedure for estimating the frictional pressure drop in two-phase vertical flows, based on single-phase pressure drop has been presented by Gomezplata et al. (228). Nancetti et d. ( 3 2 B ) report experiments of descending two-phase flow in vertical tubes from which correlations were developed for film thickness and pressure drop with volume flow rates of the two phases. Experiments on the upward flow of gas-liquid mixtures in vertical tubes have been reported by Veda (45B,46B). From the experimental results and an annular flow model, an expression is developed for the interfacial shear stress which is a function of the flow states of both gas and liquid. From the shear stress distribution in the film (obtained by assuming a turbulent layer and a laminar sublayer), an expression is obtained for the film thickness from which the eddy diffusivity and velocity distribution (in the film) are obtained by comparison with the experimental results. I n Part 11, expressions are developed for the steady-state frictional pressure drop in both annular and bubbly upward flow. Chawla (70B) claims to have developed a general relation for frictional pressure loss in the flow of liquid-gas mixtures through horizontal tubes which is valid for a wide range of conditions, excepting the simultaneous existence of very high pressure and small vapor contents. The relation is stated to be more accurate than previously reported methods. New formulas for calculating two-phase pressure drops are presented by Hodossy (24B). I t is concluded that most formulas developed in the past 4 or 5 years are valid only over a narrow range of flow. Chawla and Thome ( 7 7B) have presented general equations for the determination of the total pressure drop for refrigerants in an evaporator system. Relationships for calculating the coefficient of energy loss due to interfacial friction in a one-dimensional two-phase flow have been obtained by Anisimova et d.
(3B). Baroczy (423) has obtained pressure drop data for two-phase potassium flowing rhrough horizontal tubes. Subcooled boiling of potassium is characterized by large, periodic temperature and pressure oscillations which closely resemble those observed for low pressure water. The paper additionally describes the use of a n orifice as a quality and liquid fraction meter. Wallis (47B, 48B) has presented an interesting set of papers concerned with the use of the Reynolds flux concept for analyzing one-dimensional two-phase flows. T h e object of the work is to “exploit the simplest level of sophis tication which will describe convective transfer phenomena.” The concept is applied to a variety of two-phase problems and is claimed to yield useful accurate predictions without enormous effort. Andeen and Griffith ( 2 B ) have reported measurements of the momentum flux in twophase steam-water and air-water flow systems. The results are used to evaluate the utility of various two-phase flow models and their underlying assumptions. Brown and Kranich ( 9 B ) present an experimental study of velocity and void fraction profiles for the system air and water, and air and glycerol-water systems. A model is proposed, which in conjuction with experimental sampling data, allows the determination, by trial, of the two-phase velocity and void fraction profiles. Experiments on velocity and entrained droplet mass velocity distribution are described by Gill and Hewitt (20B). The results of previous and current studies indicate a n effect of the means of injection of liquid into a flowing gas stream on pressure gradient, film thickness, and integrated flow of entrained droplets. Lafferty et al. (28B) dcscribe experiments in which an analytical model, together with the experimental data, is used to calculate the velocity distributions in two-phase air-water vortex flow. Equations for the calculation of steam volume fraction from theoretical arguments are reported by Rouhani ( 3 9 B )for subcooled, forced flow boiling. Staub (42B) presents a detailed analysis for the prediction of the initial point of net vapor generation in subcooled flow boiling, while Leslie ( 2 9 B ) develops a theory of flashing flow VOL. 6 2
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which predicts the variation of voidage with distance down the duct for saturated flow boiling. Van Wijngaarden (43B)has derived equations describing onedimensional unsteady flow i n bubble-fluid mixtures with attention subsequently being focused on pressure waves of small and moderate amplitude propagating through the mixture. Integral mass, momentum, and energy equations are developed by Delhaye ( 7 8 B ) for a noninteracting two-phase system. T h e equations are time and space averaged to yield spatial and statistical mean equations for two-phase flow. As a n alternative solution to averaging, Deich ( 7 5 B ) proposes a probabilistic method from which the equations for two-phase media are claimed to be attainable with mathematical rigor. Wedekind and Stoecker (4QB)develop a theoretical model for predicting the transient response of the mean position of the transition point, or boundary between a two-phase region and a downstream superheat region orizontal evaporator tubes. Tests upon hydrodynamic insta s in boiling channels are reported by Hayama (236’). The stabilizing effect of a throttle i n the downcomer is investigated, as is the occurrence of reverse flow in multichannel systems. Bourt et al. ( 8 B ) review the physical processes likely to provide a theoretical explanation for oscillations observed in heated two-phase flows. Flow instabilities in subcooled and saturated forced flow boiling have been investigated by Mayinger et al. (30B, 3723). Silvestri ( 4 7 B ) discusses unconventional applications of twophase concurrent flow. Included are the magnetohydrodynamic conversion of heat into electricity by means of liquid metals; highpressure, high-temperature electrolytic cells for production of hydrogen to be converted into ammonia, etc. Rounthwaite (40B) describes water-steam heat transfer experiments for a horizontal “hairpin” type of test-section tube. Two-phase flow patterns and heat transfer mechanisms existing in the tube are deduced. Experiments describing the production of 300-@ water droplets by two types of air-water two-phase nozzles are reported by Kevorkian et al. (26B). Discharge coefficients of the water phase, based on sonic air velocity a t the throat, were calculated. Chisholm ( 7 2 B ) extends equations of the form already developed for the case of incompressible two-phase flow, to the case of compressible mixtures. T h e derived equations are compared with data for flow through venturis, nozzles, and sharp-edged orifices. Reith et al. (37B) have measured gas hold-up and liquid phase axial dispersion coefficients in water-filled bubble columns as a function of superficial gas velocity. O n e series of cxperiments was performed using a n ionic solution of 140 g of NaCl per liter of water. Hosler (25B) has observed and photographed flow patterns occurring in forced flow vaporization of water a t high pressures i n a vertical rectangular channel. The differences in the various flow regimes between two-phase adiabatic flow and twophase flow with heat transfer, were noted. Curtet and Djonin ( 7 4 B ) describe a n experiment i n which a jet of water falls freely along the vertical axis of a pipe and impinges on the surface on the water which fills the lower portion of the pipe. When the jet nozzle is a t a sufficiently high height above the water surface, the resulting emulsion reaches the bottom of the pipe and then air is transported through the pipe with the water. T h e experimental results are compared with analyses by the method of W-allis and of Zuber and Findlay with good agreement. Bolotov et al. ( 7 B ) develop a general dimensionless equation for the flow of mixtures of steam and liquids in vertical pipes which enables the true steam content to be calculated throughout almost the entire boiling zone. T h e expression is claimed to be usable also for determining the velocity head and natural circulation velocity i n evaporators with external colandrias. Dejong et al. ( 7 6 B )consider the effect of slip and phase change on sound velocity in steam-water mixtures. Experimental data and a n approximate analysis indicate that both slip and phase change occur within a wave traveling in steam-water mixtures and that the wave velocity is affected by these events. T h e need for additional experimental data is discussed. -4dditional topics of interest include a n exact equation derived by Boiko ( 6 B ) for determining the hydraulic resistance coefficient and frictional pressure drop, a theoretical study of adiabatic twophase annular dispersed flow by Biasi et al. (56’)from which the AUTHORS William N.Gill, Robert Cole, E. James Davis, Josejh Estrin, and Richard J. Nunge are all at Cladson College of Technology, Potsdam, N.Y. Howard Littman i s zeiith Rensselaer Polytechnic Institute, Troy, N.Y.
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INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
liquid film thickness is related to system parameters, a method developed by Vasil’ev (44B) for determining the boundaries of a periodic operation and the boundaries of fluctuating stable operation of heat exchangers for flow with low-steam content i n designing thermal engineering plants, and a new correlation by Gloyer ( 2 7 B ) for liquid holdup in two-phase flow in a heat exchanger tube. Instrumentation techniques have been discussed by a number of authors. Chomiak ( 7 3 B ) presents principles for the construction of simple probes for taking samples of two-phase media; the emphasis appears to be on gas-solid systems however. Alia ( 7 B ) describes the use of a probe which can operate either as a Pitot probe or as a n isokinetic sampling probe for determining phase and velocity distribution in the care of a two-phase flow. I n the wall region, use is made of a suitably-shaped isokinetic sampling probe. T h e determination of void fraction by gamma-ray attenuation techniques is discussed by Evangelisti et al. (7QB),and the results are compared with those i n the same channel using the dilation technique. Delhaye ( 7 7 B ) considers the measurement of gas content in two-phase flows by means of hot-wire anemometry. Cavilofion
Current problems of interest include studies on the inception of cavitation, collapse of cavitation cavities, cavitation research on valves, and cavitation phenomena in liquid metals. Arndt and Ippen (7C) report experiments on cavitation inception and the associated bubble dynamics i n turbulent boundary layers adjacent to surfaces roughened with triangular grooves. T h e experiments result in the significant conclusion that the cavitation inception index is directly related to the skin friction coefficient for smooth and rough boundaries. T h e circumstances under which cavitation, excited by strong velocity gradients, develops, is discussed by Duport (4C). Shear flow cavitation is experimented with by means of liquid jets i n a practically unlimited field of resting liquid. Kozirev (QC) views highly nonsymmetrical bubble collapses photographically. I t is noted that the bubbles collapse in such a fashion as to produce liquid jets, similar to shaped charges used in explosives. .4 model based on cumulative jet formation is postulated to explain the damaging power of such collapses. Such damage is examined and found to be similar to that from water jet impacts. T h e collapse of an imploding spherical cavity is considered by Hoit ( 6 C ) . A perturbation solution is found by using a series expansion in terms of the ratio of the speed of sound to the cavity radius. Boguslavskii ( 2 C ) analyzes the nonstationary convective diffusion of a gas into a rapidly expanding spherical void for both ultrasonic and hydrodynamic cavitation. A review of cavitation research on valves is presented by Tullis and Marschner ( 7 0 C ) . Included are reports on needle, gate, globe, butterfly-fixed roller-gate, radial-gate, cone, ball, straightflow and disk-type valves. Correlation of results is claimed to be severely handicapped by a lack of uniform definition and nomenclature. Work covered includes the reduction of cavitation damage by various means, the effectiveness of air injection i n delaying the onset of Cavitation, and methods of detecting cavitation including visual, aural, hydrophone, accelerometers, pressure transducers, external microphones, damage inserts, and change of flow and torque characteristics. Hammitt (5C) presents a review of the many problems associated with cavitation phenomena in liquid metals. New important interests have arisen in connection with sodium-cooled fast neutron breeder reactors. O n the basis of presently known erosion theories, Canavelis (3C) presents an analogy between jet impact and cavity erosion which is claimed to be confirmed by experiment. As a consequence of the analogy, data on jet impact experiments are used to explain some peculiar aspects of cavitation damage. Karlikov and Sholomovich ( 8 C ) suggest a n approximate method for determining the influence of the walls during cavitating flow past bodies in water tunnels. Such influence is studied by analyzing the relation between the cavitation number in infinite space and the same parameter in a water tunnel, for similar or identical cavitation patterns. Pressure fluctuations and ultrasonic shock waves originating from cavitation in a D I N Venturi tube were investigated by Ikeda (7C). TransDarent walls in the Venturi tube made it possible to observe the phenomenon directly.
huid Films This year, the literature appears to be divided fairly equally between the topics of wave motion, film stability, and mass transfer
with only minor attention being devoted to velocity profile and film thickness determinations. Berbente and Ruckenstein (30)propose a method for obtaining periodical solutions of the nonlinear equations of motion. By means of a triple series expansion, theoretical equations for the wave length, wave velocity, and film thickness are established in terms of a single dimensionless quantity 1c.. Good agreement is obtained with experiment, The relation of the increased resistance for air flowing over a wavy surface (relative to a smooth surface) to the properties of the waves has been studied by Cohen and Hanratty (50). Experimental data indicate that the drag on interfaces with three-dimensional wave structures increased with the square of the gas velocity and depended more on the height of the waves than on other parameters characterizing the interface. A linear analysis of wave flow for a viscous liquid flowing down a n inclined plane is presented by Smith (270). For a wavy bottom, a phase lag of the surface waves is found for any nonvertical inclination. The effect of inertia terms (neglected in the linear analysis) is studied by asymptotic methods and found to reduce the phase lag. Davies et d. (70)report an experimental study on surface stresses and ripple formation due to low-velocity air (24-100 cm/sec) passing over a water surface. Both filmcovered water surfaces on which the surface tension gradient exactly balanced the wind stress, and freely moving clean water surfaces were employed ; the results from both techniques yielding agreement. Craik (60)considers the stability of liquid films on a horizontal flat plate where an air stream blows over the liquid surface and the film is contaminated by an insoluble surface-active agent. At large Reynolds numbers, the presence of surface contamination enhances stability, whereas a t small liquid Reynolds numbers, a class of disturbances exists for which surface elasticity may be destabilizing. T h e stability of a viscoelastic liquid film flowing down a n inclined plane is analyzed with respect to three-dimensional disturbances by Gupta and Rai (720). Under certain circumstances, these disturbances are more unstable than the twodimensional ones. Buevich et d. (40)consider the problem of the flow stability of a fluid layer with respect to small two-dimensional disturbances of the bounding surface. T h e finite radius of curvature of the bounding surface and the nonzero hydrodynamic friction are both taken into account. The nonlinear stability of the flow of a liquid layer down an inclined plane is studied by Nakaya and Takai (780), taking into account the surface tension and the Reynolds stress due to the first periodic disturbance. Yih (250)presents a n analysis utilizing perturbation techniques, of the stability of long waves for a system consisting of a layer of viscous liquid with a free surface set in motion by the lower boundary moving simple-harmonically i n its own plane. A hydrodynamic stability analysis is formulated by Ludviksson and Lightfoot (770)for a thin “Marangoni” film flowing over (up or down) a vertical solid plate of high heat capacity and thermal conductivity, with a constant downward directed temperature gradient (cold top, warm bottom). Ford and Missen (80)present methods for determining the sign of the surface tension gradient normal to the boundary in falling liquid films. T h e numerical sign of the gradient is a criteria for film stability. Vorotilin and Krylov (240) present a solution to the problem of convective diffusion of a substance in a laminar boundary layer of gas, flowing around a thin layer of liquid, and in an outflowing laminar layer of liquid. Expressions are obtained for the partial mass-transfer coefficients in each of the phases. A nonlinear treatment of the hydrodynamics of wave motion is suggested by Ruckenstein and Berbente (200)and its results are used for the solution of the mass-transfer problem. An equation of convectivediffusion valid in the small penetration case is established and is solved exactly by means of a similarity transformation. Oliver and Atherinos (790)have conducted flow visualization studies on a small inclined plane which show that gravity-controlled roll waves cause mixing of the surface layers of liquid but have little effect on the liquid adjacent to the solid surface. Mass transfer experiments are described in which cast &naphthol and benzoic acid surfaces are dissolved by a water film. I n other tests, carbon dioxide-air mixtures are absorbed into the surface of a flowing film. A method is developed by Howard and Lightfoot (750)for predicting rates of gas absorption into laminar rippling films in terms of the surface velocities. The description is an extension of the surface-stretch model of mass transfer and is therefore useful for cases of high Peclet number. An experimental study of the effect
+
of standing waves of controlled amplitude and frequency on the steady state rate of mass transfer through thin horizontal liquid layers is presented by Goren and Mani (770). Vibration is found to increase the mass transfer rate u p to a n order of magnitude more than that predicted by molecular diffusion alone. Banerjee et al. (20)relate the mass transfer rate a t a free interface to the rate of viscous dissipation in the turbulent flow near the surface. An order of magnitude estimate for the dissipation for a wavy turbulent liquid film is derived from a consideration of the time averaged vorticity equation and the liquid phase mass transfer coefficient related to the average wave length, amplitude, and speed of the surface disturbances. Diffusion of one constituent of a gaseous mixture into a liquid film flowing down a vertical plane wall or cylinder with first-order homogeneous isothermal chemical reaction is considered by Forste (90). The concentration of the gas i n the film is described by the diffusion equation and the solution is constructed by Galerkin’s method. Thomas (220)reports a n investigation on the enhancement of film condensation rates on vertical tubes by use of longitudinal fins. A model based on the effect of fin shape on film and rivulet hydrodynamics is claimed to explain the observed behavior. An experimental investigation by Golubev (700)indicates that higher gas velocities and higher rates of heat and mass transfer occurred with transverse gas flow than with cocurrent or countercurrent flow. An analytical study describing the laminar accelerating flow of a thin film falling along a vertical wall is presented by Haugen (740). The approximate mathematical solution is given, with emphasis on the growth and decrease of the boundary layer and considers the mofilm thickness, respectively. Ackerberg (70) tion of a film of viscous incompressible liquid running down a semi-infinite vertical plate under the influence of gravity. T h e problem is to determine how the final asymptotic shear flow is achieved when the fluid is introduced a t the leading edge with uniform velocity. Additional papers of interest include the determination of dimensionless characteristics from mathematical models and measurements on industrial equipment for boiling heat transfer in a falling film by Kattanek et d. (760),a n experimental investigation of the flow of a thin fluid layer on the surface of a rotating cone by Voinov and Khapilova (230)where a new term, the “critical depth” is introduced, and a n investigation by Hartland (730)of the radius of the draining film beneath a drop approaching a plane interface through a n immiscible fluid. Stratifled Flows
Stability and wave analyses again dominate the literature on stratified flows with the appearance of several articles concerned with withdrawal, spinup, evaporation, and boiling. Gage and Reid (3E)study the stability of a thermally stratified fluid in the presence of a viscous shear flow; a situation where a n important interaction exists between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. T h e hydrodynamic stability of liquids in which the viscosity varies with distance below a free surface is examined by Craik and Smith (7E). Treated are the cases of wave-generation by a concurrent air flow a t a horizontal liquid surface, and the stability of inclined flow under gravity. The situation for which the viscosity varies with distance from the free surface may be regarded as a model of a melting surface, although the density is assumed constant. T h e stability of the flow of two layers of viscous liquids down an incline is investigated by Kao (8E). T h e presence of the upper layer is generally destabilizing compared with that of a homogeneous fluid of the same total depth. Sinkha (74E)predicts oscillations of the system consisting of a light fluid of finite depth resting over a heavy fluid of infinite depth, through standard use of a convergent power series in the small amplitude parameter. The stability of a horizontal fluid interface in the presence of a periodic vertical electric field is considered by Yih (79E).The interface can be unstable even if the electric field is a t all times weaker than that needed for instability in the case of a steady field. Zadoff and Begun (20E)analyze the gravitational instability of a system consisting of two homogeneous fldids, with horizontal interface, in the presence of a uniform horizontal magnetic field. Miles ( 7 7E, 72E) investigates analytically the generation of lee waves by, and the consequent drag on, a n obstacle in a twodimensional, steady, inviscid, stratified shear flow in which the VOL. 6 2
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upstream dynamic pressure a n d density gradient are regarded as constant. A thin barrier in either a channel of finite height or a half space is considered i n the first paper while a semi-circular obstacle in a half-space is considered i n the second. A theory of unsteady long waves in a stratified fluid over a channel of arbitrary cross section is proposed by Shen (73E). T h e theory predicts the occurrence of various types of internal waves a t the depth where the rate of change of the density profile is the largest in the increasing direction of depth, and with slight modification, to the case of a fluid with infinite height or a fixed boundary, where density stratification plays a n indispensable role. Grimshaw ( 5 E ) considers the steady two-dimensional flow of a n inviscid incompressible fluid of variable density in a long channel, bounded above by a rigid horizontal plane and below by a n obstacle. When the obstacle satisfies a certain convexity condition, there is no critical internal Froude number or obstacle height for which the problem fails to be well posed. An experimental study is reported by Stevenson (75E) in which two-dimensional internal waves were generated in a stably stratified fluid by the movement of a long horizontal circular cylinder. T h e resulting internal wave system was stationary with respect to the moving cylinder and the phase configuration of the waves compared well with Lighthill’s theory for waves in dispersive systems. Martin and Long (IOE)present a study on the slow motion of a thin flat plate moving through a linearly stratified salt-water mixture. T h e study shows, i n addition to a n upstream “wake,” a boundary layer over the plate whose thickness increases upstream from the back of the plate. Further the theory also shows that salt diffusion is important i n a second, thinner boundary layer whose thickness increases from the front of the plate. Experimental verification of the presence of the thin diffusion boundary layer was obtained. An investigation on the analogy between rotating fluids and stratified fluids is continued by Veronis (77E). Although the basic equations are the same, differentiation is made on the basis of the forces which drive the motion. Thorpe ( 7 6 E ) describes experiments in which the end of a long closed tube containing a stratified fluid is raised, resulting in a laminar accelerating flow. T h e flow i n the central portion of the tube is twodimensional and predictable until the onset of instability. T h e occurrence of instability, its nature, and the subsequent transition to turbulence, are described qualitatively. Wood (78E) presents both analysis and experiment concerned with withdrawal from a reservoir through a horizontal contraction to a channel, both the reservoir and channel containing a stable multilayered system of fluids. A variety of situations are considered, e.g., flo6 in one layer only, with the theoretical predictions being confirmed by experiment. T h e flow of stratified fluids through curved screens has been investigated by Lau and Baines ( 9 E ) . A screen placed in a flow field causes deflection of the streamlines normal to the screen as they pass through, in a similar way to the refraction of light a t a material interface. This is distinct from the pressure drop across the screen produced by the wakes of the elements. The describing equations have been solved numerically for the shape of screen required to produce a specified velocity distribution. An approximate solution is also obtained for general velocity profiles and the screen shape which produces uniform shear is derived. Experimental verification of the analysis was obtained. Janowitz (7E) considers the general nature of the flow a t large distances from a two-dimensional body moving uniformly through a n unbounded, linearly stratified, nondiffusive viscous liquid. The solution of the linear problem shows a system ofjets upstream and a pattern of waves downstream of the body. T h e effects of viscosity on these lee waves are considered in detail. Holton and Stone ( 6 E )point out that recent theoretical results for the spin-up of a continuously stratified fluid with insulating boundary conditions are a t variance with experiment. However, the apparent contradiction is not real, but results from a n inconsistent scaling assumption. T h e nature of the spin-up as shown by the experiments is briefly described. Temperature profiles and heat transfer coefficients i n twophase, liquid-liquid, stratified laminar flow down a n inclined adiabatic plane were calculated by Gollan and Sideman ( 4 E ) for the case where the thickness of the upper, volatile, liquid layer varies continuously because of surface evaporation. I n a second paper, Fortuna and Sideman ( Z E ) consider stratified flow-boiling in a stirred cell, i n which two immiscible liquids of different density are in direct contact. Heat transfer coefficients in a simultaneous boiling and stirred liquid were calculated from the coefficients determined independently for stirring and boiling alone. 54
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
Bubble Dynamics
T h e literature on bubble dynamics for the period covered by this review continues to concentrate primarily on the shape and rising velocities of gas or vapor bubbles and the growth or collapsr problem. Although no new papers have appeared which consider nonequilibrium effects, work is continuing i n this area with a t least several publications due in the 1969 literature. I n a continuation of an analogy reported in last year’s review for the rise velocity of bubbles i n liquids of infinite extent and the velocity of surface waves over infinitely deep liquids, Maneri and Mendelson (27F) find a corresponding analogy which relates bubble velocities i n bounded liquids to wave velocities over liquids of finite depth. I n what appears to be a similar approach, Chincholle ( 6 F ) , in studying foam flow, suggests that a bubble should be considered as a cavity or deformation of the medium within which it is moving; the deformation propagating itself in the same manner as a n energy wave. Kojima et al. (751,’) present experimental data for the rise velocity and shape of sirigle air bubbles rising through highly viscous liquids. The time-dependent motion of small gas bubbles in liquids subjected to a linear pressure gradient has been studied by Gutta ( 9 F ) . T h e bubble slip velocity for air in water is found to asymptotically approach zero for uniform flow and closely approach unity for highly accelerating flows. Zieminski and Raymond ( 3 3 F ) describe a method for the measurements of the rates of volume change of a rising gas bubble, its surface area, velocity of rise, and the instantaneous mass transfer coefficient. T h e principle of the method is the simultaneous use of high-speed photography and a capillary dilatometer. h theoretical analysis of the behavior of gas bubbles in vertically vibrating liquid columns is presented by Rubin ( 2 7 F ) . Agreement with published experimental work is good. Cheh and Tobias ( 4 F ) employ a boundary layer approach for the fluid flow problem i n order to predict mass transfer to spherical drops or bubbles rising steadily through a liquid a t high Reynolds numbers. I n addition to the bubble rise problem, Cheh and Tobias (5F) also consider the bubble growth problem. Theoretical calculations are presented on the dynamics of asymptotic bubble growth in a n initially nonuniform concentration field. A significant simplification is achieved by noting that the Jakob number for mass transfer is usually small so that the convective transport can be neglected in comparison with diffusive transport. Hewitt and Parker ( 7OF) present experimental growth rate and collapse data in liquid nitrogen for two areas where additional theoretical work is needed; bubble collapse in a subcooled liquid, and bubble growth with decreasing pressure. A significant advancement i n the theory of a bubble growing i n contact with a plane surface is presented by Witze et al. ( 3 2 F ) . An exact integral solution is presented for the potential distribution surrounding a sphere that is growing while remaining tangent to a plane solid surface. By using a simple analytic representation for the integrand, closed form results were obtained for both the potential and velocity distribution. For one specific growth law, the pressure distribution was evaluated and used to form a new criterion for detachment of the sphere from the surface. Shima ( 2 8 F ) considers the behavior of a spherical bubble as it collapses in the vicinity of a solid wall. Numerical examples are given for the time change in bubble size, flow velocity a t the bubble surface, and impulse pressure occurring upon collapse. T h e PLK coordinate perturbation technique is used by Jahsman (72F) for the collapse of a gas-filled spherical cavity in a n infinite compressible liquid. A numerical example is presented and curves of cavity wall position, pressure, and velocity histories are given for the period associated with collapse and rebound of the cavity. Tokuda et al. (291;) present a study of the growth or collapse of a spherical gas bubble being injected into a quiescent liquid of different composition. Of special interest is the fact that translatory motion is given particular attention. Such motion is shown to cause a significant increase i n the growth rate. ‘Kosky ( 7 6 F ) reports experimental bubble growth measurements i n uniformly super-heated water a t pressures from 0.5 to 1.2 atm. T h e use of a pressure transducer to obtain a transient pressure record along with a high speed photographic record of the bubble growth, indicates that previous workers may have inadvertently reported higher than actual superheats as a result of the growth occurring during a portion of the pressure transient. I t should be noted that the data of Kosky also fall in a transient pressure region and as a result, comparison of the data with theories assuming constant pressure is somewhat questionable. Experiments designed to clarify the difference between growth rates obtained under transient conditions, as opposed to steady-state conditions have
been undertaken by Akiyama et al. (7F). Rehm (25F) extends previous work on bubble formation, growth, and separation, to include fluids which have viscosity and surface tensions levels significantly different from water. Van Stralen (3UF, 37F)continues to apply his “relaxation micro layer” theory to the growth rate of vapor bubbles in superheated pure liquids and binary mixtures. A review, analysis, and original experimental data are presented, including data and analysis for rising bubbles oscillating both in shape and volume. Borishanskii ( 3 F ) reports a study of the fluid motion near a heated surface in the presence of numerous vaporization nuclei. Additionally, a n expression is developed to calculate the bubble radius a t separation from the surface. Departure diameters i n forced-convection boiling have been investigated by Koumoutsos et al. (77F). A consideration of the hydrodynamic stability of the neck joining the departing bubble to the surface, allows the departure-size-to-velocity relationship to be predicted. A number of papers deal with the problem of bubble formation a t a n orifice under constant flow conditions. Krishnamurthi et al. (78F, 79F) have developed expressions for the bubble volume in viscous liquids and a t low gas rates where surface tension effects are present and in the latter reference, a n expression for bubble volume from noncircular orifices into liquids of differing viscosities. A mathematical model and correlations for bubble formation are derived by Kumar et al. (ZOF), taking account of expansion and solution effects. Hobler et al. (77F) present information on the formation, size, and shape of bubbles formed a t horizontal slits, this shape being chosen as the simplest element of a grid tray. Bubbles generated from a porous plate are investigated by Koide et al. (74F). Correlations are obtained for the average bubble diameter. Additional correlations are developed for the condition of bubble generation for the cases where no coalescence is observed and when coalescence is a t the maximum rate. Pan and Acrivos (22F) consider the shape of a drop or bubble translating through an unbounded, quiescent, and viscous fluid medium for the case where the Reynolds number pertaining to the continuous phase and the Weber number are both small and the interface is clean. T h e Taylor-Acrivos solution is extended to the more general case where the motion inside of the bubble is unrestricted. The theoretical predictions indicate that the inertia forces of the internal circulation do not affect the shape of a gas bubble, but do contribute in a minor, but significant, way to the deformation of a liquid drop. The variation of overall radius of a toroidal bubble is treated by Pedley (24F). Additional considerations include the effect of hydrostatic variation in bubble volume, and a prediction of the time which will have elapsed before the bubble becomes unstable under the action of surface tension. Additional topics of interest include an analysis of the oscillations of a vapor cavity in a rotating cylindrical tank neglecting the effects of gravity by Pao and Siekmann ( 2 3 F ) , an investigation of bubble driven fluid circulations by DeNevers ( 7 F ) which is shown to be similar to the natural convection mechanism but with much larger driving forces, a n analysis for the prediction of bubble trajectories and equilibrium levels in vibrated liquid columns by Foster et el. ( 8 F ) , a study of the relationship of intermittent vapor mass behavior and liquid film consumption on the boiling crisis in pool boiling by Katto and Yokoya (73F), consideration of the drift current behind a rising vapor bubble to extend the boiling theory of Han and Griffith, reported by Beer et al. ( Z F ) , and a method for dealing with the difficulties of treating problems involving a free surface where surface tension effects are present by Richardson (26F), the problem treated being that of two-dimensional bubbles in slow viscous flows, Drop Formation and Motion
