Oscillating liquid jet
WILLIAM N. GILL ROBERT COLE JOSEPH ESTRIN RICHARD J. NUNGE HOWARD LITTMAN
ANNUAL REVIEW
FLUID DYNAMICS This review presents comprehensive coverage of the 1965-66 literature and emphasizes the development of basic fluid dynamic principles his two-year review covers papers published during T the period from January 1965 to December In constructing any review which deals with as wide 1966.
ranging and difficult a subject as fluid dynamics, at the outset one has to determine whether to concentrate on breadth or depth of coverage. I n the present review we have attempted to strike a balance in this regard. However, since we found it necessary to report approximately 1100 references, some preference has been given to the breadth of coverage. It 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. I t is felt that such work is most likely to have permanent value. No doubt, some very significant papers relevant to fluid dynamics have not been reported. Single-phase laminar Flow of Newtonian Fluids in Channels
Entrance region. The importance of accurate descriptions of entrance region flows for stability analyses and for furthering the theory of other convective trans-
port processes is reflected by the numerous analytical studies refining and extending previous boundary layer and theoretical treatments. Finite difference solutions to the equations of motion predominate. Besides the work on isothermal, incompressible flows for various geometries summarized in Table I, the work of Rosenberg and Hellums (57u) is notable in that it accounts for radial convection, inertia, and temperature dependence of the viscosity. Also, Horton and Yuan (36u) found that the entrance length was reduced considerably by injection through porous walls. A series of short discussions on the effect of the velocity entrance condition on the validity of the boundary layer approximation ( I & , 34u) and the nonuniqueness of the entrance length ( 6 4 ~point ) to further research. Bibliographies of hydrodynamic entrance region flow (25u, 3%) (reference 25a has over 130 references) should prove useful to workers in this area. Stability. Experiments providing important new information on the stability of pipe flow were performed by Scheele and Greene (SOU) for heated vertical systems with transition induced by small disturbances of the velocity profile through natural convection. Disturbances by salient rings or sharp corners at the inlet were used in ~
TABLE I.
Circular tubes Annulus Flat plates Rectangle Asymmetric gap Porous walls
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294, 77a)
(44a, 58a)
(374) (8a) (4Oa ) (350 )
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experiments by Cumo et al. (74a) for horizontal pipes. Other experiments involving horizontal heated systems of flowing hydrogen with transition flow are described (2a, 68a). Important additions to the theory of stability of Poiseuille flow were made by Gill (27a, 2 8 a ) who proposed a new relationship between the mean velocity profile and the occurrence of disturbances. I n further theoretical treatments the lower critical Reynolds number for laminar-turbulent transition in rectangular ducts was successfully predicted by Hanks and Ruo ( 3 2 a ) . Barnett ( 3 a ) considered stability in a parallel plate system. The critical Reynolds number in pulsating pipe flow was determined experimentally by Sarpkaya (59~1, and Conrad and Criminale (72a) analyzed the stability of time-dependent parallel flows. Other analytical transition studies include the stability of plane Couette-Poiseuille flow with ( 2 6 a ) and without (55a)temperature gradients. Isothermal channels. Extensive analytical friction factor results for fully developed laminar flow in eccentric annular ducts have been given by Snyder and Goldstein (67a) and by Jonsson and Sparrow ( 3 8 a ) . Vaughn ( ~ O U )significantly, , has pointed out that this and several other problems in steady, axial, irrotational flow of Newtonian fluids through impervious ducts are already known from analogous problems in the theory of elasticity. Other analytical solutions for steady laminar flow in straight ducts with impervious walls include a study of confined wakes (522) (which is to be noted primarily for a discussion of numerical techniques) and a study of a stepped rectangular duct (79~).Converging steady flow in axisymmetric passages including inertia effects was treated by Ackerberg (la),Goldstein ( 3 0 a ) , and Nishimura and Oka (50a)using the boundary layer approach. Sutterby (63a)studied a similar problem using a finite difference technique. The stability of Jeffery-Hameltype solutions for divergent flow was investigated by Eagles ( 2 2 ~ ) . Unsteady flows have been considered analytically for elastic cylindrical shells (53a), for rectangles (ZOa, N u ) , and for parallel plates under various types of pressure gradients (45a). Experimental data for the radial flow of water between parallel disks were obtained by Chen ( I O U ) . Further experimental and analytical treatment of the radial flow problem has been given ( 3 7 ~ ) . Nonisothermal laminar flow. Heat transfer and friction for laminar flow of fluids in horizontal and vertical tubes have been of considerable interest because of the difficulties associated with treating temperature dependent physical properties. Experimental studies of vertical systems are given in references 5a and 77a. The definitive work of Bergrnan and Koppel ( s a ) is noteworthy in pointing out the inadequacies of some existing theoretical and experimental efforts. Experimental work on horizontal systems includes the measurement of velocity distributions in heated straight (48a)and curved pipes ( 4 9 a ) by Mori and coworkers. Other experimental efforts are reported by Mikhailov et al. (47a) for horizontal flow of water. Analytical stud70
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ies of the variable physical property problem in tubes and other geometrics are given (sa,7a,33a, 54a,69a,73a, and 74a).
Flow rates, pressure drop and friction factorsh ave been correlated empirically for isothermal (73a) and nonisothermal flows (47a-43a) for laminar, transition, and turbulent flows in coiled tubes. Porous walls. Theoretical solutions for laminar isothermal flow through ducts with porous walls have been extended to include different geometries and different boundary conditions. Das ( 7 % ) in an interesting extension of previous work has treated circular cylinders with variable injection and suction rates and ) uniform arbitrary injection and suction annuli ( 7 6 ~with velocities. ‘Terrill (65a) considered analytically slow flow in a converging or diverging duct, in a flat plate system with difyerent uniform velocities at each wall (67a), and also treated the nonisothermal case (66a). Experimental pressure drop data for air in a porous tube with uniform suction rates were taken by Wallis (72a). Peng and Yuan (52a) considered nonisothermal pipe flows with hydrogen as a coolant. Slip flow. Several experiments have been reported on slip flow and the transition between free molecule and continuum flows. Barrer and Kicholson (4a)studied several rare gases in long capillaries. Experiments including the transition regime were made by Carley and Smetana (9a)and Sreekanth (620). Ebert and Sparrow ( 2 3 a ) in analyzing slip flow in rectangular and annular ducts, found that the effect of slip flow was to flatten the velocity profile and reduce the axial pressure gradient. McComas and Eckert (46a)in experiments on entrance region flows for both slip and continuous flow determined the slip flow corrections. The transport of air-water vapor mixtures through freeze dried meat, determined experimentally by Dyer and Sunderland (27a) was favorably compared with a capillarl- tube model constructed from an analysis of the transport of a binary gas mixture valid for the range of flow from molecular to viscous. Reynolds and Richley (56a)considered free molecule gas flow and surface diffusion anall-tically for single tubes and indicated applications to porous media. Two-Phase Flow
The current literature on two-phase flow is concerned with the prediction and description of flow patterns, determination of two-phase pressure drop, void fraction and slip ratio, investigations of critical flow, and two-phase flows in particular geometrics such as pipe elbows, tees, or contractions. -4 book by L. S. Tong (466)on boiling heat transfer and two-phase flow should be of considerable value to anyone interested in this field. Review articles have been published by Govier (766) and Lottes et al. (28b). Berenson and Stone (76) in a photographic study of the vaporization of Freon 113 flowing inside a horizontal pyrex tube conclude that the annular-mist flow transition results from the vaporization of the liquid annulus while liquid droplets remain dispersed throughout the vapor core. An analytical model for the prediction of two-
TABLE II. ADDITIONAL REFERENCES ON TWO-PHASE FLOW Design (361 Hydraulic resistance ( 1l b , 306, 436,456) Two-phase pressure drop (46, 146, 226, 236, 486) Liquid metals Stability Void fraction, slip ratio
( 146 (66, 236) (96, 726, 736, 196, 336, 346, 416)
Interfacial area
(76)
phase annular flow with liquid entrainment has been presented by Levy (266). Quandt (366, 376)) using dye injection techniques, has investigated two-phase annular flow with droplet interchange and developed dimensionless criteria for predicting when the major flow patterns will exist. T h e flow regimes occurring during the evaporation of Freon 12 and Freon 22 were investigated by Lavin and Young (246). A correlation was obtained for the transition between annular and mist flow. Brill et al. (8b) have concluded that vertical and horizontal two-phase flow correlations can be used to predict accurately production rates possible from both flowing and gaslift wells. The unsteady behavior of gas-liquid slug flow through a vertical pipe has been investigated by Street and Tek (446). The accompanying analysis was successful in predicting the two-phase pressure drop, provided gas bubble lengths and frequencies of generation were known. Leonov et al. (256) have reported a visual investigation for two-phase dispersed and slug flows in an annular passage with a rotating inner cylinder. Within the conditions of the experiment, neither the flow structure nor gas content was affected by the rotation, however the hydraulic resistance of the annulus was substantially increased. A correlation considering fluid properties, mixture quality, and mass velocities has been obtained by Baroczy (56) for the prediction of two-phase friction drop in both single- and two-component flow. Van der Walle et al. (476) have presented a theoretical study of twophase flow characteristics, in which they ascribe the increase in frictional pressure drop and heat transfer in twophase flow, relative to single-phase flow, as due to an increased turbulence level resulting from the motion of vapor bubbles relative to their surroundings. The effect of void fraction, velocity profile, phase distribution, and fluctuations, on the momentum flux in two-phase flow are discussed by Andeen and Griffith (26).
William N . Gill is Chairman of the Department of Chemical Engineering, Robert Cole and Joseph Estrin are Associate Professors, and Richard J . Nunge is Assistant Professor of Chemical Engineering at Clarkson College of Technology, Potsdam, N. Y . Howard Littman is Professor of Chemical Engineering at Rensselaer Polytechnic Institute, Troy, N. Y. AUTHORS
Zuber and Findlay (576) have developed a general expression, useful for either predicting the average volumetric concentration in a two-phase flow system, or for analyzing and interpreting experimental data. Pressure drops and void fractions in horizontal two-phase flows of potassium have been measured by Smith et al. (42b). The two-phase friction factors for potassium were found to be substantially lower than values predicted by the usual correlations. Void fractions were measured by x-ray attenuation. Pike et al. (356)report on the details of the measurement of void fractions in two-phase flow by x-ray attenuation. T h e paper is mainly concerned with experimental techniques and error analysis. I n this respect the paper should be of considerable value. T h e experimental data have for the most part been deposited with the ADI. A probe designed to measure the detailed radial distribution of two phases and their local velocities has been developed by Shires and Riley (406). Ishigai, et al. (276) describe the measurement of the component flows in a two-phase system by means of a strain gage. T h e effect of pipe length to diameter ratios on twophase critical flows has been investigated experimentally by Min et al. (296) and analytically by Moody (326). Additional theoretical models for the prediction of two-phase critical flow have been presented by Moody (376) and by Levy (266). Isbin (206) has prepared graphs showing the application of the Moody model for estimating the two-phase critical flow rates of the liquid metals Cs, Na, K, Li, Rb, and Hg. Two-phase flows in particular geometries have received notable attention recently. Geiger and Rohrer (756) have reported a n experimental study of sudden contraction losses in two-phase flow. The results mainly indicate the fog flow model to yield the best agreement with experiment, and as pointed out in the discussion, the possibility thus exists that the flow can be treated as singlephase, using the wet steam density. Zaloudek (50b) has investigated the effects of elbows and tees on the critical flow of steam-water mixtures from a piping system when the system is opened to the atmosphere adjacent to a tee or elbow. Two-phase flow in a cavitating venturi has been considered by Hammitt and Robinson (776). Chen (706) has investigated two-phase flow through apertures. The interaction of a two-phase gas liquid flow in a venturi scrubber has been studied photographically by Yugai and Volgin (496),while Rippel et al. (386) have presented data on two-phase flow in a coiled tube. VOL. 5 9
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Additional two-phase flow papers are classified in Table 11. Cavitation
Current problems of interest fall into two main areas : effects of cavitation in specific types of equipment and the effects of various parameters on cavitation damage. An analytical approach for the calculation of the effect of cavitation on flow past a flat plate has been presented by Zwick (306). Young and Holl (296) report an experimental investigation on the effects of cavitation on periodic wakes behind symmetric wedges. A theoretical analysis showing the bending of ship propeller blades at the trailing edge to be a result of high pressure and collapsing cavitation bubbles is presented by von Wijngaarden (276). Troskolanski (25c) reports a cavitation theory of jet pumps in which relationships are deduced between pressure and cavitation number. Cavitation studies in pipe bends have been presented by Stonemetz (23c) for bends of 6 0 ° , 90", and 120" with pipe diameters of 1.5, 2.0, and 4.0 inches and by Kamiyama (77c) for a 90" bend of square cross section. Spraker (226) uses water, gasoline, fuel oil, and crude oil to investigate the effects of fluid properties on cavitation in centrifugal pumps. Visual studies of cavitation and flow patterns within a rotating blade row have been described by Soltis (276). Numachi and Kobayashi (766) have reported experiments to improve the design of venturi nozzles with regard to cavitation effects. Investigations on cavitation and energy dissipation for water jets associated with conduit expansions are presented by Rouse and Jezdinsky (79c). The wall effects of a current on the cavitation flow around a cylinder have been studied by Shalnev (206). I n this study, six cylinders having diameters ranging from 5 to 20 mm. were arranged in a channel having a cross-section of 20 X 50 mm. The analysis of the test results was referred (by extrapolation of the experimental data) to a flow of infinite width. Ratner and Bukrinskii (78,) develop a procedure for the prediction of the occurrence of cavitation in a gate valve. Comparisons with existing experimental data are carried out for various valve openings. Vasvari (26c) postulates that the cavitation damage process consists not only of momentum and thermal effects, but also chemical reduction, electrolytic corrosion, fatigue, and electrical spark discharges. Detailed investigations were carried out by Thiruvengadam and Waring (24c) to determine the cavitation damage resistance of 11 metals in distilled water. Results indicate the correlation between strain energy and cavitation damage resistance to be a direct indication of the intensity of cavitation damage. Plesset and Devine (776) present some extremely interesting experiments on the effect of exposure time on cavitation damage. The authors conclude that severe cavitation damage changes the flow properties in the neighborhood of the damaged surface so that the formation of clouds of cavitation bubbles are greatly inhibited. Leith (74c) has presented a prediction 72
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
for cavitation damage in the alkali liquid metals. The effect of the speed of flow of water on cavitation damage has been investigated by Kozyrev (73c). A model for cavitation destruction is postulated by Korobeinikov (726) assuming the main reason for cavitation erosion to be a shock wave from the implosion of gas bubbles with the source of the wave being a detonation at a point on the surface of the interface between solid and liquid. Johnson ( 7 0 6 ) presents what is essentially a review paper with the purpose of directing attention to research models which can be more easily scaled to full size units. Initial cavitation damage to test specimens in a cavitating venturi has been investigated by Hammitt et al. (66). Eisenberg and Preiser ( 4 6 ) present an up-to-date review on the mechanisms of cavitation damage and methods of protection. A review on cavitation phenomena in hydraulic systems, including selected experimental data to complement the discussion, is presented by Daily ( 3 6 ) . Many interesting papers, presented at an ASME symposium on cavitation in fluid machinery have been bound and are available from ASME, New York (286). The volume deserves serious attention from anyone concerned with cavitation phenomena. Other papers of interest include a study of visible cavitation in liquid helium by Finch and Wang (5c), experiments on luminescence during cavitation by Brauns (IC), a description of physical effects of acoustically induced cavitation by Jarman and Taylor (Sc), an experimental study of accelerated cavitation induced by ultrasonics reported by Numachi (75c), experiments and analysis on cavitation hysteresis in high speed liquid flow by Holl and Treaster (&), a discussion of some physical effects of cavity collapse in liquids by Hickling ( 7 c ) , and a useful bibliography on cavitation in fluids covering the 2-year period 1963-65 by Chadwick ( 2 6 ) . liquid Films
A substantial portion of the literature on liquid films is devoted to the prediction and measurement of velocity profiles, film thicknesses, surface waves, and film stability. For the case where cocurrent or countercurrent gas-liquid flow exists, additional interest is focused upon interfacial resistance and prediction of the two-phase pressure drop. Semenov e t al. (394 report a modification of the method of feed stoppage for determination of the film thickness on a wall of known area. For this method, the liquid film contains NaCl at a known concentration. After feed stoppage, the film is washed down by distilled water and the amount originally present on the wall determined by titration of the C1 ion with AgN03. The experimental error is claimed to be within 57,. Chernobyl'skii e t al. ( 7 I d ) , describe an experimental apparatus for the evaluation of heat transfer in liquid film flow. A needle thermocouple, micrometer, and oscilloscope, together with a knowledge of the wall temperature, are used to determine the film thickness and temperature profile. Cohen and Hanratty (744 take into account the effect of surface waves on film thickness by use of a dimensionless
film height defined in terms of the friction velocity and kinematic viscosity of the liquid. Experiments were performed with gas-liquid flows in a horizontal channel. White (49d) has investigated the effect of surface tension on the relationship between flow rate and mean film thickness, while Chien (72d) has obtained a convenient parameter for determining liquid film thickness by rearrangement of the variables in the solution for laminar gravitational film flow. An analytical solution relating the film mass flow rate and film thickness, based on a modified Dukler theory has been reported by Davis (17d). Mouradian and Sunderland (30d) have analytically investigated the velocity and temperature distribution in a liquid film appearing on a flat surface due to either transpiration through a porous medium or the melting of a solid material. Atkinson and Caruthers (4d) have employed hot wire techniques to obtain the velocity profile for both laminar and turbulent liquid film flows. Turbulent velocity profiles for vertical film flow have been predicted analytically by Lee (27d) using the eddy viscosity expressions of Deissler and von Karman. Upward annular flow of air-water mixtures has been investigated by Shearer and Nedderman (4Od), Willis (504,and Gill et nl. (244. The primary emphasis has been that of predicting the two-phase pressure drop, although film thickness, interfacial velocities, and wall shear stress variations have also been reported. Pike (36d) has considered the problem of downward annular flow of air-water mixtures. A theoretical model for the velocity distribution is proposed which essentially replaces the liquid film with a rough wall moving at some finite velocity in the same direction as the gas stream. Horizontal stratified flows of air-water mixtures have been considered by Smith and Tait (42d) with the primary emphasis being upon interfacial shear and momentum transfer. Ostrach and Koestel (354 discuss the different physical phenomena that lead to instability of a liquid film. Anshus and Goren (2d) have presented a method of obtaining approximate solutions to the Orr-Sommerfeld equation (obtained by substituting the assumed stream function into the Navier-Stokes equation and eliminating the pressure by cross differentiation) for flow on a vertical wall. The method consists of replacing the velocity (normally a function of distance from the wall) by its free surface value, while the second derivative is kept at its true value. Whitaker and Jones (47d) consider the stability of falling liquid films by a perturbation solution of the Orr-Sommerfeld equation. Taylor’s instability of a liquid film on a solid is investigated by Zaitsev (52d). Massot et al. (29d) describe the wave motion and streamlines in a falling liquid film by means of steady state periodic solutions of the complete Navier-Stokes equations. Epstein (78d) has presented mathematical corrections to an article by Portalski which however do not affect any of the conclusions of that paper. The effect of wave flow on mass transfer has been considered by Ruckenstein and Berbente (38d). Experimental measurements of wave motion have been reported by Cohen and Hanratty (744, who consider the
