BRIEFS Summary of papers published in this month’s research quarterly, I&EC Fundamentals
FLUID AND PARTICLE MOTION IN TURBULENT STIRRED TANKS. FLUID MOTION
MASS TRANSFER TO FALLING WAVY LIQUID FILMS IN TURBULENT FLOW
Using streak photography, fluid velocities were measured in baffled tanks turbulently agitated by six-bladed turbine impellers. Outside of the impeller discharge zone (in most of the tank) the average and r.m.s. fluctuating velocities were found to be proportional to the rotational speed of the impeller, and its diameter squared, and inversely proportional to the cube root of the tank volume. This relationship was verified over a wide range of impeller and tank sizes, fluid properties, and agitator speeds, and agrees in form with predictions based on dimensional analysis of the power dissipation in such tanks. The similarity of the turbulent power-dissipation correlation-Le., constancy of the power number-for a large number of impeller-tank arrangements suggests that the present fluid velocity correlation may have broad applica3 bility, and that maintaining a constant value of NO2/(T 2 H ) 1 /may be a suitable scale-up criterion in many instances. Further, this fluid velocity correlation helps explain various anomalies in correlations which have been proposed for heat and mass transfer in stirred tanks.
The mass transfer rate at a free interface can be related to the rate of viscous dissipation in the turbulent flow near the surface. An order of magnitude estimate for the dissipation for a wavy turbulent liquid film is derived from a consideration of the time-averaged vorticity equation. The liquid phase mass transfer coefficient is then related to the average wave length, amplitude, and speed of the surface disturbances. The predicted values of the transfer coefficients agree well with experimental observations.
Henry G. Schwarttberg and Robert E. Trtybal, New York University, New York, N . Y . IND.ENG.CHEM.FUNDAMENTALS 7,l-6 (1968)
Sanjcy Banerjee, D. S. Scott, and Edward Rhodes, University of Waterloo, Waterloo, Ontario, Canada
IND.ENG.CHEM.FUNDAMENTALS 7,22-27 (1968) EXCESS PRESSURE DROP IN LAMINAR FLOW THROUGH SUDDEN CONTRACTION. NEWTONIAN LIQUIDS
The Couette and Hagenbach contributions to the excess pressure drop in laminar flow through sudden contractions are discussed. Values reported in the literature are theoretical, and have been obtained on the basis of assumptions which are seriously violated i n the case of a sharp-edged contraction. Experimental data show that the excess pressure drop is much larger than predicted by published analyses, over the entire range of Reynolds numbers,
Gianni Astarita and Guido Greco, Zstituto di Elettrochimica, University of Naples, Italy FLUID AND PARTICLE MOTION IN TURBULENT STIRRED TANKS. PARTICLE MOTION
Differential equations describing particle motion in a turbulent fluid were applied to a flow model for turbulent stirred tanks, and solutions obtained for particle slip, over-all, and fluctuating velocities a t various particle sizes and densities, agitation speeds, viscosities, Lagrangean integral scales, and drag coefficient levels. Particle velocities were measured by streak photography and compared with the calculated values. The particle motion followed the fluid motion very closely; settling velocities were markedly lower than in still fluid. This behavior appears to be due to abnormally high drag. Because of the intense turbulence the threshold of detection for slip velocity was too high to permit other than order of magnitude verification of the calculated slip velocities. T h e calculated slip velocities indicate that particle to fluid mass transfer in stirred tanks cannot be explained in terms of a simple slip velocity model.
Hengv G. Schwartzberg and Robert E. Treybal, New York University, New York, N . Y . 10453 IND.
ENG.CHEM. FUNDAMENTALS 7,6-12 (1968)
IND.END.CHEM.FUNDAMENTALS 7,27-31 (1968) DRAG REDUCTION IN A MODEL OF SHEAR-FLOW TURBULENCE
A Burgers-type system of equations exhibits behavior typical of weak steady shear-flow wall turbulence. The essential new feature is a nonlinear turbulence production term which is absent from the original Burgers model. Approximate inclusion of triple correlations gives realistic friction factor-Reynolds number curves. The viscous term in the model equations is then replaced by a linear viscoelastic term. I n weak turbulence a n increase of friction factor then occurs, even though dissipation is reduced, both in the model equations and in the corresponding threedimensional equations. When triple correlations are included, however, drag reduction occurs above a critical shear stress. Thus we have a possible model of some of the observed turbulent drag-reduction phenomena in polymer solutions.
R. 1. Tanner, Brown University, Providence, R. 1. IND:
ENG.&EM.
FUNDAMENTALS 7, 32-38 (1968)
LONGITUDINAL DISPERSION FOR TURBULENT FLOW IN PIPES
A linearized stability analysis based on the equations of motion developed previously shows the state of uniform fluidization to be unstable against small perturbations in voidage. The instabilities take the form of rising and growing voidage fluctuations whose propagation characteristics are related to the physical properties of the fluidized system. The rate of growth of instabilities is much larger when particles are fluidized by air than by water.