The current literature on drop formation and motion is primarily concerned with such topics as shape, coalescence, terminal velocity, contact angle, vaporization, burning, and breakup. Baumeister and Hamill (2G)determine the shapes of liquid drops resting on flat surfaces by a Runge-Kutta solution of the Laplace capillary equation. Additionally, asymptotic solutions for small and large drops were combined to give explicit expressions for the maximum drop height and radius. Robertson and Lehman (22G) present an alternative analysis for the shape of a liquid drop under the influence of gravity and surface tension. A variational technique is used to minimize the total energy of the drop, and the resulting expressions for the drop shape are evaluated numerically. T h e new treatment is claimed to make more direct use of the ac-
curately measurable drop width than do the classical treatments based upon the analysis of Bashforth and Adams. Halligan and Burkhart (75G) determine the profile of a growing droplet with a pressure balance similar to that derived by Laplace for the static droplet. An additional term was addrd to the balance to account for the pressure on the interface due to the motion of the fluid within the droplet and the entire balance combined with the differential equations describing the droplet geometry. A second paper by Halligan et al. (74G)determines the profile of a separating droplet whose volume exceeds the critical value by a small amount. The basic assumption is that the separation process results from a pressure difference across the droplet interface. The agreement with experimental results is claimed to be good until the droplet develops a minimum i n its profile. Lee and Hodgson (27G) derive expressions relating the profile of a thin film formed between a drop and a n interface, or between two drops, to the pressure drop due to flow. T h e influence of interfacial tension gradients on tangential movement of the interface is examined, equations showing how mobility is controlled by mass transfer of a third component are considered, and the effect of mobility on drop coalescence times in dispersions is discussed. The effect of a surface active agent on the drainage and rupture of the film beneath a liquid drop a t a deformable liquid-liquid interface has been investigated photographically and electrically by Hartland (77G). Brown and Hanson (5G) study the influence of high-energy oscillating electric fields on the coalescence of single drops a t a plane oil-water interface and discuss various possible mechanisms. I n a second paper, Brown (4G) discusses the mechanism of coalescence, the role of secondary droplet formation, and the promotion of coalescence by electric fields. Thorsen et al. (24G) report experimental terminal velocity data for drops of organic liquids falling through water which show that when extreme care is taken to avoid contamination, the values obtained for nonoscillating drops greatly exceed those previously reported in the literature. An empirical equation is presented to represent the data, developed from the seven high interfacial tension systems of the work. An equation for the terminal velocities of ensembles of solid particles, drops, or bubbles is developed by Gal-Or and Waslo (73G). The exprrssion is a generalization of Levich’s solution for a single drop in the presence of surfactants, of Happel’s solution for a n ensemble of solid particles, of the Hadamard-Rybczynski solution, and of Stokes solution. Using the free surface cell model employed in developing the above-mentioned expression, the velocity components of the solution can b r employed to predict heat, mass, and charge transfer rates in various particulate systems. A method for determining interfacial tension from sessile drop measurements is presented by du Prey (9G). The method consists of measuring (by microscope) the diameter of the drop and the contact angle between the supporting surface and the interior contour of the drop. With the known density of the liquid and the unknown interfacial tension, a dimensionless number is formed. A chart of two intersecting family-of-curves is formed on a coordinate system with contact angle as the abscissa and the dimensionless number as ordinate. T h e point of intersection defines the interfacial tension. Ehrlich (7UG) has developed a method for computing the contact angle from the dimensions of a small sessile drop. A perturbation solution is obtained to the differential equation resulting from the application of Newton’s second law to the surface of a sessile drop. From this result, the contact angle may be calculated from the maximum radius and the radius a t the plane of contact on a smooth horizontal surface. The accuracy of the calculation is claimed not to be critically dependent on the accuracy of the required interfacial tension. Dickinson and Marshall (8G) report a computational study of the evaporation rates of sprays of pure liquid drops having negligible velocity with respect to the air, and for drop velocities great enough to affect the evaporation rate. I n a series of two papers, Cuino et al. (6G, 7G) in the latter consider the amount of cooling which results from drops of liquid colliding with heated walls. The paper reports a n experimental study of interaction using a synthetic two-phase mixture with high-drop dispersion and employing high-speed photography and the Schlieren method. I n the former paper, the evaporation of liquid droplets from a heated wall is studied by means of high speed photography. T h e incidence of droplets on liquid films is also discussed in relation to the thermal crisis in annular two-phase adiabatic flow. A correlation for predicting drop volume for injection a t low velocities of one Newtonian liquid into a second stationary immiscible VOL. 6 2
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Newtonian liquid in the absence of surface active agents is presented by Scheele and Meister (23G). T h e analysis predicts drop volumes within a n average error of 1170for fifteen liquid-liquid systems covering a wider range of variables than any previous study. Karam and Bellinger (2UG) discuss the theoretical factors involved in deformation and breakup of liquid droplets in a simple shear field. Experimental verification is claimed to be excellent and the practical significance of rhe results is discussed in terms of the formation of suspensions. Fendell et al. (77G, 72G) consider first, a small rigid spherical droplet taken to undergo quasisteady adiabatic vaporization, and second, a rigid spherical droplet undergoing first order decompositional burning. The vaporization and burning rates arc obtained in the form of matched inner and outer expansions. T h e motion of a spherical drop a t Reynolds numbers high enough for boundary layer theory to hold, but small enough for surface tension to keep the drop spherical is considered by Harper and Moore (IGG). Drag coefficients calculated from the theory are claimed to agree quite well with experimental values for liquids satisfying the conditions of the theory. No experimental results were found to test the resulting prediction of the internal circulation. Balashov et al. ( 7 G ) give the results of a theoretical and experimental investigation into the unsteady flow of gas about dispersed fluid particles. Other papers of interest include a study of maximum drop diameters from photomicrographs of atomized jets of ethanol or water injected into nitrogen or helium gas streams by Ingebo (79G), a study of some general laws governing droplet size distribution during atomization of a liquid by Blokh et al. ( 3 G ) , and a study on the effect of plate wettability on droplet formation by I-Iimmelblau et ul. (78G) showing that contact angle plays a n important role in governing the size of droplets a t a horizontal orifice. Fluidizied Beds
From a fluid mechanical point of view, fluidization is a branch of multiphase flow. Most chemical engineers will use this review as the basis for understanding bed properties, heat and mass transfer, and chemical reactions taking place in fluidized beds. A review of fluidized beds for the year 1967 appeared as part of last year’s fluid mechanics review (33H). An excellent review of the literature (240 references) covering theoretical and applied aspects of fluidization is given by Botterill (73H). Papers which have appeared since his last review in 1965 ( 7 2 H )are covered. Several reviews and summaries of particular aspects of fluidized beds have appeared this year (211, 75H, 24H, 37H, 42H, 43H, 72H, 87H, 96H, 713H). General Theory, Hydrodynamic Stability, a n d Viscosity. Panton ( 7 7 H ) derives continuity, energy, and momentum equations for the nonequilibrium two-phase flow of a gas-particle mixture. A quasi-one-dimensional flow containing an arbitrary volume of particles is examined. A feature of this paper is the time and area averaging procedures. The equations of motion describing one-dimensional unsteady flow of bubble-fluid mixtures is presented by Van Wijngaarden (704H). I n addition, the propagation of pressure waves of small and moderate amplitude is discussed theoretically. Anderson and Jackson (%?H) perform a linearized stability analysis to their previously derived equation of motion and show that the state of uniform fluidization is unstable to small perturbations in the voidage. These instabilities take the form of rising and growing voidage fluctuations whose propagation characteristics are related to the physical properties of the fluidized system. T h e propagation of density disturbances is also considered by Ruckenstein and Tzeculescu-Filipescu (84H). Equations are derived for the wave velocity and the frequency of oscillation. A statistical theory of local phase pulsating motions in uniform flows of a two-component system of gas and uniform particles is presented b y Buevich ( 7 7 H ) . I t assumes that deviations of the mean flow induced by pulsations are comparatively small. The variation in pulsation velocities of the individual particles cannot be considered Markoffian. A statistical theory of particulate fluidization is put €orward and assessed experimentally by Saxton (86H). Several papers deal with drag in sphere assemblages ( 5 8 H ) , in fluidizing and sedimenting systems ( 2 6 H ) , i n suspensions (27H, ZRH), and in a vertically oscillating water column (707H). I n a paper by Jones ( 4 5 H ) ,departures from Darcy’s law are interpreted as due to secondary flow. An analysis of fluid bed inhomogeneity from the standpoint of particle energies shows that inhomogeneity increases with initial bed height and decreasing density of the fluidizing fluid (103H). 56
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
es in deep beds can be reduced by dividing the bed into shallow sections and redistributing the gas (78H). T h e interrelation of bed instability, bed expansion, and solids carried overhead is given by Chernov et ~ l (22H). . Momentum transfer i n a threephase fluidized bed has been studied and experiments have been performed comparing movement phenomena with that of a &vophase bed (64“). Two short notes ( X H , 6 3 H ) discuss the disiribution of gas flow in a fluidized bed between the bubble and dense phases. T h e effective viscosity of a fluidized bed measured using a frequency response technique has been reported by Gel’perin et ul. (32H). The viscosity of a heterogeneous liquid--solid system is estimated quantitatively from momentum increase considerations by Kurgaev (56H). Velocity pulsations in the flow due to The particles cause more intensive momentum transfer and a11 increase i n viscosity. Bubble Properties. Stewart (YZH) has summarized the assumptions made in theoretical analyses of the motion of a single bubblr in an infinite medium and compared the various theoretical predictions with experinient. H e disciisses in particular the cloud size and pressure field along the vertical axis of symmetry and the importance of bed viscosity 0x1 bubble shape and rise velocity. Sl’hile the significant aspects of bubble niotiori can be determined, no existing theoretical analysis fits all the facts. I n another paper ( 9 3 H ) , he offers a n explanation for the discrepancy between theoretical analyses and the experimental results of Lockett and Harrison for measurements of changes i n voidage near bubbles rising in two-dimensional gas fluidized beds. Toei et a1. (Y9H) shows that when the distance between bubble centers arranged vertically is less than the sum of their diameters, the lower bubble accelerates toward the upper one causing coalescence. Two bubbles arranged horizontally do not immediately coalesce. Their mathematical model explains their results fairly well. Toei and co-workers ( 9 8 H ) also analyze the coalescence of bubbles generated spontaneously from a simple nozzlc. Changes in bubble diameter and frequency with bed height were measured and then calculated using their model (9911). Agreement of calculated and experimental results were found. These results were presented a t the Eindhoven symposium last year (3311). Shichi, Mori, and Muchi (891-1)find that the relative position of two bubbles in a fluidized bed has a n important influence on gas and solid flows and 011 coalescence. ‘They find as Toei et al. ( W H ) did that coalescence is greatest when bubbles are verrically in line. Gas interchange between bubble and dense phase has been studied by Toei and co-workers (9711). A model for calculating the gas interchange coefficient is given. Yeh and Yang ( 7 0 9 1 ~present ) a n analytic study of the dynamic behavior of a moving gas bubble induced by the flow field with a nonuniform pressure gradient. Experimental data for bubble growth i n superheated liquid nitrogen, bubble collapse i n subcooled liquid nitrogen, and bubble growth with decreasing liquid nitrogen pressure are compared with theoretical solutions for noncryogens ( 3 9 H ) . A method of measuring the rate of change of a rising bubble in liquid, its surface area, rise velocity, and instantaneous mass transfer coefficient is described by Zieminski and Raymond ( I 75H). Surface wave theory is used by Maneri and Mendelson to describe the motion of bubbles in tubes and channels (66H). Rise velocity data for argon bubbles in mercury ( 8 8 H )are reported and shown to be in good agreement with Mendelson’s equation derived using wave theory. The rise velocity of bubbles in viscous liquids is reported by Kojima, Akehata, and Shirai (57H). Nicolitsas and Murgatroyd (7.5H) give a precise method for measuring slug speeds in air-water flows. T h e structure of the wake of a large three-diinensional gas bubble rising in a liquid has been photographed by Slaughter and Ft‘raith ( 9 7 H ) . The wake appears to consist of a toroidal vortex moving with the bubble and a streaming cylindrical tail extending along the rise path. Bubbles generated in liquids from orifices ( 5 3 H ) and from porous plates ( 4 9 H ) are described. The effect of a magnetic field on bubbling processes for a binary mixture where one of the components is magnetic is discussed by htroschenko ( 5 H ) . Two theoretical works describe the properties and history of the toroidal bubble ( 7 9 H )and two-dimensional bubbles in slow viscous flows (83H). Gas Mixing a n d Models. Chung and Wen (2311) have studied longitudinal liquid mixing in packed and fluidized beds using frequency response techniques. Variables include particle size, fluid velocity, void fraction, and particle density. ,4 general-
ized correlation is given for packed and fluidized beds for the Reynolds number range 10-3 to 103. Dispersion coefficients obtained from heat transfer and adsorption studies are reported by Miyaguchi, Kaji, and Saito (68H). The concept of backmixing is discussed by Klinkenberg (50H). He distinguishes between two kinds of backmixing ideas in use. The first is the mechanism by which a phase can move against its main flow (refers to a fixed plane or point) and the second is any mechanism that can cause a spread in residence time distributions (axial mixing-defined with reference to the mean velocity). A model which can be applied to fluidized bed reactors and takes into account backmixing in the first sense above is given by Latham, Hamilton, and Potter (57H). Afschar and Schugerl ( I H ) measure backmixing (second sense) and interphase mass transfer coefficients for three-phase fluidized beds with parallel flow of air and water. The effect of floating spherical packing (polyethylene spheres) on the residence time distribution of gas in a fluidized bed is reported by Goikhman et al. (36H). A helium tracer in an air-sand fluidized bed was used and a two-phase model used to interpret the data. Longitudinal and lateral turbulent mixing in a three-phase fluidized bed has been investigated by Vail, Manakov, and Manshilin (702H). A new bubbling bed model has been introduced by Kunii and Levenspiel (54H)which requires only the effective bubble diameter to describe the gas flow through each phase of the bed and the extent of gas interchange between phases. The model is used to discuss gas-particle heat and mass transfer and catalytic reactions (55U) and to predict residence time distributions (110Zf). A model for dealing with slug flow fluidized bed reactors is given by Hovmand and Davidson (40H). Imai and Miyaguchi (47H) present a generalized equation of continuity useful for two-phase flow accompanied by rate processes. A theory of contact time distributions (73H) and means for measuring such distributions (74H) are discussed in two papers by Nauman and Collinge. The authors point out the fact that in correlating reaction data the contact time rather than the residence time is the really important piece of information required. The measurement technique for contact time distributions is based on using a weakly absorbed species on the solid surface. The flow pattern of fluid in a spouted bed is studied theoretically and experimentally by Mamuro and Hattori (65H). Their theoretical equations give information on spoutability and the efficiency of fluidsolid contact in a spouted bed. Note also the review of mixing in packed and fluidized beds by Gunn (37H). Particle Motion a n d Mixing. Woollard and Potter (708H) have studied the movement of solids associated with the rise of a single gas bubble in ah incipiently fluidized bed. Their results agree in general with those of Rowe but the total volume of material displaced was only 30 to 40y0 of the bubble volume. A pulse technique with particle temperature as the tracer is used for investigating solids mixing by Sandbloom (85H). A diffusion coefficient describes the mixing. Solids mixing diffusivities measured using a radioactive tracer are reported by Pippel et al. (8022). A critical survey of different models proposed for solids transport in gas fluidized beds by Verloop, DeNie, and Heertjes (105H) shows that many models can fit the residence time distribution curve quite well, despite the fact that the physical behavior of the bed is not recognized in these models. Particle movement in a two-dimensional gas fluidized bed is analyzed statistically by Mori and Nakamura (70H). Circulation up the center and down the sides was confirmed. The autocorrelation coefficient of the particle velocity with respect to space and time are given. The spectral density exhibits a maximum in the range of 1 to 2 Hz. Individual particle velocities in liquid fluidized beds are reported by Carlos and Richardson ( I S H ) . Histograms of the velocity components of the tracer particles were Maxwellian and the kinetic energy associated with axial motion was twice that associated with horizontal motion. Circulation was up the center and down the sides. Axial mixing coefficients are given in another paper by these investigators (2OH). An equation for particle velocities in beds of mixed particle sizes is given by Soma and Kondukov (95H). A simplified method for calculating particle trajectories is proposed (48H). Freezing of a fluidized layer with paraffin is used by Jinescu as a method of studying particle movement and for measuring diffusion coefficients of solid particles in the laver (43H). The contact resistance of elecirodes placed in fluidized beds is discussed by Glidden and Pulsifer (34H).
Particle mixing in a packed fluidized bed is reported by Aoki and Yamazaki (4H). Solid circulation in a large fluidized bed is discussed by Zablotny et al. (174H). The effect of the gas distributor on solids mixing and heat transfer in a fluidized bed was investigated by Oigenblik et 01. (76H). Radial mixing is largest near the distributor plate and the axial mixing in the upper part of the bed is 5 to 20 times the radial velocity. Other mixing studies are reported (6H, 8 H ) . Two articles deal with elutriafion of particles from fluidized beds (7H, 38H). Other articles deal with fluidized beds for solids flow control (25H), residence time of dried material in a fluidized bed (70H), and the general mass balances for processing of solids of changing size in bubbling beds (60H). Two mechanisms for particle suspension are described by Shook et 01. (90H)and the similarity of the physical laws of fluidization and suspension are discussed by Weinspach (106H). A method for determining the holdup of granular material in a continuously operating fluidized bed reactor is given by Kalinowski et al. (47H). Pressure a n d Density Fluctuations, Bed Expansion, Holdup, Pressure Drop, a n d Minimum Fluidizing Velocity. Density and pressure fluctuations in gas fluidized beds with and without screened inserts were investigated by Winter (1O7H). The frequency distributions of the density and pressure fluctuations were determined as functions of several variables. They could be described by dimensionless coefficients and +distributions with different degrees of freedom. Variables included distance from the gas distributor plate, bed height, particle size, gas flow rate, and internal screening. A radiometric density meter for fluidized beds which uses a Srao p source and a solid-state detector is described by Schuricht (87H). The relation of density fluctuations to kinetic characteristics in the SO$ oxidation is discussed by Gabuchiya et al. (29H). No dependence was found to exist between bed heterogeneity and the degree of bed fluidization, height of the bed layer, diameter of bed, or tube diameter in a fluidized bed with vertical tubes inside (31H). Geldart (3UH) studied the expansion of bubbling gas fluidized beds and concludes that there is evidence to support the view that visible bubble flow is less than the difference between the total flow and that necessary for minimum fluidization. Bed expansion in screen-packed fluidized beds was found to give straight lines on a Richardson-Zaki type plot by Capes and McIlhinney (18H). Bed expansion in three-phase fluidized beds were studied by Levsh et 01. (61H). The bed height increases with the square of the gas velocity. The maximum spoutable bed depth for beds of mixed particle size reached a peak a t a critical Reynolds number of around 70. The theoretical value is 68 (82"). Pressure drops of screens without packing as a function of gas velocity and liquid rate are given by Levsh et 01. (62H)in connection with the calculation of pressure drop in three-phase fluidized beds ( 6 I H ) . Equations for pressure drop, minimum fluidization velocity, and bed expansion for power law fluids flowing through fixed and fluidized beds are presented by Yu, Wen, and Bailie ( 1I IH). Pressure gradients and velocities of trains of slugs in pipeline flow are reported (16H) as are pressure drops and friction factors for the turbulent flow of gas-solid suspensions (67H). Several papers are concerned with the prediction and measurement of the minimum fluidizing velocity (IOH, I IH, 14H, 71H, 94H, IUOH). Liquid holdup and the minimum fluidization velocity in a turbulent contactor were measured by Chen and Douglas (21H). Conditions for the onset of fluidization for a heated fluid flowing through a tube are described by Kafengauz and Fedorov (46H). Lehmann et al. (59H) describe an experimental investigation in which the dimensions of equipment are varied and the parameters related to the bed depth, amount of turbulence, pressure drop, and bed consistency. Experimental and design data obtained from commercial cap-type gas distributors are reported by Kozin and Basakov (52"). The best gas distribution in a conical bed is obtained with a cone angle of 20 ', An equation for the pressure drop and density change amplitude in terms of the bed properties is given ( I 72H). A method for estimating the apparent and bulk densities of fluidized beds of plastic particles is reported by Belyi and Yurkevich
Porous Media
The question of describing the geometry of a packed bed has been explored and, because of their usefulness in experiment, spherical packings have been popular in these studies. A theoretical treatVOL. 6 2
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men? of the voidage variation with distance normal to a solid boundary i n a bed of uniform spheres has been given by Ridgway and Tarbuck (291). Ben Aim and LeGoff ( 7 1 , 21) have considered binary mixtures of spheres. The effect of particle size distribution on packing density has been investigated by Sohn and Moreland (321) who found that the packing density increases if the particle size distribution is extended. The problem of deviations from Darcy’s law in single phase flow through porous media has received attention during the past year. The mechanisms causing a less than proportional increase i n the volumetric flow rate with pressure gradient in granular media above bed Reynolds numbers of approximately unity have been studied by Wright (381) and Jones (751),while Swartzendruber (361)has commented upon the lesser appreciated phenomenon of a greater-than-proportional increase which has been observed in waterflows through clay soils. Turbulence measurements using a hot wire for air flow in a granular bed by TVright have indicated in a qualitative manner that the onset of turbulence in such media is delayed to Reynolds numbers approaching 100. I n a packed bed with convergent boundaries, the friction factor was determined to be lower than that for a parallel bounded bed a t the same conditions. T h e intermediate flow regime between Reynolds numbers of 2 and 100 was found to have characteristics similar to those in a coiled pipe. Jones has also presented evidence that the departures from Darcy’s law in this range may be due to secondary circulatory flows such as occur in coiled or “S” shaped pipes. I t was demonstrated that an empirical formula describing the increased resistance to flow in a curved pipe relative to a straight pipe can be modified to describe the increased resistance of a granular bed outside the range of Darcy’s law. A theoretical approach to the problem of flow through porous media a t Reynolds numbers where the creeping flow assumption is no longer valid has been taken by LeClair and Ilamielec (782). A surface interaction model for assemblages of spheres with flow a t intermediate Reynolds numbers is solved and the predictions are tested against experimental results. Raats (251) and Raats and Klute (261, 271) have discussed the mechanics and dynamics of porous media using the continuum theory of mixtures wherein all concepts and relationships that are ordinarily introduced in describing single component media are defined for each constituent present. The forces acting on the solid phase with any number of fluid phases are considered (251). Specific consideration is also given to the balance of mass (261)and the balance of momentum (271)in transport through soils using the continuum concept. Liquid Distribution a n d Holdup i n Packed Columns. Porter and coworkers have presented a n extensive investigation of liquid flow in packed columns. I n a series of papers, a theoretical model, termed the rivulet model, is developed and tested against experimental measurements. I n the model (221)rivulets of liquid which have a stable but randomly oriented flow path run over the packing and sometimes run into one another and coalesce and sometimes break up. The model illustrates that diffusion theory may be used to predict the spread of liquid through the packing. At the wall, rivulets from the packing coalesce with liquid on the wall and new rivulets are generated a t a rate proportional to the wall velocity. This assumption leads to the conditions that flow builds up a t the wall to some equilibrium value and that a n exchange of liquid between the wall and the packing takes place. Measurements (231) nf the liquid spread show that the agreement between theory and experiment depends on the sample size, the agreement being best for large samples. Thus the liquid spread can be described by a probability mechanism. Further measurements confirm the existence of a stable rivulet pattern. I n the third paper (241),measurements of the amount of liquid flowing down the walls of two different sized columns packed with Raschig rings were correlated on the basis of the theoretical description of the wall region given in the rivulet model. T h e large exchange of liquid between the wall and the packing postulated in the model was confirmed by means of salt. tracer solutions. Further postulations concerning the nature of the liquid behavior i n the wall region of packed columns have been given by Dutkai and Ruckenstein (81). Here a new boundary condition in which penetration of the liquid into the wall region takes place via an adsorption-desorption mechanism is used. Coupled with the diffusion equation for the spread of liquid, this condition yields a sulution in good agreement with experimental results. Liquid irrigated packed columns have been investigated by 58
INDUSTRIAL AND E N G I N E E R I N G CHEMISTRY
LeGoff and coworkers in an attempt a t determining the nature of the liquid phase, that is, the fraction of liquid trickling through the packing in the form of films, rivulets, and drops (41),as a function of the system parameters. A large number of measurements with both cocurrent and countercurrent gas flow and wettable and nonwettable packing of various sizes have been reported (54,and the evolution of the “texture” of the liquid stream with flow rates and relative flow direction has been described. I n addition, the vclocities and flow rates of the various fractions have been noted (41). Flooded columns with countercurrent gas flows have also been compared (271) with irrigated columns by measuring the holdup and other quantities of interest. The dynamic and static holdup in packed beds with liquid metals have been investigated by Standish. For the dynamic holdup (331)it is found that nonwetting liquids require a different correlation than liquids which wet the packing, but the results for some liquid metals may be predicted from nonwetting water experiments. I n regard to the static holdup tests (351)with liquid metals and aqueous solutions, it is concluded that n o unique relationship between the static holdup of metallic and nonmetallic materials exists. Standish (341)has also described a device for investigating transient changes in liquid holdup in packed beds. Filtration. LeLec (191) has discussed the factors such as overall and local resistances, initial resistance, porosity, and interior flow rate which can cause practical filtration results to differ from classical filtration theory because they may not be considered as constant except in cases of dilute slurries and prolonged processes. Attention is then directed to concentrated slurries and new theories for treating the filtration of such slurries are outlined. Ives (741) has raised several questions concerning the validity of a theoretical model for deep bed filters developed previously by Heertjes and Lerk [see (729)of the Annual Review for 19671 and the experimental procedures used to obtain their data. The reply (721) to these comments helps to clarify several aspects of the earlier work. Dispersion a n d Displacements. A good deal of attention has been focused on dispersion and miscible displacement phenomena during the past year. .4summary of much of the existing experimental data for the longitudinal dispersion coefficient in packed and fluidized beds has been given by Chung and Wen (GZ) and a correlation relating the Peclet number and the Reynolds number has been developed. New experimental measurements of longitudinal dispersion using air as a carrier gas and an Argon tracer have been reported by Edwards and Richardson ( S I ) for a range of Reynolds numbers between 0.008 and 50. An interesting result is the maximum in the Peclet number us. Reynolds number plot which approaches the theoretical value of 2 asymptotically but shows a maximum of about 3. The overshoot may be attributable to radial diffusion which plays an important role in the Reynolds number range near the maximum. Further reports of new miscible displacement data for liquids have been given by Krupp and Elrick (771) for water displacements i n an unsaturated media. The displacement curves show a long “tail” and a shift from normal break-through curves and these are explained on the basis of the stagnant liquid zone concept. T h e idea of bed capacitance has also been investigated theoretically in a number of recent papers. Buffham and Gibilaro (31) have given a solution to the cell model with communicating stagnant zones for any finite number of cells. €Ian (701)has also dealt with a capacitance type model. He considers porous parallel plates and a porous cylindrical tube, the outer boundaries of which are impermeable, and formulates a two-region problem consisting of fluid flowing through the main channcl and displacing solute in the pores of the bounding material by molecular diffusion. Dayan and Levenspiel (71)have further generalized the stagnant pocket model of Turner to include adsorption on the solid surface and were thus able to simulate the separate effects of holdup in the dead end pores and surface adsorption. For the case of a linear equilibrium relation between phases and interfacial equilibrium, they demonstrate that adsorption increases dispersion. A discussion of the scaling laws for comparing experiments on miscible displacements in porous media whose internal geometries are similar has been given by Raats and Scotter (281). Microscopic similarity allows one to compare experimental results even when the macroscopic model does not describe the dispersion adequately. Data from experiments on dispersion in geometrically similar porous media with a n oscillating flow (301)show that the dispersion due to oscillating flow depends strongly on the amplitude of the oscillations.