transition from two- to three-dimensional wave structure. Charles and Lilleleht (7Od) report a stereophogrammetry technique used to measure amplitudes and wave lengths for wave patterns generated at the interface between two liquid layers flowing cocurrently in a closed rectangular conduit. A light absorption technique was employed by Stainthorp and Allen (434 for measuring the thickness, velocity, and frequency of waves at the surface of a liquid film. Ponter and Davies (374 have investigated the effect of surface alignment on hydrodynamic stability for falling films. Goodridge and Gartside (25d) report that ripplefree films are produced with Reynolds numbers up to 900, provided the angle of inclination is less than 20 minutes; a rather surprising result, which, if true, could explain the many existing discrepancies in existing surface velocity data. Other papers of interest on stability and wave motion include a theoretical treatment of the stability of a liquid layer adjacent to a high speed gas stream by Chang and Russell (9d), an experimental investigation of the instability of a liquid surface during slippage of detonation and shock waves along it by Borisov et al. ( 7 4 , an analysis of the effect of surface films in damping eddies at the free surface of a turbulent liquid by Davies (76d), and an analysis of the propagation of long waves on liquid films flowing down iiiclined planes by Benney (6d). The breakdown of liquid films on a heated wall is of interest as a result of the possibility of local boiling and potential “burnout.” Papers concerned with this or related problems have been presented by Hsu et al. (23d), Hallett (ZOd), Hewitt and Lacy (27d), Hewitt et al. (ZZd), and Murgatroyd (374. White and Tallmadge (48d) have presented an analysis for estimating the film thickness and flow rate at which a liquid is dragged out of a bath by a moving flat plate. An experimental investigation of films adhering to large wires upon withdrawal from liquid baths was reported by Tallmadge et al. (44d). O’Loughlin (334 has considered the problem of predicting the thickness of a laminar liquid film draining from a vertical surface with evaporation at the film surface. Other papers of interest include the problem of free flow over the leading edge of a vertical surface in the ena study of viscous flow down trance region by Bruley (84, an inclined plate in which stream width and upper surface are prescribed by surface tension (rivulet flow) by Towel1 and Rothfeld (454,and a consideration of the radial spread of a liquid stream on a horizontal plate by Olsson and Turkdogan (344. Nuttall (32d) has presented an analysis of the flow of a viscous incompressible fluid in an inclined uniform channel, with application to the flow on a transporter belt. An experimental investigation of waves in a thin liquid layer on a rotating disk has been reported by Espig and Hoyle (79d). Wave formation on a thin rotating liquid layer on the inner surface of a cylindrical tube has been investigated experimentally by Alimov ( I d ) . Rotation of the liquid layer was achieved by feeding air into the cylindrical tube by means of a tangential slit orifice. VOL. 5 9
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Non-Newtonian flow in f i l m s . . . . Stratified flows.. . . Dynamics of bubbles., . . Non-1Vewtonian flows have been investigated either analytically or experimentally for a variety of problems. Yih (574has considered the stability of a special form of Reiner-Rivlin liquid flowing down an inclined plane under gravity. Listrov (28d) and Bagchi (5d) have analyzed the same problem for the flow of a viscoelastic Maxwell liquid. Film flow of power-law model non-Newtonian fluids over rotating surfaces has been considered by Vachagin et al. (42). Astarita (3d) has experimentally investigated the dissolution of slabs of benzoic acid into a falling film of aqueous carboxymethylcellulose solution, while Gutfinger and Tallmadge (26d) consider the problem of films of non-Newtonian fluids adhering to flat plates for withdrawal and drainage. Experimental data using viscoelastic and inelastic fluids were obtained for the withdrawal problem. Stratified Flows
Stability and wave analyses appear to dominate the current literature on stratified flows. Kao (Se, 6e) considers the stability of a two-layer viscous stratified flow with upper free surface down an inclined plane under gravity. Two modes of disturbance are found, the first resulting in instability at the free surface and the second yielding instability at the interface. Density and thickness ratios of the upper and lower fluid layers determine which of the two modes will initially cause instability. Charles and Lilleleht ( l e ) report an experimental investigation of stability and interfacial waves in the cocurrent flow of oil and water in a rectangular conduit. Manolescu (9e) presents a simple, but interesting investigation in which two immiscible liquids are placed in a vertical tube with the denser liquid underneath. Upon inverting the tube the system becomes unstable, with the meniscus being influenced by gravity and surface forces. The time of persistence of the meniscus was measured and correlated in terms of a dimensionless parameter involving among other quantities, the triphase contact angle. The motivation for the investigation was to aid in clarifying the mode of action in the process of displacing petroleum crude by water. Willson (73e) has presented an analysis of the stability of two superposed fluids by means of the dispersion equation. An analysis of the stability of two fluid wheel flows is presented by Reshotko and Monnin ( 7 0 e ) . Townsend (72%) reports an analysis of internal waves produced in a stably stratified fluid by means of a convective layer. The predictions are compared with observations of temperature fluctuations in the stable region of an ice-water convection system and with observations of 74
INDUSTRIAL AND E N G I ~ E E R I NC GH E M I S T R Y
clear air turbulence over stratocumulus clouds. A mathematical description of resonant interactions between surface and internal waves in a stratified fluid has been presented by Thorpe ( 7 7 e ) . Yih (14e) has published a note on edge waves in an inviscid stratified fluid. Information concerning velocity profiles, pressure gradients, and drag coefficients for stratified flows in rectangular conduits has been reported in several papers. Charles and Lilleleht (Ze, 3 e ) present pressure gradient flows, while experimental results on velocity profiles, wall shear, interfacial shear, interfacial velocity, and interfacial position for turbulent cocurrent flows have been presented by Darby and Akers (de). Bubble Dynamics
The current literature on bubble dynamics continues to include such topics as bubble growth and collapse, information and analyses on frequencies and rise velocities, deformation, size distribution, and wake formation. Bankoff (5f)and Plesset (25f)present discussions of some of the theoretical aspects of bubble dynamics. Yang and Yeh (371, 32f) analyze growth and collapse rates in terms of pressure distribution and rate of energy dissipation for both h-ewtonian and non-Newtonian fluids. Hsieh (1st)treats both the growth-collapse problem and that of bubble oscillation. An experimental investigation of the dynamics of vapor bubbles in propane is presented by Wanninger (30s)at pressures from 8 to 29 atmospheres. I n addition to the growth data, information is also presented on frequency, departure diameter, and rise velocity. Arpaci et al. (2f)report an approximate analysis and experimental investigation of the dynamics of gas-vapor bubbles in binary systems. The collapse of a cavitation bubble, assuming an adiabatic compression of gas within the bubble, is analyzed numerically by Ivany and Hammitt (77f). Elliott (77f) has considered the transient growth or collapse of a spherical bubble by numerical solution of the Lagrange equations. Papers concerning various aspects of gas or vapor bubbles rising through a liquid medium seem to occupy a major portion of the literature on bubble dynamics. has investigated rise velocities and wake Collins (8f-70f) behavior for two-dimensional gas bubbles. An expression relating the rise velocity of a swarm of spherical bubbles to that of a single bubble has been developed by Marrucci (22f). Moore (23f)has investigated the velocity of rise of distorted gas bubbles. Condensation of single vapor bubbles rising in an immiscible liquid medium has been the subject of an experimental project by Sideman and Hirsch (28’). Bubble rise velocities in the presence
of mass transfer with Newtonian fluids have been the subject of an investigation by Redfield and Houghton (26f) and with non-Newtonian fluids by Barnett et al. (Sf). Rise velocities in non-Newtonian fluids without mass transfer have been studied by Astarita and Apuzzo (3f). The rise of large gas bubbles in tubes, often referred to as “slug flow,” has received attention from several sources. Brown (7f)has considered the rise velocity of such bubbles, Hughmark (76f)has treated the subject of holdup, while Street and Tek (29f)have employed an integral method to predict the bubble shape. Zukoski (33f)has investigated the effect of inclination angle on the motion of long bubbles in closed tubes. Bubble motion in a vertical turbulent water stream has been investigated by Baker and Chao (4f).I t was found that the bubble relative velocity in a turbulent stream is similar to the rise velocity of single bubbles through a quiescent liquid. Gal-Or and Resnick (72f)have reported that the relative velocity in an agitated gas-liquid dispersion increases as the impeller rotational speed is increased. Il’ichew and Neuimin (27f) have found the size distribution of gas bubbles in an impeller-created turbulent flow to vary like a log-normal distribution. Other topics of interest include the motion of a bubble in a vertically oscillating inviscid liquid by Jameson and Davidson (78f)and Jameson’s (79f) treatment of the same problem but for a viscous liquid. Padmavathy et al. (24f)have considered the effect of orifice submergence on bubble formation in horizontal orifices. The deformation of gas bubbles and liquid drops in an electrically stressed insulating liquid has been investigated by Kao (ZO j) while Gleim and Vilenskii (73f)have considered the physiochemical conditions for the generation of bubbles and drops in boiling liquids. Schraub et al. (27f)have discussed in detail the use of hydrogen bubbles for quantitative determination of time-dependent velocity fields in low speed water flows. Drop Formation and Motion
The current literature on drop formation and motion is primarily con cerned with such topics as shape, oscillation, internal motion, stability, entrainment, breakoff criteria, and size distribution. Wellek et al. (72g) develop expressions for the eccentricity of nonoscillating liquid drops moving in liquid media with Reynolds numbers from 6 to 1354. Twenty-eight dispersed phase-continuous phase systems were investigated and 17 additional ones were obtained from the literature. The effect of small amounts of various powders on the shape and terminal velocity of water drops falling in air has been investigated by Lihov (3g). Raghavendra and R a o (7g) have investigated the motion of drops of nitrobenzene through water in various modes such as isolated assemblages of drops in the same horizontal plane, isolated assemblages one behind the other, sprays, and single drops one behind the other. The effects of local variations in interfacial tension on the motion of drops in liquid-liquid systems have been reported by Valentine et al. ( IOg). Schroeder and Kintner (Sg) have investigated the os-
cillations of drops falling in a liquid field. They conclude that oscillations begin at or near the drop size corresponding to the peak velocity and that a vortex trail (requiring NRe> 200) is a necessary condition. Circulation velocities inside drops for Reynolds numbers from 3.76 to 19 were measured using dark-field trace photography by Horton et al. (2g). Francis et al. (Ig) have employed photographic techniques to investigate the entrainment of air into a liquid spray created in single-hole fan spray nozzles. The mass of entrained air, decay of air velocity along the spray axis, and spread of the drops normal to that of the spray sheet were related to the operating conditions. Conditions for the formation of uniform 0.5- to 1-mm. drops obtained from the breakup of liquid jets have been reported by Vivdenko and Shabalin (779). Parvatikar (5g) has published a method for determining the surface tension of fluids by measuring the equatorial radius and depth of a pendant drop at the end of a conical tip. An analysis of the process of drop formation in the extrusion of a Newtonian fluid from a capillary has been presented by Manfre (4g). Additional papers of interest include a stability analysis by G. I. Taylor (99) which considers the two situations of a drop being torn apart by an electric field and a drop of fluid of small viscosity in a fluid of much greater viscosity flowing in one direction with a uniform rate of shear and an investigation by Paul and Sleicher (6g), tentatively indicating the effect of pipe diameter on the maximum stable drop size in turbulent flow. Jets
Jet flow problems in the current literature cover such topics as wave formation, stability, shape, deflection, impingement, swirl, and screening. Two new texts have appeared: one by Gurevich (8h) concerning the theory of jets in ideal fluids, which is a translation from the Russian and has received excellent reviews, the second by Vulis and Kashkarov (25h)is concerned with the theory of jets of a viscous fluid and is in Russian. Wave motion on jet surfaces is considered by Middleman and Gavis (74h), where the motion results from sinusoidally, transversely vibrating nozzles, by Crowley (3h), who considers the effect of steady and nonsteady applied electrical fields, and by Ivanilov (7 7h), who shows that a necessary condition is an increasing energy with increasing distance from the jet axis. Grant and Middleman (7h) and Middleman (75h) consider Newtonian jet stability and stability of a viscoelastic jet, respectively. Goren and Wronski (5h)present both analysis and experiment on the shape of a jet of Newtonian liquid issuing from a capillary needle into air. The deflection of a jet injected into an entraining stream, where the jet is initially perpendicular to the stream has been treated numerically by Vizel and Mostinskii (24h) and compared with experiment. Agreement depends upon choosing the proper value of an unknown coefficient which appears in the term describing the force acting on the jet. Banks and Bhavamai (2h) VOL. 5 9
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present an experimental study of a liquid jet impinging normally on the surface of a heavier immiscible liquid. T h e results are analyzed in terms of dimensionless quantities obtained from an analytical model. The configuration of the free surface above a vertical jet was investigated by Hunt and Hsu (70h). The problem of a swirling radial jet has been considered by both Das ( 4 h ) and O'Nan and Schwarz (7%). Rarity (20h) presents analyses for flow produced by a jet impinging centrally on an unbounded rotating disk, flow on a rotating disk in an unbounded fluid without impinging jet, and flow produced by a jet on a nonrotating but oscillating disk. Tsel'nik (2%) reports the study of an interesting problem in which a two-dimensional jet impinges at right angle on a flat plate but, instead of dividing and turning toward each side, it is turned to one side by the action of a difference in pressure in the background gas from one side of the jet to the other. I n a series of three papers, Ziabicki et al. (26h-28h) report on the hydrodynamics of a free steady jet subject to axial tension. McNaughton and Sinclair (73h) present an experimental investigation of liquid-into-liquid submerged jets in short cylindrical flow vessels using aqueous blue tracer solution. Shashidhara and Seetharamiah (27h) consider the energy dissipation in a submerged high velocity water jet in a stilling basin. The problem of horizontal jets in stagnant fluids of other densities is considered by Abraham (7h). Other topics of interest include the treatment of laminar free jets with arbitrary Prandtl number by Hug and Warder ( S h ) , an analysis applied to the formation of capillary jets by Goren ( 6 h ) , a photoviscous analysis of two-dimensional laminar flow in an expanding jet by Peebles and Liu ( 7 9 h ) , an analysis of curved free jets by Uchida and Watanabe (23h), an investigation of discharge coefficients of fire nozzles by Murakami and Katayama ( 7 6 h ) , and an investigation of the expansioncontraction behavior of laminar liquid jets by Oliver ( 7 7 h ) . The flashing of highly superheated free jets has been studied experimentally by Lienhard and Stephenson (72%). Open Channel Flow
I n an experimental investigation, Asthana (22) has employed boundary layer control techniques to improve upon the pressure distribution for flow over a Creager profile weir. Harleman and Elder (8i)have considered the problem of withdrawal from two-layer stratified flows such as occurs when a sluice gate is utilized for withdrawing cold water from the lower levels of thermally stratified rivers and reservoirs. A review paper on two-layer stratified viscous flow systems has been presented by Harleman (9i)in which the natural circumstances and basic mechanism of such flows are discussed. T h e problem of sediment transport is analyzed by Gradowczyk and Folguera (72) utilizing mathematical models and applied to erosion around a square pile and 76
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establishment of uniform flow in an open channel with a sand bed. Suryanarayana (729 has investigated the effect of one type of roughness upon the Chezy resistance function. The computation of velocity fluctuations in a stream with rough bed has been considered by Deineka (4). Rouse ( 7 7 i ) published a critical review on open channel resistance which considers surface effects, bends, obstructions, and roll waves. T h e problems of nonsteady and of nonuniform open 62) channel laminar flows are treated by Eremenko (5, while Clyde and Einstein (3i)apply an oscillating plate type of analysis as a model for experimentally measured total head fluctuations in the viscous sublayer below turbulent open channel flows. Other topics of interest include : boundary dissipation of oscillatory waves by Van Dorn (732) where it is assumed that in addition to viscous damping at the bottom and walls of a channel, additional dissipation results from a boundary layer at the free surface; an analysis of hydraulic jump in trapezoidal channels by Advani ( 7 4 ; and experimental research of a dividing flow in an open rectangular channel with the branch at 90" to the main channel, by Law and Reynolds ( 7 0 i ) . Fluidized Beds
Progress toward understanding the behavior of fluidized beds on a rational basis really begins with the symposia in 1961 and 1962 (Zj, 3j). A good review of the literature from mid-1962 to mid-1964 is given by Botterill (7j) and covers all aspects of fluidization. See also the review by Weintraub (4j). I n the period covered by this review remarkable progress has been made on many of the basic ideas developed earlier. From the fluid mechanical point of view, fluidization is a branch of two-phase flow. Most chemical engineers, however, will use the work covered in this review as the basis for understanding bed properties, heat and mass transfer, and chemical reactions taking place in fluidized beds. General theory and hydrodynamic stability. Murray ( 7 6 k ) derives the conservation momentum and energy equations for dispersed two-phase flow and specializes them for application to fluidized beds. The linearized equations are used for a stability analysis which shows that isothermal beds of incompressible phases are unstable to small internal disturbances. Jackson ( 8 k ) and Pigford and Baron (77k) reached similar conclusions in their analyses of this problem. Presumably these initial disturbances develop into the complete voids (volumes containing no particles) present in most gas- and some liquid-fluidized beds or into the turbulent motion prevalent in most liquid-fluidized beds. Once the disturbances have grown large the nonlinearity of the basic equations of motion must be dealt with (a problem which is as yet unsolved). The rate of growth of these disturbances increases very rapidly with particle to fluid density ratio and determines whether a given bed will be observed to fluidize aggregatively or particulately.