The applicability of the one-dimensional turbulent dispersion equation as a model to describe longitudinal dispersion in pipe flow was investigated by pulse testing for the NaC1-water system in test sections of 72 feet of glass pipe of different diameters and with conductivity measuring cells located a t varying distances from the base cell. The model was valid for describing longitudinal dispersion for Reynolds numbers greater than 4 X 104. I n fully developed turbulent flow the dispersion coefficient was independent of pipe length; the effects of diameter and fluid velocity were in agreement with the model.
T . B. Anderson and Roy Jackson, University of Edinburgh, Edinburgh, Scotland
C. N . Sittel, Jr., W . D . Threadgill, and K . B. Schnelle, Jr., Vanderbill University, Nashville, Tenn.
IND. ENG.CHEM.FUNDAMENTALS 7,12-21 (1968)
IND.ENG.CHEM.FUNDAMENTALS 7, 39-43 (1968)
FLUID MECHANICAL DESCRIPTION OF FLUIDIZED BEDS. THE STATE OF UNIFORM FLUIDIZATION
STABILITY OF
VOL. 6 0
NO. 2
FEBRUARY 1 9 6 8
81
BRIEFS HYPERFILTRATION. CONCENTRATION POLARIZATION IN TUBULAR SYSTEMS WITH DYNAMICALLY FORMED MEMBRANES
INFLUENCE OF DROPLET SIZE-AGE DISTRIBUTION O N RATE PROCESSES IN DISPERSED PHASE SYSTEMS
The Sherwood, Brian, Fisher, and Dresner equations for concentration polarization with turbulent flow in tubes have been verified by data obtained with dynamically formed, salt-filtering membranes. Magnesium chloride and sodium chloride concentrations from 0.01M to 0.26M were tested with hydrous Zr(1V) oxide membranes. Allowance was made in computer analysis of results for variation of rejection with salt concentration. The converged value of a parameter related to the mass transfer coefficient agreed well with the generally accepted figure. Graphical extrapolation of a function of observed rejection gave values of rejection corrected for concentration polarization in good agreement with those obtained by least-squares analysis.
The performance characteristics of continuous dispersed phase reactors are related to the size-age distribution of the droplet population. A comparison is made between completely mixed and segregated systems to establish the effect of the droplet size distribution for various rate processes occurring in the droplet phase. Droplet breakage is shown to influence conversion, even for firstorder rate processes when mass transfer is involved.
Arthur J. Shor, Kurt A. Kraus, and J . S. Johnson, Jr., and W m . T . Smith, Jr., Oak Ridge National Laboratory, Oak Ridge, Tenn., University of Tennessee, Knoxville, Tenn. IND.ENG.CHEM.FUNDAMENTALS 7 , 4 6 4 8 (1968)
MASS TRANSFER TO SPHERICAL DROPS OR BUBBLES AT HIGH REYNOLDS NUMBER
Mass transfer to single spherical drops or bubbles rising steadily through a liquid a t high Re was calculated by using a boundary layer approach for the fluid flow. The result can be written as
42
Nu = Z(Re,p,p)Pe*/2, where I(Re,p,p) is a given func47r tion of Re and two physical properties-namely, the viscosity and the density of the system. The result reduces to the potential solution as Re +a. Comparison with the experimentally observed behavior of two systems shows satisfactory agreement.
K. J. Valentas and N. R. Amundson, University of Minnesota, Minneapolis, Minn. IND.ENG.CHEM.FUNDAMENTALS 7,66-72 (1968)
DENDRITIC CRYSTALLIZATION OF ICE FROM AQUEOUS SOLUTIONS
Dendritic aggregate of ice formed upon freezing of aqueous solutions has been studied; chlorides of sodium, potassium, lithium, and hydrogen were used as solutes. The spacing between ice platelets or dendrites was experimentally found proportional to the square root of freezing time when freezing rate was constant from beginning to end of solidification. During unidirectional freezing from a constant temperature chill, dendrite spacing increases linearly with distance from the chill surface and is inversely proportional to the square root of the maximum freezing rate. In binary solutions dendrite spacing increases linearly with solute concentration; above a critical solute concentration ice platelets develop side branches. .4t a given solute concentration, spacing between ice dendrites increases linearly with solute diffusivity. In ternary and quaternary solutions, dendrite spacing is a function of the concentrations and diffusivities of each of the constituent solutes.
H. Y . Cheh and Charles W . Tobias, University of California, Berkeley, Calif.
P. K. Rohatgi, S. M . Jain, and C. M . Adams, Jr., International Nickel Co., Inc., Sujern, N.Y., and Massachusetts Institute of Technology, Cambridge, Mass.