Hassinger and von Rosenberg ( 7 7 1 ) have given a n extensive numerical and experimental study of dispersion transverse to the main flow in porous media. A numerical solution of the convective-diffusion equation is performed and the transverse dispersion coefficient is determined by matching the mean concentration a t the outlet of their experimental apparatus with a theoretical curve. I t is concluded that the transverse dispersion coefficient decreases as particle size increases for constant values of the product of velocity and packing size. Simultaneous longitudinal and transxrse dispersion between two miscible fluids has also been considered theoretically by Li and Yeh (201). A model for immiscible displacement of water by air in random sphere packings as a function of pressure has been developed by Iczkowski (132). Non-Newtonian Flow. Experiments on non-Newtonian fluids, which can be represented by the power-law model, flowing through fixed and fluidized beds have been reported in (391). Data on the pressure drop, minimum fluidization velocity, and bed expansion are well correlated by extensions of empirical equations for similar quantities for Newtonian fluids. White (371)has considered non-Newtonian flow through stratified porous media by deriving the differential equations linking pressure distributions within a three-dimensional porous bed and a n equation for the stream function in axisymmetric flow in a similar bed. The utility of the model developed is tested against experimental results. Filtration experiments for non-Newtonian fluids have been reported in (762). After developing the equations characterizing the flow of incompressible, time independent fluids exhibiting the “anomalous surface effect,” constant pressure and constant rate filtration expressions are obtained and used to correlate experimental results. Slattery (372) has extended his previous treatment of local averaging in porous media to multiphase flows under the assumption that the phases are in equilibrium with no mass transfer across the phase boundaries. Considering a No11 simple material, the conditions under which inertial effects are negligible compared to viscous effects are discussed. I t is found also that when Darcy’s law is extended to multiphase flow, a n additional relationship is required. Rotating Flows
A closed form solution of the equation of motion for the axial development of the tangential velocity profile in flow through an annulus with the inner cylinder rotating at speeds such that the Taylor number is less than the critical value has been made ( I J ) . The critical assumption involved is that the radially distributed axial velocity component can be replaced by its mean value. A satisfactory comparison with past and new experiments is made but as a consequence of the main assumption, theoretical predictions agree better with experiment for higher Reynolds number flows. For Couette flow with two concentric cylinders rotating i n opposite directions, experiments reported by Snyder ( 2 8 J ) have demonstrated that the waveform a t the onset of the instability may be asymmetric about the axis of rotation. This substantiates earlier theoretical predictions. Some previous experimental results on this system which had seemed contradictory are explained with the new experimental results. I n some cases, Snyder’s experiments show the onset of instability to be three-dimensional rather than two-dimensional in nature. I n another paper ( 2 Q J ) concerned with the stability of Couette flow, Snyder has presented new experimental evidence for wide and medium gap widths and compared his results along with previous data to the recent theoretical predictions of several investigators who avoided the narrow gap approximation. Excellent agreement is obtained in most cases. An analysis of the stability of Taylor vortex flow in a concentric cylinder system with only the inner cylinder rotating, subject to disturbances which are periodic in the azimuthal and axial coordinates and which travel with some phase velocity in the azimuthal direction, has been made by Davey et al. (85). Such a flow is stable under disturbances with the same axial wavelength and phase but is unstable against perturbations differing in phase by 7r/2. The critical Taylor number is predicted to be 8 % above the value a t which vortices appear; experimentally the comparable result is 5 to 2091,. Linear theory has also been used to investigate the stability of flow between eccentric cylinders with the inner cylinder rotating
and under the narrow gap approximation (25J). Over a considerable range of eccentricity ratios, the flow is less stable than in the concentric case. Experiments have been performed on rotating Couette flow in several modified forms ( 2 7 J ) including a square outer cylinder and a square inner cylinder. I n all cases vortices much like Taylor cells are observed when the angular velocity exceeds a critical value. The critical Taylor number is changed by as much as a n order of magnitude by changing the shape of the boundaries. A previous solution to the eigenvalue equation arising from the Kelvin-Helmholtz stability problem in a rotating fluid has been shown to be incorrect by Huppert ( 7 5 J ) . This problem and other theoretical results derived by the same erroneous analysis are corrected and extended. The instabilities caused by small axial shears induced by rotation have been considered by Pedley ( 2 2 J ) and Busse ( 6 J ) . Pedley has demonstrated that a n arbitrarily small axial shear is sufficient to destabilize systems which approximate solid body rotation without invoking the narrow gap approximation which had been done in the past. As an example of the theory developed, Poiseuille flow in a rotating pipe is treated. Busse has considered instabilities occurring as a wave propagating in the azimuthal direction since his work is restricted to Rossby numbers small enough (differential rotation) to ensure that the Rayleigh criterion is satisfied. Critical Rossby numbers are predicted and compared favorably with some available experimental evidence. Unsteady rotating flow has been considered for several different configurations. The unsteady flow period during the spin up of a viscous fluid partially filling a right circular cylinder which is impulsively accelerated from rest to a constant angular velocity has been treated by Goller and Ranov (70J). Expressions for the transient angular velocity and the configuration of the free surface were obtained after making assumptions which could be verified qualitatively in the experiments. Experimental measurement and theoretical prediction of the free surface were in good agreement. I n the spin-up process, Stewartson layers play a n unimportant role. However in other rotating flows, such as the system considered by Barcilon ( 3 J ) ,the nature of such layers contribute to a n understanding of the resulting flow. The transient Stewartson layers formed as boundary layers on vertical solid boundaries in a rotation of a cylinder of fluid about the vertical and the problem of a cylindrical annulus filled with fluid rotating as a rigid body to which additional fluid is introduced by uniform injection and withdrawal along the inner and outer vertical boundaries are considered. Attention is focused on the small Rossby number region. Hide ( 7 7 5 ) has conducted a n extensive general theoretical and experimental study of the nature of source-sink flows introduced through permeable boundaries. Experiments involving a variety of source-sink distributions inside a rotating cylinder of liquid are reported and new areas of investigation suggested. A general analysis of unsteady flow in concentric rotating cylinders with arbitrary initial velocity distribution and time varying boundary conditions has been presented by Farnsworth and Rice ( Q J ) . The result is a closed form expression for the velocity distribution which reduces to known solutions for special cases. The problem of dusty gas flow between coaxial cylinders rotating impulsively from rest has been considered ( 2 0 J )for the cases of both cylinders rotating and one cylinder rotating and the other stationary. Solutions for the mass concentration of the dust are given. Unsteady rotating flows in semi-infinite systems have been treated analytically for different stimuli. Homsy and Hudson (72J) have studied the approach to the steady state for time-dependent flow due to the impulsive starting of a single infinite disk. Particular emphasis is placed on the nature of the overshoot of the steady state by the tangential velocity component while the secondary flow is still developing. The flow generated in a semiinfinite region bounded by an infinite disk when both fluid and disk are in solid body rotation and the disk performs nontorsional oscillations in its own plane has been considered by Thornley (305). The question of the resonant frequency, which is twice the angular velocity of rotation, is given special emphasis. The effect of the presence of a second disk is discussed. I n a similar study, Singh and Sathi ( 2 6 J )have considered the transients caused by the impulsive translation a t a steady velocity of an infinite flat plate in its plane in a system which is initially rotating a t a constant speed about an axis perpendicular to the plate. I t is shown that rotation tends to increase the drag on the plate. I n a study of a rotating disk system with an imperfect semipermeable membrane interface, Zeh and Gill (375) have deVOL. 6 2
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d o p e d a new and versatile technique for solving the momentum equations which is not restricted to constant property systems. T h e problem of laminar flow between two parallel disks, supplied with fluid a t the outer radius having a pressure greater than that a t the circular exhaust hole a t the inner radius, has been considered by Rice and co-workers (4J, 76J). In ( 7 6 J )a n analytical asymptotic or fully developed solution valid a t values of the radius interior to the entrance region is developed and i n ( 4 J ) the entrance problem is solved by numerical means. Laminar flow between two disks, one rotating and the other stationary, has been studied experimentally for both a finite encased system (7J)and a n infinite disk system ( 7 8 J ) . Conover ( 7 J ) considered cases with and without radial inflow and determined velocity patterns by means of a dye tracer. Stability limits as a function of system parameters were noted. Mellor et ai. ( 7 8 5 ) provided numerical solutions for comparison wirh the velocity and pressure distribution data and also discussed edge effects. Rotational effects in a rarefied gas created by a n infinite disk have been studied ( 2 5 ) using a Karman-Pohlhausen type boundary layer analysis with a slip flow boundary condition. As the gas becomes increasingly rarefied, the dimensionless boundary layer thickness increases with increasing slip coefficient. Bretherton and Turner (5J)have reported a n experimental and theoretical study of the mixing of angular momentum in a stirred rotating fluid. Experiments on a flat tank of liquid rotating and stirred by vertical oscillations of a grid showed that the motion generated was dominated by the loss of angular momentum to the walls and to the grid. T h e theory of flow over a corrugated boundary i n a rotating fluid system has been extended (731) to the case where the crosscorrugation velocities are greater i n order of magnitude than the thickness of the boundary layer. Various theoretical and experimental aspects of the problem of rotating fluids with imposed temperature gradients, particularly rotating annuli with the temperature gradient imposed across the annular gap, ha.ve been considered. Pfeffer and Fowlis (24J)report data on wave dispersion observed by traces of aluminum powder. Dye injections were employed by Petree et al. (23J) to measure velocities between a n inner heated cylinder and an outer rotating one. iVhen the upper surface was free, the motion i n the annulus due to rotation tended to counteract natural convection but when the upper surface was i n contact with a solid boundary, a different situation, one in which upper and lower circulation patterns were present, was observed. T h e case of a rotating annulus of a large Prandtl number fluid where the heat transfer across the annulus is dominated by convection has been treated by McIntyre (77J). Merilees ( 7 9 J ) reports an investigation of the effects of depth, annulus size, and Prandtl number on the transition between the Hadley and Rossby regimes i n a rotating annulus of fluid. Axially symmetric flow between rotating disks with heat transfer caused by the upper disk being a t a higher temperature than the lower has been discussed by Hudson ( 7 4 J ) . An analysis of laminar flow through a circular pipe rotating rapidly about a n axis perpendicular to the pipe axis has been made in conjunction with a study of heat transfer in such a system ( 2 7 J ) . T h e secondary flow produced by rotation dominates in the central core and a boundary layer approach is taken for regions near the wall. Non-Newtonian Fluids
In this section we restrict our attention to developments i n the non-Newtonian area which are general and thus find application in many of the specialized areas covered i n this review. I n particular, we describe developments i n property measurements and models for fitting such measurements. Also we include here papers on steady and unsteady laminar flow i n bounded geometries and stability. X series of refresher articles on non-Newtonian fluids including discussions of rheology ( 2 6 K ) , instruments for viscometry ( 2 7 K ) , isothermal laminar (28K), and turbulent flow ( 3 0 K ) i n pipes and flow i n noncircular ducts ( Z S K ) have been given by Wohl. Pipkin ( 7 8 K ) has begun a general description of viscometric flows by defining a special category of such flows which includes all previously known exact solutions. These flows are termed “partially controllable” and include all cases wherein the geometry is such that the normal stress functions do not influence the velocity field, although the viscosity function does. X rectilinear flow generator of a n oscillating type has been de60
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
veioped ( 7 7 K ) for studying the rheological properties and flow characteristics of Kewtonian and non-Newtonian liquids. T h e apparatus consists of two concentric tubes; the inner can be oscillated axially while the external, which is mounted on elastic supports, moves i n response to the motion of the fluid caused by the inner tube. Movement of and drag on the external tube are measured. T h e capabilities of the Maxwell orthogonal rheometer for measurement of material functions have been set forth by Bird and Harris ( 3 K ) . The limitations of a previous analysis of the steady-state operation of this device are pointed out and the analysis is repeated with a different rheological model. A modification of the Zimm-Crothers viscometer for operation a t very low shear levels has been made ( 2 7 K ) wherein the shear stress can be varied without disturbing the set-up. A development significant to the use of torsional viscometers has been reported by Johns (12K). In a n investigation of the equations of motion, it was found that, i n general, curvilinear solutions d o not exist for arbitrary members of the class of fluids described as incompressibIe and simpie for cone and plate and parallel disk geometries. X procedure for fitting non-Kewtonian viscosity data which provides a uniform basis for comparison of parametric information from different experiments has been suggested by Cramer and Marchello (8K). These same authors ( Q K )report the results of fitting several different simple models of non-Newtonian behavior to a wide variety of data a n d thus provide a basis for establishing the relative accuracy which can be expected of each model. Bird and co-workers have proposed ( 2 K ) and tested ( 4 K ) a nonlinear rheological model for polymeric fluids which removes several of the inconsistencies found in previous models. T h e model is expanded to yield a generalized Kewtonian model along with second order correction terms, one of which involses a time derivative of the normal stress that may be important in some unsteady flows. T h e model has been tested with experimental data on six viscoelastic fluids of various properties and found capable of fitting the data over several decades of the shear rate or frequency. Further verification of the ability of the rate equation concept for correlating the rheological properties of polymer solutions has been given by Kim and Brodkey ( 7 4 K ) . Solutions of polymethymethacrylate in diethylphthalate u p to 55‘53 concentration are investigated and tested with the kinetic approach. Additional experimental data on the stresses in concentrated polymer solutions have been tested by Williams (25K)against a previous approximate molecular theory developed by the same author. Further discussion of dimensional considerations in viscoelastic flows regarding the topics of the definition of a characteristic time, the significance of the 1,Veissenberg and Deborah numbers, and the solidlike behavior of viscoelastic fluids during rapid accelerations has been presented by Slattery ( 2 0 K ) . I n particular, it is pointed out that the definition of a characteristic time may be made arbitrarily as long as it is appropriate to the fluid. T h e stability of plane Poiseuille flow of second- and thirdorder fluids has been considered using linear stability theory. Chun and Schwarz ( 5 K ) working with the Coleman-No11 model of a second-order fluid found that the critical Reynolds number dccreases as the fluid becomes more non-Ncwtonian. Although Chun and Schwarz report no experimental stability studies, Jones and ‘CValters ( 7 3 K ) note that stability investigations do exist and are i n disagreement concerning the effect of elasticity on the flow. T o add further information, they investigate the stability of plane Poiseuille flow for a third-order fluid which exhibits a variation in viscosity with rate ofshear in simple shear flow. In agreement with previous analyses for second-order fluids, elasticity effects tend to destabilize the flow. Craik ( 7 K ) has attributed the result of a previous analysis that a layer of elasticoviscous fluid between parallel plate boundaries may be in unstable equilibrium to the inadequacy of the constitutive equation used i n the analysis, An improved treatment shows that the equilibrium is stable whenever the fluid has a fading memory. A theory of the growth and decay of surfaces of discontinuity i n nonlinear, incompressible, viscoelastic fluids has been developed by Coleman and Gurtin ( 6 K ) . A practical application is a new explanation for melt fracture in the extrusion of molten polymers through narrow passages. I t is postulated that such a phenomenon may be due to the propagation of shear-acceleration waves into regions with high rates of shear.
T h e problem of flow development from rest due to a constant axial pressure gradient in a tube for a power law fluid has been studied (79K) by approximate techniques, a n approach necessitated by the nonlinearity of the equation of motion. The dimensionless time required to reach the fully developed state is determined as a function of the power law index. I t has been suggested that one of the reasons that impact tube measurements of non-Newtonian flows with instruments calibrated for Newtonian fluids give erroneous readings in some cases is that a n additional correction for non-Newtonian behavior is needed. Some experimental data (23K) are presented to support this contention. Additional values of the two geometric parameters required for a general relationship between the flow rate and pressure drop for laminar flow of any time-independent incompressible nonNewtonian fluid in a duct of arbitrary cross section have been obtained (22K). These are for infinite and finite square and triangular arrays of cylinders and for regular polygonal and starshaped conduits. Steady non-Newtonian flow in annular ducts ( 7 7 K ) and rectangular and triangular ducts has also been investigated (76K). T h e excess pressure drop above the losses due to well developed flow in the laminar flow of several non-Newtonian liquids, both viscous and viscoelastic, through a sudden contraction connecting two tubes of different diameters has been studied ( 7 K ) . For the data obtained, no conspicuous non-Newtonian or elastic effects are noted. Steady laminar flow of a n elasticoviscous fluid i n a rotating annulus with porous walls where the rate of injection a t one wall equals the rate of suction a t the other wall has been considered ( 7 5 K ) . Two types of approximate solutions are obtained by perturbation expansions, one in the elastic number and the second i n the cross-flow Reynolds number. Ehrlich and Slattery (7OK) have determined the condition under which a power-law liquid is a better lubricant than a Newtonian fluid in a n infinite journal bearing. For a fixed ratio of bearing radii, the results depend upon the power law index and the eccentricity ratio. Natural Convection and Related Flows
I n this section, to a greater extent than in any other section of this review, flow behavior is intimately related with energy and mass transfer. This is so because the more interesting aspects of body force effects very often involve a coupling between momentum transport and the convective transfer of energy and mass. Several analytical studies of laminar free convection around vertical plates have been reported. Brodowicz ( 5 L )has considered the effect of the flow outside the boundary layer and of the leading edge upon the flow inside the boundary layer. A method of approximation, for natural convection a t small Prandtl numbers, which employs transcendental functions to represent the profiles has been described by Cygan and Richardson (QL).Sparrow and De Mello (45L) investigated the effects of transverse pressure gradient and the axial diffusion of momentum pn natural convection on a vertical plate. Catton ( 6 L ) analyzed free convection heat transfer from a vertical plate where the strength of the gravitational field varies linearly along the plate. Vertical and horizontal plate systems a t small and moderate Grashof numbers were studied by Suriano and Yang (46L)by using finite difference techniques. Takhar (48L) has considered a partially heated vertical plate; the heated portion starts a t the bottom edge and the rest of the plate is insulated. Dent ( 7 7 L )has studied a vertical plate vibrating transversely and Kelleher et al. ( 2 7 L ) analyzed the effects of surface temperature oscillations on laminar free convection boundary layers. K a t 0 et al. ( 2 0 L ) have performed a n analysis which predicts turbulent temperature and velocity profiles by using forced convection eddy diffusivity functions. Experimental results of a study of turbulent natural convection on a vertical plate have been reported by Lock and deB. Trotter (30L). Another experimental study by Warner and Arpaci ( 5 4 ) has shown good agreement with a correlation of the Nusselt number with the one-third power of the Rayleigh number for Rayleigh numbers u p to loL2. CheeseWright (7L) reported data on mean velocity and mean temperature distributions in turbulent free convection boundary layers for Grashof numbers between 101o and 10". Laminar free convection from a tapered downward projecting
fin has been studied by Lock and Gunn (37L). They assumed quasi-one-dimensional conduction in the fin and matched this solution to that of the convection system which is treated as a boundary layer problem. Kierkus ( 2 2 L ) has calculated the temperature and flow fields about a n inclined isothermal plate. Several studies of heat and mass transfer in free convection boundary layer have been reported. Lightfoot (2%) analyzed the limiting case where the Schmidt number is much larger than the Prandtl number, which is frequently encountered in liquid phase systems. De Leeuw Den Bouter et al. (70L) measured simultaneous heat and mass transfer in laminar free convection boundary layers on a flat plate. An electrochemical method was used to measure mass transfer and differential thermal analysis was used to measure heat transfer rates. Adams and Lowell (715) analyzed sublimation from a vertical plate and varied the ambient temperature and pressure to determine their effect on the Nusselt and Sherwood numbers. Vanier and Tien (57L) have analyzed the influence of a n anomalous density-temperature relationship and solid-liquid phase changes on natural convection heat transfer from a vertical plate. The ice-water system was studied in detail. A theoretical analysis for combined laminar free and forced convection heat transfer to non-Newtonian fluids is presented by Kubair and Pei (25L). The effect of a magnetic field on the laminar convective flow of a n electrically conducting fluid adjacent to a vertical plate has been studied by Papailiou and Lykoudis (37L). T h e existence of similarity solutions was confirmed by experiments. Evans, Reid, and Drake ( 7 3 L ) studied transient natural convection in a vertical cylinder which was partially filled with liquid and subjected to a uniform heat flux a t the walls. T h e experiments were compared to an analytical model and were found to be i n good agreement. Goldstein and Aung (77L)have studied heat transfer by free convection from a horizontal wire with 0.015-inch diameter to carbon dioxide in the critical region. They concluded that the usual free convection correlations probably can be used even near the critical state providing the properties are suitably evaluated. An experimental study of the effect of vibration on natural convection heat transfer from a horizontal fine wire was conducted by Mabuchi et d.(32L). Three fluids, air, water, and ethylene dioxide, were studied over a range of the relevant parameters. T h e effects of radial curvature on free convection boundary layers on vertical cylinders and cones has been investigated analytically for some special nonuniform temperature differences between the surface and ambient fluid by Kuiken (26L). T h e solution is given as a power series expansion. Laminar natural in a n enclosed rectangular region has been studied by de Vahl Davis (72L) by solving the governing equations numerically. A theoretical analysis of laminar nature convection about a n isothermally heated sphere a t small Grashof number has been done by Fendell (74L). The method of matched asymptotic expansions was employed. Schenk and Schenkels (47L) performed experiments by melting ice spheres in water which was varied in temperature between 0 and 10°C. I n particular, the effect of anomalous thermal expansion was studied. An experimental study of the effects of natural convection on forced convection mass transfer was carried out by Pearson and Dickson (38L). Drops were made of methyl acetate and 2-ethoxyethyl acetate and distilled water was the continuous medium. Lipkea and Springer (29L)measured heat transfer in vertical annuli with walls a t different temperatures. Below certain Rayleigh numbers and length to diameter ratios, heat was transferred from the hot to cold boundary by conduction only. Combined free and forced convection has been studied by several authors. Shannon and Depew (42L) studied combined free and forced laminar convection in a horizontal tube with uniform heat flux. A range of parameters was investigated experimentally and Nusselt numbers 2.5 times those for the uniform property case were found far downstream from the inlet. Mitsuishi et al. ( 3 5 L ) conducted a theoretical and experimental study of convection in vertical tubes with constant wall heat flux. T h e effect of natural convection was much greater i n pure water than i n aqueous sugar solutions and for the latter variable, viscosity was the dominant factor. Sherwin (44L) calculated the effects of natural convection on forced convection in vertical annuli. Reejhsinghani et al. (4OL) studied the effect of natural convection on convective diffusion (miscible displacement) in vertical tubes. Experiments were carried out with heavier fluid on the top and bottom, and VOL. 6 2
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the former are analogous to heat transfer with a colder tube wall than fluid temperature whereas the latter relate to heat transfer from the tube wall to the fluid. Takhar (47L) considered flow in the inlet region on a vertical pipe with significant natural convection effects. Mack and Hardee (33L) studied natural convection between concentric spheres a t low Rayleigh numbers. They considered both velocity and temperature profiles. Sherman (43L) considered the possibility of inducing a convective secondary flow in the fully developed channel flow of a Boussinesq fluid. Instabilities of this type can only occur when the temperature gradient in the direction of the body force exceeds a certain critical value. Birikh et al. ( 3 L ) analyzed the stability of steady flow between parallel plates which are a t different temperatures. The flow becomes unstable to perturbations a t relatively small values of the Grashof number. Gage and Reid ( 7 6 L )considered the stability of thermally stratified plane Poiseuille flow. They showed how the mechanism of instability due to stratification and the Tollmien-Schlichting mechanism due to shear interact. Brian (4L)studied the Marangoni effect, surface tension driven flow, in vertical falling films and horizontal stagnant liquid layers. Zadoff and Begun (55L) investigated the gravitational instaes of a system consisting of two homogeneous fluids with a horizontal interface in the presence of a uniform horizontal magnetic field. Jakeman (79L) investigated the problem of the onset of thermal instability in heated layers of fluid using a linear approximation and assuming the confining planes to be surfaces of constant heat flux. I n the steady state, a limiting point of the neutral stability curve can be obtained exactly by analytical methods. Plows (39L) made numerical calculations for two-dimensional steady laminar Bernard convection. Results are reported of calculations of the Nusselt number for laminar convection in a thin fluid layer between parallel conducting plates. Musman (36L)studied convection in a horizontal layer of water which had its lower boundary maintained a t 0°C and its upper boundary above 4°C. Tien (50L) also has considered the thermal instability of a horizontal layer of water near 4°C. I n such a case the density achieves a maximum within the layer and the author compares experiments with stability theory. Krishnamurti (23L, 24L) predicted that when the mean temperature of a fluid layer is changing a t a constant rate q, hexagonal flows are stable in a range of Rayleigh numbers near the critical and the direction of flow depends upon the sign of q. The theory is tested by experiment. Flow patterns adjacent to horizontal surfaces have been discussed by Husar and Sparrow (78L) and by Leontiev et al. (27L). Husar and Sparrow employed a flow visualization technique which creates changes i n color resulting from changes in pH. Leontiev et al. (27L) showed that near the horizontal heat transfer surface a n eddy cell flow exists. Tanger et ~ l (49L) . studied heat transfer to sulfur hexafluoride near the thermodynamic critical region i n a natural circulation loop and test results were correlated i n a simple formula. Beaver and Hughmark ( 2 L ) obtained heat transfer and circulating data for a single tube thermosiphon reboiler operated under vacuum and a t atmospheric pressure and the results were compared with existing correlations. Finlayson (7515)has applied Galerkins method in a new way to convective instability problems. Both stationary and oscillatory instability can be studied using the same trial functions and the method is illustrated for convective instability of a rotating fluid layer transferring heat. Mitchell and Quinn (34L)studied convection induced by surface tension gradients in heated fluid layers. With steady heating from below a t a point, oscillating temperature and velocity fields i n thin horizontal films were observed. Chen and Whitehead ( 8 L ) describe a n experiment designed to examine the band width of finite amplitude steady state modes in a fluid heated from below. Veronis (5315)has investigated numerically large amplitude Bernard convection i n a rotating fluid. Two Prandtl numbers, 0.2 and 6.8 were used for the range of Taylor numbers from 0 to l o 5 and Rayleigh numbers extended a n order of magnitude from the critical value of linear stability theory. Veronis (52L) also studied the effect of a stabilizing gradient of solute on thermal convections. H e found that a finite amplitude instability may occur first in fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to or greater than unity, in62