Jackson et al. and Baron all give expressions for calculating the growth rate. The growth rate of the instability of the Pigford and Baron model is independent of the horizontal component of the wave vector of the perturbation (dk). Murray (76k) considers also the stability of hot, centrifugal, and electromagnetic beds in which the density of the solids is much larger than that of the fluid. Levich and Myasnikov (75k) propose a simplified kinetic theory of a fluidized bed. Viscosity and drag coefficients. Viscous effects are important in understanding the mechanics and stability of fluidized beds. Expressions for the shear and bulk viscosities in the particulate phase are derived by Murray (76k). Vakhrushev (79k) gives equations for the absolute and kinematic viscosities of suspensions. Drag force equations are derived (76k, 79k),drag coefficient data in liquid-fluidized beds have been reported (78k). Apparent viscosity measurements in air-fluidized beds using a torsion pendulum viscometer are reported by Hagyard and Sacerdote (7k). At low rates of shear, Newtonian behavior is observed. Axial viscosity gradients have been determined on the basis of the velocity of fall of a sphere of given size and dimension (6k). Bed fluctuations and bed uniformity. A generally valid index for bed uniformity is not available as yet ( 3 k ), although these inhomogeneities are visually obvious and related to the bubbling in a gas fluidized bed. Pressure and density fluctuations are easy to measure, and a variety of transducers are described in the literature. Studies of bed fluctuations and bed uniformity are reported ( 7 k , Zk, 5k, 9k-74k, 20k). Bubble properties, fluid- and particle-flow patterns around bubbles. Once bubbles form in a bed they profoundly affect the fluid and solids flow patterns and therefore bed properties, heat and mass transfer, and the efficiency of fluidized bed reactors. As a result of the work of Rowe and co-workers on individual bubbles and Davidson’s analysis of the steady motion of a spherical bubble in an incipiently fluidized bed (51,61),a more fundamental understanding of the mechanics of gas-solids contacting in fluidized beds exists today. Collins (21) has extended Davidson’s two-dimensional theory by considering a bubble shape closer to that found experimentally; he showed that the gas and particle flow patterns are significantly altered only close to the rear of the bubble. The particle streamlines there are sharply bent suggesting the formation of trailing vortices behind the bubble [a result found by Rowe and Partridge (321)l. Murray (791) considers several cases of the steady motion of fully developed bubbles using equations which differ from Davidson and Harrison (61) in respect to the solids momentum contribution. I n two dimensions, he considers the circular bubble, the kidney-shaped bubble, the fully developed bubble with a cusped free streamline wake, and in three dimensions, the spherical bubble. The results in the first and last cases agree qualitatively with Davidson and Harrison (6l) and with experiment. Good pictures of actual bubbles in both gas and liquid
fluidized beds, showing their shape and wake, are available in several papers, for example (281, 351). Theoretical and experimental pressure distributions around bubbles are available (191, 201, 281). Several papers consider the rise velocities of single bubbles in a gas fluidized bed (291,351). The results agree in general with previous work in fluidized beds. Murray (791, 201) predicted the rate of a rise of a spherical bubble as
where C is an unassigned coefficient arising in the linearization of the convective term in the solids momentum equation. Murray suggests 1 or 8 / ~ as possible values, but calculation of C from Rowe and Partridge’s experimental data (351) shows that C is not constant and takes values averaging about 0.4 (341). Collins (31)shows that the rise velocity of the Davidson and Harrison (61)fluidization bubble cannot be arbitrarily specified but is in the form of the Davies and Taylor (701)equation. A second approximation for the velocity of a large gas bubble in an infinite liquid gives a result only slightly different from the first approximation by Davies and Taylor (701). The velocities of slugs in fluidized beds do not differ markedly from Nicklin’s (271) except that the 1.2 factor varies between 1 and 2 (221). Several articles (751, 781, 351, 391) discuss bubble size, velocity, frequency, and/or coalescence in continuously bubbling beds. X-ray photographs of splitting and coalescence of single bubbles are given by Rowe and Partridge (351). Zenz (421) proposes a mechanism for the formation and destruction of bubbles in fluidized beds. Botterill et al. (71) simulate a distributor plate by continuous local gas injection into an incipiently fluidized bed. Data on bubble size, frequency, volume, and volume flow rate are given for various injection conditions. Ostergaard (231) gives bubble frequency and shape data for bubbles formed at a single orifice in a water-fluidized bed. Turner (401) and Davidson and Harrison (81) d’lSCUSS the question of how the velocity of a bubble in an array (cloud) of bubbles is related to the velocity of the same bubble in complete isolation. Three possible relations are discussed but the question is left unsettled. Davidson also discusses the problem of how to divide the total flow between the bubble and particulate phases in a continuously bubbling bed. Partridge and Rowe (251) consider the case where “cloud” flow occurs and divide the flow between the “cloud” associated with a rising cloud and the interstitial flow. This division of the flow is important in determining the performance of catalytic reactors (241, 261). Hassett’s (741)suggestion that the two-phase theory of fluidization is inadequate is rebutted by Davidson and Harrison (71) and by Rowe and Partridge (331) who beVOL 5 9
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lieve his results are due to his low pressure drop distributor which allows the gas to maldistribute itself. Romero and Smith’s (301)work supports the two-phase theory. The stream functions inside the spherical (3D) and circular (2D) void, assuming Davidson’s theory to be valid outside the void, are derived by Pyle and Rose (261) and by de Kock and Judd (771). The latter authors use a “vorticity” boundary condition in addition to those used by Pyle and Rose, the validity of which is discussed by Collins (371). T h e experimental data of de Kock and Judd, obtained using a hot wire anemometer probe, support their assumption that for the flow in a 2D hole, the “vorticity” in the hole is proportional to the distance from the axis of symmetry. Rowe (371) repeated de Kock and Judd’s experiments using NO2 traces and concluded that the assumption of radial velocity conservation is better than de Kock and Judd’s “vorticity” assumption. Raso et al. (271) observe that solids falling into the void affects the ascent of the bubbles and the flow of gas around and inside them. The net transfer between an injected bubble and the continuous particulate phase for a bed at the minimum fluidizing velocity has been measured by Davies and Richardson (91). An expression for diffusive exchange across a “cloud” boundary is given by Partridge and Rowe (241). An interesting finding by Davies and Richardson (91) is that a bubble injected into a nonbubbling bed grows in volume at the ,expense of the continuous phase as it rises. I n a related study, Rowe and Partridge (351)find that the size of an injected bubble varies linearly with the volume injected. I n addition, the bubble size is larger than the injected volume when the excess gas is greater than 8% above the minimum and of smaller size when the excess gas is below the 8yofigure. Clearly a bubble injected into an incipiently fluidized bed is of snialler volume than the volume injected. A complete study of the mechanisms of solids mixing in gas- and liquid-fluidized beds is given by Rowe et al. (361). They note that gas-fluidized beds of particles greater than 100 p are mixed solely by bubbles which pick up particles in their wake at formation and regularly shed them from the wake as the bubble rises. No particle diffusion in the sense of interparticle diffusion akin to molecular or eddy diffusion was found, in spite of appreciable gross movement. They were not, however, able to relate the gross mixing rate quantitatively to the number and size of bubbles passing through a random bubbling bed. In liquid beds, mixing occurs by a diffusive mechanism because bed expansion is appreciable and the spaces between particles are large enough. Gas-fluidized beds with particles less than 60 p behave in an intermediate way. Bubbles produce displacement but, in addition, some eddy diffusive mixing occurs because of the disturbances set up by bubbles. Handley et al. (731) describe two types of particulate fluidization in liquid-fluidized beds. The first type, called uniform fluidization, is characterized by homogeneous random motion of particles and by constant mean fluid velocity and momentum distribution through78
INDUSTRIAL A N D ENGINEERING CHEMISTRY
out the bed. The motion of the particles is described by the equation for diffusion in isotropic turbulent flow. The second type of fluidization is characterized by large scale fluid channeling and bulk circulation of solids when a nonuniform flow field is introduced at the distributor. For large particles they find a tendency toward aggregative fluidization. Scarlett et al. (381),on the other hand, note fluid channeling and bulk circulation of solids even when there is no maldistribution of fluid by the distributor plate. A stochastic model for homogeneous fluidization by Houghton (761) concludes that particle and interstitial fluid diffusion should be characterized by a single directional diffusivity sensitive to void fractions and particle-fluid properties. I t is worth mentioning that liquid beds are unstable to small internal disturbances but that the growth rate of the disturbances is small. Observations of inhomogeneities in solid-liquid fluidization are not uncommon (771, 281, 471). From their measurements on drag coefficients for liquids flowing through regular arrays of spheres, Gunn and Malik (721) conclude that the particles in a liquidfluidized bed tend to form arrangements that offer low resistance to flow but that this tendency is opposed by inherent perturbations originating in the fluid system. The hydrodynamic equilibrium of a sphere suspended in a fluid stream has been considered (371). Gas mixing. Work is reported on axial, radial, back and cross mixing obtained from gas tracer experiments. IVinter (38m) determined the effect of gas velocity and particle size on the axial gas mixing coefficient from residence time distribution data. He shows that nonuniformity of the superficial gas velocity in the radial direction, not bubbles, is mainly responsible for the axial mixing. As also found by Schuegerl (32m), axial mixing by diffusion is dominant in beds of small particles and convective axial mass transfer between the phases becomes more important as the particle size and gas velocity increase. Different mixing zones in beds of fine particles are reported by Schuegerl (33m). Sectioning a bed with sieve plates (6m) or by staging (77m) reduces axial mixing. Mensing e t al. (27m) discuss mixing in fluidized packed beds. Other gas mixing studies are reported (25m, 27m, 37m). Radial mixing is discussed in several papers (76m, 37m, 3 8 m ) , and mixing coefficients are given in the latter two references (37m, 38m). Winter’s (38m) radial mixing coefficients are small compared to the axial ones and are unaffected by placing screens horizontally in the bed. An increase in mixing with gas velocity and height to diameter ratio is also reported (23m). Heimlich and Gruet (7m) determine interchange or cross flow coefficients from residence time data. They find the assumption of perfect mixing in the dense phase satisfactory. Solids mixing. No generally accepted means exists for predicting or describing solids mixing rates in practical beds. A diffusion coefficient should be applicable in most liquid-fluidized beds but not for gas-
fluidized beds. I t is used, however, in the latter case as a measure of the gross mixing rate. Rowe et al. (2Qm) give approximate diffusivities for copper and copper-nickel shot in water which increase rapidly with fluid velocity above the minimum. Kennedy and Bretton ( 7 7m) find Fick’s law applicable to the self-diffusion of spheres of a single size, but for mixed sizes, axial dispersion coefficients and particle size gradients require consideration simultaneously of diffusion (random particle motion) and classification (segregation). The diffusivity increases with liquid velocity above the minimum. Solid particle movement from the kinematic and dynamic standpoint is discussed by Musil and Prochaska (24m). Ruckenstein (30m) derives an equation for predicting the diffusion coefficient of solid particles in a gas-fluidized bed assuming the bubbles are homogeneously distributed. Experimental data ( 8 m ) testing this equation show that it gives only the correct order of magnitude for the mixing. Its failure is attributed to wall effects which make the bubble distribution over the cross section of the tube nonuniform. Alfke et al. ( 2 m ) found the mixing dependent on solids flow and bed height but independent of the gas flow. Kang and Osberg (Qm) were unable to apply the diffusion model to an unpacked bed but successfully correlated their mixing data for a screen packed gas-fluidized bed using a voidage dependent diffusion coefficient. Gabor (4m, 5m) used the diffusion model to describe axial mixing rates in fluidized packed beds. Generally his mixing rates increased with his velocity, bed height, diameter of fixed packing, and column diameter. The diffusion coefficient for radial mixing is proportional to the gas velocity and to the square of the bubble diameter (22m). Nicholson and Smith (26m) give an alternate method of describing particle motion in a fluidized bed, For short times, the blending of two layers of dissimilar particles are correlated in terms of a decay constant of the concentration variance. Solid particle flow patterns measured using a thermistor anemometer probe by Marsheck and Gomezplata (78m) appear to confirm that bubble flow establishes the particle flow pattern. Particle circulation rates were correlated as a function of the gas velocity. Several investigators (72m, 73m, 78m) report three particoncentration zones axially. Radially the solids concentration is higher at the wall than in thecenter (8m). The particle size distribution is almost constant throughout the bed including the low bulk density zone near the top (35m). Lateral solid circulation rates based on the method of Katz and Zenz (70m) were obtained by Lochiel and Sutherlana (76m). The particle flux was proportional to the excess flow above the minimum and inversely proportional to the 0.81 power of the particle diameter. Reducing the free area for lateral transfer by screens had the expected effect of reducing the flux. Gabor (#m, 5m) gives diffusivities for lateral transport of particles in a fluidized packed bed which are proportional
to the fixed packing diameter and the excess flow above the minimum. His diffusivity equation is derived using the random walk theory and relating the average particle velocity to the fluidizing gas velocity. The velocities of particle motion (7m, 7#m, 3 4 m ) and residence time distribution studies (28m, 39772) are reported., Solids motion resulting from pulsing the gas flow is reported by Massimilla et al. (79m, 2Om). Volticelli et al. (36m) discuss nonhomogeneities in gasliquid fluidization. Minimum fluidizing velocity, pressure drop, and bed expansion. Correlations for the minimum fluidizing velocity are reported (5n, ?3n, 23n, 24n),and a method for determining it from heat transfer measurements ( 7 % ) is also recorded. Three references concentrate on predictions for binary mixtures (#n, 20n, 27n). The effect on the minimum fluidizing velocity of gas temperature (3n), vibration of the distributor plate (25n), partial vacuum (26n), fine particle interparticle forces (In), and conical vessel shape (2n) are all given. The effect on the pressure drop of the bed height (79n), air velocity (Qn, 79n), distributor plate (77n-79n), vessel shape (Qn), and particle size distribution (5n) are reported. Pressure drop and bed expansion data in fluidized packed beds have become available (672) as have some new data on spouted beds (70n, 7 In). Work is reported on bed expansion in homogeneous fluidization (7n),on expansion of the bed before bubbling (8n),and on the effect of the walls on expansion (74n). One paper predicts the limiting conditions under which fluidized beds exist (8n). Bed porosity (75n, 76n) and holdup (22n) in gas-liquid fluidized beds and porosity and porosity fluctuations in liquid-fluidized beds have been studied. 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. The various aspects of natural convection attract researchers from a wide variety of disciplines. Indeed, one striking aspect of a broad review of the literature is the remarkable variety of studies which range from very practical to completely theoretical investigations. A number of investigations of free convection boundary layers have been reported. I t was found that longitudinal oscillations of a flat plate in its own plane had little effect on heat transfer (270). I n another investigation of time-dependent phenomena, attention was directed at determining the range of validity of the pure conduction region which describes early stages of the transient response to a disturbance (990). The effects of variable fluid properties on convective flow and heat transfer have been studied in detail for VOL 59
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Flows where behavior is intimately connected with heat and mass transfer . . . hydrogen, helium, carbon dioxide, and water vapor blown into air (360). Condensation of steam from a steam-air mixture was shown to be significantly influenced by the presence of a small concentration of the noncondensable gas (620). I n the case of the horizontal plate, boundary layer solutions exist for a heated plate facing upward and a cooled plate facing downward (350). The relative magnitude of free convective currents in water at 4” and 20” C. has been analyzed in detail (390). Similarity solutions of natural convection flow of non-Newtonian fluids were studied (670), as were the effects of variable fluid properties on free convection in a steam environment (630). The interaction of thermal radiation with free convection has been treated as a singular perturbation problem (13’0). Low Prandtl number analyses of a nonisothermal cone (430) and a uniformly heated vertical plate (140) have been reported. Leading edge effects have been considered in two studies (770,890). Analytical boundary layer studies of combined free and forced convection were made for the wedge geometry (400) and for horizonatal and vertical flat plates (420). Another study of combined free and forced convection (70) determined the parameters which reflect the relative importance of forced flow and body force effects for different Prandtl number ranges. Film condensation with a gravitational-type body force and a forced flow parallel to the surface was analyzed by an integral method (470). Laminar film condensation on a nonisothermal vertical plate also has been studied (jrU00). An analysis of film free convection in Helium I1 on a vertical plate and horizontal cylinder has been studied mathematically (760). A variety of boundary layer-type experimental studies has been reported. Natural convection to horizontal and vertical plates immersed in non-Newtonian Carbopol, for which viscosity data were assumed to obey the Ostwald de Waele power-law model, has been studied (740, 750). A binary stagnation point boundary layer on a horizontal cylinder was created by the effusion of water vapor from the porous surface of the cylinder; at high mass transfer rates the higher than predicted heat transfer rates were attributed t o a fluctuating motion in the boundary layer (870). An interferometer was used to study the sublimation of p-dichlorobenzene into a heated boundary layer (20) and, as one would expect, the mass transfer tended to inhibit the heat transfer. Heat and mass transfer to a vertical plate were measured with the plate held at cryogenic temperatures (30). Natural convection in 80
INDUSTRIAL A N D ENGINEERING CHEMISTRY
low pressure air, wherein heat conduction and therefore body shape are particularly important, has been investigated (720). Experiments with vibrating cylinders have been reported (280,570). With transverse vibrations it was found that heat transfer increased with frequency and amplitude and some increases were of an order of magnitude. When the vibrating surface is normal t o the direction of gas movement past the surface, then the disruption of the boundary layer is greater than when both motions are parallel (700). A discussion of the correlation of the influence of sound on heat transfer in the boundary layer 011 horizontal cylinders is given in reference (290). Velocity and temperature distributions were measured for free convection flow above a horizontal wire in air (90). Free convection from a wire to CO2 near its critical point evidences three general types of flow patterns: (1) the usual free convection flow, (2) a highly turbulent flow in which fluid aggregates similar to bubbles were seen to appear and disappear at the wire, and (3) an oscillating flow in which the usual free convection flow alternated with the “bubble-like” pattern (520). Turbulent free convection in near critical water has been studied both theoretically and experimentally and the analytical integral technique involved assumptions dealing with Reynold’s analogy, the Blasius wall shear stress, and suitable velocity and temperature profiles (540). A study of transient natural convection from thin vertical cylinders revealed that the conduction solution agreed better than the quasistatic with the data (210). An asymmetrical turbulent swirling natural convection plume in a motionless ambient fluid has been studied both theoretically (550) and experimentally (560). The experimental results agree well with the analysis. The effect of chemical reactions in the boundary layer on a horizontal wire has been studied ( 8 0 ) . It was found that very large temperature differences, on the order of several hundred degrees, were needed to cause an increase in heat transfer due to chemical reaction. Free convection flow of a Bingham plastic between two vertical plates has been studied analytically (1070). Similar solutions were found for the momentum and energy equations for systems with unequal mass transfer rates at the wall of both vertical and horizontal parallel plate ducts (370). As part of a continuing study, finite difference calculations, assuming two-dimensional flow, were made for transient natural convection in a long horizontal enclosure of rectangular cross section with