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stability occurs first as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with large heat flux. laminar Flow Pas1 Submerged Objects
Inviscid Flow. Relatively little new work on potential flow per se, which is of interest to chemical engineers, was reported in the literature in 1968, but applications of inviscid flow theory continued to occur associated with heat and mass transfer and hydromagnetic flow. I n a theoretical vein, Von Simmgen (735M) and Von Kraemer (734M) provided mathematical analyses of flow over slender bodies or wings. The former studied the periodic flow over an underwater wing and obtained integral representations of the acceleration potential and the velocity potential. For large Froude numbers, the integrals developed can be expanded in series. Von Kraemer considered the momentum integral of the Trefftz-plane, finding that the total momentum of any incompressible potential flow decaying a t infinity is indeterminate within the limits established in the paper. A problem of some interest in nucleate boiling and some mass transfer operations, that of the growth of a sphere in a semi-infinite liquid and attached tangentially to a plane solid surface, was examined by Witze et al. (744.U). They developed a solution for the potential flow in the flow field around the bubble. Cole and Aroesty ( 2 0 M ) analyzed the inviscid flow field over a slender body when the boundary layer associated with hypersonic or supersonic flows is blown off the body by high rates of mass addition a t the surface. Applying their two-layer model, they matched the inviscid flow outside the injectant layer to the inviscid flow within it. Fan and Ludford (33M) studied the effect of a magnetic field on the potential flow of a conducting fluid past a slender body and determined that a steady hydromagnetic flow occurs. Kidd ( 6 5 M ) developed a solution for potential vortex flow adjacent to a stationary surface. Michael ( 8 3 M ) used potential flow theory to determine the effect of a dusty gas on the steady flow past a sphere. He computed the drag force on the sphere and determined a criterion for the collision of particles with the sphere, i.e., that the Stokes number is greater than 1/2. Slow Flows. The interest in the motion of solid and fluid spheres in bounded and unbounded viscous fluids remains high, and attention has been given to various aspects of the motion of spheres. I n addition, some studies of unsteady flows around spheres and associated with oscillating plates have been reported. I n a fundamental study, Ockendon ( 9 7 M ) determined the drag associated with the unsteady motion of a sphere, finding that any small Reynolds number expansion of the unsteady flow field will only be valid for all times provided that the time scale upon which the velocity depends is sufficiently large. The drag for such large time scale values was shown to differ significantly from the drag predicted by the unsteady Stokes equations. Unfortunately, the theoretical predictions are not supported by experimental evidence to resolve the differences. Steinberger et ~ l (779M) . studied the drag and other characteristics of the motion of two falling spheres, one falling directly above the other in a viscous fluid for Reynolds number in the Stokes regime. The upper sphere fell faster and had associated with it a drag coefficient significantly smaller than the lower sphere. The results are partly in conformity with Oseen’s theory, but not consistent with other theoretical considerations. Van der Leeden and Van den Hurk (737iU)used slow flow approximations to explain and predict their experimental results on the drag associated with flow past an ordered stack of spheres. I n another paper dealing with drag on a particle with Stokes flow, Edwards and Papadopoulos ( 3 7 M ) developed a perturbation solution to calculate the effects of del iations from the spherical shape. O’Brien ( 9 6 M ) studied the Stokes flow solutions for a variety of shapes by semianalytic numerical methods to determine the effects of the deviation from a sphere on the drag. I t should be pointed out that Pan and Acrivos ( 7 O O M ) dealt with the related problem of the deviations of a fluid bubble from spherical. Rimmer (7~181.4)extended Acrivos’s earlier theoretical analysis of heat transfer from a sphere to obtain the Nusselt number as an expansion i n the Reynolds and Prandtl numbers valid for N R , < 1 and for ArFr of O ( 1 ) . O’Neill (99.44) derived an exact solution of the linearized Stokes flow equations for viscous flow about a sphere attached to a plane wall when the undisturbed fluid motion is a uniform linear shear flow. Several papers dealing with the motion of bodies confined in a cylindrical tube have been published recently. Greenstein and
Happel ( 4 6 M )considered the rotation and translation of a spherical particle in a fluid in a circular cylinder, the undisturbed flow being Poiseuillian, and they presented approximations for the frictional force, torque, and permanent pressure drop caused by the obstacle. Hide et al. ( 5 5 M ) , Moore and Saffman ( 8 8 M ) , and Maxworthy ( 8 0 M )all considered the motion associated with the Taylor column, the disturbance due to the slow rotational motion of a n object in a fluid. The former work is an experimental and theoretical study of the phenomenon; Moore and Saffman calculated the drag of axisymmetric bodies rising through a rotating fluid, and Maxworthy provided experimental work in partial support of the theoretical results of Moore and Saffman. I n a pair of papers dealing with the effects of mass transfer on the fluid mechanics Fendell et al. ( 3 8 M ) and Fendell ( 3 7 M ) performed theoretical investigations of flow past a vaporizing droplet and past a burning droplet. They predicted that the Stokes drag is affected by the mass transfer and depends on the Schmidt number. Richardson (707M)described a method of transforming a problem of two-dimensional Stokes flow into a boundary-value problem in analytic function theory, but the application of his analysis of a two-dimensional bubble is probably quite limited. Gupta ( 4 7 M ) studied the rotation of axisymmetric bodies in a conducting fluid to elucidate the coupling of viscous and magnetic effects, and Nakano and Tien ( Q 7 M )used Galerkin’s method with a variational principle to obtain an approximate solution of creeping flow of a non-Newtonian power law fluid over a Newtonian fluid sphere. The unsteady viscous flow past a circular cylinder was studied by Ingham ( 5 9 M ) who used a numerical method to study the starting flow. Blumberg and Mohr ( Q M )reviewed the theory of Stokes flow around cylindrical particles, and they presented data showing the effect of particle orientation on the settling velocity of cylindrical particles in a bounded fluid. Hodnett ( 5 7 M ) and Dennis et al. ( 2 3 M )examined the problem of heat transfer between a circular cylinder and a viscous flow. Hodnett treated compressible variable property flow with small temperature differences between the cylinder and the free stream, and the latter investigators compared experimental data with the theoretical solution based on the Oseen type linearization of the heat transfer equation. The comparison was unfavorable. Better results were obtained using the correct velocity distribution and numerically integrating the energy equation. A number of studies involved Stokes flow or Oseen flow near flat plates. Liu and Jasper ( 7 6 M ) solved the Stokes equations for laminar flow normal to a plane surface with mass injection or removal at the surface. Miyagi ( 8 7 M ) studied Oseen flow past a flat plate set perpendicular to the undisturbed flow, calculating the drag coefficient by numerical methods. Nicoll et al. ( 9 3 M ) and Panton (707M)studied fluid motion near an oscillating plane. The former investigation included the effects of mass transfer at the surface. Flow around Drops and Bubbles. The literature related to flow around fluid objects such as drops and bubbles is becoming quite extensive. Of particular interest to the chemical engineer are studies of heat and mass transfer to drops or bubbles, the growth of such objects and the basic fluid mechanics. This section will be primarily concerned with the latter, for the bubble dynamics are covered elsewhere in the review. The problem of the behavior of a fluid sphere in the vicinity of a flat fluid interface was examined by Bart ( 5 M ) . He analyzed the motion of a solid or fluid sphere settling in the presence of a plane interface between immiscible fluids. Several limiting conditions were examined. Shima ( 7 7 7 M ) provided a potential flow analysis to determine the behavior of a spherical bubble collapsing in the vicinity of a solid wall, but the tedious computations involved are probably not justified because of the neglect of viscosity and surface tension effects. Three papers deal with oscillatory motion of bubbles or drops. Foster et al. ( 3 9 M ) and Rubin (770M) developed theoretical analyses of bubble behavior in vertically vibrated liquid columns; however, there is little overlap between the analyses, and virtually no overlap in their references. Both theoretical analyses claim agreement with experimental data and predict bubble trajectories. In a fundamental study, Miller and Scriven ( 8 4 M ) theoretically considered the oscillations of a droplet of one viscous fluid immersed in another viscous fluid for the cases of a free interface and a n inextensible interface between the fluids, and they discussed the mechanism of damping of the oscillations in some detail. Pan and Acrivos ( 700M) provided a theoretical analysis and experimental
data on the deformation of a spherical drop or bubble in translational motion through an unbounded, quiescent, viscous fluid. They developed a perturbation scheme expanding the stream function in the liquid phase Reynolds number, the zeroth order terms being the Hadamard-Rybczynski solution. For the smaller bubbles observed, the agreement between theory and experiment is remarkably good. I n yet another fundamental study of bubble shapes, Pedley (703M) considered the motion of a toroidal bubble through a quiescent liquid. The bubble model is that of a ring vortex, and the author examined the bubble motion for both an inviscid ambient fluid and a viscous fluid. The capture of small diffusing charged particles by a much larger charged droplet in an electric field was investigated theoretically by Zebel (747M). Equations for the capture efficiency of the droplet were developed. Thorsen et al. (727M) and Gal-Or and Waslo ( 4 2 M ) studied the terminal velocities of drops; the former experimentally measured terminal velocities of circulating and oscillating drops; and the latter developed a theoretical analysis of the effects of neighboring particles and surfactants on the terminal velocity of an ensemble of falling drops. Harper and Moore ( 5 0 M )developed a boundary layer analysis for the motion of a spherical liquid drop which applies when the viscosity of the fluids inside and outside of the drop are small, the interfacial tension is high, and the interface is free of surface active agents. The flow outside the sphere is taken as potential flow and that inside the sphere is Hill’s spherical vortex flow. Balashov et al. ( 4 M ) , in a rather nonrigorous analysis, developed equations describing the unsteady flow of gas about dispersed droplets and they compared the analysis with data on flow from an atomizer. Among the work dealing with heat and/or mass transfer from drops or bubbles are four papers of a somewhat fundamental nature. LeBlond ( 7 7 M ) dealt with the problem of diffusion from an ascending gas bubble, and he elaborated on the effects of diffusion, translation of the bubble, and the surface tension on the volume and pressure of the bubble. Another primarily qualitative analysis of transfer between a fluid sphere and the ambient fluid is Abraham’s theoretical analysis of heat transfer to a falling water drop ( I M ) , but Abraham does not adequately account for the effects of translation on the radial temperature gradient. Tokuda et al. (72QM)dealt more rigorously with the similar problem of heat transfer to or from a spherical gas bubble in translational motion in a gas stream. Their theoretical predictions compared favorably with experimental data. Winnikow (743M)derived analytical expressions for the steady rate of heat or mass transfer from a fluid sphere at large Reynolds and Peclet numbers, and he compared the results with the transfer rates predicted from the model of an ideal fluid. The transfer rates predicted from viscous flow theory were considerably less than those obtained assuming an ideal fluid. Lastly, Cheh and Tobias ( 7 5 M ) used a boundary layer approach to study mass transfer to rising drops or bubbles in a viscous liquid. The Nusselt number was shown to be a function of the Reynolds and Peclet numbers and the viscosity and density of the system. Favorable agreement between the theory and experiment was found. Their solution reduces to the potential flow solution for large Reynolds numbers. Particle-Fluid Relative Motion. In many applications of interest, the laminar flow past a solid object involves Reynolds numbers in a range between the Stokes or Oseen flow regimes and inviscid flow. This section deals with such flows. Jenson et al. ( 6 0 M )reviewed some of the elementary theory related to the calculation of drag coefficients, and they discussed the Reynolds analogy for momentum, heat, and mass transfer. They compared and summarized experimental and theoretical results for the pressure distribution on a sphere, drag and skin friction coefficients, and the Chilton-Colburn heat and mass transfer j-factors. Selberg and Nicholls (773M) measured the drag coefficient of spherical particles accelerating in the convective flow behind the shock wave in a shock tube, and they compared their results with conventional drag coefficient curves. Not surprisingly, the results for smooth sapphire balls compared favorably with well established results, but the H P 295 ball power used proved to be rough, yielding high drag coefficients. I n another paper on drag coefficient measurement, Alger and Simons ( 3 M ) correlated their experimental data for a series of irregularly shaped particles of gravel and other material for Reynolds numbers up to 400,000. They introduced a new shape factor which produced better correlations than the Corey shape factor. Cercignani and his associates (72M, 7 3 M ) used a variational principle to obtain the drag on an axisymmetric body with a rarified gas flow, and their calculations for the drag
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on a sphere were shown to be i n good agreement with available experimental data. I n a n empirical study, Noordsij and Rotte (95M)measured and correlated mass transfer data for a simultaneously rotating and translating sphere. They correlated their data for the Sherwood numbcr in terms of the Reynolds and Schmidt numbers. Berg et al. (7M) measured local rates of mass transfer from composite spheroids i n a wind tunnel by a photographic technique involving multiple exposures of the film to provide the history of the shape. Feldman and Brenner ( 3 6 M ) reported measurements of the pressure drop created by a sphere settling in a fluid confined in a large diameter circular cylinder for 0.2 5 N R 5 ~ 21,000. The results agreed with theoretical results in the Stokes and Oseen regimes, decreasing asymptotically to the limiting value predicted by elementary momentum principles for an unbounded fluid when A'R~ > 125. Doig ( 2 6 M ) studied the suspension of spheres in a vertical tube for 100 5 ATRe 4 25,000, and his results are consistent with those of Feldman and Brenner, for he found that the drag coefficient for suspension in a bounded fluid is greater than that for suspension in a n unbounded fluid. Maxworthy (79M)reported his experimental studies of the pressure distribution around a sphere placed in aligned magnetic and velocity fields. He developed a model of the phenomenon consistent with his findings that the drag is increased because of a loss in total pressure along streamlines just outside the surface boundary layer. Houghton ( 5 8 M ) and Tunstall and Houghton (730M) considered particle retardation by sinusoidal fluid oscillations. T h e latter paper concerns experimental measurements of the retardation of falling spheres in a n oscillating water column. For larger spheres, the measured velocities were lower than theoretically predicted values; and for smaller particles, experimental values were higher than predicted values. I n another paper on oscillating flows, Chen and Wirtz theoretically studied the flow past an oscillating Joukowski foil ( 7 6 M ) , and Sugano and Ratkowsky ( 7 2 0 M ) presented the results of an experimental study of mass transfer from vertically vibrated horizontal cylinders of naphthaline and phenol to air. The mass transfer coefficlent is increased greatly over that associated with a nonvibraring cylinder. Hartland ( 5 7 M ) continued his studies of bubble and particle motion in the vicinity of an interface with an experimental study of the approach of a rigid sphere to a deformable liquid-liquid interface. He observed that the draining film a t the interface is symmetrical and has nearly constant dimensions during the draining period. Odar (98M)performed careful and mechanically well-designed experiments on the tangential and normal forces on a sphere moving along a circular path in a viscous fluid, finding that the conventional drag coefficient can be applied to the rotational motion. Smith (778M) measured the spatial distribution of freely falling spheres a t low Reynolds number and for the rather small solid volume fraction of 0.025. The observed spatial distribution agreed well with a random distribution analysis. I n a short note, Hasinger (52M) reported on some observations of particle motion i n a free vortex. A number of papers related to the motion of suspended particles have been published recently. Bourgeois and Grenier (TOM) attempted to correlate incipient fluidization with the terminal velocity of the particles in the fluidized bed, and they provided a semi-theoretical correlation. Sengupta ( 7 7 4 M ) made estimates of the deviations from Stokes flow for a spherical nonconducting charged particle and suspensions of such particles settling in an electrolyte solution by applying Booth's theory. The effect of the charge is to decrease the sedimentation rate. Nienow ( 9 4 M ) developed a graphical correlation for determining the impeller speeds required to suspend particles in turbine-agitated vessels, and he commented on the fluid flow pattern and particle distribution throughout the vessel. A correlation for the air velocity required to suspend spheres i n a vertical tube using an air stream containing smaller dispersed particles was developed by Doig and Roper ( 2 7 M ) and compared with a theoretical model. A theoretical study of the motion of arbitrarily-shaped solid particles was provided by Cox and Brenner ( 2 7 M )to determine the lateral migration of particles in laminar flow through vertical tubes. Some qualitative aspects of the theoretical results are elaborated upon. Wang et al. (737M), assuming that the motion of suspended particles can be described by a diffusion model, solved the convective diffusion equation to describe the distribution and deposition of particles suspended between parallel plane surfaces and undergoing simultaneous settling and diffusion. I n a series ofpapers, Mason and his associates studied various aspects of 64
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particle motion in suspensions (74M, 2 2 M ) and the flow of suspensions through tubes (IZZM, 723M). The latter two papers deal with particle interactions and radial migration in pulsatile flow, and the former pair treat particle motion in shear suspensions and in electric fields. Several papers on flow fields and heat and mass transfer associated with the flow around cylinders were published in 1968. Zdravkovich (, 7 4 6 M ) obtained some remarkable photographs of the laminar wake behind a group of three cylinders. The vortex streets are clearly shown. Hieber and Gebhart (56M)used the method of matched asymptotic expansions to study forced convection heat transfer from a circular cylinder a t low Reynolds numbers. The theory for moderate Prandtl numbers agreed well with available experimental data. Sarpkaya ( 7 7 7 M ) provided a theoretical analysis of the forces acting on a circular cylinder with separated flow about the cylinder. The results of the theoretical analysis compared favorably with data on the characteristics of symmetric vortex separation. I n another theoretical paper, Chen et al. (78iM) developed the equations for the flow in a falling cylinder viscometer with eccentricity, and they solved the equations to determine the effect of eccentricity on the fail velocity. They also developed an approximation method to apply the results to power law non-Newtonian fluids. I n a somewhat related paper, Blumberg and Mohr ( 9 M ) discussed the theory and presented new data on the effect of particle orientation on the terminal velocity of cylindrical particles settling i n the Stokes flow regime. Boundary Layer Flows. T o effectively review the extensive literature related to boundary layer theory of interest to chemical engineers, it seems proper to exclude large segments of the literature that might be of only tangential interest to that audience. I n this discussion papers dealing primarily with hypersonic and supersonic flow, turbulent boundary layers, the interaction of boundary layers with shock waves, and flow over wings and airfoils have been excluded, but research on the application of boundary layer theory to problems involving heat and mass transfer is considered to be germane to chemical engineering. The recent papers that are primarily concerned with mathematical techniques and the mathematics associated with boundary layer theory include work by Serrin (775M),Lew (73M),Kendall and Bartlett ( 6 4 M ) , and Lee and Fan (72,M). Serrin provided a detailed analysis of Jeffrey-Hamel flow, Lew briefly discussed finite difference schemes in the solution of the boundary layer equations, and Kendall and Bartlett used a numerical method to predict the behavior of laminar boundary layers within a general equilibrium chemical environment. The latter used an integral approach for the primary dependent variables, relating other terms analytically to the primary variables, and eventually solving the resultant set of algebraic relations by a Newton-Raphson technique. Lee and Fan developed a quasi-linearization technique to solve the boundary layer equations. A number of papers involved the quest for a similarity solution, and a note on the application of group theory to similarity analysis was written by Moran and Gaggioli (89M). They applied their analysis to a class of compressible boundary layer flows. Mills (85M) investigated the similarity of laminar incompressible boundary layer flow numerically by treating the problem as a n integral equation in Crocco variables. Comparisons of the numerical results were made with Polhausen's exact analytical solutions to the incompressible boundary layer equations. Hansen and N a (49M)examined two-dimensional laminar boundary layer flo* of a general non-Newtonian fluid, and they found that, in general, similarity solutions exist only for a 90' wedge. Numerical results were presented for a Powell-Eyring fluid. I n another study of non-Newtonian boundary layer flow, White (740M) determined that for the motion of a rigid sheet moving through a stagnant non-Xewtonian fluid a similarity transformation is sometimes possible, l'v'illbanks ( 147M) discussed similarity solutions for a problem of diffusion and flow in a free boundary layer. Variations on the theme of boundary layer flow on a flat plate continue to be of interest. Varzhanskaya (732M) solved the boundary layer equations for a flat plate numerically and calculated the flow in the laminar wake behind the plate, and ilckerberg (2.44j, using Mises variables, obtained numerical solutions for the flow of a film down a semi-infinite vertical plate to determine how the final asymptotic shear flow develops. Tokuda (728121) analyzed the unsteady laminar boundary layer flow induced by the impulsive motion of a semi-infinite flat plate. Contrary to earlier investigations of the problem, he found a power series solution using suitable variables which avoided the essential singularity in the solution encountered previously. Schetz and O h ( 1 7 2 M )
developed an approximate analysis of the same impulsive motion problem but for a nonisothermal system for a Prandtl number of one. They developed a solution by using the unsteady momentum integral equation. Martin and Long ( 7 8 M )provided a theoretical and experimental study of the slow motion of a flat plate through a linearly stratified salt-water mixture and obtained a similarity solution when the diffusivity of the salt and inertial forces were neglected. Experimental results validated the similarity solution for the cases in which the theoretical assumptions applied, and the results indicated the presence of a thin diffusion boundary layer. Taitel and Hartnett (727M)theoretically studied the interaction of convection and radiation associated with laminar boundary layer flow over a flat plate when the net heat flux a t the wall is specified. They obtained a thin boundary layer approximate solution, a similarity solution for an optically thick boundary layer, and a n exact solution. Several papers involved boundary layer problems associated with cylinders. Cebeci et al. ( 7 7 M )studied the second-order effect of the transverse curvature on skin friction ahd heat transfer i n laminar flows past a slender circular cylinder. When the radius of the body is of the same order as the boundary layer thickness, they found the effects to be very significant. Dunham (Z8M) applied Spalding’s unified boundary layer theory to compute the lift generated on a circular cylinder with its axis normal to the flow by slot-blowing round the upper surface. Experimental results partly confirmed the analysis. Kuiken ( 6 8 M ) provided a theoretical investigation of the radial curvature effects on axisymmetric free convection boundary layer flow for vertical cylinders and cones for certain temperature differences between the surface and ambient fluid. For the Prandtl numbers considered, the surface temperature on a cylinder was found to be lower than on a flat plate for the same heat flux. Dimopoulos and Hanratty ( 2 5 M ) and Furuya and Nakamura ( 4 7 M ) carried out experimental investigations related to boundary layer theory for cylinders. The former measured velocity gradients a t the wall for flow around a circular cylinder confirming the front stagnation point and separation predicted by boundary layer theory for Reynolds numbers greater than 150. They used Hanratty’s electrochemical technique involving a rectangular electrode embedded in the surface of the cylinder. Furuya and Nakamura measured boundary layer velocity distributions on a rotating cylinder with a hemispherical nose in the axial stream. They made some comparisons of their results with momentum integral boundary layer calculations. Some technical notes and a longer paper have been published recently on the subject of compressible boundary layers. McLeod and Serrin ( 8 2 M )discussed the mathematical properties of similar solutions including the effects of heat transfer, blowing, suction, and slip a t the wall on velocity overshoot i n the boundary layer. Devan ( 2 4 M ) used integral relations rather than the more computer-time-consuming finite difference techniques to solve the compressible laminar boundary layer equations, and Lewis (74M) extended an earlier analysis relating high speed and low speed flow through a general transformation of the boundary layer equations. Wells and Blumer (739M) reported the results of velocity profile and displacement thickness measurements on a blunt body, and their comparisons with existing theoretical results for the static pressure distribution and the displacement thickness were very favorable. Recent interest i n three-dimensional boundary layer flow consists i n methods of obtaining accurate solutions to the boundary layer equations and applications to problems involving heat and mass transfer. Dwyer (3OM) used a n implicit finite difference technique to study flow over a flat plate on which stands a semiinfinite circular cylinder with its axis normal to the flow. Large cross flow effects were predicted. Fannelop ( 3 5 M ) reduced the three-dimensional boundary layer equations to a sequence of twodimensional equations by means of a perturbation scheme, and he applied his technique to study the boundary layer flow on a blunted cone a t angle of attack. He also predicts large crossflow effects. Karabelas and Hanratty ( 6 2 M ) used Hanratty’s electrochemical technique to measure the direction of surface velocity gradients in three-dimensional boundary layers. T h e electrochemical technique has proved to be very accurate and useful i n numerous fluid mechanical applications. A considerable amount of work has been devoted recently to the study of boundary layer flows with heat and mass transfer. By mass transfer, it is usually meant that suction or blowing occurs a t the solid surface associated with the boundary layer flow. Elliott
(32M)developed a general theory of the two-dimensional boundary layer with strong blowing, dividing the boundary layer into a nearly inviscid inner layer and a thin viscous outer layer. H e applied the analysis to flow past a circular cylinder with a constant blowing velocity, claiming better results than previous analyses. Kubota and Fernandez (67M) used the same general method as Elliott to study boundary layer flows with large mass injection and heat transfer. They constructed matched asymptotic expansions for each layer, and they reported simple expressions for heat transfer rates, skin friction, and derived approximations for integral properties of the boundary layer. I n one of the few papers dealing with mass transfer in the usual chemical engineering sense, Hayday et al. ( 5 4 M ) treated the problem of flow past a catalytic surface for certain surface reaction rate models. T h e gas in the boundary layer was assumed to be a chemically frozen mixture. Results were compared with other theoretical predictions i n the literature. Laganelli et al. ( 7 0 M ) used the model of a laminar incompressible boundary layer on a flat plate to study the effect of the boundary layer growth preceding a transpiration surface. Although significant effects on the skin friction coefficient over the transpiration surface were predicted, further study was concluded to be needed. Using a n integral method, Bethel ( 8 M ) developed a n approximate solution for compressible laminar boundary layer flows around two-dimensional and axisymmetrical bodies with mass transfer, and the results were i n good agreement with predictions from other methods. A general asymptotic suction solution for the axisymmetric boandary layer problem and a new class of exact solutions of the axisymmetric boundary layer equations with mass transfer were discussed by Morduchow and Libby (QOM),and Kassoy ( 6 3 M ) published a note on the calculations of mass transfer (blowing) effects in high Prandtl number boundary layer flows. Their results are somewhat related to those of Kubota and Fernandez mentioned above. Fox et al. (4OM) reviewed various methods of solution to the boundary layer equations for flat surfaces including the effects of suction and mass injection, and they compared the methods quantitatively. Granville ( 4 5 M ) developed an iterative method to obtain approximate analytical solutions for suction and mass injection problems, and Mirels and Welsh ( 8 6 M ) obtained numerical solutions for the velocity and thermal boundary layer a t the stagnation point with a large wall-to-freestream enthalpy ratio. I n addition to mass transfer effects, Wazzan et al. (738M)studied the effects of slip a t the surface, and they presented results for various combinations of the two parameters. For the boundary layer flow of water past a flat plate and over a circular cylinder, Poots and Raggett ( 7 0 5 M ) carried out numerical solutions of the appropriate equations for flows with heat transfer and adverse pressure gradient. They took into account variable fluid properties. Libby and Sepri ( 7 5 M ) extended a n earlier analysis of multicomponent laminar boundary layers to describe the conditions a t a n axisymmetric stagnation point, and the results of exact calculations were compared with a “cold wall” approximation and on a model for massive blowing. T h e influence of temperature dependent viscosity on the stability of a laminar boundary layer was analyzed by Hauptmann ( 5 3 M ) using a perturbation procedure for small variations in viscosity. Thompson and Snyder (726M)published a note on the laminar boundary layer flow of a n Ostwald-de Waele non-Newtonian fluid with uniform wall suction. Pergament and Cess (704M)examined the problem of the boundary layer flow of a gray radiating gas across an isothermal black plate, and they developed limiting solutions for the heat transfer. A few studies of the laminar boundary layer associated with magnetohydrodynamic flow have been made recently. Ramamoorthy (706M) used the Karman-Pohlhausen method to examine the influence of the electromagnetic forces within the boundary layer, resolving problems already treated by other techniques. Gilman and Benton ( 4 4 M ) theoretically studied the effect of an axial magnetic field on the Ekman suction velocity. T h e magnetic field was found to inhibit Ekman suction. In the magnetohydrodynamic flow over a rotating disk, Pao (102M) predicted a thickening of the flow boundary layer and a reduction of the strength of the axial flowfield. H e used similarity assumptions to reduce the boundary layer equations to ordinary differential equations which were solved numerically. Another paper involving mass transfer associated with a magnetohydrodynamic flow was that of T a n and Wang (724M) who examined flow past a flat plate. For blowing and mild suction, the effects of the magnetic field were to decrease skin friction and heat and mass transfer coefficients. VOL. 6 2
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Velkoff and Ketcham (733M) analyzed the effect of an electrostatic field on boundary layer transition. Numerous other studies of boundary layer flows have been made that do not fall into the groupings given above. Among the more fundamental studies is the analysis of Chen and Libby (77’M) of a boundary layer with a constant pressure gradient parameter and an initial velocity profile close to a Falkner-Skan profile. Rott and Ohrenberger (709M)studied the boundary layer equations for rotating flow about the axes of symmetry over an axisymmetric surface, developing transformations of the governing equations to permit certain generalizations. Nickerson ( 9 2 M )analyzed boundary layer adjustment of a neutrally stratified boundary layer to a sudden change in surface roughness as a n initial value problem, and he used a numerical technique to solve the appropriate nonlinear equations. The boundary layer flow over a longitudinally curved oscillating surface was examined by Von Teipel (736M) using a perturbation technique and similarity solutions. Gyadguadatte et ~ l (.4 8 M ) used an integral method to solve the equations for a wedge-shaped surface, taking the variable viscosity into account. Sheridan ( 7 7 6 M ) used a finite difference scheme to solve the equations for flow about a sphere and an ellipsoid, and Fannelop ( 3 4 M ) dealt with the problem of streamwise vortices on laminar boundary layer flow. Gaster ( 4 3 M ) and McCormick ( 8 7 M )theoretically examined the stability of the laminar boundary layer by considering the development of wave packets or patches under the boundary layer. Kuiken (69M) applied boundary layer theory in his study of free convection flows a t extreme Prandtl numbers, and Johns ( 6 7 M ) used boundary layer methods for the determination of the viscous damping of small amplitude gravity waves. Among the numerous shorter papers on boundary layer flows are the note by Williams ( 7 4 2 M ) on nonsteady flow in the vicinity of either a two-dimensional or an axisymmetric stagnation point, the finite difference calculation of unsteady leading-edge boundary layer flow of Dwyer ( 2 9 M ) and notes by Clayton and Massey ( 7 9 M )and Taulbee and Pate1 (725M)on flow over curved surfaces ; I n another note, Zakkay and Alzner (745M) studied the problem of the external pressure distribution associated with a constant wall shear stress boundary layer flow. Although most of the recent boundary layer studies have been theoretical, a few experimental papers have been published. Lyamshev and Rudakov ( 7 7 M ) measured the wall pressure fluctuations in the boundary layer on a rising submerged structure; Bellhouse and Schultz ( 6 M ) measured fluctuating skin friction in air with heated thin-film gauges, and Knapp and Roache (66iM) used a smoke visualization technique and hot wire anemometry to study boundary-layer transition. The experiments of Hanratty and his colleagues have been mentioned above. Jets
Ackerberg ( 7 N ) considers the injection of a two-dimensional jet into a uniform stream, the fluids being assumed inviscid and incompressible. TYhen the total head of the jet is much larger than that of the uniform flow, the motion is characterized by two disparate length scales, and uniformly valid asymptotic solutions can be found by the method of matched expansions. A novel feature of the analysis is the necessity of imposing a logarithmic singularity as an ‘inner’ boundary condition for the outer solution in the external flow. I n a second paper, Ackerberg (ZAV)describes the injection of a high speed jet into a uniform flow of lower total head when the jet injection angle is small. An investigation of the interaction of a jet of air impinging normally on a liquid surface has been carried out by Rosler and Stewart (751V). In this study, the interaction between the jet and the liquid is characterized up to jet velocities causing dispersion. The influence of the surface tension of the liquid is determined and compared with the model analysis. Clarke ( 6 N ) reports an analysis concerned with the steady, symmetric, two-dimensional flow of a viscous, incompressible fluid issuing from a n orifice and falling freely under gravity. T h e emergence of a jet of liquid from a two-dimensional channel in which there is Poiseuille flow far upstream, the flow being driven by a n applied pressure gradient is treated by Tillet (761%‘). Vallentine (18N)analyzes the radial spreading of a vertical j e t released some finite distance above a horizontal plane. The problem is subdivided into flow with and without surface resistance. Experiments by Tobolskii ( 7 7 N ) on dispersion of viscous jets under the cooling effect of a high speed j e t of air intersecting them a t 66
INDUSTRIAL A N D ENGINEERING CHEMISTRY
a right angle shows that the air may (a) chop off the viscous jet, (b) split it into fibers, or (c) disperse it into scalelike particles. T h e experiments were performed with jets of liquid blast-furnace slag. Lindow and Greber (73” analyze the flow of a laminar, twodimensional, incompressible jet flowing over a curved surface by a similarity transformation and a perturbation about the straight surface solution. The laminar flow of a wall jet over a curved surface is considered by Wygnanski and Champagne (20,V). A unique similarity solution is obtained for both concave and convex surfaces when the local radius of curvature is proportional to $14. Chervinsky and Lorenz (5&V)analyze the flow of a compressible or heated rotating jet far downstream from the orlfice and in the neighborhood of the axis of symmetry. Expressions for the axial and radial velocity and enthalpy distributions are derived. Similarity solutions of the boundary layer equations are obtained by IVyganski and Fiedler for the flow ofjets in a n external stream and tailored pressure gradients (27iV). The solutions apply to jets in both coflowing and counterflowing streams. Crowley ( 7 N ) studies the growth of waves on a falling jet under the assumption that the fractional change in the jet velocity is small. Several approximate dispersion relations for the limit of very fast and very slow growth of the instability resulting from a normal electric field are given. The stability of a cylindrical jet of a perfectly conducting, incompressible, and inviscid fluid in the presence of an axial magnetic field is considered by Chakraborty (4iV)for both axisymmetric and asymmetric disturbances. \Vang ( 7 9 N ) has analyzed the effect of finite amplitude on the stable and unstable states of a column of an ideal fluid of circular cross-section under the action of surface tension. A third-order theory has been developed by Yuen (22X) to study capillary instability of a liquid jet. Lienhard (72N) has developed a method for approximating the transverse velocity component in the incompressible twodimensional boundary layer equations which makes it possible to include gravity and surface tension effects in the problem of liquid jets leaving Poiseuille tubes. Kozicki and Tiu ( 7 I N ) present a n analysis of the expansioncontraction behavior of laminar viscoelastic non-hTewtonian jets which extends previous analyses by consideration of the anomalous behavior, as characterized by an effective velocity of slip, exhibited by the fluid a t the solid-fluid interface. The effect of slot height and slot turbulence intensity on the effectiveness of the uniform density, two-dimensional wall jet is investigated experimentally by Kacker and Whitelaw ( 7 O N ) . I t is concluded that the ratio of slot lip thickness to slot height is the most significant parameter and that the result of a n increase in this ratio is to decrease the effectiveness. Mikami and Takashima (74X) report an interesting study on the separation of a gas mixture in an axially symmetric supersonic jet. The supersonic jet is separated into the peripheral stream and the core stream by a skimmer and the total diffusion flux of the lighter component into the peripheral stream is obtained. The experimental results are presented for the binary gas mixture of hydrogen and nitrogen. The dynamics of a heated free jet of variable viscosity liquid a t low Reynolds numbers was investigated, both analytically and experimentally by Glicksman ( 9 N ) . Predictions of the jet shape, the temperature distribution, and the tension in the jet as a function of the material properties and the process variables were obtained. Measurements of the jet shape and the tension distribution in the jet were made for various values of the flow rate, the collecting drum speed, and the nozzle temperature. Duda and Vrentas (8“vr) present a rigorous analysis of jet hydrodynamics in order to develop a technique for determining diffusion coefficients from laminar liquid jet absorption experiments. It is concluded that the laminar jet experiment is a rapid, accurate method of obtaining diffusion coefficients of dissolved gases in liquids, The mean drop size generated by a liquid jet penetrating a gaseous environment is estimated by Rdelbcrg ( 3 N ) . Primary drop generation is treated for the case when the unstable waves grow and are dominated by capillary action as well as acceleration forces. The theoretical results are reasonably consistent with experimental data in the literature. Turbulence
A book by Beran ( 7 7 0 ) entitled “Statistical Continuum Theories” serves to bring together several areas of current activity -turbulence, flow through porous media, theory of heterogeneous materials, and theory of partial coherence-by developing the general physical concepts and mathematics common to all. His last chapter on turbulence provides a summary of the older theories
based on mixing length ideas through Kolmogorov‘s hypothesis to the newer approaches of Kraichnan, Wyld and Shut’ko, and of Hopf. Some experiments were carried out which involved turbulent phenomena and are not readily located in the categories which follow; these are listed here. Turner (7300) employed a n apparatus which permitted mixing across a density interface between two layers of liquid. A direct comparison, by dimensional analysis, between heat and mass (salinity) transport is performed in the paper. Cooper and Wolf (300) measured the velocities of the fluid (air and water) emanating from a turbine-type impeller by using a hot wire anemometer probe and two- and threedimensional pitot tubes. Effects upon these velocities of the impeller rotational speed and dimensions were investigated. Tucker and Reynolds (7280)measured turbulent energies along the centerline of a wind tunnel in which uniform straining occurred. They analyze their results and make comparisons with the earlier work of Townsend. Mobbs (800) found that the unstrained free turbulent flow generated by a composite grid possessed a distinct boundary of virtually homogeneous intensity which separated nonturbulent from turbulent fluid. Arguments are given which suggest that the mean flow of a turbulent fluid is irrotational far from the boundaries. Gough and Lynden-Bell (460) attempt to demonstrate the phenomenon by an experiment which involved the generation of turbulence by dissolving reacting Alka-Seltzer in a floating, rotating beaker. Other experiments include those of van Atta and Chen (1320) who used recent, fast computing methods to determine that the joint probability density function for velocity fluctuations a t two points separated i n time are nonGaussian; Grant et al., in two papers, (470, 480) discuss measurements made to study turbulent flow and temperature structure in the open sea. Liu ( 7 2 0 ) comments upon the earlier measurements of Chuang and Cermak who used a n electrokinetic probe technique. Theoretical investigations and applications of recent theory are present in the literature of last year. Philip (950) considered the influence of a known velocity structure upon particle locationi.e., the probability density distribution function governing particle position. He makes use of the statistical notion of ensemble of particle releases subdivided into subensembles characterized by the particle a t the instant of release. He considers the application to the problem of the individual plume. Banerjee, Scott, and Rhodes ( 8 0 ) relate free liquid interface mass transfer to the rate of viscous dissipation in the turbulent flow near the surface. Kovlsy (640) discusses “life-time’’ distribution functions associated with eddy surface renewal models. Baldwin and Haberstroh (60) tested Phillips mechanism for the manner in which turbulent components support Reynolds stresses in turbulent shear flow. They made use of previously published statistical properties a t the centerline of a pipe to evaluate a constant appearing in Phillips’ formulation. Gibson (420, 430) makes use of the idea of production, destruction, and motion of points of zero gradient and surfaces of minimum gradient magnitude to analyze turbulent mixing of passive scalar properties, e.g., temperature, concentration. H e then develops similarity hypotheses for describing the fine structure of such scalar quantities mixed by turbulence. Orszag and Kruskal (880) formulate a theory of homogeneous, isotropic turbulence in an incompressible fluid. Chemical reactions i n turbulent fields are discussed by Corrsin (370) in a note on some properties of the Taylor diffusion problem and by O’Brien (830-850) who discusses closure, the direct interaction hypothesis, and the Lagrangian history direct interaction approximation in treating a second-order chemical reaction. Discussions of theory involving small scale structure of turbulence include those in the following papers. Sutton ( 7 7 7 0 ) gives a relation for the decay rate of passive scalar fluctuations in terms of the decay rate for velocity fluctuations for isotropic turbulence with a - 5 / 3 spectrum. Leith (690, 700) presents a diffusion approximation for the transfer of inertial energy and for spectral components of mean-square fluctuations of a scalar quantity in a n isotropic turbulent field. Tennekes (7240) gives a general approximation for the spectral transfer of turbulent energy a t high wavenumbers. I n another note (7250), he models small scale structure as vortex tubes which undergo stretching. Pao (900) reformulates a continuous spectral cascading concept for turbulent energy and scalar quantities which was presented earlier. Crow (330) seeks to determine under what conditions shear flow phenomena may be regarded as occurring i n a continuous viscoelastic
medium. Deissler (350) analyzes turbulence and longitudinal heat transfer for axisymmetric accelerating or decelerating flows by neglecting triple correlation terms in the basic equations. He (340) also shows that under certain conditions of weak turbulence and stable stratification the eddy conductivity and viscosity can become negative. Wall Region Flow. An accurate description of the nature of the flow very near to the solid boundary continues to be a challenging problem for researchers. Clark (270) measured fluctuating velocity components and frequency spectral distributions of fluctuating velocity components in a wind tunnel of aspect ratio 12 for Reynolds numbers 10,000 to 130,000. Particular attention was given to viscous sublayer behavior for which he found
.$;
= 0.4 toy’ = 7
and which he considers to be in accordance with the results of Mitchell and Hanratty. Sherwood, Smith, and Fowles ( 7090) measured instantaneous axial and circumferential components of both velocity and turbulent intensity in the wall region t o y + = 0.2. A flow visualization technique was employed using microphotography of tracer particles. Differentiation of the axial component data gave values of eddy viscosity which are several times greater than are commonly assumed to apply. Implications of this are discussed a t length giving rise to a modified momentum-heat transfer analogy. Armistead and Keyes ( 4 0 ) measured fluctuations in heat transfer in the wall of a pipe in which water flowed with Reynolds numbers ranging from 11,000 to 170,000 by using a hot film sensor. Results compared well with the data of Hanratty using wall mounted mass transfer sensors. Bellhouse and Schultz (700) calibrated a thin film heated-element probe for the determination of fluctuating skin friction. Karabelas and Hanratty (590) describe how the direction of surface velocity gradients, such as may be encountered in three-dimensional boundary layers, may be measured by employing the electrochemical techniques previously used by Hanratty and coworkers. Dimopoulos and Hanratty (360) show how these techniques may be used in the determination of wall velocity gradients for studies of flow about solid objects-no calibration is required and the object is rasily fabricated. Achenbach (70) measured local pressure and skin friction distribution around a cylinder in cross flow to a Reynolds number of 5 x 106. Total drag, friction drag, and pressure drag were calculated. Shear stress was determined by making use of a small solid, transverse projection into the boundary layer and pressure differences were measured. Other discussions concerning turbulence and transport a t the wall are those by Clamen and Gauvin (260) and by Hughmark (550). T h e former authors review experimental studies concerned with the influence of turbulence on heat and mass transfer from a particle in a fluid, and the latter discuss some published mass transfer data in relation to the wall region eddy diffusivity expression of Son and Hanratty. Boundary Layer Flow. Studies of the turbulent boundary layer which involve ordinary two-dimensional and three-dimensional geometries and also surface roughness include the following. Telles and Dukler (7220) show that the law of the wall and Coles law of the wake are derivable from a small perturbation type of solution of the general boundary layer equations for turbulent flow. Also, Spalding’s approximation is the zeroth order solution of the energy equation. Prahlad (970) discusses the validity of the similarity function, f, in incompressible, three-dimensional turbulent boundary layer theory. H e describes experiments in which Preston tubes of varying diameter and yaw sensitive probes were used. Lockwood (730) numerically solves the momentum equations by considering a variation of the Prandtl mixing length across an equilibrium turbulent boundary layer. Nickerson (820) studies the problem of a neutrally stratified boundary layer subject to a sudden change in surface roughness. This is similar to the problem treated earlier by Panofsky and Townsend. Blackadar and Tennekes (730) consider surface roughness in the flow of a large (planetary) boundary layer. Singular perturbation methods are used. Compressibility characteristics not including shock phenomena are included in the work of Maise and McDonald (750) who determined the effect of Mach number upon mixing length and eddy viscosity for the turbulent flat plate boundary layer. They emVOL. 6 2
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ployed the generalized velocity method of van Driest. For Mach numbers zero to 5, the effects are small. Bradley (760) solves the governing equations for a compressible, turbulent boundary layer. H e assumes that the component of flow normal to the direction of the inviscid streamline is small which decouples the cross flow momentum equation. Lewis (770) analyzes compressible boundary layer flow involving a pressure gradient and heat transfer by employing a notion of Coles for establishing a direct correspondence between a high speed and a low speed flow. Miles and Kim (780) evaluated the Coles theory for predicting drag on smooth plates in a compressible flow system and found it to be competitive with any other method. Komar ( 6 2 0 ) presents a correlation for the virtual origin and for variation of momentum thickness with length for the turbulent compressible boundary layer. Studies of turbulent boundary layers with finite flow occurring a t and normal to the wall follow. Simpson and Whitten ( 7 7 2 0 ) employed Preston tube techniques for turbulent flow on a flat plate with blowing and found that the wall shear stress may be determined even though the wall velocity is not accurately known. Stevenson (7760) discusses differences i n the results of two research efforts to measure skin friction as a function of Reynolds number for flow over porous surfaces with transpiration. H e suggests that different types of porous surfaces should be studied. Bradshaw (770) suggests a refinement of the work of Rosenbaum and Margolis to learn more of the \kcous sublayer i n transpired flow. T h e vorticity in asymptotic boundary layer flow generated by prescribing a spanwise variation in suction velocity is considered by Kelly (600). Tl’all shear stress and heat transfer were computed. This type of flow is important in unstable boundary layers phenomena. Boundary layer transition region phenomena are involved in the work of Knapp and Roach (670). They employed smoke visualization and hot wire anemometry to study the transition to turbulence of a boundary layer on a pointed nose, axisymmetric cylinder. Interesting photographs and explanatory diagrams are included. McCormick (770) analyzes the formation and motion of turbulent patches in the transition flow of zero pressure gradient boundary layers. T h e fluctuating properties are space and time dependent. Patel and Head ( 9 7 0 ) deduce a criterion for the onset of reverse transition in terms of the mean shear stress gradient in the wall region. Other studies which may be considered as related to turbulent boundary layer flows are those of Cheesewright ( 2 2 0 ) who experimentally measures temperatures and velocities i n turbulent natural convection boundary layers; Sandborn and Liu (7070) experimentally and analytically study separation and report that the separation model of Sandborn and Kline was demonstrated; Good and Joubert (450)measure pressure distributions in a boundary layer containing a bluff flat plate (fence); and Crow ( 3 2 0 ) solves the vorticity equation as applied to the Rayleigh problem (infinite, flat accelerating plate immersed in a n infinite fluid) and finds that certain wall speed programs generate boundary layers that correspond to Clauser’s equilibrium boundary layer. Duct Flows. Studies of duct flows of homogeneous fluids usually attempt to improve our understanding of the transport properties of the turbulence. Rosler and Prieto (7000) investigated the flow of air and of water, a t a Reynolds number of 50,000 (based on centerline velocity) through the same PVC pipe to check upon the extent of correctness of Reynolds similarity. The axial turbulent intensity profiles were found to diverge as the wall is approached. Clark (270) studied flows for Reynolds numbers from 10,000 to 130,000 in a wind tunnel of aspect ratio 12. This work is discussed further i n the section FI7allRegion Flow. Webb (7340)gives a brief, elegant analysis showing the minimum information necessary to calculate the flow fieid over a wide range of Reynolds numbers for both pipe and channel flows. Turbulent flow in paral!el geometries is considered by Szablewski ( 7 2 0 0 ) who used dimensionally consistent formulas for the eddy viscosity for the region near the wall and a constant eddy viscosity-matched with the former-in the outer region. Experimental data from the literature are well described. Capps and Rehm ( 2 7 0 ) give an empirical equation to calculate the velocity distribution and eddy momentum diffusivity profile for turbulent flow in smooth and rough pipes. Results compare well with the data of hTikuradse. Tennekes ( 7230) describes attempts to use singular perturbation methods to analyze turbulent pipe flow; he concludes that the crucial part of the analysis involves the description given to wall layer flow. 68
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Investigations of turbulent flows in geometries other than those of the straight pipe or parallel plates include that of Quarmby (980). He used von Karman’s similarity hypothesis and Deissler’s sublayer profile to predict the Reynolds number and radius ratio dependence of the u + - y’ profile. T h e analysis is carefully carried out and the dependence of the location of maximum velocity as it depends upon the radius ratio and Reynolds number is obtained. Clump and Kwasnoski ( 2 8 0 ) also used the eddy diffusivity relationship of Deissler and von Karman with parameters modified as required. When the location of maximum velocity ( r m ) obtained by Brighton and Jones was used, velocity distributions from experiment were obtained. This was not the case when the rm location for laminar flow was used. Bourne, Figueiredo, and Charles (740) obtained friction factor data experimentally for the flow of water through a n annulus of unit eccentricity for Reynolds numbers from 200 to 20,000. They conclude that the transition to turbulence is not sharp and a minimum i n friction factor occurs a t a diameter ratio of 0.75. Bowlus and Brighton (750)analytically considered the entrance region for a n incompressible, turbulent fluid. They employed the 1/7 power mean velocity distribution and their resulting equation agreed with published data to within lO”/C. Tunstall and Harvey ( 7 2 9 0 ) found experimentally that turbulent pipe flow through a mitred (sharp) right-angle bend produces a downstream circulation which does not conform to the twin-circulatory flow usually observed in pipe bends. Rather, the flow was dominated by a single circulation about the axis with a low frequency switching from one rotational direction to the other. Korkegi and Briggs ( 6 3 0 ) develop a turbulent flow model for plane Couette geometry starting with von Karman’s mixing length model. They considered both compressible and incompressible flows. Consideration of turbulent flow in which the transition region is involved is the subject of studies described i n the follcwing papers. Badrinarayanan (50) carried out experiments to study the reverse transition i n two-dimensional channel flow. Velocity fluctuation properties were measured over the height of the channel and with distance downstream-where the duct is wider i n the lateral direction-he concludes that a definite critical Reynolds number of 1400 3z 50 exists for reverse transition. In the experiments of Eckert and Rodi (380),air was injected into a porous tube into which a fully developed turbulent flow entered. T h e flow appears to become laminar and then, further dowcstream, reverts back to turbulence. They explain this as due to the stabilization effects of the pressure drop in the axial direction being of greater influence than the destabilization effect of the fluid injection. Patel and Head ( 9 7 0 ) deduce a criterion for the onset of reverse transition in terms of the mean shear stress gradient. Even though their reasoning was based on earlier measurements of boundary layer flow, they believe the criterion may be applicable to duct and channel flows. Hershey and Im ( 5 2 0 ) determined critical Reynolds numbers for water i n sinusoidal flow in glass tubes. They found that the critical Reynolds number decreases for increasing frequency parameter values. Hanks ( 4 9 0 ) modifies a constant in the Gill and Scher expression for mixing length and gives it additional significance. T h e expression shows excellent friction factor and average to maximum velocity ratio behavior. T h e roughness of ducts enters into the considerations of Robertson, Martin, and Burkhart ( 9 9 0 ) who carried out experiments using 8-in. “natural” pipe and 3-in. sand-roughened pipe. Pressure drop, near-wall velocity profiles, and turbulence profile are presented for pipe Reynolds numbers in the range 1.3 X l o 4 to 28 X 104. These experimental results offer new information on turbulence structure. Cohen and Hanratty ( 2 9 0 ) studied air and liquid cocurrent flow in an enclosed 12-in. X 1-in. channel. They describe the effects of the wave nature upon the drag on the interface and note results in terms of roughness parameters. Hicks and Mandersloot (530)make use of the momentum-heatmass transfer analogy to consider heat and mass transfer in systems with turbulence promoters. Smith, Gowen, and Charles ( I 740) measured turbulent flow temperature profiles and heat transfer coefficients i n rifled pipe and considered the efficiency as based on Stanton number-friction factor ratio.