one vertical wall heated and the pther cooled (980). Predictions were made of the lower limiting conditions of free convection in a vertical heated tube, closed at the bottom and open at the top to a cooler environment (600); this is known as a n open thermosyphon. Combined frce and forced convection in tubes inclined at various angles to the horizontal was analyzed by using a perturbation inethod (460). A review of woik carried out at Perm, USSR, since 1946, has been @\en by Ostroumov (770)in a n article which lists 113 references. Natural convection in horizontal liquid layers is examined; the results compare favorably with experiments (120). Experimental studies of natural convection in confined spaces have been fairly numerous. An investigption of natural con\.ection between concentric spheres indicates three distinct types of flow patterns exist (70). Therniosyphons (40)have been studied in detail. The closed therinos>plioii, u hich consisted of a clased tube heated from below and cpoled at the top and filled with carbon dioxide, was studied in vertical and inclined positions ( 4 7 0 ) ; it was found that near the critical state the effective thermal conductivity achieved unusually high values of 5000 to 10,000 times that of copper. The flow and heat transfer characteristics of fluid layers confined by t u o parallel plates inclined at various angles to the horizontal have been considered in studies which covered a Prandtl number range for 0.02 to 30,000 and a Rayleigh number range from lodto 7 X lo8 (220, 260, 860); it was obserced that the fluid flow in the laminar regime consisted of a combination of rolls and square planform cells (860). Natural convection in a rectangular cavity with internal heat generation, cooled along a pair of vertical side walls and simulating the channels in an internally cooled homogeneous 'nuclear reactor was studied both theoretically and experimentally (850). Combined free and forced convection in conduits has been investigated experimentally in several papers. I t has been found for flow between two horizontal flat plates that the flow and temperature fields are affected by vortex rolls, and are therefore three dimensional, when the temperature difference between the plates exceeds a critical value (650). Flow in a horizontal tube with constant heat flux has been studied experimentally (610),and it was shown that the experimental Nusselt numbers were significantly greater than predicted by perturbation theory (380). Combined free and forced convecticn results in vertical systems with natural convection aiding and opposing the main flow have been given ( 1 0 0 , 1 7 0 ) . I t was shown that free convection can inhibit or enhance dispersion in a horizontal tube; if the flow is very slow so that radial variations in the axial pressure gradient are significant, dispersion is enhanced, whereas at higher flow rates circulation effects predominate and dispersion is inhibited (730). Experiments related to convective instability presented some interesting results. The interaction of buoyancy and shear forces in the free convective flow of a liquid in a rectangular cavity, across which a temperature difference is produced by maintaining the two vertical
walls at two different temperatures, has been studied by measuring velocity and temperature fields (230). This work was continued (240)in the Rayleigh number region greater than lo6, and the wavelike motions which become increasingly random and lead to turbulence were described. Numerical experiments with such a vertical slot have also been carried out (2500). A layer confined between horizontal plates was studied both experimentaIly and theoretically, and steady rolls with wavelengths twice the plate separation were obtained in both cases. As the two-dimensional restraint is relaxed, turbulent fluctuations develop ( 7 8 0 ) . Measurements were made of the temperature field within a horizontal rotating layer of mercury heated from below and cooled from above to explore the flow field and temperature distributions at various Taylor numbers (330). A horizontal layer of fluid cooled from above by surface evaporation exhibits a nonlinear time-dependent vertical temperature distribution, and this instability problem has been studied theoretically and experimentally (300, 379). Data are presented on the RayleighJeffreys instability in air, argon, and carbon dioxide ; the experrmentally determined value of the critical Rayleigh 80, and the theoretical value is 1708 number is 1793 (900). Measurements which extend over a wide range of Rayleigh numbers and encompass several flow regimes, including stable periodic flows, have been reported (640). Two regimes of free convection for a fluid layer heated from below, which depend on Rayleigh number, are reported and the transition Rayleigh number is equal to (2.2 f 0.4) lo4 (200). Other interesting studies involve natural convection in pools of evaporating liquids (60) and free convection in a horizontal layer in which a mean flow velocity exists (450). I n reference 60 certain flow patterns were identified as being induced by surface tension-driven instability and others as being due to buoyancy-driven convection; still other patterns seem to be associated with the presence of surface active agents. I n reference 450 the agreement between Kraichnan's mixing length theory of turbulent thermal convection and experiments lends strong support to the theory. The motion of turbulent jets of heavy salt solution injected upward into a tank of fresh water has been compared with that of plumes which are initially buoyant but become heavy as they mix with the environment (920). The reversal of buoyancy in the latter case is produced by using fluids having a nonlinear density change on mixing. Salt jets reach a steady height above which only small fluctuations occur, whereas plumes with reversing buoyancy exhibit violent regular oscillations. A considerable amount of theoretical work has been done on the problem of convective instability, or the onset of convection, under various conditions. For the case when the temperature gradient is large in a layer which is narrow by comparison with the overall depth of solution, the stability and growth of disturbances in a fluid with time-dependent heating have been investigated (580). The onset of convection in an
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electrically conducting fluid confined within an infinite cavity of rectangular cross section has been investigated for the case when the fluid is subjected to both a vertical temperature gradient and a vertical magnetic field (530). Forced motion of a liquid heated nonuniformly from below and subject to a nonconstant basic temperature profile has been examined (940), as has the stability of the laminar flow of an idealized induction furnace (760). It is interesting to note that the system may be unstable when a layer of cold water is on top of a layer of hot salty water even though the differences in density caused by concentration are far greater than those due to temperature (930). The linearized stability problem for steady cellular convection resulting from gradients in surface tension has been examined by retaining the dynamical equations in both phases in order to avoid the use of an assumed coupling mechanism at the interface (830). The effect of surface active agents on surface tensiondriven convection has a strong stabilizing effect (50). A new method, based on energy considerations, deals with nonlinear and time-dependent disturbances (480). A general, necessary criterion for stability has been developed for the case in which a homogeneous vertical magnetic field acts on a horizontal fluid layer (590). A method has been given which yields the finite amplitude steady solutions of the Boussinesq equations by successive approximations (780). Not every solution of the linear problem is an approximation to the nonlinear problem, yet there are still an infinite number of finite amplitude solutions. One may also determine the velocity and temperature field after the onset of instability by finite difference analysis (320). The structure of nonlinear cellular solutions to the Boussinesq equations has been investigated and it was shown that a whole class of equations leads to hexagonal cells (790). Nonlinear stability analysis of a linear temperature profile in a layer of fluid heated from below aims at clarifying the mechanisms leading to the hexagonal convection cells observed in controlled experiments (800). The effect of Coriolis force on the onset of thermal convective instability in a shallow layer has been considered (690). Gravitational instability in flows between horizontal plates also has been considered (770). With Prandtl numbers greater than five, the stability of boundary layer flow on a vertical plate varies as the third power of the Prandtl number (880). The buoyant motion within a hot gas plume in a horizontal wind has been studied by a perturbation method on the assumption that excess temperatures are small and buoyant motions are slow (750). Cloud physicists, and others interested in air movement and the behavior of the atmosphere, have considerable interest in buoyant motions. The effect of a constant source of heat on the development of a buoyant element has been investigated by integrating the hydrodynamical equations numerically (680). The effect of radiant heat transfcr on a buoyant axisyminetric turbulent plume indicates that if lapse rate variation and radiative heat 82
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transfer are considered simultaneously they act in unison when the lapse rate is stable in the ambient air and thus they make the plume die out at a lesser height than it would if only one were present. When the lapse rate is unstable these two effects act in opposition (660). Two-dimensional convective motion in a rectangular cavity, the two vertical walls of which are maintained at different temperatures, has been studied for the special case in which the temperature difference between the two walls is so large that the transfer of heat from one vertical wall to the other is almost entirely by convection (340). Heat transfer by conduction is assumed to be important only in thin boundary layers near the walls. Further work on the analogy for the onset of convection in a fluid confined between two horizontal plates and for the stability of viscous flow between two c)-linders rotating at almost the same angular velocity has been reported for cases in which the primary temperature distribution is parabolic (790). The stability to infinitesimal disturbances of plane Couette flow with negative vertical temperature gradient has been studied (440). The energy method is used for fluid layers to deduce that the critical Rayleigh number is a monotonically decreasing function of the Nusselt number (490). This method is applied further to examine the effect of internal heat generation and a spatial variation of the gravity field on the onset of thermal convection in spherical shells (500). I t has been shown that, under suitable conditions, periodic oscillations are formed in a one-dimensional model of thermal convection (570). The model consists of a tube filled with fluid bent into rectangular shape and standing in a vertical plane. T h e fluid is heated at the center of the lower horizontal segment and cooled at the center of the upper horizontal segment. Convective stability of a fluid confined within a rigid horizontal circular cylinder whose wall is nonuniformly heated has been studied (820). The exchange principle has been examined for fluids completely confined in an arbitrary region with rigid bounding surfaces that are good electrical conductors with respect to the fluid (840). I n the magnetohydrodynamic thermal stability problem, the exchange principle was found to be valid if the total kinetic energy associated with an arbitrary disturbance is greater than or equal to its total magnetic energy. Internal waves produced by a convectivc layer have been studied (970). Large amplitude Bernard convection results, in terms of Nusselt number versus Rayleigh number, have been reported over the range of Prandtl numbers from 0.01 to 100, and they show that heat flux increases slightly with decreasing Prandtl number (950). Buoyancy effects in an ideal dissociating gas have been studied by a perturbation technique (7020). It has been shown (970) by solving the characteristic value problem numerically that the neutral state is a stationary rather than an oscillatory one. The effect of uniform rotation on surface tension-driven convection in an evaporating fluid layer has been considered both
theoretically and experimentally (960). The theoretical analysis to a relationship between the Marangoni and Taylor numbers, and in the limit of rapid rotation the velocity and temperature fluctuations, is confined to a thin Ekman layer near the surface. Non-Newtonian Fluids
Developments in non-Newtonian fluid mechanics were characterized by reviews of existing problems, a re-evaluation of the constitutive equations, new equations of state, novel means of evaluating the nonNewtonian parameters, and investigations of the corrections needed for more accurate rheological measurements, Metzner et al. (3@, 38p), in carefully written and definitive studies of the proper choice of constitutive equations, reviewed the complexities associated with short time (large Deborah number) processes wherein polymeric materials can behave as solids. Many existing studies were reconsidered from this viewpoint and some anomalous results resolved. Their review contains 79 references. A book by Coleman et al. (72p) on viscometric flows should prove useful to theoreticians and experimentalists working in this area. Papers on boundary layer flows, drag reduction, viscous heating, and normal stress measurements among others were given in the proceedings of the Fourth International Congress on Rheology (44p). Rheological measurement devices received a great deal of attention in the literature with contributions including an extension of the falling cylinder viscometers to non-Newtonian fluids (2p) and a coaxial cylinder viscometer with oscillations (Sp, 7p, 27p). Other viscometers for special applications were described in references 74p, 78p, 2Op, 2Qp, 32p, SZp, 44p, 58p, 5Qp,62p. Tanner (SS), 56p) suggested a means of using conical flows for parameter estimations and applied the technique to some experimental measurements. Williams (SSp), who used a plane and cone viscometer, reported normal stress and viscosity measurements for nine polymer solutions. Some effects which can cause anomalous results in viscometry were considered. Gerrard et al. (75p, 76p) investigated viscous heating in capillaries, and Wellman et al. (63p) studied kinetic energy and hydrostatic head corrections. Morrison and Harper (47p) found wall effects in Couette flow, and Markovitz (33p)and Savins (48p) studied normal stress effects. Kapoor et a!. (24p), Vela et al. (Sir&), and Spriggs and Bird (52p) modified the Oldroyd-type model, and experimental data reported in references 24p and 67p were correlated with some success. Bogue and Doughty (Qp) reviewed the constitutive equations for theoretical approaches involving time derivatives of the stress (Oldroyd) and involving integrated deformation histories, and Bogue ( 8 b ) presented a new explicit constitutive equation of the integrated type. Seely (SO@) discussed the means of computation used in his earlier paper which presented a simplified, three-constant equation of state. This empirical equation appears more accurate in some cases and easier to use than some of the more common
ones. Sutterby (53p,54p) proposed a new semiempirical model and applied it to correlating data for laminar converging flows. A wide variety of specialized problems in the laminar flow of non-Newtonian fluids were studied experimentally and analytically. Analytical treatments of inlet region flows were reported by Kapur and Gupta (25p-27p) and Metzner and White (37p),who found that viscoelasticity increased the entry length. White and Metzner (37p, 65p) included the effects of the normal stresses in the boundary layer equations which apparently had not been done previously. Wheeler and Wissler (64p) reported friction factorReynolds number relationships for steady flow through rectangular ducts and also reported an approximate critical Reynolds number. Experiments for the critical Reynolds number in tube flow were described by Carlomagno ( 7 0 ) ) . Beard et al. (36) and Rubin and Elata (46p) commented upon the stability of different types of non-Newtonian fluids in Couette flow. Adams et al. (7p) investigated experimentally the stresses in converging and diverging flows and found that the results agree with the second-order Coleman-No11 theory. Vaughn and Bergman (SO#) were unable to predict the pressure loss and flow rates of their experimental data on concentric annuli using a power-law model. McEachern (35p), using the Ellis model, successfully correlated his experimental data on shear thinning annular flow. Savins and Wallick (49)) found that coupling effects between discharge rates, pressure gradients, angular velocity, and torque in helical flow allowed the axial discharge rate to be higher than in annular flow. Polyisobutylene was shown to be a simple fluid by Dierckes and Schowalter (73p) by correlating their helical flow data by the theory of simple fluids. Griffiths and Thomas (77p) discussed the stability problem in helical flows. Kozicki et al. (30p) suggested methods of predicting the transition point and also generalized Fanning friction factor relationship for ducts of arbitrary cross section. For nonisothermal transitional and turbulent flows, correlations of pipe flow data by Petersen and Christiansen (43p) should prove useful for design calculations. Further references are summarized in Table
111. Porous Media
Convective transport processes in packed beds and other porous media have received a great deal of attention in the recent literature thus attesting to the importance of an understanding of the fluid dynamical characteristics of such systems. Applications in absorption, reaction kinetics, filtration, heat transfer, and miscible and immiscible displacements are numerous and hence many studies seeking fundamental information, such as the experimental, two-phase, countercurrent flow and axial or radial mixing studies reported in references 2q, 70q, and 78q, have been made. Voids in packed beds. A local knowledge of the structure of packed beds is necessary if other than bulk charVOL. 5 9
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TABLE I I I , NON-NEWTONIAN FLOW I N CHANNELS Straight circular tubes Curved pipes, ellipticai cross section Annuli
acteristics of the momentum and heat arid mass transfer problems in such systems are to be investigated. Promising approaches to predicting the local voidage characteristics using statistical methods have been made by Debbas and Rumpf (6q) and Haughey and Beveridge (8q). Thadani and Peebles (23q) used a no\,el radiographic technique for determining point void fraction data, and Griffith (7q) proposed immersing the porous medium to be studied in a liquid of the s a m e refractive index, entrapped air preventing the liquid from filling the macrovoids, thus rendering the voids visible for measurement. Newtonian fluids. Fundamental studies of the singlephase flow of Newtonian fluids through packed beds have been concerned with the local conditions of this complex problem. Whitaker (24q) conducted a searching inathematical investigation of the flow of fluids through porous media with the goal of establishing theoretically the conditions under which the commoiily used empirical extensions of Darcy’s law are valid. I n an interesting experimental study using visual observations and measurements of the instanteous rate of mass transfer, Jolls and Hanratty ( 1 7 q ) were able to establish the approximate range of Reynolds numbers in which transition from laminar to turbulent flow occurred. Mickley et al. (17q) measured velocity profiles and turbulence parameters in a rhombohedral array of table tennis balls in a square bed. By use of this array and sections of the balls, the effect of the wall on the void fraction near the wall was eliminated. Yet, the velocity peaks observed previously in random packings, and generally attributed to wall effects, were still present. Snyder and Stewart (27q) performed an analysis of the pressure and velocity distributions in dense cubic and simple cubic packed beds of spheres for creeping flow. Approximate solutions to the equations of motion obtained using Galerkin’s method predicted pressure drop data to within 57,. Other analytical studies which, although not directly related to packed beds but which may be useful in describing the internal flow patterns, were made by Schmid (799) and Leonard arid Lemlich (74h 759).