Turbulent Jets and Wakes. Experiments involving turbulent jets include those of Wood (7380),who carried out an approximate analysis based on the known properties of a free turbulent jet. Experiments were performed in conjunction with this analysis in a turbulent jet of water flowing horizontally through a cylindrical or rectangular vessel. At zero time, the fluid was changed from a dilute sodium chloride solution to pure water, and outlet coneen-
tration was obtained as a function of time. Foss and Jones (400) measured flow velocity profiles and total and static pressures for an air j e t issuing from a nozzle of aspect ratio 6 . Results indicate a s condary flow structure. Kacker and Whitelaw (580) determined the impervious-wall effectiveness of a two-dimensionless wall jet. They used a constant value of lip thickness (0.032 in.) and four slot heights (0.5 to 0.074 in.). They conclude that the ratio of lip thickness to slot height is the most significant parameter-an increase in this ratio decreases the effectiveness. Miles and Shih (790) describe experiments to obtain values of the similarity parameter required in the analysis of two stream j e t mixing. Lamb and Bass (680) review recent experimental results for developing free shear layers and compare theories. Wygnanski and Fiedler (7390) obtained similarity solutions of the boundary layer equations for flow of jets i n a n external stream with tailored pressure gradient. These solutions apply to both coflowing and counterflowing streams. Agreement with the experiments of Gartshore is good. Barcilon ( 9 0 ) obtained solutions which describe buoyant jets in calm stratified atmosphere. Plume behavior is discussed in terms of the base Froude number. Chow (250) discusses the solution of the j e t mixing problem which is applicable to the turbulent j e t with eddy diffusivity as a function of streamwise coordinate only. He uses Meksyn’s observation that flow properties are rapidly decaying functions of the coordinate normal to the solid surface; for the jet, the coordinate normal to the jet boundary is used and solutions on either side are matched. Schetz (7030) discusses eddy viscosity models i n mixing jets. He modified Clauser’s model for axiqymmetric flow and compares with experiments i n which the streams were of different temperature and density. T h e diffusion processes occurring in the mixing of coaxial streams is considered by Zakkay, Sinha, and Fox (7470). A different diffusion model should be used for each of thc three mechanisms involved in the mixing process. Discussions of jets in rotating systems and along curved streamlines are included in the following papers. Chervinsky (230) treats similarity in swirling jets. Two scales are possible depending on whether the flow is controlled by linear or angular momentum. Pedley (940) obtains similarity solutions for turbulent jets and plumes directed along the axis of rotation of a body of fluid ill solid-body rotation. Wyngaard et al. (7400) revise conclusions p~. eviously drawn from studies of the production, dissipation, and advection characteristics of curved mixing layers. Uchida and Suzuki (7370) analyze the mixing of a turbulent half-jet along a curved streamline. Equations of motion referred to streamline coordiiiates are simplified by boundary layer approximations and integr 4 ted on the assumption of similarity. A maximum velocity in the profile across the stream results for convex flows. Heidmann and Groeneweg (570) infer certain dynamical properties of liquid j e t breakup. Adelberg ( 3 0 ) presents a n analysis for the mean drop size for primary drop generation resulting from the injection of a liquid j e t into a high speed stream. Two expressions are derived: one for the capillary wave regime and one for the acceleration wave regime. Acrivos (20) et al. present results of experiments performed to study steady separated flow past a variety of bluff objects. These were carried out in a closed loop oil tunnel with a test section of Plexiglas which permitted visual observation of the generated wakes. I n all cases the experimental data were consistent with the theoretical model of Acrivos, Snowden, Grove, and Peterson. Chevray (240) studied the turbulent wake behind a 6 to 1 prolate spheroid for a Reynolds number of 2.75 X 106 a t 12 sections downstream-O.25 to 18 diameters downstream. Reynolds stresses were measured. The wake was established within a distance of 3 to 6 diameters. Gibson, Chen, and Lin (440) employed hot film anemometry and conductivity and thermistor probe techniques to measure temperature and velocity fluctuations i n the wake of a sphere in a water tunnel. T h e “central continuous turbulent region” was confined to a small recirculation zone just behind the sphere. Energy dissipation and temperature variance decrease approximately as x-2.4 for 60 diameters downstream. Smith (7750) conducted experiments with a circular cylinder in two-dimensional flow to show the influence of the unsteady wake upon the flow and heat transfer near the forward stagnation point. T h e velocity fluctuations found were large and heat transfer was not influenced greatly. Hanson and Richardson (500) used hot wire anemometry to study the flows in the zone of separation for two spheres with Reynolds numbers of 10,600 and 53,000. Baldwin and Sanborn ( 7 0 ) present some hot wire results of far wake turbulence for low speed axisymmetric flows. Intermittency is
identified as due to large eddy ‘%potiness” rather than “gross fluctuations of the wake plume.” Sarpkaya (7020) studied the forces acting on a circular cylinder by time-dependent cross flow making use of a potential flow model. Trentacoste and Sforza (7260) review the similarities between three dimensional wakes and jets-particularly centerline velocity, half width growth, and velocity irregularities. Sutton ( 7 7 8 0 ) considers the effect of chemical reaction fluctuations i n the theory of species density fluctuations in turbulent wakes. The treatment employs the bimodal model of a chemically reacting turbulent wake which assumes immediate partial mixing of stream tubes as they enter the wake. He also studies the rate of mixing of the reactants to the molecular scale (7790). T h e visualization methods used in the following three references are worthy of note even though they relate to turbulent wakes only marginally. Magarvey and MacLatchy (740) photographed the fall of single drops of organic liquids containing finely powdered water soluble dye through quiescent water. Zdravkovich (7420) used a smoke visualization technique to study the laminar wake and transition to turbulence behind a group of three cylinders. Slaughter and Wraith ( 7 730) employed a shadowgraph technique based on refraction properties of glycerol solutions. They studied the structure of the wake behind large gas bubbles. D r a g Reduction a n d Related Phenomena. An excellent review of the state of our knowledge and recent research in the topic of drag reduction due to polymer solutions, soap solutions, and solid particle suspensions is provided by Patterson, Zakin, and Rodriguez (930). Another discussion of this topic concludes that our understanding is incomplete and involves only that viscoelasticity is the major particular property of the fluid involved
(540). Wells et al. (7360) carried out measurements of turbulence intensity and spectral distribution of turbulent energy for a pipe flow of O.O5a/, C M C solution and for pure water. They found differences in the distribution of frequency-dependent turbulent energy over the range of Reynolds numbers studied, approximately l o 4 to 1.5 X 106. Patterson and Zakin (920) assumed a linear Maxwell model to approximate the viscoelastic response of a polymer solution and developed an expression for the reduction of turbulent energy dissipation when drag reduction is displayed. They tested their hypothesis by obtaining normal stress data measured with a jet thrust apparatus and suggested a reduction would occur i n heat and mass transfer in viscoelastic fluid flow systems. Wetzel and Tsai (7370) made impact tube measurements using a laboratory towing facility which contained a solution of polyethylene oxide. T h e tube coefficient depended upon concentration beyond a critical concentration which, i n turn, depended upon impact tube geometry. Oliver and MacSporran (870) determined the flow a t which transition from laminar to turbulent flow occurred for pure liquids and for solutions of water soluble polymer. Transition delay was dependent upon tube diameter. Explanation of the drag reduction phenomenon is offered in several papers. Vleggaar, Dammers, and Tels (7330) theorize that the energy associated with small scale eddies is converted to recoverable elastic energy rather than being degraded into heat. Tanner (7270) uses a Burgers-type system of equations approach to describe drag reduction. He replaces the viscous term i n the model equations by a linear viscoelastic term to demonstrate the technique as offering a possible model of the drag reduction phenomenon. Kozicki and T i u (650) attribute a n effective velocity of slip a t the wall to a n increased laminar sublayer thickness. They compute sublayer thicknesses from available experimental data and show this to be a possible description and suggest a reason for the increased thickness to be associated with preferred orientations of the polymer molecule. Wells (7350) makes use of the dimensionless velocity dependence upon shear stress attributed to Meyer and of the analogy between momentum and energy transport. H e describes heat transfer characteristics of drag reduction systems, and his results compare well with published results. Flow of Suspensions of Particulates. T h e work described in this section relates to pneumatic and hydraulic conveying, separation methods which deal with solids--fluid flows and other processing methods involving a suspended particulate phase other than fluidization. Panton (890) formulates the flow properties of gas-particle mixtures in which mass transfer may be occurring. H e shows that meaningful definitions of the flow properties of each phase can be constructed in terms of certain averaged flow properties. Reynolds VOL. 6 2
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stresses are important and provide a significant particle phase viscous effect. Trezek and France (7270) carried out experiments i n a blow-down apparatus to study accelerating particulate, choked flow in a circular duct. Results are presented as a correlation of dimensionless pressure us. dimensionless downstream distance. Selberg and Nicholls (1070) used shock tube techniques to study the drag of small, accelerating spherical particles in laminar incompressible continuum flow. They found Mach number to be involved even though it was low (.,
(41A) Srinivasan P S., Nandapurkar, S. S., and Holland F. A “Pressure Drop and Heat Tranifer in Coils,” The Chemical Engrneer, 218, d E 113Ll9, (May 1968). (42A) Terrill, R . M., “Fully Developed Flow in a Permeable Annulus,” J . Appl. Mech., 35E, 184-6 (1968). (43A) Test, F . L.,“Laminar Flow Heat Transfer and Fluid Flow for Liquids with Temperature Dependent Viscosity,” J . Heat Transfer, 9OC, 385-93 (1968). (44A) Vermeulen, J. R., and Holmes, D. B., “Creeping Flow in a Duct of Rectangular Cross Section Provided with a Partition,” Cham. Eng. Sci., 23, 77-8 (1968). (45A) Wadhwa, Y . D., and Wineinger, T. W., “Linear Time-Dependent Fluid Flow Problems,” Quart. AppI. Math., 26, 1-9 (1968). (46A) Williams F A “Linearized Analysis of Constant-Property Duct Flows,” J . Fluid Mech.: 34,243-61 (1968). (47A) Zielke, W., “Fre uency-Dependent Friction in Transient Pipe Flow,” J . BasicEng., 90D, 109-19 (1968).
Two-Phase Flow (1B). Alia, P., “Phase and Velocity Distribution in Two-Phase Adiabatic Annular Dispersed Flow,” Energia Nucl. (Milan), 15, 241-54 (1968). (2B) Andeen, G. B., and Griffith, P., “Momentum Flux in Two-Phase Flow,” J . Heat Transfer,90,211-22 (1968). (3B) Anisimova, M. P., et al., “Energy Losses in a Two-Phase Flow D r e to Mechanical Interaction of the Phases,” Inzh. Fiz. Zh., 15, 436-43 (1968) (Russ.). (4B) Baroczy, C. J., “Pressure Drop for Two-Phase Potassium Flowin Throu h a Circular Tube and a n Orifice,” Chem. Eng. Progr. Symp. Ser., No. E%,, 64, 12-25 (1968). (5B) Biasi L., et al., “ A Theoretical Approach to the Analysis of a n Adiabatic Two-Phke Annular Dispersed Flow,” Energia NucI. (Milan), 15, 394-405 (1968). (6B) Boiko, L. D., “Hydraulic Resistance with Condensation of Pure Steam and Steam from Steam-Gas Mixture in a Horizontal Tube,” ibid., p p 49-56. (7B) Bolotov, A. A,, et al., “Research into the Flow of Mixtures of Steam and Liquids in Vertical Pipes,” Thermal Eng. (Teploeneigetika), 14, 113-19 (1967). (8B) BourC, I.,et a!., “Self-Sustained Oscillations in Heated Two-Phase Flows,” La Houille Blanche, 22, 551-58 (1967). (9B) Brown, F. C., and Kranich, W. L.,“A Model for the Prediction ofvelocity and Void Fraction Profiles in Two-Phase Flow,” A.1.Ch.E. J . , 14, 750-53 (1968). (10B) Chawla, J. M., “Frictional Pressure Loss in the Flow of Liquid-Gas Mixtures through Horizontal Pipes,” Forseh. Ing.-Wes., 94, 47-54 (1968) (Ger.). (11B) Chawla, J. M., and Thome, E. A., “Total Pressure Drop of Refrigerants Flowing through Evaporator Tubes,” Kaltetechnik, Klimatisierung, 19, 306-9 (1967) (Ger.). (12B) Chisholm, D., “Flow of Compressible Two-Phase Mixtures through Throttling Devices,” Chem. Process Eng., 48,73-8 (1967). (13B) Chomiak, J., “ O n the Problem of Sampling in a Two-Phase Flow,” Prace Inst. Maszyn PrzepIywowych, 31, 241-52 (1966) (Pol.). (14B) Curtet, R., and Djonin, K., “Study of a Vertical Downward Mixed Water and Air Flow: Flow and Concentration Conditions,” La Houille Blanche, 22, 531-50 (1967). (15B) Deich, M. E., et al., “Conservation of Momentum Equations for the Flow of a Two-Phase Medium with Phase Conversions,” High Temperature (TepI. Vysokikh Temp.), 6,105-9 (1968). (16B) De’ong, V. !., et al., “Effect of Slip and Phase Change on Sound Velocity in Steam-bater Mixtures and the Relation to Critical Flow,” IND. END. CHEM., PROCESS DES.DEVELOP., 7,454-63 (1968). (17B) Delhaye, J.,,M., “Hot-wire Anemometry in Two-Phase Flows. Measuring of Gas Content, C.R. Acad. Sci., Parir, 266, 370-73 (1968) (Fr.). (18B) Delhaye, J. M., “Spatial and Statistical Mean Equations for Two-Phase Flow,” La Harille Blanche, 22, 559-66 (1967) (Fr.). (19B) Evangelisti, R., et al., “ T h e Gamma Ray Attenuation Method for Determining Void Fraction in a Vertical Adiabatic Boiliqg Channel,” Galore, 39, 35-45 (1968) (It.). (20B) Gill, L. E., and Hewitt G. F “Sampling Probe Studies of the Gas Core in Annular Two-Phase Flow-’111. histribution of Velocity and Droplet Flowrate After Injection through an Axial Jet,” Chem. Eng. Sci., 23, 677-86 (1968). (21B) Gloyer, W., “ A New Look at Two-Phase Flow,” Chem. Eng., 75,93-5 (1968). (22B) Gomezplata, A., et al., “Estimating Pressure Drop in Vertical Two-Phase Flow,” ibid., 74,1758-59 (1967). (23B) Hayama, S., “ A Study on the Hydrodynamic Instability in Boiling Channels,” Bull. J . S . M . E . , 10,320-27 (1967). (24B) Hodossy, L., “Hydrodynamics, Heat and Mass Transfer in Two-Phase Flow,” Magyar Kem. Lapin, 29, 29-40 (1968) (Hung.). (25B) Hosler, E. R . , “Flow Patterns in High Pressure Two-Phase (Steam-Water) Flow with Heat Addition,” Chem. Eng. Progr. Symp. Ser., No. 82, 64, 54-66 (1968). (26B) Kevorkian, V., et al., “Mass Production of 300-Micron Water Droplets by Air-Water Two-Phase Nozzles,” IND.END. CHEM., PROCESSDES. DEVELOP, 7,586-90 (1968). (27B) Kriegel, E., “Calculation of Two-Phase Flow of Gas-Liquid Systems in Tubes,” Chem.-Ing.-Tech., 39, 1267-74 (1967) (German). (28B) Lafferty, J. F., et nl., “Velocity Distributions in Two-Phase Vortex Flow,” J . BUSGEng., 90,368-72 (1968). (29B) Leslie, D. C., “ T h e Development of Flashing Flow from Existing Nucleation Sites,” Biit. Chem. Eng., - . 1% , 512-19 (1968). (30B) Mayinger F. et a/., “Hydrodynamic Processes and Flow Stability During Supercooled B’oilihg,’’ Chem.-Ing.-Tech., 40, 515-21 (1968) (Ger.). (31B) Mayinger F t t al. “Mathematical Calculation for Instability in Two-Phase Flow,” ibid., p i li’85-9i. (32B) Nancetti, G. F., et al. “Two-Phase, Gas-Liquid Flow Descending in Vertical Tubes. Pressure Drop i n d Film Thickness a t the Walls in the Annular, Dispersed Regime,” Quad. Ins. Chim. Ita/., 4, 147-51 (1968) (Ital.). (33B) Oliver, D. R., and Hoon, A. Y o y g , “Two-Phase Non-Newtonian Flow. Part I: Pressure Drop and Hold-Up, Trans. Inst. Chem. Eng., 46, p p T 106-15 (1968). (34B) Oliver D. R and Hoon, A. Young “Two-Phase Non-Newtonian Flow. Part 11: h e a t T;ansfer,” ibid., p p T 118-22. (35B) Owhadi, Ali, et a/., “Forced Convection Boiling Inside Helically-Coiled Tubes,” Int. J . Heat Mass Tranrfer, 11, 1779-93 (1968). (36B) Pletcher R . H and McManus H . N J r “Heat Transfer a;d Pressure Dro in Hdrizontai Annular Two-bhase, ‘?Twb)-Component Flow, ibid., p p 108?-1104. (37B) Reith, T., et al. “Gas Hold-Up and Axial Mixing in the Fluid Phase of Bubble Columns,” &hem. Eng. Sci., 23, 619-29 (1968).
(38B) Rogers, John D “Two-Phase Friction Factor for Parahydrogen Between One Atmosphere and’the Critical Pressure,” A.I.Ch.E. J . , 14, 895-902 (1968). (39B) Rouhani, Zia S., “Calculation of Steam Volume Fraction in Subcooled Boiling,’’ J . Heat Transfer, . . 90., 158-64 (1968). (40B) Rounthwaite C., “Two-Phase Hear Transfer in Horizontal Tubes,” J . Inst. Fuel, 41,66-f6 (1968). (41B) Silvestri M “Unconventional Applications of Two-Phase Cocurrent Flow,” Energk N h . (Milan), 15, 38-47 (1968). (42B) .Staub, F. W., “ T h e Void Fraction in Subcooled Boiling-Prediction of the Initial Point of Net Vapor Generation,” J . Heat Tranrfer, 90, 151-57 (1968). (43B) Van Wijn aqyden, L., “ O n the Equations of Motion for Mixtures of Liquid and Gas Bubbkes, J . Fluid Mech., 33, part 3, 465-74 (1968). (44B) Vasil’ev, P. P., “Determination of Hydraulic Characteristics of Local Resistances of a Flow with Low Steam Content,” Thermal Eng. (TepIoencr&ika), 14, 98-102 (1968). (45B) Veda A “ O n Upward Flow of Gas-Liquid Mixtures in Vertical Tubes. Part I. $ x p h n e n t and Analysis of the Flow State,” Bull. J.S.M.E., 10, 98999 (1967). (46B) Veda, A., “ O n Upward Flow of Gas-Li uid Mixtures in Vertical Tubes. Part 11. Consideration of Frictional Pressure 8 r o p and Void Fraction,” ibid. p p 1000-1007. (47B) Wallis G. B., “Use of the Reynolds Flux Concept for Analysing OneDimension& Two-Phase Flow. Part 1. Derivation and Verification of Basic Analytical Techniques,” Int. J . Heat Mass Transfer, 11, 445-58 (1968). (48B) WaHis, G. B., “Use of the Reynolds Flux Concept for Analysin On:; Dimensional Two-Phase Flow. Part XI. Applications t o Two-Phase S l o w , ibid. DD 459-72. (49B) Wedekind G. L., and Stoecker W. F “Theoretical Model for Predicting the Transient ’Rcsconse of the Mixhre-Vi’ or Transition Point in Horizontal Evaporating Flow, J . Heat Transfer, 90, 16?-74 (1968). . I *
Cavitation (1C) Arndt R E A., and I p en A T., “Rough Surface Effects on Cavitation Inception:” j . B&c Eng., 90, &49;61’(1968). (PC) Boguslavskii, Y u . Ya., “Diffusion of a Gas into a Cavitation Void,” SOU. Phyr. Accoust., 15, 18-21 (1967). (3C) Canavelis, R., “ Jet Impact and Cavitation Damage,” J . Basic Eng., 90, 35567 (1968). (4C) Duport, J. P., “Shear Flow Cavitation,” Reu. Franc. Mecanique, 24, 79-88 (1967) (Fr.). (5C) Hammitt, F. G., “Cavitation Phenomena in Liquid Metals,” La Houille Blanche, 23, 31-35 (1968) (Eng.). (6C) Holt M., “ T h e Collapse of an Imploding S herical Cavity” Rev. Roumaine S a . Teclniques, Serte de Mecanague Appliquee, 12, 4Op-15 (1967) (EAg,). (7C) Ikeda R “Ultrasonic Shock Waves Emitted by Cavitation in Venturi Tubes,” ‘feccnology Reports of T h e Iwate University, 2, 1-10 (1966) (Eng.). (8’2) Karlikov V. P., and Sholomovich G. I., “ A n Approximate Method for Determining) the Influence of the Wall; During Cavitating Flow Past Bodies in Tunnels,” Izv. Akad. Nauk SSSR, Mekhan. Zhidkosta i Gaza, 4, 89-93 (1966) (Russ.). (9C) Kozirev, S. P., “ O n Cumulative Collapse of Cavitation Cavities,” J . Basic Eng., 90, 116-24 (1968). (1OC) T$lis J P and Marschner B. W., “Review of Cavitation Research on Valves, J: k y d r y D i u . , Proc. A . S . C . k . , 94, HYI, 1-16 (1968).
Liquid Films (1D) Ackerber , R. C., “Boundary-Layer Flow on a Vertical Plate,” Phys. Fluids, 11,1278-91 8968). (2D) Banerjee, Sanjoy, et GI., “Mass Transfer to Falling W a Liquid Films in Turbulent Flow,” IND.END.CHEM.,FUNDAM., 7, 22-27 (1963. (3D) Berbente, C. P., and Ruckenstein, Eli, “Hydrodynamics of Wave Flow,” A.1.Ch.E. J . , 14,772-82 (1968). (4D) Buevich, Yu.A., et al., “Stability of a Laminar Film Flow,” Fluid Dynamics (Mekh. Zhid.i Gaza), 1,72-77 (1968). (5D) Cohen, L. S., and Hanratty T. J “Effect of Waves a t a Gas-Liquid Interface on aTurbulent Air Flow,” j.Flu;; Mech., 31,467-79 (1968). (6D) Craik A D. D “Wind-Generated Waves in Contaminated Liquid Films,” ibid.,pp 121-61 (1928). (7D) Davies J. T et al., “Surface Stresses and R i ple Formation Due t o Low Velocity Air hassin; Over a Water Surface,” Chem. h g . Sci., 23, 331-37 (1968). (8D) Ford J. D and Missen R . W., “ O n the Conditions for Stability of Falling Films Sib’ect ‘ i o Surface ?ension Disturbances; T h e Condensation of Binary Vapors,” &an.J . Chem. Eng., 46,309-312 (1968). (9D) Forste, J., “ O n the Diffusion into a Downward-Flowing Liquid Film,” Monatsber. Deut. Akad. Wiss. Berlin, 8, 851-69 (1966) (Ger.). (10D) Golubev, L. K., “ T h e Flow of Thin Liquid Films on Vertical Tubes in Transverse Gas Streams, Int. Chem. Eng., 8 , 626-28 (1968). (11D) Goren, S. L., and Mani, R. V. S., “Mass Transfer Through Horizontal Liquid Films in Way Motion,” A.1.Ch.E. J . , 14, 57-61 (1968). (12D) Gupta, A. S and Rai Lajpat “Note on the Stability of a Visco-Elastic Liquid Film Flowing Down An Inclided Plane,” J . Fluid Mech., 33, 87-91 (1968). (13D) Hartland S.,“ T h e Radius of the Draining Film Beneath a Drop Approaching a Plane In‘terface,” J . Phys. Chem., 72, 318-20 (1968). (14D) Haugen, R., “Laminar Flow Along a Vertical Wall,” J . Appl. Mech., 35, 631-33 (1968). (15D) Howard D. W., and Lightfoot E. N., “Mass Transfer to Falling Films: Part I A ilication of the Surface-ktretch Model to Uniform Wave Motion,” A.I.Ch:E. 214,458-67 (1968). (16D) Kattanek, S., et al.,“Dimensionless Characteristics Obtained for Heat Transfer of Boiling Trickling Film,” Chem. Tech., 20, 26-29 (1968) (Ger.). (17D) Ludviksson, V., and Lightfoot, E. N., “Hydrodynamic Stability of Marangoni Films,” A.I.Ch.E. J . , 14, 620-26 (1968). (18D) .Nakaya, $;, and Takai R., “Nonlinear Stability of Li uid Flow Down a n Inclined Plane, J. Phys. So2 Japan, 23, 638-45 (1967) (Engl. (19D).Oliver, D. R., and Atherhos, T. E. “Mass Transfer to Liquid Films on an Inclined Plane,” Chem. Eng.Sct., 23,525-;6 (1968). (20D) Ruckenstein, E., and Berbente, C., “Mass Transfer to Falling Li uid Films at Low Reynolds Numbers,” Int. J . Heat Mass Transfer, 11, 743-53 (1928). (21D) Smith P “ A Linear Analysis of Steady Surface Waves on a Viscous Liquid Flowing DAwZan Inclined Plane,” J . Eng. Math., 1,273-84 (1967). (22D) Thomas D. G., “Enhancement of Film Condensation Rate on Vertical Tubes by Lohgitudinal Fins,” A.I.Ch.E. J . , 14, 644-49 (1968).
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(23D) Voinov, A. K., and Khapilova, N. S., “Ex erimentdl Investigation of the Flow of a Thin Fluid Layer on the Surface of a c o t a t i n g Cone,” P M T F : Z h . Prikl. Mekhan. Tekh. Fiz., 2, 107-109 (1967). (24D) Vorotilin, V. P., and Krylov, V. S., ‘.