Of practical interest in scaleup and pilot plant work are the initial experimental efforts reported by Templeman and Porter (22q) who measured the ratio of wall flow rate to total flow rate us. the packed height of small diameter columns. The ratio was dependent on the total flow rate and a significant fraction of the total flow occurred very near the wall, even using recommended design procedures. In an analysis of solid deposition for cornprcssible flow, Kuo and Closmann (739) found that reservoirs containing sulfurs can exhibit significant solids buildup in 84
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Straight elliptical. tubes and the space between elliptical tubes Flat plates Rectangular tubc He 1ic a 1 Converging and diverging ducts Coaxial cones Porous walls
porous media and lience inhibit gas flow. Recornmendations for overcoming this plienorrienon are gilwi. Boehlen et al. ( 3 q ) , in making pressure test.s up to 21 atmospheres, found that the relations for calciilating pressure drops for gases flowing in packed columns at normal pressures were valid at higher pressures. Other papers of practical interest inclitde friction factors in sand beds ( 7 q ) , a. review of the usc of the tortuosity factor in granular beds ( Y q ) , and an experimental determination of the apparent axial and radical diffusion coefficients for air in a packed bed ( 2 O q ) . Non-Newtonian fluids. The importance of aqueous polymer soliitions in secondary oil recovery operations has motivated some interesting experimental research on the flow of these materials through porous media. The modification of the Blake-Kozcny- correlation using a power-law model made by Christopher and Middleman ( 5 q ) successfully correlated their data. and existing data on dilute polymer solutions ovcr five orders of magnitude of the Reynolds number and should prove useful i n design. The succcssful correlation of the data refuted an earlier conclusion that viscoelastic effects are present in the flow of such solutions through porous media. In a later study of a siiriilar system, McKinley et ai. (76q) sought a simpler means of correlation, but were only partially successful. Rurcik (4q) has verified experimentally that the hydrophilic nature of partially hydrolyzed acrylic polymers restrict the flow of water in porous media since the polyrner molecules are retained in the pore structure. Jones and Maddoek (72q) pointed out a similar phenomenon in that increased drag in porous media of polya.crylamide solutions makes them less advantageous than ordinary fluids as oil-displacing mcdia. Rotating Flaws
Rretherton ~t al. ( 7 r ) prrsented an informative review of the proceedings of the I.U.T.A.M. symposium on rotating fluids. Topics covered at the symposium included steady and transient flows, thermally drivcn flows, planetary waves, gravity waves, and inertial oscillations. Iighthill ( 2 7 r ) , iii a paper prmeiitcd at
the symposium, surveyed the field of the dynamics of rotating fluids from the point of view of vorticity. An extension of the suspended particle method of flow visualization described by Coles (Qr) and the development of an ion technique for sensing the radial component of a Taylor vortex by Donnelly ( 7 2 7 , 737) are promising contributions toward improved experimental studies of transition in flow between rotating cylinders. Analytical studies of instability in Couette flow included a discussion of the nonlinear aspects by Di Prima and Stuart (77,) and a derivation of the smallgap equations for nonaxisymmetric disturbances by Krueger et al. (20r). Johnson (79r) treated a viscous vortex ring as an initial disturbance to an unbounded rotating fluid. Bentwich and Elata (57)described the formation of eddies in an eccentric annulus for steady flow with the inner cylinder rotating and correlated theory with experiment. I n a careful experiment Coles and Van Atta ( 7 0 r ) found measured velocity profiles in circular Couette flow to be distorted by end effects despite the large distance of the measurements from the ends of the equipment. Snyder and Lambert (24r) developed a method of measuring quantities of interest for determining the nature of flows in the transition region be) tween concentric rotating cylinders. Hill et d.( 7 6 ~reported experimental studies on a disk-and-cylinder system which suggested that the direction of the circulation in secondary flow may depend on the relative importance of the normal stresses exhibited by a fluid. Greenspan (757) developed a linear theory to describe the establishment of rigid fluid rotation from prescribed initial conditions. Unsteady flows for specific geometries were treated analytically by Benton (27, 37) who considered flows around both an impulsively started rotating sphere and rotating disk, and by Pearson ( 2 2 , 23r) who solved for flow between two rotating disks. Theoretical analyses of nonisothermal rotating flows for several different geometries were given. Veronis (267) considered a rotating layer of fluid heated uniformly from below and cooled above and found that steadyfinite-amplitude motions could exist for values of the Rayleigh number smaller than the critical value for overstability. Duncan ( 7 4 ) analyzed the conduction and convection processes between horizontal rotating disks with a nonuniform temperature distribution on the upper disk. Thermal stability problems in a rotating sphere (6r) and in Couette flow (277) were also treated analytically. Carrier ( 8 r ) investigated the influence of geometric features and density gradients on fluids in rotating systems. Benton and Boyer ( 4 r ) observed flow in a horizontal circular tube rotating rapidly about a vertical axis and
presented a theoretical treatment which agreed well with experiment. Flow properties of a fluid in the space between two concentric rotating spheres with slightly different angular velocities were considered by Stewartson ( 2 5 r ) . Ball ( I T ) , concerned with rotational effects on shallow water motions, worked on flow in an elliptic paraboloid. Horlock and Wordsworth ( 1 7 r ) derived and solved numerically the equations describing boundary layer flow on a rotating helical blade. T h e variation of the maximum size of drops of condensate and the number of drops per unit area with speed for a cylinder rotating in a steam atmosphere were observed by Hoyle and Matthews ( 7 8 7 ) . Solid-Fluid Systems
The fundamentals of the fluid mechanics of solids suspended in a fluid phase are virtually completely covered in two books of direct interest to chemical engineers. One by Happel and Brenner (70s) includes their thorough treatment of fluid motion about one particle with possible interaction effects. Their subject matter evolves to include concentrated systems. The book by So0 (28s) deals largely with concentrated systems and also adheres to as much rigor as is possible in treating this complex topic. So0 (29s) continues to develop his continuum mechanics approach to gas-solids systems to include the presence of a distribution in sizes of particles. He illustrates the theory with examples of adiabatic potential flow, boundary layer motion and flow of a charged cloud in a two-dimensional grounded channel. Julian and Dukler (72s) found that the velocity distribution of Gill and Scher for the turbulent flow of homogeneous fluids in tubes may be adapted to include a factor depending on the mass ratio of solids to gas. These authors demonstrate the success of their method which relates Reynolds number, friction factor, and solids loading by employing literature data of gas-solids flow in pipes. A modification of the laws of transport of granular solids is noted (75s). Experimental work involving suspension flow in straight channels includes that of So0 and Trezek (30s) who used a working fluid of 30-micron magnesium oxide particles in turbulent air in a 5-inch pipe of 38 feet in length. They observed interesting wall deposition effects due to particle charge. They determined friction factors, deposition rates, and concentration and velocity profiles using techniques described in earlier work. Rall and Riedel (24s) describe a method for measuring solids velocities by using the radiation of radioactive lanthanum. Velocity profiles were measured in solids-gas mixtures by Goto and Iinoya (8s) in which transverse effects were noted with variation in the mixture ratio. I n a series of papers Michiyoshi et al. (77s-19s) describe VOL. 5 9
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pressure drop measurements in the pipe flow of a water suspension of alumina powder in both laminar and turbulent flows. Part 8 of a series on the flow of capsules includes a study of free flowing single spherical and cylindrical particles whose size is comparable to the diameter of the pipe in laminar, transition, and turbulent ranges (25s). Other experiments include : suspensions of particles of flexible and irregular shape (in plug flow), of particles of high density, and of buoyant spheres (4s); sand-water mixtures in identical vertical tubes with upflow and downflow (6s) ; flocculated suspensions (74s) ; coke particles of 125-200 microns in a vertical tube 15 mm. in diameter and 100 cm. long in which axial solids concentrations were determined ( 7 3 s ) ; paste slugs of coal-water suspended in mineral oil (7s) ; flow of suspensions of solids in a duct with a stable stationary deposit making use of dilatant suspension behavior for interpretation (27s) ; particle-liquid flow through a restriction (3s); the determination of data for preparation of charts and monographs for design purposes (27s); and turbulent flow characteristics of fiber suspensions (2s). I n a continuing series of flow studies in tubes (7s’ 26s), inertial effects were noted, and oscillatory flow, in which was transverse migration of rigid suspended spheres, was compared with theory. Houghton obtained drag coefficients for a particle in vertically oscillating fluids ( I 7s), and Denson et al. noted spherical particle migration in a Poisseuille flow field (5s). Nubar noted migration of the red blood cells in blood (23s). The viscometry of fiber suspensions reveals apparent wall slip and time-dependent behavior (20s). Viscosities of fluidized spheres were determined with a torsion pendulum viscometer (9s); the viscosity dependence upon the Reynolds number (based on rate of shear) was derived using a cage model and a series expansion technique (22s); and effects of surfactant upon the apparent viscosity of several solids suspended in a hydrocarbon mixture were noted (76s). Settling fundamentals and applications. Discussions of the theory of all stages of the settling process were relatively numerous. Tory and Shannon (38t) investigated the history of “settling in compression” and offer a reappraisal of older concepts. They examined data from calcium carbonate slurries and found that the settling velocity is determined largely by the local solids concentration even when they are considered to be “in compression.” Fitch ( 7 2 ) suggests that in addition to the mechanisms of free settling and settling in compression, a third mechanism exists. I t consists of a channeling effect which permits a short-circuiting of the escaping fluid. Scott (30t) discusses two models based on the Carman equation and that due to Michaels and Bolger. The latter appears to describe the data of Gaudin and Fuerstenau reasonably well. Famularo and Happel ( 7 7 t ) used the equations for creeping motion for a particle and various models to approximate the settling characteristics of dilute suspensions and gave results as settling velocity as a function 86
INDUSTRIAL A N D ENGINEERING CHEMISTRY
of concentration. Experimental rates are generally faster, possibly owing to agglomeration effects. Goldman et al. (73t) employ bipolar coordinates to determine linear and angular velocities of two identical spheres at small Reynolds numbers as a function of their separation. Saffman (29t) theoretically studied the motion of a sphere in very viscous liquids in unbounded shear flow and showed that the particle experiences a lift force perpendicular to the flow direction. Cox ( Q t ) calculates both the force and couple on a particle of arbitrary shape moving with both translation and rotation in an infinite fluid. A simple physical model is given by Brauer and Kriegel (3t) for sedimenting uniform grains which they compare with data from five different sources. Tory (37t) follows up some of his previous work in settling theory by considering the relationship between the Reynolds number of the settling sphere and the occurrence of concentration gradients. Davies (70t) approaches the problem of aerosol deposition from streams in turbulent pipe flow by making use of the eddy diffusivity concept. He also makes use of the RMS value of the fluctuation velocity in the direction normal to the wall and discusses difficulties in obtaining consistency in experimental conditions for experiments intended to investigate aerosol deposition. Theoretical work and companion experiments are discussed in the following: Jovanovic ( 2 2 ) included a correction for flocculation effects in dilute sedimenting systems; Smith (33t) investigated a dispersion of sizes of particles of uniform density; and Johne (20t) described the effects of concentration upon settling of a monodisperse suspension. Small spherical particles suspended in a viscous liquid in Newtonian flow were observed by Jeffrey and Pearson (77t) to display the SegrC and Silberberg effect when they are neutrally bouyant. Trajectories for individual particles in the dilute suspeiision were cbtained. Obiakor and Whitmore (26t) imposed a mechanical vibration on settling suspensions of polyvinyl chloride spheres and kaolin clay. A marked improvement in rate was obtained. Christiansen and Barker (7t) measured drag coefficients for cylinders, prisms, disks and spheres in air, and several liquids. The data were in the Reynolds number range 1000 to 300,000, and the drag coefficient was given as a function of particle and fluid densities, shape, and Reynolds number. Jayaweera and Mason (76t) measured drag coefficients and orientations during the fall of single cylinders (0.01 < Re < l O O O ) , two long thin cylinders, and cones. Photographs of wake forms are included. Other experiments (27t) yielded terminal velocities of particles between the Stokes and Newton law ranges. These were determined for nonspherical particulate solids in the Reynolds number range of 12 to 460 in a variety of gas-solid systems. There were also experimental studies of quartz powder in benzene and xylene (28t) and barium sulfate and quartz in carbon tetrachloride-alcohol mixtures (35t). Sedimentation studies involving flow systems include : open channel transport of sand-water slurries ( I t ) in an attempt to describe conditions corresponding to a clean
Settling in practice.. . . Flow past submerged solids . . Inviscid flows . . . . slow flows . . . , bed and a sediment bed; velocity variations in rectangular sedimentation tanks (8t); and diffusion and sedimentation of particles (aersols) in flow through ducts of arbitrary cross-section (5t). An analysis is given by Shannon and Tory (37t) which accounts for the behavior of slurries in batch and continuous thickeners. T h e basic assumptions include uniform solids and that local settling velocity is a function only of local solids concentrations. These factors also are involved in some correspondence (4,3 2 ) . The operating principles of ideal classifiers and a discussion of e s ciency including test data are offered by Treasure (36t). Jernqvist (78t, 79t) gives graphical procedures for determining maximum steady-state capacity of thickeners. Bramer and Hoak (2)estimate the degree of flocculation which occurs in a sedimentation basin. Particle entrainment mechanisms are aspects of settling processes. Entrained liquid collection in packed beds is described by an inertial impaction theory, and experimental observations using several common packing materials are discussed (75t). A theory of clarification ( 2 4 ) is discussed in light of observations made earlier with dilute suspensions of particles (20 to 1100 p ) in three different filter media (232). A similar approach has been employed with experiments involving the filtration of dilute particulate suspensions (1.3 p ) in two different particulate media (74t). Other studies which may properly be noted here are a discussion of the limitation of model studies in predicting gas velocity patterns in Cottrell precipitators (27t); an experimental study of flow patterns ( 3 4 ) and oscillatory motion (6t)in cyclone separators; and an experiment on the slow motion of a sphere in a rotating viscous fluid (25t). laminar Flow Past Submerged Objects
Inviscid flow. Hussey and Reynolds (2u) derive a n expression for the added mass of a cylinder due to the two-dimensional motion of two cylinders at right angles to their line of centers within a circular boundary. Comparison is made with experimental results obtained on finite cylinders moving on a pendulum in liquid helium. Tuck (72u) has presented an analysis on plane potential flow past a circular cylinder beneath a free surface under gravity, the objective of the analysis being to determine the effect of the free surface boundary condition. An analysis for nonequilibrium flow past an infinitely long cylinder whose axis is parallel to the mainstream flow and whose surface contains regular corrugations, has been studied by Nanda (6u). Pavlikhina (9u)
has presented velocity and pressure distributions for flow past an oscillating cylinder. Two-dimensional unsteady subsonic flow past an oscillating thin profile has been treated by Meister ( 4 ~ ) .A simple analytic technique involving a “hyperpotential” has been proposed by Oswatitsch (7u) to evaluate approximately the threedimensional flow patterns past slender bodies moving in an inviscid compressible medium. T h e motion of a slender axisymmetrical body in a vortex-free flow of an ideal incompressible liquid with arbitrary angle of attack has been investigated analytically by Taits ( 7 724). Michael (524)has considered the problem of an ideal flow parallel to the axis of an infinite row of spheres; the separation of the spheres varying from one diameter to infinity. Parsons (824) has investigated the related problem of the flow past two spheres for the cases where the spheres are and are not in contact. Wang and Wu (73u) have studied the problem of a two-dimensional cavity flow of an ideal fluid with small unsteady disturbances in a gravity-free field. Pressure distributions for cavitated flows, as applied to sections having rounded noses, have been calculated numerically by Lurye (324). Georgescu (724) has determined, to the second approximation, the subsonic flow of an inviscid compressible fluid around a von Mises profile. Three-dimensional flow patterns generated by an inviscid compressible fluid in the presence of an irregularly shaped wall have been studied by Zeitunyan ( 7 4 ~ ) .T h e model is an idealization of an air flow about an obstacle located on Earth when the unperturbed air is moving horizontally far above the obstacle. Suchkov (70u) has presented an analysis for two-dimensional flow with stationary streamlines. Slow flows. Problems considered in the current literature include those concerned with Stokes flow, Oseen flow, flows with rotation and vibration, and flows associated with freely falling objects. Of interest to all concerned with slow flows is a review article by Brenner (30)on the hydrodynamic resistance of particles at small Reynolds numbers. Ranger (280) has added to the existing solutions for uniform external viscous flow past objects of various geometrics by treating a solid elliptic limicon of revolution. Stokes flows around deformed spheres have been considered by O’Brien (2221) and Matsunobu (7821). The latter is concerned with the flow about a liquid drop which deviates slightly from a sphere. Stokes resistance and motion for particles of arbitrary shape are discussed by Cox (62)) and Brenner (40). VOL. 5 9
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TABLE IV.
The Oseen approximation has been employed by Olmstead (24v-26v) to investigate viscous flow past semiinfinite flat plates. The first paper treats the case of continuously distributed injection or suction at the plate surface. The second is concerned with the presence of a line of concentrated horizontal force located on the y-axis ahead of the plate, while the third paper (which is not limited to the flat plate) develops a general velocity field past a boundary of general shape. Oseen flows past a paraboloid of revolution, a tangential flat plate, and an ellipsoid of large aspect ratio have been treated by Kap, and Miyagi (350), and Shi (33u), lan ( ~ Z V ) Tamada respectively. Bruyatskii (Sa) has considered the flow around an infinitely long cylinder whose axis is perpendicular to the direction of the undisturbed flow. The results of the analysis are shown to yield satisfactory agreement with existing test results for Reynolds numbers less than twelve. Experimental measurements, reported to be accurate to better than h 20/& have been presented by Maxworthy (190) and show the formulas of Oseen and Goldstein to represent accurately the drag of a sphere for low Reynolds numbers. Flumerfelt and Slattery (9u) have employed a variational method for the solution of Newtonian flow past a sphere. The results are presented as drag coefficients over a wide range of Reynolds numbers. Numerical techniques have been used by Dennis and Shimshoni (Bv) to obtain drag coefficients for steady viscous flow past a circular cylinder. Nuttall (210) has considered the problem of slow rotation of a circular cylinder in a viscous fluid. An experimental study of drag on a sphere moving in a viscous rotating fluid has been reported by Maxworthy ( 2 0 ~ ) . Kelly (14v) has considered the problem of a disk performing torsional oscillations in its own plane with strong suction at its surface. Two-dimensional oscillating flows have been studied by MacCainy (77v)while Williams (39v)has presented a note on the drag experienced by an axially symmetric obstacle vibrating slowly in a viscous fluid. Davis (7v)has reported an experimental study of the steady rise of a solid polyethylene sphere along the axis of a uniformly rotating viscous liquid, for translational Reynolds numbers less than unity. I t is shown that rotation affects the terminal velocity through the presence of the Coriolis force. Jayaweera and Mason (100) have presented an experimental study of the behavior of freely falling cylinders and cones. I n addition to flow pattern, attitude, and stability observations for the singly falling objects, the interaction effect at low Reynolds numbers has been investigated for two cylinders with either horizontal or vertical separation. The velocities of spheres falling under gravity in a viscous fluid have been used by 88
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
SLOW FLOWS
Compressible flow Injection or suction
(13v,1 6 ~ )
Cavity flow Wake flow Lift Array of cylinders
(38V )
-4symmetric hydromechanics
(Tu)
(ZV,
I l v , 34v)
(15~)
Odar (23u) to verify a proposed expression for the calculation of forces on a sphere undergoing acceleration. Interaction effects for two spheres falling through a viscous fluid have been examined analytically by Wakiya ( 3 7 ~and ) experimentally by Shafrir (320). In the latter paper, the spheres were water droplets with radii up to 40 I* falling through air. Non-Newtonian flows have been considered by Palekar (270) for the slow rotation of a sphere in a slightly viscoelastic fluid contained in an infinite cylinder, by Tanner (36v)who treats plane creeping flows of incompressible second-order fluids, and Sat0 et al. (2921)who have detcrmined drag coefficients from experiments on the terminal velocities of falling spheres in both Newtonian and nonNewtonian fluids. Other papers of interest are listed in Table IV. Boundary layer flow. Among the many recent topics considered in boundary layer theory are unsteady flows, flows with suction or injection, flows with heat and/or mass transfer, compressible flows, and nonNewtonian flows. Many papers deal almost exclusively with analytical or numerical methods of solution. Of basic interest to workers in the field are two papers by Fife (33w,34x1) concerning the validity of Prandtl’s approximation in a boundary layer. Fundamentall)., the papers are further attempts to put the boundary layer approximations on a firmer theoretical basis. Schilz (9320) has presented an experimental investigation on boundary layer control by sound. I t is shown that when flexural surface waves are set up, under certain conditions. amplification of disturbanccs in a downstream direction takes place. The position of transition is found to depend upon the amplitude and phase of the waves. An experimental investigation of the effect of a twodimensional roughness element on boundary layer transition has been reported by Klebanoff (527.0). Flow problems involving oscillatory, pulsating, or vibrating flows ) Wang ( 1 1 1 ~ who ) are treated by Stuart ( 7 0 4 ~ and consider the case of flow generated by a circular cylinder oscillating in an infinite fluid, Yeh and Yang (118,) who investigate the vibrating sphere, and Messiha (7Uw) who reports on oscillating flow along an infinite flat plate. Problems on impulsive motion are treated by Akamatsu and Kamimoto (5w)for the semi-infinite flat plate and by Benton ( T O W ) for the rotating sphere. Duric (25w) has presented a one-parameter method for calculations of nonsteady boundary layers, with the results for flow
around a cylinder following an impulsive start compared with other approximate solutions. Catherall et al. (77w) consider the problem of flow past a flat plate with uniform injection. Libby (67w, 64w) analyzes the problem of uniform injection or suction in laminar boundary layers for the cone geometry. Twodimensional viscous flow past an infinite fixed plane with time-dependent suction is considered by Kelly (50w), while Erickson et al. ( 2 8 ~study ) the laminar boundary layer on an infinite flat plate with suction or injection where the plate is moving with constant velocity through 55w) presents analyses for a fluid at rest. Kozlov (54~1, a flat plate with slot suction, and for suction through a porous surface. Peube (BOW) discusses similar solutions for constant density, variable physical property, and laminar boundary layer flow over a heated or cooled flat plate. Expressions for the displacement thickness in the presence of surface mass transfer have been reported by Hayasi (4520) and by Fannelop (37w). The latter considers the effects of significant transverse curvature. Sarma (87w) has presented a general method for the integration of the twodimensional incompressible unsteady boundary layer energy equation. Thermal boundary layer characteristics on a rotating, uniformly heated sphere have been reported by Banks (8w). Wanous and Sparrow (772w) have considered longitudinal flow along a circular cylinder with continuously distributed mass injection or removal. T h e problem of coupled heat and mass transfer at the wall for small mass transfer rates has been analyzed by Fannelop (30w). Libby and Liu (62w) report a method of analysis for the case of a multicomponent boundary layer with catalytic reaction at the wall. I t is suggested that comparison of experimental results with theory may yield information on physiochemical data relative to surface catalysis. Smith and Clutter (98w) describe a practical and accurate method of machine calculation for solving the complete equations of steady two-dimensional and axisymmetric laminar boundary layer flow of a compressible real gas subject to equilibrium dissociation. Iterative techniques are employed by Oliver and McFadden (79w) for the laminar flow of a Compressible gas over an ideally radiating isothermal flat plate with arbitrary irradiation at the edge of the boundary layer. Sarma (8620,88w) considers the general theory and solution of unsteady compressible boundary layers with and without suction or injection. Unsteady reacting compressible boundary layers with vaporization at the wall have been treated by Strahle ( 7 0 2 ~ ) . Chan ( 7 9 ~ )describes an integral method for compressible laminar boundary layer flow, while Libby and Fox (6Ow) develop a method of moments. T h e theory of laminar boundary layer flow at
stagnation points is expanded by Poots (87w) to include compressibility and heat transfer effects. Fussell and Hellums (37w) apply finite difference methods to the boundary layer equations. Central differences are used and the method appears to be rapid and accurate. An exponential function of a n exponential function is used by Hanson and Richardson (43w) to approximate the velocity profile in laminar boundary layer flow, thus removing the need to specify an artificial boundary layer thickness as a boundary condition. Volkov ( 7 7 0 ~ )reports a refinement of the KarmanPohlhausen integral method. A parametric method of integrating the boundary layer equations is presented by Loitsyanchkii (66w) and is claimed to occupy an intermediate position between the numerical and analytical methods. Luk’yanov and Sharaya (67w) report on the use of a static electrointegrator for obtaining solutions to boundary layer problems. Schetz and Jannone (90w) discuss methods of linearization applied to the problems of free jet Bow and wall-slot injection. Acrivos (2w) has presented an analysis on the effect of combined forced and free convection heat transfer in laminar boundary layer flows. The problem of isothermal diffusion in a variable molecular-weight binary gas mixture is considered by Hanna ( 4 2 ~ ) .Solutions are obtained for both forced and free convection a t large Schmidt numbers and large mass transfer rates toward the surface. Massey and Clayton (69w) have studied the development and separation of laminar boundary layers on highly curved surfaces. T h e influence of Reynolds number on the position of separation from a cylinder has been examined experimentally by Tsinober et al. (70720). The boundary layer on a rotating helical blade has been analyzed by Horlock and Wordsworth (44w), while Banks (9w) has discussed the rotating sphere. Rotating flows over a rotating disk have been treated by Schwiderski and Lugt ( 9 5 ~ )and , twisted flows have been considered by Bogdanova ( 7 2 ~ ) . Acrivos et al. (71.) present a n asymptotic method for solving approximately the laminar boundary layer equation for a power-law non-Newtonian fluid where the external flow has a general form. T h e flow of viscoelastic fluids past flat plates has been treated by Sharma and Gupta (97w), White ( 7 7 6 w ) , and Gulati and Gulati (39x1,40w). The latter consider both porous plates and plates in torsional oscillation. White and Metzner (7 7520) have considered the constitutive equations for viscoelastic fluids with application to rapid external flows. Axially symmetric stagnation point flow of power-law fluids was treated by Maita (68w), viscous wake flow was discussed by Rotem (83w), and wedge flows were considered by Sestak (96w) and Lee and Ames ( 5 7 ~ ) . VOL. 5 9
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TABLE V.
L A M I N A R BOUNDARY LAYER FLOWS
Compressible flows Injection or suction Unsteady flows
Other papers of interest in laminar boundary layer flows are presented in Table V. Turbulence
Chemical engineers have displayed interest in the consequences of turbulence in fluid flows for some time; however, it is relatively recently that they have become involved with the fluid mechanical fundamentals. T h e literature of turbulence is vast, including the work of the physicist as well as the considerations of application of the practicing engineer. Some degree of personal judgment is necessary in preparing this section of the Annual Review, and a major restriction imposed on coverage here is that of incompressible, electrically neutral flows. A book which has appeared on the theoretical aspects of turbulence is in Russian ( 7 3 x ) with descriptions in English available (74x, 75x). rapers which contain some useful review of theory are those of Pao ( 7 7 x ) and Favre (3x1
New ideas or methods in statistical theory are discussed by the following authors. Monin ( 7 2 x ) demonstrates the useful properties of a joint characteristic functional involving the velocity field and a random force field. Kraichnan ( 9 x ) continues his study of the finite closure problem and points out the shortcomings of Eulerian moments-Le., leading to an infinite sequence of differential equations-he introduces a generalized velocity which he uses to explain Lagrangian flow concepts. Shut’ko ( 7 t h ) makes use of some of Kraichnan’s earlier expressions to obtain the Kolmogorov inertial spectrum which Kraichnan also does in the above paper. Orszag and Kruskal (76x) suggest a theory of isotropic turbulence based on some of Kraichnan’s earlier work-the “maximal randomness principle”-which deals with the closure of the equations for moments of the velocity field. Herring ( 6 x ) applies perturbation theory to an equation for a probability distribution function rather than dealing with moment equations which offers certain mathematical benefits. Zhigulev (27x) considers the problem from the viewpoint of the Liouville equation for a gaseous ensemble and considers averages in phase space of coordinates and velocities. Deissler ( 7 x ) solves some twopoint nonlinear correlation equations by neglecting highest order correlations to obtain reasonable steady state solutions. Lee ( 7 0 x ) and Koga ( 8 x ) have also carried out studies which properly should be noted here. The work of Meecham ( I l x ) , Eschenroeder ( Z x ) , and the experimental work and discussion of Gartshore (5x), which involve energy distribution characteristics of turbulent flows, are noted. Frenkiel and Klebanoff (4x) demonstrate the use of a two-dimensional probabilty distribution based on their wind tunnel hot-wire measurements. Heskestad ( 7 x ) derives an approximate relation 90
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Chemical reaction Mass transfer Heat transfer Heat and mass transfer Slip flows Shear flows Falkner-Skan equation (solutions)
-
(89w,703~0, 106w, 713w, 774w, 119w) (46w,47w,5.5w,9Ow, 720w) (6w,7w,ZOw, 26w, 27w, 32w, 38w,51w, 77w, 94w ) (35w,85w) (3w,4w, ZSw, 82w) (16~0, 56w,58w) (7620,84w, 109w) ( Z l w , 72w) (22w,24w, 47w,65w) (14w,15w, 71w)
Separation Perturbations about the Blasius s o h tion Initial value problem Matching with algebraic decay Quasi-linearization Variational methods Polynomial approximations Solutions with line or area similarity
( 78w 1 (36w,59w)
Asymptotic behaviors Paraboloid of revolution Rankine half-body
( 73w )
(49, 1 (23w)
between time and space derivatives which permits the computing of mean space derivatives from experimental time derivatives for high Reynolds number shear flow. An eddy viscosity model is considered by Squire ( 7 9 x ) based upon the circulation around a vortex. The tensor nature of the eddy viscosity is discussed by StanisiC and Groves (20x) who show it to be at least of the fourth order. Turbulence at the wall. The nature of turbulence in close vicinity of the wall has naturally been of deep interest to chemical engineers. There are recent experiments on this aspect of turbulent flow phenomena. Hettler (5y) set about to demonstrate the existence of the laminar sublayer along smooth walls. He photographed minute drops of suspended immiscible fluid by successive exposures through a microscope very close to a tube wall and does conclude that a sublayer exists. Corino ( 4 y ) also photographed particles near the wall of a tube and observed that fluid elements are ejected outward toward the center line periodically from a region 5 5 y f 5 15. Orlov ( 5 )also uses photography and suspended particles in a rectangular transparent channel. Willmarth and Roos (70y) determined the power spectrum of wall pressure fluctuations using four transducers of different size and extrapolating to zero size. They found the root-mean-square fluctuations to be 2.66 times the wall shear stress. Mitchell ( S y ) used a diffusion-controlled electrode to measure fluctuations in wall shear stress. Askew and Beckmann (7y) determined velocity profiles in the vicinity of the wall of an
agitated vessel using an impact tube and varied impeller type, speed, and size. Worley ( 7 7 ~ ) used hot-wire anemometry to study roughness effects in a rectangular channel. Other experiments are those of T u and Willrnarth (@) using correlation measurements in a thick boundary layer. Black (5)discusses a theory of wall turbulence which depends on observations of jets or plumes which erupt periodically from the viscous sublayer. He suggests that these are the means by which momentum transfer occurs. These are apparently noted in Corino’s work meiltioned above. A simple velocity profile expression is offered by Burton (3y) to describe the velocity to u+ = 23. Willmarth (9y) discusses some measurements of wall turbulence carried out in earlier work. Turbulent boundary layer flow. Rotta (232) reviews calculational methods for turbulent boundary layers with pressure gradients and heat transfer. He surveys papers published after 1958. Mellor and Gibson (772) briefly introduce the “equilibrium” boundary layer of Clauser and hypothesize an eddy viscosity which is independent of pressure gradient and valid in the defect layer. The theory shows good agreement with experiment. I n a second paper (1.52)’ Mellor presents a dimensional argument leading to a functional form for the effective viscosity which includes the previous hypothesis and gives a way of determining the effective viscosity in the viscous sublayer when the flow occurs in the presence of mainstream pressure gradients. Mellor (76.z) continues with numerical integrations of the mean partial differential equations of motion (rather than the ordinary equations of equilibrium flows). I n a series of papers, Townsend continues his discussion of the notion of “self-preserving” characteristics of turbulent flow. He discusses first ( 2 9 ~ ) the self-preserving development of changes in velocity and temperature due to a discontinuolls change in surface roughness and heat flux in a very deep boundary layer. Then he refines the approach to include boundary layers more commonly encountered (302). Finally he considers the effect of strong adverse pressure gradients by making use of a two-layer model (372). Other theoretical work involving incompressible turbulent boundary layers on an impermeable surface include that of Escudier and Spalding (52) who employ an empirical formula relating the dissipation integral, the drag coefficient, and shape factor. Felsch ( S t ) makes use of the dissipation integral for similar purposes. Hudinioto (722) assumes that the turbulent boundary layer thickness is proportional to the turbulent velocity fluctuation due to the vorticity in the layer to treat boundary layers with pressure gradient. Townsend (282) describes an induced fluctuating potential flow in