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(14G) Halligan, J. E., et al., “ T h e Profile of a Separating Droplet,” J . Colloid Interfac. Sci., 25, 127-32 (1968). (15G) Halligan, J. E and Burkhart, L. E. “Determination of the Profile of a Growina ~-Droplet,” 2.I.Ch.E. J., 14, 411-11 (1968). (l6G) Harper, J F and Moore, D. W . , ” T h e hlotion of a Spherical Liquid Drop a t High Reynaids”Number,” J . Fluid Mech., 32, 367-91 (1968). (17G) Hartland S “ T h e Coalescence of a Liquid Drop a t a Liquid-Liquid Interface. Pa’rt $: T h e Effect of Surface Active Agent,” Trans. Insl. Chem. Eng., 46, T 275-82 (1968). (18G) Himmelblau, et al., “Effect of Plate Wettability on Droplet Formation,” IND.ENG.CHEM.,PROCESS DES. DEVELOP.,7, 508-11 (1968). (19G) Ingebo, R. D. “Maximum Drop Diameters for the Atomization of Liquid Jets Injected Co-durrently into Accelerating or Decelerating Gas Streams,’’ NASATN D-4640 (1968). (20G) Karam H. J., and Bellinger, J. C., “Deformation and Breaku of Liquid 7, 5 7 8 8 1 (1968). Droplets in A Simple Shear Field,” IND.ENG.CHEM.,FUNOAM., (21G) Lee J. C and Hodgson, T. D “Film Flow and Coalescence-I: Basic Relationis, Fil; Shape and Criteria b r Interface Mobility,” Cher~.Eng. Sci., 23, 1175-07 (1968). ~..~., (22G) Robertson, W. M., and Lehman, G. W., “ T h e Shape. of a Sessile Drop,” J . Aflpl. Phys., 39,1994-96 (1968). (23G) Scheele, G. F., and Meister, B. J. “Drop Formation a t Low Velocities in Liquid-Liquid Systems: Part I. Prehiction of Drop Volume,” A.I.Ch.E. J., 14.9-19 (1968). (24G) Thorsen, G . ,el al.,“On the Terminal Velocity of Circulating and Oscillating Liquid Drops,” Chem. Eng. Sci.,23,413-26 (1968). ~
Fluidized Beds (1H) Afschar A . S., and Schugerl, K., “Properties of Three Phase Fluidized Beds with Parallil Flow ofWater and Air,” Chem. Eng. Sci., 23, 267-78 (1968). (2H) Akehata T., “Flow in Equipment I. Fixed Bed, Moving Bed and Fluidized Bed,” Kagak; Koguku, 32, 417-21 (1968) (Jap.). (3H) Anderson T. B and Jackson, R., “Fluid Mechanical Description of Fluidized Beds. itabili;; of the State of Uniform Fluidization,” IND.END. CHEM., FUNDAM., 7,12-21 (1968). (4H) Aoki, R., and Yamazaki, R., “Particle Mixing in a Packed Fluidized Bed,” Funsai (13) 3-7 (1968) (Jap.). (5H) Atroschenko, L . S., and Voronina, S. M., “Hydrodynamics of Processes in Bubbling Reactors with External Potential Fields,” Inzh. Fiz. Z h . 15, 416-21 (1968). (6H) Basakov 4. P., and Gim el’man, E. Ya.,“Mixing Solid P,micles in Adjacent Fluidized Bkbs as Studied g y an Unsteady-State Method, Khim. Prom., 44, 412-14 (1968) (Russ.). (7II) Basakov, A. P., and Gubin, I. V., “Investigation of Solid Particle Entrainment from a Fluidized Bed,” ibid., pp 534-6. (8H) Basov, V. A,, Markhevka, V. I., Melik-Aklmazarov, T. K., and Orochko, D. I., “Structure of a Heterogeneous Fluidized Bed,” ibzd., pp 619-22.
(9H) Belyi V. A., and Yurkevich, I., “Method for Estimating the Apparent and Bulk D e k t i e s of Fluidized Beds of Plastic Particles,” Inzh. Fiz. Zh,, 14, 55-60
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(10H) Bena J Havalda, I., Bafrnec, M., and !lavsky J “ T h e Velocities a t Inci ient ’ FlAidization of Polydisperse Materials. I: $heoretical,” Collect. Czecf. Chem. Commun., 33,2620-35 (1968) (Eng.). (11H) Bena, J Havalda, I., Ilavsky, J., and Bafrnec, M., “Velocities a t Incipient Fluidization ’bf Polydisperse Materials. 11. Experimental Results and Design Equations,” ibid., pp 2833-54. (12H) Botterill, J. S. M., “Progress in Fluidization,” Brit. Chem. Eng., 10, 26-30 I1 5 )- , . \ _0_6 _ (13H) Botterill, J. S. M., “Progress in Fluidization,” ibid., 13, 1121-6 (1968). (14H) Bourgeois, P., and Gernier, P., “Ratio of Terminal Velocity to Minimum Fluidizing Velocity for Spherical Particles,” Can. J . Chem. Eng., 46, 325-8 (1968). (15H) Bowling, K. M., and Waters, P. L., “Fuel Processing in Fluidized’ Beds,” Brit. Chem. Eng., 13,1127 (1968). (16H) Brown, R . A. S., and Jensen, E.J.,“The Pipeline Flow of Paste Slugs. Part 11. Pressure Gradients and Velocities of Trains of Slugs,” Can. J . Chem. Eng., 4fi - (3). 157-61 (1968). (17H) Buevicp; In. A., “ O n the Statistical Mechanics of Particles Suspended in a Gas Stream, J . Appl. Math. Mech., 32, 85-93 (1968). (18H) Ca es C. E and McIlhinney A. E “Pseudoparticulate Expansion of Screen-FaGked G.?i-Fluidized Beds,” h.I.Ch.2. J., 14, 917-22 (1968). (19H) Carlos C. R., and Richardson, J. F.,“Solids Movement in Liquid Fluidized Beds. I-$article Velocity Distribution,” Chem. Eng. Sd., 23, 813-24 (1968). (20H) Carlos, C. R., and Richardson, J. F., “Solids Movement in Liquid Fluidized Beds. 11-Measurements of Axial Coefficients,” Chem. Eng. Sd.,23, 825-31 (1968). (21H). Chen, B. H., and Dou las, W. J. M., “Lisuid Hold-Up and Minimum Fluidization Velocity in a T8urbulent Contactor, Can. J . Chem. Eng., 46, (4), 245-9 (1968). (2;H) Chernov, V. D., Reitman, G. A,, Serebryakov, B. R., and Dalin, M. A., Experimental Study of the Instability of a Fluidized Bed,” Khim. Tekhnol. Top!., Masel, 13 (9), 12-15 (1968) (Russ.). (23H) Chung, S . F., and Wen, C. Y . ,“Lon itudinal Dispersion of Liquid Flowing Through Fixed and Fluidized Beds,’’ A.I.8h.E. J . 14,857-66 (1968). (24H) De Jonge, J., “Fluidization,” Mugyar Kem. Lapja, 23, 581-7 (1968) (Croat). (25H) De Nevers N and Grimmett, E. S., “Fluidized Beds for Solids Flow Control;’ IND.ENO;&EM., PROCESS DES.DEVELOP.,7, 101-6 (1968). (26H) Dobrinescu D ‘ICoefficient of Friction in the Sedimentation and Fluidization ofBeds of Sihe&al Particles,” Rev. Chtm. (Bucharest) 19 (3), 166-71 (1968) (Rom.). (27H) Doig, I D “Suspension of S heres in a Vertical Tube a t High Reynolds Numbers,” dhem: Eng. Sci. 23, 794-8 (1968). (28H) Doi I. D., and Roper, G. H., “Contribution of the Continuous and Disersed Pfases to the Suspension of S heres by a Bounded Gas-Solid Stream.” ~ N D ENQ. . CHEM.,FUNDAM. 7.459-71 (A68). (29H) Gabuchiya,‘?. G Todes 0. M., Mikhalev, M. F., Mukhlenov, I. P., and Platonov P. N., QuaAtitativ6 Measurements of the Inhomo eneity of a Fluidized Catalyst Bed in Apparatus of Different Sizes,” Kinet. Kataf 9,889-94 (1968) (Russ.). (30H) Geldart D “ T h e Expansion of Bubbling Fluidized Beds,” Powder Technol., I, 355-68 (id68jl (31H) Gel’perin, N. I Ainstein, V. G., and Kofman, ‘P. L., “Uniformity of Fluidization in a B e d h i t h Bundles of Vertical Tubes,” Zh. Vses. Khim. Obshchest., 13 (5), 592-3 (1968) (Russ.). (32H) Gel’.perin, N. I., Ainstein, V. G., Lapshenkov,, G. I., and,,Mikhailov, V. A., Determination of the Effecuve Viscosity of a Fluidized Bed, Teor. Osn. Khtm. Technol., 2, 615-22 (1968) (Russ.). (33H) Gill, W. N., Cole, R., Estrin,’ J., Nun e, R . J., and Littman, H., “Fluid Dynamics,” IND.END.CHEM.,61 (l),41-78(1969). (34H) Glidden, H. J., and Pulsifer, A. H., “Electrode Contact in a Fluidized Bed,” Can. J . Chem. Eng., 46,476-8 (1968). (35H) Godard, K., and Richardson, J. F., “Distribution of Gas Flow in a Fluidized Bed,” Chem. Eng. 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Solids-Gas Flow,” Kagaku Kogaku, 32, 411-4 (1968) ($ap.). (43H) Jinescu G. I “Methods of Analyzing a Fluidized Layer,” Rev. Chim. (Bucharest) i 9 (9), 322-5 (1968) (Rom.). (45H) Jones, W. M., “Viscous Drag and Secondary Flow in Granular Beds.” Brit. J . Appl. Phys.,Ser. 2, 1, 1550-65 (1968). (46H), Kafen auz, N. L., and Fedorov, M. I “Conditions for the Onset of Fluidization in &eat Transfer wIth Turbulent h i d Flow,” Inzh. Fiz. Zh., 14, 923-4 (1968). (47H) Kalinowski, B., Wlodarski,, R., and Bor sowski, J., “ A proximate Determinatlon of the Hold-Up Time of Granular daterial in the Fkidized Phase in a Continuously Operating Reactor,” Intern. Chem. Eng., 8, 224-9 (1968). (48H) Kheifets,,K. I., Dzha atspanyan, R . V., and Shcherbunov, A. I., “Particle Paths in Fluidized Beds,” fnzk. Fir. Zh., 15, 642-7 (1968) (Russ.). (49H) Kiode, K., Ka;?, S., Tanaka Y . , and Kubota, H “Bubbles Generated from a Porous Plate, J . Chem. Eng.’Jap., 1, 51-6 (1968) (Zng.). 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(52H) Kozin, V. E., and Basakov, A. P., “An Investigation of the Grid Zone of a Fluidized Bed Above Cap-Type Distributors,” Intern. Chem. Eng., 8, 257-60 (1968). (53H) Krishnamurthi, S., Kumar, R., and Kuloor, N. R.,“Formation of Bubbles,” Chem. Procerr Eng., 49, 91-7; 100 (1968). (54H) Kunii, D., and Levenspiel, O., “Bubbling Bed Model. Model for the 7, 446-52 Flow of Gas Through a Fluidized Bed,” IND. END. CHEM.,FUNDAM., (1968). (55H) Kunii, D., and Levenspiel, O., “Bubbling Bed Model for Kinetic Processes in Fluidized Beds. Gas-Solid Mass and Heat Transfer and Catalytic ReacDES.DEVELOP.,7,481-92 (1968). tions,” I N D . ENO.CHEM.,PROCESS (56H) Kurgaev, E. F “Viscosity of Heterogeneous System Fluid-Solid Particles,” Inzh. Fiz. Zh., 15, 74’84 (1968) (Russ.). (57H) Latham, R., Hamilton, C., Potter, 0. E “Back-Mixing and Chemical Reaction in Fluidized Beds,” Brtt. Chem. Eng., 13,’666-71 (1968). (58H) LeClair,.B. P., and Hamielec, A. E., “Viscous Flow Through Particle Assembla ea at Intermediate R e nolds Numbers. Steady State Solutions for 7, 542-9 Flow 3hrough Assemblages of Jpheres,” IND.ENG. CHEM.,FUNDAM., (1968). (59H) Lehmann W., et al., “Influence of the S:i of Equi ment and the Perforated Plate on the $ormation of a Fluidized Bed, Chem. h c h . (Berlin) 20, 414-8 (1968) (Ger.). (60H) Levenspiel, O., Kunii, D., and Fitzgerald, T “ T h e Processing of Solids o f Changing Size in Bubbling Fluidized Beds,” Powdw’jkchnol., 2 ( 2 ) , 87-96 (1968). (61H) Levsh, I. P., Krainev, N. I., and Niyazov, M. I., “Calculation of the 311-2 (1968). Pressure Drop and Heights of Three-phase Fluidized Beds,” Int. Chem. Eng., 8, (62H) Levsh I. P Krainev, N. I and Ni azov M . I., “Hydrodynamic Calculations of Abiorbe;; with Fluidized’Beds,’’ irid., {p 619-21. (63H) Lockett, ,M. J., Dazidson, J. F., and Harrison, D., “Distribution of Gas Flow in a Fluidized Bed, Chem. Eng. Sa.,23, 661 (1968). (64H) Mach, W., “Momentum Transfer in a Three-phase Fluidized Bed,” Chm.1ng.-Tech., 40 (21-22), 1045-50 (1968) (Ger.). (65H) Mamuro T., and Hattori, H., “Flow Pattern of Fluid in Spouted Reds,” J . Chem. Eng.jap., 1,l-5 (1968) (Eng.). (66H) Maneri, C. C., and Mendelson, H. D., “ T h e Rise Velocity,of Bubbles in Tubes and Rectan ular Channels as Predlcted by Wave Theory, A.1.Ch.E. J., 14, 295-300 (1968y; same article-Chsm. Eng. Prog. Symp. Ser., 64 (821, 72-80 (1968). (67H) . M c p r t h y , H. E., and Olson, J. H., “Turbulent Flow of Gas Solid Sus7,471-83 (1968). pensions, I N D . EN& CHEM., FUNDAM. (68H) Miyaguchi, T., Kaji H and Saito K., “Fluid and Particle Dispersion in Fluid-Bed Reactors. E$per?mental Inbestigations by Steady Heat-Transfer and Steady Backmixinq of Adsorptive Gases,” J . Chem. Eng. Jap., 1, 72-7 (1968) (En&). (69H) Moneman, E., “Determinapn of the Avera e Residence Time of Dried Material in a Fluidized Column, Chem. Plum., 18 81-12), 582-6 (1968) @lo). (70H) Mori Y Nakamura K “Particle Movement in a Gas-Solid Fluidized Bed,” J . dhem’,’Eng. Jup., 1: 18z-6 (1968) (Eng.). (71H) Motamedi, M., and Jameson, G. J.,“A New Method for the Measurement of the -~~ Incinient Fluidizinz Velocitv.” Chem. Ena. Sci., 23, 791-3 (1968). (72H) Naik, S. C “Performance of Fluidized Bed Reactors,” Chem. Age India, 19 (194),276-80’?1968). (73H) Nauman E. B and Collinge C N “ T h e Theory of Contact Time Distributions in &as FlGidized Beds,” be;. Jng. Sci., 23, 1309-16 (1968). (74H) Nauman E. B and Collinge C. N., “Measurement of Contact Time Distribution in d a s F d d i z e d Beds,’ !bid., p p 1317-26. (75H) Nicolitsas, A. J., and Murgatroyd W “Precise Measurements of Slug Soeeds in Air-Water Flows,” Chem. Enp. Sk., 25, 934-6 (1968). (76H) Pigenblik, A. A., Kaganovic?: Yu. Ya Goikhman, I. D., Babenko, V. E., Teminkov V. I and Genin, I. S., Effect oythe Gas Distributor on Solid Phase Mixin a i d on Heat Transfer in Fluidized Bed,” Khim. Prom., 44, 615-8 (1968) ~
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(77H) Panton R “Flow Properties from the Continuum Viewpoint of a NonEquilibrium’Ga~-ParticleMixture,” J . Fluid Mech., 31, Part 2, 273-303 (1968). (78H) Paul R . J. A AI-Naimi T. T and Das Gupta, D. K.,‘‘Stability of Coupled Fluid Bed;,” Brit. ?hem. Eng., \3,66-4 (1968). (79H) Pedley, T. J., i‘The Toroidal Bubble,” J . Fluid Mech., 32, Part I, 97-112 (1968). (80H) Pi pel, W., R u n e, K., Geyer, H., Lehmann, W., and Mueller F., I ‘ Mixin of Soli$ in Gas-Soli2 Fluidized Beds,” Chem. Tech. (Berlin) 20, $50-5 (19687 (Ger.) (81H) R a havendra, N. M., “Multiphase Flow,’’ Chem. Age India, 19, 580-7 (1968) (kng.). (82H) Reddy K. V. S Fleming R . J and Smith J. W “Maximum Spoutable Bed Depths’of MixedlParticle-&e Bdds,’’ Can. J.’Chem. %ng., 46, 329-39 (1968). (83H). Richardson, S “Two-Dimensional Bubbles in Slow Viscous Flows,” J . Fluid Mech., 33, Pal.; 111, 476-93 (1968). (84H) Ruckenstein, E., and Tzeculewu-Filipescu, M. “ Hydroa namics of Fluidized Beds with Incipient Nonhomogeneities,” Chem. kng. Sci., 2 2 1121-5 (1968). (85H) Sandbloom H., “ T h e Pulse Technique for Investigating Solids Mixing in a Fluidized Bed,” h i t . Chem. Eng., 13, 677-9 (1968). (86H) Saxton J. A., Jr., “Statistical Thermodynamics of Particulate Fluidization,” Ph.D: Thesis, Univ. of Calif., Berkeley (1966). (87H) Schurhht, V., “Radiometric Density Meter for Fluidized Bed Studies,” Isotopenpraxir, 4, 324-6 (1968) (Ger.). (88H) Schwerdtfeger K., “Velocity of Rise of Argon Bubbles in Mercury,” Chem. Eng. Sn’., 23,637-8 (1968). (89H) Shichi, R., Mori, S and Muchi I “Interaction Between Two Bubbles in Gaseous Fluidization,” Xugaku Kogaku: 3?, 343-8 (1968) (Jap.). (90H) .Shook, C..A.,,Daniel S. M Scott J. A and Holgate J. P., “Flow of Suspensions in Pipelines. $art 2:’ T w d Medhanisms of $article Suspension,” Can. J.Chem. Eng., 46,238-44 (1968). (91H) Slaughter, I., and Wraith, A. E., “ T h e Wake of a Large Gas Bubble,” Chcm. Eng. Sa.,23,932 (1968). (92H) Stewart P S. B “Isolated Bubbles in Fluidized Beds-Theory and Experiment,’’ Trun;. i n s t . d e m . Eng., 46 ( Z ) , T60-66 (1968). (93H) Stewart P. S. B. “Prediction of Void Fraction Near Bubbles in Fluidized Beds,” Chem.)Eng.Sd., i3,396-7 (1968). (94H) Sosna M. Kh., and Kondukov N . B “Criteria and Formula for Calculating Fluidization Rate, Polydisperse Beds,” In;h. Fir. Zh., 15, 73-8 (1968) (RuSS.). (95H) Sosna, M. ,Kh., and Kondukov, N. B., “Rate of Fluidization of Polydisperse Beds,” Khim. Prom., 44, 410-12 (1968) (Russ.).
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(96H) Subramanian, N., “Fluidization and Pneumatic Transport,” Chem. Age India.. 19.588-94 (1768) (Eng.). . (97H) Toei R Matsuno, R., Miya awa, H., Nishitani, K., and Koma awa Y “Gas Intkch’kge Between a BubbTe and Continuous Phase in Gas-Sofid Fiuidi ized Bed,” Kagaku Kogaku, 32, 565-70 (1968) (Jap.). (98H) Toei, R., Matsuno, R., and Mori, M., “ T h e Coalescence of Bubbles in GasSolid Fluidized Bed,” ibid., pp 104-5. (99H) Toei R Matsuno, R., Sumitani T. and Mori, M. “Coalescence ofBubbles in a Gas-hol;h Fluidized Bed,” Int. Chm.’Eng., 8, 351-? (1968). (100H) Tuichief I. S., Riazaev N. U., Merenkov K . V., and Yusipov, M. M., liThe Hydrod;namic Propertiks of Ion Exchangd Resins During Fluidization,” ibid., pp 221-3. (101H) Tunstall, E. B., and Houghton G., “Retardation of Falling Spheres by Hydrodynamic Oscillations,” Chem. E&. Sei,, 23, 1067-81 (1 968). (102H) Vail, Yu K., Manakov, N. Kh. and Menshilin V. V., “Turbulent Mixing in a Three Phase Fluidized Bed,” Irk Chem. Eng., $, 293-6 (1968); essentially the same article, tbid., pp 516-19. (103H) Vakhruchev, I. A., and Basov, V. A,, ‘“on-Homogeneous Fluidization,” Khim.Prom., 44 ( 6 ) , 401-5 (1968) (Russ.). (104H). Van Wijn aarden, L “On the Equations of Motion for Mixtures of Liquid and Gas fubbles,” J:’Fluid Mech., 93, Part 3, 465-74 (1968). (105H) Verloop, J., DeNie L H . and Heertjes P. M. “Residence Time of Solids in Gas-Flutdized Beds,” ;ouhr Fechnol., 2, 32i42 (19k8). (106H) M‘einspach, P. M., “Fluidizatlon and Suspension,” Chem.-Zng.-Tech., 40 (18). - 884-9 ,1968). (107H) Winter 0 “Density and Pressure Fluctuations in Gas Fluidized Beds,” A.I.Ch.E.J., i 4 426-34 (1968). (108H) Woollard, I. N. M., and Potter, 0. E., “Solids Mixing in Fluidized Beds,” ibid., (31, 388-91 (1968). (109H) Yeh H and Yang W “Dynamics of Bubbles Moving in Liquids with Pressure Gradjen;:” J . Appl. khyi,’, 39,3156-65 (1968). (llOH) Yoshida, K . , and Kunii, D “Stimulus and Response of Gas Concentration in Bubbling Fluidized Beds,” J . ‘ h e m . Eng. Jap., 1, 11-16 (1968) (Eng.). (111H) Yu, Y. H., Wen, C. Y., and Bailie, R. C.,“Power Law Fluids Flow Through Multiparticle System,” Can. J . Chem. Eng., 46 (3), 147-54 (1968). (112H) Yufa, M.,: Dokuchaev Y. N Lakshin Y. K., Korovkin, E. V., and Traber D. G., ierodynamic’Struct:re of Cohical Fluidized Bed and Determinatidn of the Optimum Taper of the Cone,” Khim. Prom., 44 (6), 408-9 (1968) (Russ.). (113H) Zablotny, W., “ A Survey of Isotopic Methods Used in the Study of Fluidized Beds in the Petroleum Industry,” Przem. Chem. 47, 7-9 (1968) (Pol.). (114H) Zablotny, W., Akerman, K.,g45sniewski,J., Winnicki, R., Lelental, M . , Michalski, B., and Chrubasik, A Tracer Studies on Circulation of Solids in Fluidized Beds,” Nukleonrka, 13, 775-86 (1968) (Pol.). (115H) Zieminski, S. A,, and Raymond D. R “Experimental Study of the Behavior of Single Bubbles,” Chem. Eng. ki.,29 yl), 17-28 (1968). ~~
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Porous Media (11) Ben Afm R and LeGoff P “Effect de Paroi dans les Empilements Desordonnfs de &hires et App1i;atib)n a la Porosite de Melanges Binaires,” Powder Techno/., 1,281-90 (1767/68) (in French). (21) Ben Aim, R. and LeGoff P., “ L a Coordinance des Empilements Desordonnes d e Spheres. kpplication a;x Melanges Binaires de Spheres,” ibid,, 2 , 1-12 (1768/69) (inFrench). (31) Buffham, B. A,, and Gjbilaro, L. G., “ T h e Analytical Solution of the DeansLevich Model for Dispersion in Porous Media,” Chem. Eng. Sci., 23, 1399-1401 (1768). (41) Charpentier, J.-C., Prost, C and LeGoff P., “Liquid Trickle Flow in a of the Partial Films, Rivulets and Drops VePacked Column-Determinatiol locities and Flow Rates,” Chim. Ind. Genie Chimique, 100, 653-65 (1968) (in French). (51) Charpentier, J.-C., Prost, C., Van Swaaij, W., and LeGoff, P.,“ Hold-Up in a Packed Column in Co-Current and Counter-Current Gas-Liquid Flow. Representation of the Texture by a Model with Films, Rivulets, and Drops,” ibid., 99, 803-25 (1968) (in French).’ (61) Chung, S. F., and Wen, C. Y., “Longitudinal Dispersion of Liquids Flowing through Fixed and Fluidized Beds,” A.I.Ch.E. J., 14, 857-66 (1968). (71) Dayan, Joshua, and Levenspiel Octave “Longitudinal Dis ersion in Packed Beds oiPorous Adsorbing Solids,” k h m . Eng. Sci., 23, 1327-34 6968). (81).Dutkai, E., and Ruckenstein, E., “Liquid Distribution in Packed Columns,” rbrd., pp 1365-73. @!),Edwards, M. F., and Richardson, J. F., “Gas Dispersion in Packed Beds,” tbtd.,pp 109-23 (1968). (101) Han C. D., “Axial Dispersion,,in Fluid Flow through Porous Plates in Parallel k d through a Porous Tube, Appl. Sci. Res., 19, 1-13 (1968). (111) Hassinger, R . C.,~andvon Rosenberg, D . U., “A Mathematical and Experimental Examination of Transverse Dispersion Coefficients,” SOC.Pet. Eng. J . , 8,195-204 (1968). (121) Heertjes, R . M., and Lerk C F “Rep1 by Professor Heertjes and Dr. Lerk,” Trans. Inst. Chem. Eng., 46, T‘282-6 (196l). (131) Iczkowski, R . P., “Displacement of Liquids from Random Sphere Packings,” IND.END.CHEM.,FUNDAM., 7,572-6 (1968). (141) Ives, K . J., “ T h e Functioning of Deep-Bed Filters,” Trans. Inst. Chem. Eng., 46,T283-4 (1968). (151) Jones W. M “Viscous Drag and Secondary Flow in Granular Beds,” Brit. J.AjpI.P~ysicr,S~;.Z, 1,1559-65 (1968). (16I),Kozicki, W., Tiu, C., and Rao, A. R . K., ‘(Filtration of Non-Newtonian Fluids,” Can. J . Chem. Eng., 46, 313-21 (1968). (171) Krupp, H. K., and Elrick, D. E “Miscible Dis lacement in a n Unsaturated Glass Bead Medium,” Water Resourc;; Res., 4, 809-lf (1968). (181) LeClair B. P. and Hamielec, A. E , “Viscous Flow through Assemblages a t Intermedia& Reyholds Numbers,” IND.’END. CHEM.,FUND.AM.,7, 542-9 (1968). (191) LeLec, P., “ T h z Compressibility of Filter Cakes and Its Influence on the Filtrahon Equation, Filtration Separation, pp 114-23, 126, Mar/Apr 1968. (201) Li, W.-H and Yeh G.-T “Dispersion a t the Interface of Miscible Liquids in a Soil,” W,a?er Resources)Res., 4: 369-77 (1968). (211) Musil, L., Prost, C., and LeGoff, P., “Hydrodxnamics of Flooded Packed Columns wlth Counter Current Gas-Lquid Flow, Chim. Ind. Genie Chimique, 100,674-82 (1968) (in French). (221) Porter K . E. “Liquid Flow in Packed Columns. Part I : T h e Rivulet Model,” ?ram. I d t . Chem. Eng., 46, T 69-73 (1968). (231) Porter, K. E., Barnett, V. D., and Templeman, J. J., “Part 11: T h e Spread of Liquid over Random Packings,” ibid., pp T 74-85.