stably stratified surrounding fluid due to the instantaneous flow just outside of the boundary layer. Experimental work was conducted by Favre et al. (72) who measured hot-wire signals of longitudinal velocity fluctuations to generate correlation coefficients in zero pressure gradient, flat plate boundary layer flow. They determine the relative convection rates of large scale and small scale eddies with distance into the layer. Bellhouse and Schultz ( I z ) developed a thin metallic film which when heated could be used in measuring skin friction and heat transfer. Two types of the gages were developed. T h e authors discuss the accuracy of the method and demonstrate it by measuring the fluctuating skin friction in turbulent flow over a flat plate. Other work involving two-dimensional, incompressible, subsonic boundary layers flowing over impermeable surfaces are given in references 32, 42, 92-772, 732, 742, 792, 242, 25’2, 322, 332. Three-dimensional turbulent boundary layers were theoretically investigated : So0 (262) deals with flow over a body of revolution which is spinning about its axis or about which the fluid spins; Perry and Joubert (202) study the effect of an obstruction which causes rapid yawing in a developing boundary layer; Pierce (212) examines Coles’s model for turbulent boundary layer flows in collateral and skewed three-dimensional boundary layers. Rotta (222) discusses the velocity distribution of turbulent flow in the vicinity of a porous wall through which homogeneous sucking or blowing occurs. The sublayer is discussed in detail. Mickley et al. (782) compare two velocity defect laws in the boundary layer with smoothly varying injection. Tennekes uses a common velocity scale to relate the “law of the wall” and a velocity defect law for application to boundary layers with suction or injection (272). Other studies involving mass transfer at the wall are given in 22, 62,342. Turbulent conduit flows. There appear some basic experimental investigations into the turbulence characteristics of pipe flow. Martin and Johanson ( 7 7 A ) used hot-film anemometry to determine Eulerian integral scale and relative intensity information a t the center of a 6-inch pipe through which water flowed. Reynolds numbers ranged from 19,000 to 160,000. Rust and Sesonske (76A) employed a fast response thermocouple to measure turbulent temperature fluctuations in mercury and ethylene glycol in a 0.925-inch diameter test section of a heat transfer loop. Root mean square values, amplitude, and frequency distributions were the characteristics measured as functions of position and Reynolds numbers. A Soviet group also measured the intensity of temperature fluctuations in mercury flow with heat transfer occurring (3A). VOL. 5 9
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Turbulence.. . mixing, jets and wakes., . , Gessner and Jones (5A) report upon an experimental program to investigate thoroughly secondary flow in fully developed turbulence in square and rectangular channels. Tracy (78A) describes similar phenomena based on his experiments in a rectangular duct. Jonsson and Sparrow ( 6 4 7A) report on experiments carried out with air flow in eccentric annular ducts. W-olffe reports on a similar configuration (ZOA). Meksyn (72A) uses flow between parallel plates to illustrate a concept of treating turbulence as a macroscopic viscosity. Seagrave ( 1 7 A ) discusses the notion of the eddy viscosity concept and turbulent velocity profiles in pipe flow and the conditions these impose on each other. Macagno and McDougall (IOA) extend to annular conduits velocity profile information for circular cross sections to obtain friction factor and maximum velocity location information for smooth and rough walls. They compare the results of their analyses with available data. Barrow et al. (7A) discuss the annular geometry by using Goldstein’s description of the similarity hypothesis and derive expressions for the velocity defect in the inner and outer sections of the annulus. Some discussion of their procedure occurs in a subsequent communication ( 2 4 ) . I n other analyses, entrance flow in a convergent two-dimensional channel (QA), parallel Couette flow (79A)and developing boundary layers in conical diffusers ( 4 4 SA) are discussed. Turbulent flow in round tubes with heat transfer and the temperature field-momentum transfer interdependence are discussed ( 7 3 4 7 4 4 ) . Reynolds ( 75A) analyzes, from dimensional considerations, Townsend’s modification of Prandtl’s description of steady velocity distribution near a wall in channel flows. Mixing, jets, and wakes. The mixing due to turbulence and equipment utilized to generate turbulence for mixing purposes has long been of interest to chemical engineers. Much of this subject is covered in the Mixing Review of this IND.ENG.CHEM.series (ZQB,30B). The mixing due strictly to fluid mechanical aspects of the turbulent field falls readily into other subheadings within this review. Some references which may properly be listed here because of emphasis on a mixing notion or apparatus follow. Cutter (8B) measured velocity distributions in a stirred tank by photographing suspended lycopodium particles. H e presents results as Eulerian correlation coefficients and scales of turbulence as functions of location within the vessel. Brodkey (4B) evaluates earlier data of the decay of concentration of an injected dye stream in pipe flow with Corrsin’s modified isotropic turbulent mixer theory. Lee (23B, 24B) studies statistically stationary isotropic turbulent mixing making use of various closure approximation theories. Rosensweig (34B) considers isotropic turbulent mixing with boundary sources and Crocco (7B) 92
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
introduces a mixing coefficient to describe the spreading of a boundary layer into the nonturbulent outside stream. General, fundamental descriptions of wake phenoinena are those of Burggraf (5B) and Acrivos et al. (7B). The former reference presents a nuinerical study of the flow properties of a free eddy; the latter describes the characteristics of steady separated flow past a circular cylinder in the limit of 1-anishingviscosity. Other studies demonstrate the powerful use of flow visualization techniques for studies of Lvake behavior (6B, 78B, 25B, 4ZB). Study of turbulent wakes not entirely fluid mechanical include chemical reaction studies (OB, 35B). Other discussions of theory and experiments are gil en in references 72B, 13B, 20B-22B, 37B, 33B. The analysis ofjets in which mixing effects are emphasized includes the work of Escudier and Nicoll (7OB) who examine the entrainment function i n boundary layer and wall-jet calculations. Szablewski (38B) considers jet mixing of gases of different iiiolecular weights and temperature and mixing of streams of nearly equal density and slightly lower velocit) for the outside stream (39B). The exchange coefficients for heat and momentum transfer are considered to be different from each other. Schetz (36B) gives ana1)tical results based on earlier theoretical work. He presents \docit), stagnation enthalpy, and concentration distributions in t e r m of integral transforms of the space variables for a twodimensional jet mixing with a uniform stream. Takada (40B, 41B) considers free jet mixing in which the jet is initially laminar but undergoes an instantaneous transition to turbulence at a specific location. Spaldiiig (37B) derives general conservation equations by making assumptions concerning the nature of the profiles and the entrainment from the mainstrani. He applies his analysis to several systems including the wall jet in stagnant surroundings. Florent ( 7 7B) makes use of the Navier-Stokes equations with negligible viscous terms to describe three flow zones in tandem for a jet entering a motionless atmosphere. Hill (75B)uses empirical turbulent free jet data to describe the mean velocity field of the bounded jet by assuming self-preservation. Lee (26B) assumes similarity of profiles to describe free turbulent, sivirling jets. Experimental studies of jet f l o s~include the hot-wire measurements of Heskestad for plane jet? (76B) and radial jets (17B). The plane inea?iirernents werc made sufficiently downstream so that thc flou \vas self-prescrving. Data on intermittenc) and for calciilating turbulent motion energy balance information were obtained. Beguier (2B) carried out ineasureinents of fluctuating velocity components for parallci iiii\iiig of zx 11 all jet and a slower flow. The mean \ elocity pi ofilc \L as found to be unsymmetrical with two maxiilia and a riiiniinum. Bradbury (3B) compares turbiilciicc iiic‘isLirriiiciit>,carried out in plane jet flow in a slow parallel stream. The
structure, when the flow is self-preserving, compares well with the structure of a plane wake. Intermittency and the turbulent energy balance show differences. Ragsdale and Edwards (32B) and Michalke and Wille (28B) carried out experiments which involve photographic illustrations and transition zone phenomena. Hetsroni et al. (74B) studied a plane air jet obtaining velocity and temperature distributions in an asymmetric temperature environment. Air jets at different temperatures intersecting at 30” and at 60” were studied (27B), and water jets containing aluminum dust were injected through a 2-mm. orifice and observed (79B). Turbulent non-Newtonian flows. Astarita ( I C ) offered a possible mechanism responsible for drag reduction in viscoelastic liquids in turbulent flow. He suggested that the frequencies of the smaller eddies, considered to be dissipative, are larger than the inverse of the stress-relaxation time associated with the viscoelastic liquid. These smaller eddies undergo shape oscillations which are less dissipative (more conservative) than occurs in purely viscous liquids. Astarita and Nicodemo (3C) further developed a theory to explain the phenomena which occur when viscoelastic fluids flow through sudden enlargements. Meyer (8C) derived an equation describing the frictional effects of dilute viscoelastic fluids in pipes which contain a parameter dependent upon solute concentration and a constant. Slattery discussed the scale-up problem for viscoelastic fluids ( I IC) based on Noll’s theory of simple fluids. Skelland (70C) used a Blasius-type relationship to derive velocity distribution expressions for power-law fluids in turbulent boundary layers. He discusses transport of heat and mass and the transition region. Tennekes (72C) modified the mixing length theory to discuss velocity profile behavior with non-Newtonian characteristics. McDonald and Brandt (7C) discussed the skin friction for power-law fluids in turbulent flow on a flat plate and give appropriate equations and curves. Zandi and Rust (74C) treated this problem, and their analysis shows good agreement with empirical mean velocity profiles for a variety of fluids in pipes. Experimental effort in this area included the measurements of Ernst (5C) who obtained velocity data in both sublayer and turbulent core regions in fully developed flow in straight tubes. He used C M C solutions with power-law exponents of 0.93 to 0.95. Astarita (2C) discussed the use of the pitot tube in making velocity determinations which give “apparent” velocity results and may differ significantly from the actual velocity in viscoelastic media. He gave discussion and results of experimental distributions obtained with viscoelastic media in a 0.96-cm. i.d. test section. Tesarik (73C) measured velocity profiles in a tube and obtained good velocity defect us. radial location results. Manin and Vinogradov (6C) carried out experiments with 12% A1 naphthenate in hydrocarbons and discuss two types of turbulence-the ordinary, inertia turbulence, and elastic turbulence which is found at low rates of deformation. R a m and Tamir (QC), in studying the viscosities of polyisobutylene solutions in kerosene and mineral oil,
reported that turbulence effects were observed at lower shear rates as molecular weight increased. Cheng et al. (4.7)determined the laminar and turbulent frictional characteristics of bentonite suspensions in straight pipes and pipes containing sudden expansions or contractions. Turbulent diffusion, Experiments emphasizing the diffusive properties of a turbulent field include those of Becker et al. ( 7 0 ) who injected a smoke tracer stream into the core of a fully developed pipe flow. Mean and root mean square fluctuating point concentrations were determined, mapped for several Reynolds numbers, and spectrally analyzed. The maximum ratio of root mean square point concentration fluctuation to mean concentration exceeded 100%. Leach and Walker (3D)used N 2 0 tracer gas in a 20-cm. diameter duct and in a 20 X 20 cm. sq. section to model the distribution of dust concentrations in larger tunnels. The diffusion of steam in turbulent air in a smooth tube was studied ( 6 D ) . A trough was used to determine photographically the instantaneous velocity field ; mass transfer studies investigating the turbulent diffusion were carried out ( 7 0 ) . Morkovin ( 4 0 )and Patankar and Taylor (5D) discussed the experimental results of Poreh and Cermak for the diffusion of ammonia from a line source into a two-dimensional turbulent boundary layer. Hjelmfelt and Mockros ( 2 0 ) made use of Basset’s equation for the motion of a viscous particle to deduce the behavior of dispersion of particles in a turbulent field. Instability and turbulence in flows with free surfaces. The stability of liquid films flowing due to the effect of gravity has been discussed by Benney (3E) who studied long waves of arbitrary amplitude. Whitaker and Jones ( 2 I E ) discussed the effects of surface active agents on surface stability through use of a perturbation solution of the Orr-Sonimerfeld equation. This effect was considered also by Smith (79E),Graef ( 7 E ) ,and Anshus and Goren (7E) who employed the surface velocity rather than a parabolic profile in the Orr-Sommerfeld equation. Stability of two-layer stratified flow down an inclined plane has been studied by Kao (73E, 74E) by extending the work of Benjamin and Yih. Yih (22E) extended his theoretical instability studies to film flow of non-Newtonian liquids. Massot et al. (78E) used Kapitsa’s approach to describe ripple flow. Lee (76E) derived turbulent velocity profiles in falling films using Deissler’s and von Karman’s eddy viscosities. Experiments related to the above noted stability problem were carried out by Jones and Whitaker ( 7 7E) who compared data with results of the solution to the Orr-Sommerfeld equation and Kenning and Cooper (75E) who describe and explain steady circulatory motion on the surface of water flowing past an air bubble. Jepsen et al. (7OE) noted the effect of waveinduced turbulence in films upon mass transfer with the aid of an interferometric technique. Ueber obtained interfacial turbulence at steady state with a special cell and interferometric measurements (20E). Atkinson and Caruthers ( 2 E ) measured laminar and turbulent velocity profiles in liquid films. Schlieren techniques VOL. 5 9
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were used to demonstrate Marangoni instability (4E, 77E). Liquid jet stabiiity, aci affected by ambient pressure and nozzle turbulence and velocity profile development, was investigated by Graht and Middleman (8E). Turbulence effects u p m a free surface were observed by Davies and Khan ( 5 E ) who studied the frequency of clearances which occur whe-- powder is dusted onto the surface of a turbulmt liquid.. Further studies in velocity distribution (GE) and turbulence characteristics (LIE) in open channel fldw are available; also turbulence experiments iri channel flow with a defined roughness pattern have been carried out (72B). REFERENCES Single Phase Laminar Flow of Nowioniaii Yluids in Channels ( l a ) Ackerberg, R. C., “ T h e Viscobs Incdmpressible Flow Inside a Cone,” J. Fluid M e c h . 21, 47 (1965). (2a) Bankstrom, 0. A., Sibhill, W. L.,_Skogl,,np, V. J., “Stability of Gaseous Distribution Among Parallel Heated harn-Is,” AEC Rept. No. LA-DC-7656 (1965). (3a) Bar:&, P. G., “Stability of Laminar l l o w 0 1 a Liquid Between Parallel Planes, AEC Rept. .4EEW-R-41/. (1965). (4a) Barrer, R . M., Nicholson, D., “Slip Flrw in Long Single Capillaries. 11. Steady-State Flow and Specular Reflection,” Can. J . Chem. 43, 896 (1965). (5a) Bergman, P. D., Koppel, L. C., “Cn:fo~m-flux Heat Transfer to a Gas in Laminar Forced Convection in a C i i c d a r Tube,” A I C h E J. 12,648 (1966). (6a) Bostandzhiyan, S . A., Mcrzhacov-, .A. ., “Longtitudinal Particle Mixing in a ScreenPacked Gas-Solid Fluidized Bed,” Can. J . Chem. En,?. 44 (3), 142 (1966). (10m) Katz, S., Zenz, F. A . “ A Mathematical Approach to Mixing and Internal Circulation,” Petrol. Refine; 33, 203 (1984). ( I l m ) Kennedy, S.C., Bretton, R. H., “Axial Dispersion of Spheres Fluidized with Liquids,” AIChE J . 12, 24 (1966). (12m) Kislykh, V. I., Chirkov, Y. S “Distribution of the Prohahihty of FluctuaOver the Height of a Fluidized Bed,” tions in the Number of Solid-Phasc’~articles Khim. Prom. 4 2 (6), 414 (1966) (Russ.). (13m) Kobulov, V . E . , Todes, 0. M., “Structure of a Fluidized Solids La er in Relation t o the Type of Gas Distribution,” Zh. Prikl. Khtm. 39 (5), 1075 6966) (Russ.) (14m) Kornilov A. N Kondukov, N. B., “Parameters of Particle Motion in a Fluidized Bed’as Stuzied h i t h Radioactive Isotopes. 111. hlean Velocities of Particle Motion,” Inzh.-Ftz. Zh., Akad. iVouk Belorusrk. SSR. 10 (6), 764 (1966) (Russ.). (15m) Levin, B. D., Lyandres, S. E., Planovskii A. N Akopyan L. A., “Radial Mixing of Gas in a Fluidized Layer,” Izv. V y h i k h fichebn. Zavkdenii, A’eft i Go2 9 (51, 55 (1966) (Russ.). (16m) Lochiel A. C., Sutherland J. P. “ T h e Lateral Mixing of Fluidized Solids Through Mished and Unmeshed Apexhres,” Chem. Eng. Sci. 20, 1041 (1965). (17m) Lyandres S E Planovskii, A. N Akopyan L. A Levin B. D Inozemtseva, E. E. ‘ ‘ M i i i n i of Gas Phase in $Fluidized’Bed,”’ Khim.’i T e c h . Topi, i M n s e l 11 (41, 21 (1966) (Russ.). (18m) ,Marsheck R. M Gomezplata, A., “Particle Flow Patterns in a Fluidized Bed,” A I C h E J : 11,167 (1965).
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INDUSTRIAL A N D ENGINEERING CHEMISTRY
(19m) Massimilla, L., Volpicelli, G., Raso, G., “A Study of Pulsing Gas Fluidization ofBeds of Particles,” Chem. Eng. Progr. Symp. Ser. 62 (621, 63 (1966). (20m) Massimilla, L., Volpicelli, G., Raro, G., “Note on Dynamics of l i q u i d Solid System Expansion and Sedimentation,” A I C k E J . 11, 548 (1965). ( 2 1 4 Mensing, FV., et al., “Eigenschaften von kornhinierten fliersbclt-fesstbettsysiemen, Chem. Ing. Tech. 38 (1966) (Ger.). (22m) hlori, Y., Nakamura, K., ”Solids Mixing in Fluidized Beds,” Kagnku Komku (Ahridzed). ” . u ~ , 4I (1). 154 11966). , (23.m) Mukhlenov, I . P., Kozlova, I . D.. “Mixing Gasses in Equipment with a Eluidized Bed of Granular Material,’’ Irc. VyJysshrkh lichebn. Zarredenii, K h m . i Khim. Techno/. 9 (31, 494 (1966) (RUSE.). (24m) MMusil, L., Prochaska. J., “Solid Particle Movement in a Fluidized Bcd,” Genie Chitn. 95 (11, 28 (1966) (Fr.). (25,) Nam-Ko,o?g, S.,Sai-ki, hi., “A Dynamic Res onse Study of Gas Residence Time in a Fluidized Bed,” Intern. Cheni. Eng. 6 , 668 6966). (26m) Nicholson, 1%’. J., Smith, J. C “Solids Blending in a Fluidized Bed,’‘ Cliem. Eng. Progr. Sjmp. Ser. 62 (62), 83 (ib66). (27m) Ravindram, M . , “Significance of Residence Time Distribution Studies in the Design of a Fluidized Bcd Reactor,” Chem. Age India 17,447 (1966). (28m) Ridgway, K., Sim, H . K., “Tapered Fluidized Brds-Passage of Spheres,” Chem. Proc. Eng. 4 7 (6): 281 (1966). (2?$ Rowe, P. N., Partridge, B. A . , Cheney, A . G., Henwood, G . A , , Ly,ilI, E., Mechanisms of Solids Mixing in Fluidized Beds,” Trans. Inrt. Chem. Engrs. 43, T271 (1965). (30m) Ruckenstein E. “Nonhomogeneous Fluidization,” IND. ENU. CHLM. FUNDAMENTALS 5,’ 139’(1966). (3;m) Schuegerl, K., “Radial Mixing in Gas-Fluidized Beds,” Chem. Tech. 18 (9), 344 (1966) (Ger.). (32m) Schuegerl, K., “Backmixing in Gas-Perfused Fluidized Bcd,” Ihid., p. 547. ( 3 3 4 Schuegerl, K. “Vorweilzeitverteilung das Anstrom Gases in Fluisslxten,” Chem. Ing. ?ech. 38: 1169 (1966) (Gcr.). (34m) Todes, .4.K., Bondareva, A. K., Grinbaum, M . B., “ T h r Motion and Mixing of Solid Particles in a Fluidized Bed,’‘ Khim. Prom. 42, 408 (1966) (Russ.). (35m) Urabe, S., Hiraki, I., Yoshida, K., Kunii, D . , “Behavior of Particle Movcmen1 in a Fluidized Bed,” Kngnku Kogaku 29 (111, 863 (1965) (Japan.). (36m) Volticelli, G., Massimilla, L., Zenz, F. .4.,“Nonhomogeneities in SolidLiquid Fluidization,” Chem. Eng. Progr. Sjmp. Ser. 62 ( 6 7 ) , 42 (1966). (37m) Val:har J “Theoretical .4nalysis of Agitation in a Multi-Srage Fluid Bed Reactor, Br;t. ?hem. Eng. 10 (P), 532 (1965). (38m) ‘iVinter, O., “Gas Mixing in Fluidized Beds: Model Experiments for ScalingU p Purposes,” Chem. Eng. Progr. Symp. S’er. 62 (67), 1 (1966). (391111 Wolf, D., Resnick, W., “Experimental Siudy of Residence Time Distribution 4 ( l ) , 77 (1965). In a Multistage Fluidized Bed,” IND.END.CHEM.FUND.AMENTALS _
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Minimum Fluidizing Velocity, Pressure Drop a n d Bed Expansion (In) Baerns, M., “Effect of Interparticle Adhesive Forces of Fluidization of Fine Particles,” IND. E N G . C H E M . FUNO4MENTALS 5 , 508 (1966). (2n) Basakov, A. P., et nl., “Critical Resistance and Critical Vclocity for Fluidization of Fine-Grained Material in Conical Cylindrical Vessels,” Khim. i Tekh. Tofizu i Mare/ 11, 14 (1966) (Russ.). (3n) Bhat, G . N.. Weingaertner, E., “ T h e Gaseous Fluidization of Solids. IV. Effect ofTemperature on Fluidized Beds,” Brit. Chem. Enion, I ~ D END.