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(241) Porter, K . E., and Templeman, J. J., “Part 111: Wall Flow,” ibid., pp T 86-94 (1968). (251) Raats, P. A. C., “Forces Acting upon the Solid Phase of a Porous Medium,” Z A M P , 19,606-13 (1968). (261) Raats, P. A. C., and Klute, A., “Transport in Soils: T h e Balance of Mass,” SoilSci.Soc. Amer. Proc., 32,161-6 (1968). (271) Raats, P. A. C., and Klute, A., “Transport in Soils: T h e Balance of Momentum,” ibid., pp 452-6. (281) Raats, P. A. C., and Scotter, D . R., “Dynamically Similar Motion of Two Miscible Constituents in Porous Mediums,” Water Resources Res., 4, 561-8 (1968). (291) Rid way K and Tarbuck, K. J., “Voidage Fluctuations in RandomlyPacked Seds’of spheres Adjacent to a Containing Wall,” Ckem. Eng. Sci., 23, 1147-55 (1968). (301) Scotter, D. R., and Raats P. A. C. “Dispersion in Porous Mediums Due to Oscillating Flow,” Water Reso;rces Res., i,1201-6 (1968). (311) Slatter J. C., “Multiphase Viscoelastic Flow through Porous Media,” A.I.Ch.E.1: 14,50-6 (1968). (321) Sohn H. Y and Moreland C “ T h e Effect of Particle Size Distribution on Packing bensit;:” Can. J . Chern.’Eni., 46, 162-7 (1968). (331) Standish Nicholas “Dynamic Hold-Up in Liquid Metal Irrigated Packed Beds,” Chem.’Eng.Sci., i3, 51-6 (1968). (341) Standish, Nicholas “Measurement of Transient Changes in Liquid Hold-Up . FCNDAY., 7, 312-14 (1968). in Packed Beds,” IND.~ N C CHEM., (351) Standish, Nicholas, “Some Observations on the Static Hold-Up of Aqueous Solutions and Liquid Metals in Packed Beds,” ibid., pp 945-7. (361) Swartzendruber, Dale, “ T h e Applicability of Darcy’s Law,” Soil Sci. Soc. Amer. Proc., 32, 11-18 (1968). (371). White, D. A,, “Non-Newtonian Flow in Stratified Porous Media and in Axisvmmetric Geometries.” Chem. Ene. Sci.. 23. 243-51 (19681. (381) Wright, D . E. “Nonlinear Flow through Granular Media,” J . Hydraul. Diu., Plod. A.S.C.E.,’94, 851-72 (1968). (391) Yu, Y. H., Wen, C. Y and Bailie R. C “Power-Law Fluids Flow through Multiparticle Systems,” Cd;. J . Chem. kng., 46,149-54 (1968). ~~
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Rotating Flows ( I J ) Astill K . N Ganley, J. T and hlartin, B. W., “ T h e Developing Tangential Velocit; Profilk) for Axial Fl& in an Annulus with Rotating Inner Cylinder,” Proc. Roy. SOC.,307A, 55-69 (1968). (25) Barbee D . G . and Shih, T.-S.,“Incom ressible Flow Induced by an Infinite Isothermai Disk ’Rotating in a Rarefied Zas,” J . Heat Tianrfer, 9OC, 359-61 (1968). (35) Barcilon, Victor “ Stewartson Layers in Transient Rotating Fluid Flows,” J . HuidMech., 33, 815:25 (1968). (4J) Boyd K. E and Rice W “Laminar Inward Flow of an Incompressible Fluid bitween Rotating Di$ks, Lith Full Peripheral Admission,” J . A@/. M e c h . , 95E. 229-37 (1968). (55) Bretherton, F. P., and,Turner, J. S.,“ O n the Mixing of Angular Momentum in a Stirred Rotating Fluid,” J . Fluid Mech., 32, 449-64 (1968). (63) Busse, F, H., “Shear Flow Instabilities in Rotating Svstems,” ibid., 93, 57789 (1 968). (75) Conover R . A. “Laminar Flow between a Rotating Disk and a Parallel Stationary h a l l w(th and without Radial Inflow,” J . Baiic Eng.,90D, 325-32 (1968). (8J) Davey, A , , DiPrima, R . C., and Stuart, J. T., “ O n the Instability of Taylor Vortices;’ J . Fluid Mech., 31, 17-52 (1968). (9J) Farnsworth, A. V., and Rice, W., “Unsteady Laminar Motion of a Newtonian Fluid Contained between Concentric Rotating Cylinders,” J . Appl. Mech., 35E, 419-20 (1968). (1OJ) Goller, H;, and Ranov T “Unsteady Rotating Flow in a Cylinder with a Free Surface, J . Baric Eng.,’9(D, 445-54 (1968). ( I l J ) Hide, R., “ O n Source-Sink Flows in a Rotating Fluid,” J. Fluid Mech., 92, 737-64 (1968). (125) Homsy, G. M., and Hudson, J. L.,“Transient Flow Near a Rotating Disk,” Ah6l.Sci.Res.. 18. 384-97 (1968). . . -rr (13J) Hsueh Y. “Viscous Fluid over a Corruga.ted Bottom in a Strongly Rotating System,” $hyr.’Fluids, 11,940-4 (1968). (14J) Hudson, J. L “Non-Isothermal Flow between Rotating Disks,” Chem. Eng. scl.,23, 1007-20 (i’m). (15J) H u pert, H . E. “ O n Kelvin-Helmholtz Instability in a Rotating Fluid,” J , FluiZMech., 33, 3.,‘14,%63-4 (1968). (24K) White, J. L., “Motion of Continuous Surfaces Throu h Stagnant Viscous Non-Newtonian Fluids,” Can. J . Chem. Eng., 46, 294-8(1968). (25K) Williams M. C “Correlation of Stress Data in Viscoelastic Polymer Solutions,” A.I.Ch.k. J . , r3,360-2 (1968). (26K) Wohl, M. H., “Designing for Non-Newtonian Fluids. Part 2. Rheology of Non-Newtonian Materials,’’ Chem. Eng., 7 5 , 130-6, 12 Feb. 1968. (27K) Wohl, M. H., “Part 3. Instruments for Viscometry,” ibid., pp 99-1049 25 March1968. (28K) Wohl, M. H., “Part 4. Isothermal Laminar Flow of Non-Newtonian Fluids in Pipes,” ibid., pp 143-6, 8 Apr. 1968. (29K) Wohl, M. H. “Part 5. Dynamics of Flow between Parallel Plates in NonCircular Ducts,” ibid., p p 183-6, 6 May 1968. (30K) Wohl, M. H., “Part 6. Isotherrtlal Turbulent Flow in Pipes,” ibid., pp 95-100,3 June 1968.
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Natural Convection and Related Flows (1L) Adams, J. A., and Lowell, Robert L Jr., “Free Convection Organic Sublimation on a Vertical Semi-Infinite Plate‘:” Int. J . Heat Mass Trunsfer, 11, 121524 (1968). (2L) Beaver P. R., and Hughmark G. A,, “Heat Transfer Coefficients and Circulatioh Rates for Thermosiphbn Reboilers,” A.I.Ch.E. J., 14, 746-749 (1968). (35) Birikh, R. V., Gershuni, G. Z., Zhukhovitskii, E. M., and Rudakov R . N., Hydrodynamic and Thermal Instability of a Steady Convective Flow,” J . Appl. Math. Mech., 32, No. 2,256-263 (1968). (4L) Brian, P. L. T., “Marangoni Instability in Vertical Fallin Films Versus Horizontal Stagnant Liquid Layers,” Chem. Eng. Sci., 23, 1513-19 (1968). (5L) Brodowicz, K., “An Anal sis of Laminar Free Convection Around an Isothermal Vertical Plate,” Int. J?Heut M u s Transfer, 11, 201-9 (1968). (6L) C y o n , I., “Effect of a Gravity Gradient on Free Convection from a Vertical Plate, Chem. Eng. Progress Symp. Series, 64, 146-9 (1968). (7L) Cheesewri ht R “Turbulent Natural Convection from a Vertical Plane Surface,” J . €feu~Tr&fer, 90, 1 (1968). (8L) Chen, Michael M., and Whitehead, John A., “ Evolution of Two-Dimensional Periodic Rayleigh Convection Cells of Arbitrary Wave-Numbers,’’ J . Fluid Mech, 31,l-15 (1968). (9L) Cygan, D. A.,, and Richardson, P. D. “ A Transcendental Approximation for Natural Convectlon a t Small Prandtl i&mbers,” Can. J. Chm. En,., 46, 321 (1968). (1OL) De Leeuw Den Bouter J. A De Munnik B and Heertiqs p. M., “Simultaneous Heat and Mass T;ansfe;)in Laminar kr2e Convection Prom a Vertical Plate,” Chem. Eng.Sn‘., 23,1185-90 (1968). (11L) Dent, J. C., “ H r a t Transfer From a Vertical Transversely Vibrating Plane Surface to Air by Free Convection,” Int. J . Heat Mars Truwfer, 11,605-7 (1968).
(12L) de Vahl Davis, G., “Laminar Natural Convection in an Enclosed Rectangular Cavity,” ibid., pp 1675-93. (13L) Evans, L. B Reid, R. C and Drake, E. M “Transient Natural Convection in a Vertical C y k d e r , ” A.I.2h.E. J . , 14,251 (i568). (14L) Fendell, Francis E., “Laminar Natural Convection about an Isothermally Heated Sphere a t Small Grashof Number,” J . Fluid Mech., 34, 163-76 (1968). (15L) Finlayson, Bruce A., “ T h e Galerkin Method Applied to Convective Instability Problems,” rbid., 33, 201-8 (1968). (16L) Ga e, K. S and Reid, W. H., “ T h e Stability of Thermally Stratified Plane Poiseuilfe Flow$tbid., pp 21-32. (17L) Gpldstein, R. J., and Aun Win “ H e a t Transfer by Free Convection from a Horizontal Wire to Carbon Bioxidd in the Critical Region,” J . Heat Transfer, 90,163 (1968). (18L) Husar, R. B., and Sparrow, E. M., “Patterns of Free Convection Flow Adjacent to Horizontal Heated Surfaces,” Int. J . Heat Mass Transfer, 11, 1206-8 (1968). (19L) Jakeman, E., “Convective Instability in Fluids of High Thermal Diffusivity,” Phys. Fluids, 11, 10 (1968). (2OL) Kato, Hiroharu, Nishiwaki, Nichi, and Hirata, Masaru, “ O n the Turbulent Heat Transfer by Free Convection from a Vertical Plate,” Int. J . Heat Mass Tianrfer, 11, 1117-25 (1968). (21L) Kelleher, M. D., et ul., “Heat Transfer Response of Laminar Free Convection Boundary Layers Along a Vertical Heated Plate to Surface Temperature Oscillations,” Z.Angew Math Phys., 19,31-44 (1968). (22L) Kierkus, W. T., “An Analysis of Laminar Free Convection Flow and Heat Transfer about an Inclined Isothermal Plate,” Int. J . Heat Mars Transfer, 11, 241-53 (1968). (23L) Krishnamurti Ruby “Finite Amplitudk Convection with Changing Mean Temperature. P;rt 1. ?heory,” J . Fluid Mech., 33, 445-55 (1968). (24L) Krishnamurti Ruby, “Finite Amplitude Convection with Changing Mean Temperature. P i r t 2. An Experimental Test of the Theory,” tbrd., pp 451-63. (25L) Kubair, V. G., and Pei, D. C. T., “Combined Laminar Free and Forced Convectlon Heat Transfer T o Non-Newtonian Fluids,” Int. J . Heat Mass Transfer, 11,855-69 (1968). (26L) Kuiken, H. K., “Axisymmetric Free Convection Boundary-Layer Flow Past Slender Bodies.” ibid,. DD 1141-53. (27L) Leontiev, A. I., and Kirdyashkin, A. G., “Experimental Study of Flow Patterns and Temperature Fields in Horizontal Free Convection Liquid Layers, ibid.,pp 1461-6 (1968). (28L) Lightfoot E. N., “Free-Convection Heat and Mass Transfer: T h e Limiting Case of G r d G r -+ 0 and Pr/Sc -+ 0,” Chem. Eng. SA., 23, 931 (1968). (29L) Lipkea, William H., and Sprin er, George S., “ H e a t Transfer Through Gases Contamed Between Two Vertical &ylinders at Different Temperatures, Int. J . Heat Mass Transfer, 11, 1341-50 (1968). (30L) Lock, G. S. H., and deB. Trotter, F. J., “Observations on the Structure of a Turbulent Free Convection Boundary Layer,” rbid., pp 1225-32 (1968). (31L) Lock? G: S. 5.. and Gunn, J. C., “Laminar Free Convection from a Downward-Projecting Fin,” J . Heat Transfer, 90, 63 (1968). (32L) Mabuchi, I., et ul., “Experimental Study on Effect of Vibration on Natural Convective Heat Txansfer from a Horizontal Fine Wire,” Bull. Jap. Soc. Mech. Engr., 10,808-16 (1967). (33L) Mack, Lawrqnce R., and Hardee, Harry C. “Natural Convection Between Concentric Spheres a t Low Rayleigh Number;” Int. J . Heat Muss Transfer, 11,387-96 (1968). (34L) Mitchell, W. T., and Quinn, J. A., ‘I Convection Induced by Surface Tension Gradients: Experiments with a Heated Point Source,” Cfiem. Eng. Sci., 23, 503-7 (1968). (35L) Mitsuishi, N. et ut., “ H e a t Transfer of Laminar Flow in Vertical Tubes with Constant Wall Heat Flux,” J. Chem. Eng. Jap., 1, 120-4 (1968). (36L) Musman, Steven, “Penetrative Convection,” J . Fluid Mech., 31, 343-60 (1968). (37L) Pa ailbu, D,. D., and Lykoudis, P. S., “ M a neto-Fluid-Mechanic Laminar Naturar Convection-An Experiment,” Int. J . Beat Mass Transfer, 11, 1385-91 (1968). (38L) Pearson, I); S., and Dickson, P. F., “Free Convective Effects on Stokes Flow MassTransfer, A.I.Ch.E. J . , 14, 903 (1968). (39L) Plows, William H., “Some Numerical Results for Two-Dimensional Steady Laminar BCnard Convection,” Phys. Fluids, 11, 1593 (1968). (40L) Ree‘hsimghani, N. S., Barduhn, A. J., and Gill, W. N., “Laminar Dispersion in Capidaries. Part V. Experiments on Combined Natural and Forced Convection in ,Vertical Tubes,” A.I.Ch.E. J . , 14, 100-109 (1968). (41L) Schenk, J., and Schenkels, F. A. M., “Thermal Free Convection from an Ice Sphere in Water,” Appl. Sci. Res., 19, 465 (1968). (42L) Shannon R. L and De ew, C. A., “Combined Free and Forced Laminar Convection i; a Hdkizontal &be With Uniform Heat Flux,” J . Heat Trunsfer, 90,353 (1968). (43L) Sherman, M., “Convective Instab es in Fully Developed Flows,” ibid., pp84-6 (1968). (44L) Sherwin, K “Laminar Convection in Uniformly Heated Vertical Concena i c Annuli,” E&. Chem. Eng., 13, 1560-5 (1698). (45L) Sparrow, E. M and De Mello F. Guinle, Luiz “Deviations from Classical Free Convection B&ndary-Layer Theory at Low ’Prandtl Numbers,” Int J . Heat Mass Transfer, 11,1403-6 (1968). (46L) Suriano F. J and Yang K.-T “Laminar Free Convection about Vertical and Horizodtal Pi& a t Smail and Moderate Grashof Numbers,” ibid., pp 47390 (1968). (47L) Takhar H. S., “Entry-Length Flow in a Vertical Cooled Pipe,’’ J . Fluid Mech., 34,641-50 (1968). (48L) Takhar, H. S., “Thermal Convection Near a Partly Insulated Vertical Flat Plate,” Z . Angew Math. Phys., 19,45 (1968). (49L) Tanger, G. E Lytle, J. H and,Vach?n, R . I “Heat Transfer to Sulfur Hexafluoride Near”the Thermod;namic Critical Regibn in a Natural-Circulation Loop,” J . Heat Transfer, 90,37 (1968). (50L) Tien, Chi “Thermal Instability of a Horizontal Layer of Water Near 4’C,” A.1.Ch.E. J., 1k,652 (1968). (51L) Vanier, D. R and Tien Chi,“Effect of Maximum Density and Melting on Natural Convectidn Heat T r k s f e r from a Vertical Plate,” Chem. Eng. Progr. Symp. Sa?., 64,240-54 (1968). (52L) Verpnis,,, George “Effect of a Stabilizing Gradient of Solute on Thermal Convectlon, J . FlurdMech., 34,315-36 (1968). (53L), Veyonis, Geor e “ Large-Amplitude Benard Convection in a Rotating Fluid,”rbtd., 31, Ill-59 (1968). (54L) Warner, Charles Y.,and Arpaci,Vedat S . , “An Experimental Investi ation of TurbuleRt Natural Convection in Air a t Low Pressure Along a Vertical deated Flat Plate, In:. J . Heut Mass Trunsfer, 11, 397-406 (1968).
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Jets (1N) Ackerberg, R . C “ O n a Non-Linear Theory of Thin Jets. Part I.,” J . Fluid Mech., 31, part 3: 583-601 (1968). (2N) Ackerberg R . C., “ O n a Non-Linear Theory of Thin Jets. Part 11. A Linear Theor; for Small Injection Angles,” 32, part 2, 261-72 (1968). (3N) Adelber M., “Mean Drop Size Resulting from the Injection of a Liquid Jet into a &gh-Speed Gas Stream,” AIAA J., 6 , 1143-7 (1968). (4N) Chakraborty, B. B., “Stability of a Perfectly Conducting Liquid Jet in the Presence of a Magnetic Field,” Phys. Fluids, 11,2402-5 (1968). (5N) Chervinsky, A., and Lorenz, D. H., “Analysis of Axisymmetric Compressible Rotating Free Jets,” ibid., pp 1648-53. (6N) Clarke N S “Two-Dimensional Flow under Gravity in a Jet of Viscous Liquid,” j.Fiuid”Mech., 31, 481-500 (1968). (7N) Crowley J. M., “Growth of Waves on an Accelerated Jet,” Phys. Fluids, 11, 2172-8 (196k). (8N)Duda, J. L., and Vrentas, J. S., “Laminar Liquid Jet Diffusion Studies,” A.I.Ch.E. J., 14, 286-94 (1968). (9N) Glicksman L. R “The D namics of a Heated Free Jet of Variable Viscosity Liquid at LodReynAlds Numters,” J . Basic Eng., 90, 343-54 (1968). (10N) Kacker S. C and Whitelaw J. H., “The Effect of Slot Hei ht and Slot Turbulence h e d t on the Effeckveness of the Uniform Density, %wo-Dimensional Wall Jet,” J . &eat Transfer, 90,469-75 (1968). (11N) Kozicki, W., and Tiu, C., “ Expansion-Contraction Behavior of NonNewtonian Jets,” C h m . Eng. Sci., 23, 1165-72 (1968). (12N) Lienhard J. H., “Effects of Gravity and Surface Tension upon Liquid Jets Leaving P o i s e h e Tubes,” J . Basic Ens., 90,262-8 (1968). (13N) Lindow, B., and Greber, I. “Similarity Solution for a Laminar, Incompressible Jet Flowing along a Curced Surface,” AIAA J., 6 , 1331-5 (1968). (14N) Mikami, H., and Takashima, Y., “Separation of Gas Mixture in an Axisymmetric Supersonic Jet,” Int. J . Heat Mass Transfer, 11, 1597-1610 (1968). (15N) Rosler, R . S.,and Stewart, G. H., “Impingement of Gas Jets on Liquid Surfaces,” J . Fluid Mech., 31, 163-74 (1968). (16N) Tillet J. P. K,, ‘,‘On the Laminar Flow in a Free Jet of Liquid a t High Reynolds kumbers,” ibid., 32, 273-92 (1968). (17N) Tobolskii G, F., “Experimental Investi ation of the Decay of a Viscous Fluid Jet Streimlined by a Gas Flow,” P M T F : Zh. Pnkl. Mekhan. Tekhn. Fiz., N0.2.113-16 (1967) (Russ.). (18N) Vallentine H. R “Radial Flow on a Horizontal Plane,” La Houille Blanche, 22,279-82 (1947) (En;.). (19N) Wang, D. P., “Finite Amplitude Effect on the Stability of a Jet of Circular Cross-Section,” J . Fluid Mech., 34, 299-313 (1968). nanski, I. J. and Champagne F. H., “ T h e Laminar Wall-Jet Over a (2?%v~y&wface,” ibid!, 31, 459-65 (1956). (21N) Wyganski I and Fiedler H. E “Jets and Wakes in Tailored Pressure Gradient,” Phis. &id$, 11, 2514-23 (1868). (22N) Yuen, Man-Chuen “ Non-Linear Capillary Instability of a Liquid Jet,” J . FluidMech., 33, 151-63 (1968).
Turbulence (10) Achenbach E “Distribution of Local Pressure and Skin Friction around a Circular Cylinier’in Cross Flow up to R e = 5 X 106,” J . Fluid Mech., 34, 62539 (1968). ( 2 0 ) Acrivos A Leal L. G Snowden, D. D., and Pan, F., “Further Experiments on Steady 6epArated’Flows)PastBluff Objects,” ibid., pp 25-48. ( 3 0 ) Adelberg M., “Mean Drop Size Resulting from the Injection of a Liquid Jet into a Hihh Speed Gas Stream,” AIAA J., 6 , 1143-7 (1968). ( 4 0 ) Armistead, .R. A., Jr,., and Keye? J. J. Jr “A Study of Wall-Turbulence Phenomena U s n g Hot-Film Sensors, Trans. A Z M E . J . Ht. Transfer, Ser. C, 90, 13-21 (1968). ( 5 0 ) Badrinarayanan, M. A., “An Experimental Study of Reverse Transition in Two Dimensional Channel Flow,” J . Fluid Mech., 31, 609-23 (1968).
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K., and Tennekes, H., “Asymptotic Similarit in Neutral Baratropic Planetary Boundary Layers,” J . Atmor. S a . , 25, 1015-20 &968). (140) Bourne, D. E., Figueiredo, O., and Charles, M . E.,“Laminar and Turbulent Flow in Annuli of Unit Eccentricity,” Can. J . Chem. Eng., 46, 289-93 (1968). (150) Bowlus, D . A., and Bri hton J A . “Incom ressible Turbulent Flow in the Inlet Region of a Pipe,” ;;mns.’A:S..d.E., J. f;aric Engr., Sei. D.,90, 431-3 (1968). (160) Bradley, R . G. “ A proximate Solutions for Compressible Turbulent Boundary Layers in T’hree-flimensional Flow,” AZAA J . , 6 , 859-64 (1968). (170) Bradshaw, P., Comment on “Pressure Fluctuations beneath an Incompressible Turbulent Boundary Layer with Mass Addition,” Phys. F h d s , 11, 1135-6 (1968). ( 1 8 0 ) Braghskii, L. N., ”Distribution of Solid Particles Along Height in Baffleless Apparatus, Theor. Found. Ch.E., 2, 126-130 (1968) (Eng. trans. of Teoret. Osnouy Khim. Tekh.). (190) Braginskii, L. N., Begachev, V. I., and Kofman, G. Z., “Distribution o Solid Particles during Mechanical Mixing,” ibid., pp 110-1 5. ( 2 0 0 ) Brown, R . A. S., and Jensen, E. J . , “ T h e Pipeline Flow of Paste Slugs, Part 11. Pressure Gradients and Velocities of Trains of Slugs,” Can. J . Chem. Eng., 46,157-61 (1968). (210) Capps, D. O., and, Re,$m, T. R., “Empirical Expression for the Turbulent DES.DEVELOP., 7, 311Flow Velocity Distribution, IND.ENC.CHEM.,PROCESS 13 (1968). (220) Cheesewright, R., “Turbulent Natural Convection from a Vertical Plane Surface,” Trans. A S M E . , J . Heat Transfer, Sei. C . , 90, 1-8 (1968). (230) Chervinsky, A,, “Similarity of Turbulent Axisymmetrical Swirling Jets,” AZAA J., 6,912-14 (1968). (240) Chevray, R., “ T h e Turbulent Wake of a Body of Revolution,” Trans. A.S.M.E., J . BorzcEng.,Ser. D,90,275-84 (1968). (250) Chow, W . L., “Study of Jet Mixing Problems by Meksyn’s Method,” AZAA J., 6,2422-4 (1968). (260) Clamen, A,, and Gauvin, W. 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G., “Production of Negative Eddy Conductivities and Viscosities in the Presence of Buoyancy and Shear,’’ Phys. Fluid, 11, 432-33 (1968). (350) Deissler, R . G., “Weak Locally Homogeneous Turbulence and Heat Transfer with Uniform Normal Strain,” Z A M M , 48, 87-98 (1968). (360) Dimopoulos, H. G., and Hanratty, T. J., “Velocity Gradients at the ?I1 for Flow Around a Cylinder for Reynolds Numbers between 60 and 360, J . Fluid Mech., 33, 303-19 (1968). (370) Doig, I. D., and Roper, G. H., “Contribution of the Continuous and Dispersed Phases to the Suspension of S heres by a Bounded Gas-Solids Stream,” I N D END. . CHEM.,FUNDAM., 7,459-71 8968). (380) Eckert, E. R . G., and Rodi, W., “Reverse Transition Turbulent-Laminar for Flow Through a Tube with Fluid Injection;’ Trans. A . S . M . E . , J . Appl. Mech., Ser. E , 35, 817-19 (1968). (390) Figueiredo, O., and Charles, M. E., “Critical Fluid Velocity and Power Requirement for the Startup of a Capsule Pipeline,” Can. J . Chem. Eng., 46, 62-6 (1968). (400) Foss, J.,?., and Jones, J. B., “Secondary Flow Effects in a Bounded Rectangular Jet, Trans. A . S . M . E . , J . Basic Eng., Sei. D.90, 241-8 (1968). ( 4 1 0 ) Gcl’perin, N. I., Ainshtein, V. G., and Kru nik, L. I., “Structure of a Theoret. Found. Chem. Eng., Horizontal Two-Phase Flow of Gas-Solid Particles! 2 . 508-16 (1968) ( E n d . transl. of Teoret. Osnouy Khirn. Tekh.). (420) Gibson C. H “Fine Structure of Scalar Fields Mixed by Turbulence, 11. SpectralThe‘ory,”$hys. Fluid, 11,2316-27 (1968). (430) Gibson, ,C. H., “Fine Structure of Scalar Fields Mixed by Turbulence I. Zero Gradient Points and Minimal Gradient Surfaces,” ibid., pp 2305-15; (440) Gibson, C. H., Chen, C. C. and Lln, S. C., “Measurements of Turbulent Velocity and Temperature Fluctuations in the Wake of a Sphere,” A I A A J., 6 , 642-9 (1968). (450) Good, M. C., and,Joubert, P. N., “ T h e Form Dra of Two-Dimensional Bluff-Plates Immersed in Turbulent Boundary Layers,” J . 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Papers from two symposia by the Division of Organic Coatings and Plastics Chemistry of the American Chemical Society. This volume includes twelve papers comprising the symposium on “The Interaction of Liquids at Solid Substrates,” chaired by Allen L. Alexander. These papers include work on “coupling agents,” afdhesion of polymers, organic/ inorganic interfaces, and ultrasonic impedometry. Also included are1 four papers concerned with heparinized surfaces at the blood/lmaterial interface which were part of the symposium on “The Medical Applications of Plastics,” chaired by R . I. Leininger.
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