;?bid., p. 1142. (32w) Farn, C. L. S., Arpaci, V. S., “Qp the wumerical Solution of Unsteady, Laminar Boundary Layers,” Zbid., p. 730. (33w) Fife, P. C., “Toward the Validity of Prandtl’s Approximation in a Boundary Layer,” Arch. Rational Mech. Analysis 18, 1 (1965). (34w) Fife P C. “The Generation of a Boundary Layer in Hydrodynapics,” Zbid., 21,’ ZF36 (lb66). (35w) Fox, H., Libby, P. A,, “Dissociated Laminar Boundary Layers with Heterogeneous Recombination,” Phys. Fluids 9, 33 (1966). (36w) Fox, H., Chen, S., “Some Remarks on Peaturbatioq Solutions in Laminar Boundary Theory,” J . Fluid Mech. 25, 199 (1966):: Solution of Bounday (37w) Fussell, D. D Hellums J D “ T h e Numerical ’. Layer Problems,” IZChE J . li, 733 ii965). (38w) Gaster, M “On the Generation pf Spatially Frowing Waves in a Boundary Layer,” J . Fluii Mech. 22, 433 (1965). (39w) Gulati,’ S. P., Gulati, S., “Shear Flow of a YwcopduGting: and Conducting Elastico-viscous Fluid Past a Pordus Flat Plate,” J . Sci. En5 Res. 9,123 (1965). ( 4 0 ~ )Gulati, S. P., Gulati, S . “Torsional Oscillations of an Infinite Flat Plate in an Elastico-viscous Fluid,” &PI. Sci. Res. 15, 359 (1965). (41w-) Hakkinen, R. J., Rptt, N., “Similar Solytions for Merging Shear Flows: Part 2,” AZAA J . 3, 1553 (1965). (42w) Hanna, 0. T , , “Diffusion in the Laminar Boupdary Layer with a Variable Density,” AZChE J . 11, 706 (1965). (43w) Hanson F. B., Richardson, P. D., “Use of a Transcendental Ap roximation in Lamina; Boundary Layer Analysis,” J . Meck.‘Eng. Sci. 7, 131 ( 1 d ) . (44w) Horlock, J. H., Wordsworth, J., I‘Tbe Three-Dimensional Laminar Bouq#ary Layer on a Rotating Hellcal Blade,” J . Fluid hfech. 22, 305 (1966). (45,“) Hayasi, N “Displacement Thickness of the Boundary Layer with Blowing,” A Z A A J . 3 , 23ib (1965). (46w) Heskestad, G., “An Edge Suction Effect,” Ibid., p. 1958. (47w) Hugeu, J. H., Wuest, W., “Similar Solutions for Comprassible Bsuqdary Layers with Heat Transfer and Suction or Blowing,” Z . Angrw. Math. Phys. i 7 , 385 (1966) (Ger.). (48w) Hunt, B. L Sibulkin M “A roximate Expression for the Boundary Layer Shape Fac;br,” AIAA’J. 2’159 (49w) Ivanov, T. F., “Laminar Flow of a Viscous Iqcom ressible Fluid Past a Paraboloid of Revolution,” Zzu. Akad. Nauk., SSSR, MeKhon. 1965 (Z), p. 11. (Russ.). (Sow) Kelly, R. E., “ T h e Flow of a Viscous Fluid Past a Wall af Infinite Extent with Time-Dependent Suction,” Quart. J . Mech. Appl. Math. 14, 281 (1965). S.,‘ “Low . Frequency, large Amplitude Fluctuation of the L a q i n a r (Slw) King, I$ Boundary Layer,” AZAA J . 4, 994 (1966). (SZw) Klebanoff, P. S., “ T h e Effect of a Two-Dimensional Roughness Element on Boundary-Layer Transition,” Proc. Eleventh Intern. Congr. Appl. Mech., Munich, 1964,” p. 803, Springer-Verlag, Berlin. 1966. (53w) Kuzlov L. F. “Calculation of Incom ressible Laminar Boundary Layer qn a Flat Plate’with &ot Suction,” Inzh. Ftz. 9,433 (1965) (Russ.). (54w) Kozlov, L. F., “On the Integration of the Equations of Laminar Boupdary Layer o p a Porous Surface,” Prikl. Mekh. 11, 119 (1966) (Russ.). (55w) Kozlov, L. F., “Integration of three-dimensional boundary layer in the presence of suction or injection;’ Zbid., 2, 110 (1966) (Russ.). (Sow) Kuznetsov, V. I “Heat Transfer of a Fiat Plate in a Laminar Heated Flow,” Inzh. Fir. Zh. 9, 148 71965) (Russ.). (57w) Lee, S. Y . , Ames, W. F., “Similarity Solutions for Non-Newtonipn Fluids,’! AlChE J . 12, 700 (1966). (58w) Li, T. Y., Kink, P. S., “An Approximate Analytical Derivation of Skin Friction and Heat Transfer in Laminar Binary Boundary Layer Flow,” Zntern. J . H e ~ MQSS t Transfer 8 , 1217 (1965). (5%) Libby, P . A. “Eigenvalues and Norms Arising in Perturbations about the Blasius Solution,’’ bZAA J . 3, 2164 (1965). (60w) Libby, P. A., Fox, H. A., “A Moment Method for Compressible Laminar Boundary Layers and Some Applications,” Intcrn. J . Heat Mass Trairjer 8, 1451 (1965). w ) Libby P A “Laminar Boundary Layer on a Cone with Unifsrm Injection,” ( 6 k h y s . Fluid; 8; 2$16 (1965). (62w) Libby, P. A Liu T M. “Laminar aoundary Layers y i t h Surface-Catalyzed Reactions,” Zbid:: 9, 436 ‘(19i6). (63w) Libby, P. A., Chen, K. K., “Remarks on Quasi-Li!+earipatian Applied in Boundary Layer Calculations,” AZAA J . 4, 937 (1966). (64w) Libby, P.A., Chen, K., “Laminar Boundary Layer with Uniform InjeFtion,” Phys. Fluids 8 , 568 (1965). (65w) Lin, C. C., Benney, D. J., “On the Instability of Shear Flows and Their Transition to Turbulence,” “Proc. Eleventh Intern. Congr. AppI. Mcch.; Mugfch, 1964,” Springer-Verlag, Berlin, 1966. (66w) Lqitsyanchkii L. G “The Parametric Method of Integratin the E uations for a Laminar Bdunda; Layer,” T r . Leningrad Polifekh. In-ts 7965 ($49, 45 (Russ.). (67w) Luk’yanov 4.T Shara a S. I “Solutions of Boundary Layer Problems with the Aid Q? Statia Modei,:’ V e s k k Akad. Nauk., KaZSSR 1965 (21, p. 63 (Russ.). (68w) Maita M. K., “Axially-Symmetric Stagnation Point Flow of Power Law Fluids,” Z h M P 16, 594 (1965). (69w) Massey, B. S., Cla ton B. R., “Laminar Boundary Layers and Their (1965). SeDaration from Curved $urf.&es,” J. BasicEne. - 87,483 . . . (70w) Messiha, S. A. S., “Laminar Boundary La ers in Oscillatory Flow Along an $roc. Cambridee Phil. Soc. 62: 329 Infinite Flat Plate with Variable Suction,” (1966). (71w) Mur hy J. S., “Extensions of the Falkner-Skan Similar Solutions to Flows I 3,2p43 . (1965). with Sur& burvature,” AZAA .
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(132) Mayer, E., Divoky, D., “Correlation of Intermittency with Preferential Trans ort of Heat and Chcmical Species in Turbulent Shear Flows,” AZAA J.4, 1995 8966). (142) hlcQuaid, J., “A Velocity Defect Relationship for the Outer Part of Equilibrium and Near Equilibrium Turbulent Boundary Laprs,” .Aeron. RPS. Council, London, Current Papers No. 885 (1966). (152) Mellor, G . L., “ T h e Effectsof Pressure Gradients o n Turbulent Flow Near .4 Smooth Wall,” J . Fluid Mech. 24, 255 (1966). (162) Mellor, G . L., “Turbulent Boundary Layers with Arbitrarv Pressure Gradients and Divergent or Convergent Cross Flows,” Princeton Uni&., Drpt. Aerospace Mech. Sci., Rept. S o . 775 (1966). (172) Mellor, G. L., Gibson, D. hl., “Equilibrium Turbulent Boundary Layers,” J . Fluid M e c h . 24, 225 (1966). (182) Mickley, H. S.. Smith, K. A , , Fraaer, hl. D., “Velocity Defect Laws for Transpired Turbulent Boundary Layers,” AZ.4A J.3, 787 (1 965). (1%) Nelson, D. M., “Turbulent Boundary-Layer Calculations Using a Law of the Wall-Law of the Wake .Method,” US. Naval Ordnance Test Station, China Lake, Calif., Rep. NOTS TP 4083, 62 (1966). (202) Perry, A . E., Joubert, P. N., “A Three-Dimensional Turbulent Boundary Layrr,” J.Fluid Mech. 22, 285 (1965). (212) Pierce, F. J., “The Law of the Wake and Plane of Symmetry Flows in ThreeDimensional Turbulent Boundary Layers,” Trans. ASh4E 88 D (J.Bnsic En,?,) 1, 101 (1966). (222) Rotta, J. C., “On the Velocity Distribution ofTurbulent Flow in the Vicinitv of Porous Walls,” Deut. Luft-und Roumfahrt, Forschungs. 66-45, 31 pp. (1 966) (Ger.). (232) Rotta, J. C., “Recent Developments in Calculation hlethods for Turbulent Boundary Layers with Pressure Gradients and Heat Transfer,” J . Appl. Mech., T r a n s . AS‘ME .&r. E 33, 429 (1966). (242) Roy, J. F., “Shear and Friction in a Turbulent Incompressible Boundary Layer,” Conrpl. Rend. Acnd. Sci. Ser. A 262 19, 1061 (1966) (Fr,). (252) Sandmayr, G., “Periodic Longitudinal Vortices in Turhulent Boundary Layers Along Concave Walls,” Deut. Zhfl- und Rourddirl, For;c/iungs. 66-41, 49 pp. (1966) (Ger.). (262) Soo, S . L., “Turbulent Auxiliarv Functions for Symmetric Three-Dimensional Boundary Layers,” Z B M P 17, 122 (1966). (272) Tennekes, H., “Similarity Laws for Turbulent Boundary Layers with Suction or Injection,” J.Fluid .\.iech. 21, 689 (1965). (28~)Tqynsend, A. A,, “Excitation of Internal !Yavcs by a Turbulent Boundary Layer, Ibid., 22, 241 (1965). ( 2 9 2 ) Towusend, A. A., “Self-Preserving Flow Inside a Turbulent Boundary Layer,” Ibid.:p. 773. (302) Townsend, 4 . A., “The Response of a Turbulent Boundary Laycr to Abrupt Changes in Surface Conditions,’’ Ibid., p. 799. (312) Townsend, A. A . , “Self-Preserving Development within Turbulent Boundary Layers in Strong Adverse Pressure Gradients,” Zbid., 23, 767 (1965). (322) Trulio, J. G.. Carr, \V. E. Siles, M . ., Rentifrow, R . 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Y . , “Mixing,” IND.ENC.CHEM.58, ( I l ) , 50 (1966). (30B) Zbid.,57 ( l l ) , 115 (1965). (31B) Proudian, A . P., Feldman S. “ A New Model for Mixing and Fluctuations in a Turbulent Wake,” AIAA J . ’ 3 , kO2 (1965). (32B) Ragsdale, R . G., ,Edwards, 0. J., “Data Comparisons and Photographic Observatlons of Coaxial Mixing of Dissimilar Gases a t Nearly Equal Stream Velocities,” NASA T N D-3131 (1965). (33B) Rom, J., Seginer, A,, Kronzon, J., “The Flow Field in the Turbulent Supersonic Near Wake Behind a Two-Dimensional Wedge-Flat Plate Model,” Technion-Israel Inst. Tech., Dept. Aeron. Engr., T A E Rept, No. 5 4 (1966). (34B) Rosensweig, R . E., “Isotropic Turbulent Mixing with Boundary Sources,’’ Can. J . Chem. Eng. 44, 255 (1966). (35B) Saidel, G. M., Hoelscher, H. E., “Chemical Reaction in the Turbulent W a k e o f a Cylinder,” AZChE J . 11, 1058 (1965). (36B) Schetz, J. A . , “Two-Dimensional Turbulent Jet Mixing,” Trans. A S M E 32 E ( J . Appl. Mech.) 1, 198 (1965). 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(42B) Thomas, D. G., Kraus, K. A , , “Interaction of Vortex Streets,” J . Appi. Phyr. 35, 3458 (1964). T u r b u l e n t Nan-Newtonian Flows (1C) Astarita, G., “Possible Interpretation of the Mechanism of Drag Reduction in Viscoelastic Liquids,” IND.ENC.CHEM.FUNDAMENTALS 4,354 (1 965). (2C) Astarita, G., Nicorlemo, L., “Velocity Distributions and Normal Stresses in ViscoelasticTurbulent Pipe Flow,” AZChE J . 12, 478 (1966). (3C) Astarita, G., Nicodemo, L., “Flow of Viscoelastic Liquids Through Sudden Enlargements,” I N D .ENG.CHEM.FUNDAMENTALS 5 , 237 (1966). (4’2) Cheng, D. C. H., Ray, D. J., Valentin, F. H. H “Flow of Thixotropic Bantonite Suspensions Through Pipes and Pipe Fitti&” Trans. Znst. Chern. Engrs. (London),43, TI76 (1965). (5C) Ernst, W. D., “Investisation of the Turbulent Shear Flow of Dilute Aqueous C M C Solutions,’’ AZChE J . 12, 581 (1966). (6C) Man;:, V. N., Vinogradov, G. V., “Typcs of Turbulence in Highly Elastic Liquids, Kolloidn. Zh. 27, 784 (1965) (Russ.). H., “Skin Friction of Power-Law Fluids i n Turbu(7C) McDonald, A. T . , Bran;:, lent Flow Over a Flat Plate, AZChE J . 12, 637 (1 966). (8C) Meyer, W. A , , “A Correlation of the Frictional Characteristics for Turbulent Flow of Dilute Viscoelastic Non-Newtonian Fluids in Pipes,” AIChE J . 12, 522 119661. (9C) R a m A. Tamir, A . “Structural Turbulence in Polymer Solutions,” J . Abbl. Po/vrne;Sci. 8. 2751 ’11964) . , iEnel.1. , ” , (IOC) Skeiland, A. H. P., “Momentum Heat and Mass Transfer in Turbulent Non-Newtonian Boundary Layers,” A I k h E J.’12, 69, (1966). (1 I C ) Slattery, J. C., “Scale-Up for Viscoelastic Fluids,” Ibid., 11, 831, (1965). (IZC), T;mekes, H., “Wall Region in Turbulent Shear Flow of Non-Newtonian Fluids, Ph~as.Fluids 9, 872, (1966). (1 3C) Tesarik, A,, “Turbulent Velocity Distributions of Kaolin-TVater Suspensions in a Smooth Pipe,” Trans. Inrt. Chmi. Engrs. (London) 43, T317 (1965). (14C) Zandi, I., Rust, R. H., “Turbulent Non-Newtonian Velocity Profiles in Pipes,” Proc. A m . Snc. Civil Engrs. HY 6 ( J . Hydr. Diu.) 91, 37 (1965). I ‘
I
T u r b u l e n t Diffusion (ID) Becker, H. A , , Rosensweig, R. E., Gwozdz, J . R., “Turbulent Dispersion in a Pipe Flow,” AZChE J . 12,964 (1966). (2D) Hjelmfelt, A. T., Jr., Mockros, L. F. “Motion of Discrcte Particles in a Turbulent Fluid,” App/.Sci. Res. 16, 149 (1’966). (3D) Leach, S. J., Wafker G L “Experiments o n Steady State Diffusion in Turbulent Pipe Flow, Min.’Po&r (Gt. Brit.), Safety Mines Res. Estab., R e p . No. 234 (1965). (4D) Morkovin M . V., “On Eddy Diffusivity, Quasi-Similarity and Difiusion Experiments :n Turbulent Boundary Layers,’’ Intern. J . H m t .Mass Transjer 8, 129 (1965). (5D) Patankar S. V., Taylor R . G. “Diffusion from a Line Source in a Turbulent Boundary Layer: Compari’son of ?heory and Experiment,” Ibid., p. 1172. (6D) Shchegolev, G. T., Varakin, V. V., “Study of the Turbulent Diflusion o Steamin Air,” Tr. Urul‘sk. Politckhn. Inst. 1965 (1431, p. 104 (Russ.). (7D] Ternovtsev, V. E., “Mass Transfer in a Uniform Turbulent Flow,” Sanit. 1 ekhn., Vodosnabrh, i Kanalir. 1965, p. 79 (Russ.). Instability a n d Turbulence i n Flows with Free Surfaces (1E) Anshus, B. E., Goren, S. L., “A Method of Getting Approximate Solutions to the Orr-Sommerfeid Equation for Flow on a Vertical Wall,” AIChE J . 12, 1004 (1966). (2E) Atkinson, B., Caruthers, P. A,, “Velocity Profile Measurements in Liquid Films,” Trans. Znst. Chem. Engrs. (London) 45, T33 (1965). (3E) Benney, D. I.,“Long Waveson Liquid Films,” J . Math. Ph>~r.45, 150 (1966). (4E) Berg, J. C., Baldwi3 D. C., “Schlieren Photograph of Interfacial Turbulence During Mass Transfer, Trend. Eng. Univ. Wash. 17, 13 (1965). (5E) Davies, J. T., Khan, W., “Surface Clearing by Eddies,” Chem. Eng. Sci. 20, 713 (1965). (6E) Gogiberidze, L. G., “Experimental Investigation of the Distribution of Velocity in a Plane Turbulent Flow,” Izv. Tblisrk. .l‘aukz In-la, Soaruih 1 Gzdroenerg. 1964 (151, p. 29 (Russ.). (7E) Graef, M., “ O n the Properties of Two- and Three-Dimensional Disturbances in Films Flowing Down Inclined Walls,” Mitt. aus Max-Plnnck Institut Stromungriorrchung 1966 (36), p. 109 (Ger.). (8E) Grant, R. P., Middleman, S., “Newtonian Jet Stabiliry,” AIChE J . 12, 669 (1966). (OE)Iwasa, Y., Imamoto, H., “Some Features of Turbulent Diffusive Processes in Open Channel Flows,” M e m . of Fnc. Engr., Kyoto Uniu. 28, 1, (1966) (Engl.). (10E) Je sen J. C. Crosser 0. K . Perry, R. H., “The Eflect of Il’ave Induced T u r b u t n c d on thd Rate of h o r p d o n of Gases in Falling Liquid Films,” AIChE J . 12, 186 (1966). (11E) Jones? L. 0 TVhitaker, S., “An Experimental Study of Falling Liquid Films,” Ibid., p. 555 (1966). (12E) Jonsson, I. G., “ O n Turbulence in Open Channel Flow Statistical Theory Applied to Micropropeller Measurements ” (in Eng.), Acta kolltech. Scand. C i d Eng. Bldg. Construct. Ser. 31, Ci 31 (1965) (Pkgl.). (13E) Kao, T. W., “Stability of Two-Layer Viscous Stratified Flow Down an Inclined Plane,’’ Phys. Fluids 8, 812 (1965). “Role o f t h e Interface in the Stability of Stratified Flow Down a n (14E) Kao, T. W., Inclined Plane,” Ibid., p. 2190. (15E) Kenning, D. B. R., Cooper M . G “Interfacial Circulation Due to SurfaceActive Agents in Steady Two-Piase Fl&s,” J . Fluid Mech. 24, 293 (1966). (16E) Lee, J., “Turbulent Velocity Profile of a Vertical Film Flow,” Chem. Eng. Sci. 20, 533 (1965). (17E) Linde H Sehrt B. “Schlieren-Photographic Demonstration of Marangoni-Instabilitg in DrAp $ormation,” Monatsber. Deut. Akad. Wisr. Btrltn 7 , 341 (1965) (Ger.). (18E) Massot C Irani F Lightfoot, E. N. “Modified Description of Wave Motion in a’ Fayling Fifm,;’AZChE J . 12, 445 ’(1966). (19E) Smith:, K. A,, “On Convective Instability Induced by Surface-Tension Gradients, J . Fluid Mech. 24, 401 (1966). (20E) Ueber, R . C., “Interfacial Turbulence in Mass Transfer: Instability Analysis and Steady State Experiments,” Univ. Microfilms, Order No. 65-5815; Dissertation Abstr. 25, 7141 (1965). (21E) Whitaker, S., Jones, L. O., “Stability of Failing Liquid Films. Effect of Interface and Interfacial Mass Transport,” AZChE J . 12, 421 (1966). (22E) Yih, C. S.,,,“Stability of a Non-Newtonian Liquid Film Flowing Down an Inclined Plane, Phjs. Fluids 8, 1257 (1965).
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