IND.ENG.CHEWFUNDAMENTALS 7,48-52 (1968)
IND.ENG.CHEM.FUNDAMENTALS 7, 72-79 (1968)
EFFECT OF NONLINEAR TEMPERATURE PROFILES ON THE ONSET OF CONVECTION DRIVEN BY SURFACE TENSION GRADIENTS
The effect of nonlinear preconvective temperature profiles on the magnitude of the critical Marangoni number is considered using linear stability theory for the case of surface tension-driven flow in evaporating liquid layers. The analysis predicts that in the presence of strong nonlinearities in the basic profile the critical Marangoni number can be much larger than that corresponding TO a linear one. Values of the Marangoni number measured a t the onset of convection in shallow evaporating layers of propyl alcohol are in good agreement with the analytical predictions.
Antonio Vidal and Andreas Acrivos, Stanford University, Stanford, Calif. IND.END.CHEM.FUNDAMENTALS 7, 53-58 (1968)
MECHANISM OF CCICO~ SCALE DEPOSITION OF HEAT-TRANSFER SURFACES
C a C 0 3 scale deposition rates were accurately measured using an annular constant heat-flux exchanger which maintains the reaction surface a t constant temperature, irrespective of scale layer thickness. The separate effects of flow velocity, scale surface temperature, and water composition were examined according to the rate model presented. Within the range of surface temperatures (67' to 85'C) and Reynolds numbers (13,000 to 42,000) investigated, CaCOa deposition is mainly controlled by the forward diffusion rate of Ca+* and HCOa- ions. In certain cases, it is also controlled by the backward diffusion rate of dissolved COn formed during the reaction.
David Hasson, Mordecai Avriel, William Resnick, T r v i Rorenman, and Shlomo Windreich, Technion-Zsrael Institute of Technology, Haifa, Israel IND. ENG.CHEM. FUNDAMENTALS 7, 59-65 (1968) 82
INDUSTRIAL AND ENGINEERING CHEMISTRY
HOMOGENEOUS NUCLEATION OF SODIUM CHLORIDE SOLUTIONS
Experimental confirmation of the theory of homogeneous nucleation for electrolyte solutions is demonstrated by a n analysis of the behavior at 25 "C of an aerosol composed of supersaturated sodium chloride solution droplets 0.01 to 0.1 micron in diameter. (The term homogeneous applies to systems devoid of foreign nuclei.) The humidity of the air with which the aerosol was in equilibrium was adjusted to attain a series of droplet supersaturation states. Nucleation and crystallization of the droplets subsequently ensued, being detected by an ion counter that permitted following the size of the aerosol particulates as a function of time at specific relative humidities. The behavior of the aerosol is attributed to homogeneous nucleation, because only a small proportion of the droplets contained foreign nuclei, when they were formed upon subdivision of the original liquid. The time for nucleation was found to be a logarithmic function of the electrolyte activity as theory predicts.
Albert0 F. Hidalgo and Clyde Orr, Jr., Georgia Institute of Technology, Atlanta, Ga. IND.ENG.CHEWFUNDAMENTALS 7 , 7 9 4 3 (1968)
INTERFACIAL AND ELECTRICAL EFFECTS ON THERMAL CONDUCTIVITY OF NEMATIC LIQUID CRYSTALS
The thermal conductivity of nematicp-azoxyanisole, as determined by a hot-wire probe method, is higher near the wire-liquid interface than in the bulk of the liquid. An electric field in the direction of heat conduction also produces higher thermal conductivity. The effect of the field is not so strong in the opposite direction. The thickness of the layer of higher thermal conductivity around the
probe appears to be 10 to 20 microns with no electric field, but considerably larger with applied fields. These phenomena are interpreted as evidence that the thermal conductivity of liquid crystals is anisotropic. A qualitative model is advanced to explain the results. J . J . C. Picot and A . G. Fredrickson, University of Minnesota, Minneapolis, Minn.
IND.ENC.CHEM.FUNDAMENTALS 7 , 8 4 4 9 (1968)
SELF-DIFFUSION IN LIQUID CARBON DIOXIDE AND PROPANE
h
The self-diffusivities of liquid carbon dioxide and propane are determined as functions of temperature and density, by radioactive tracer measurements. Results are given for reduced temperatures of 0.80, 0.90, and 0.97, and pressures up to 182 atm. The data are correlated in corresponding-states form, and are used to test several theories of diffusion in liquids.
R. C. Robinson and W. E. Stewart, University of Wisconsin, Madison,
METHOD FOR PHASE EQUILIBRIUM CALCULATIONS BASED O N GENERALIZED BENEDICT-WEBB-RUBIN CONSTANTS
A method based on the generalized Benedict-\Vebb-Rubin equation of state was used to develop the B\VR constants for cis-2-butene, I-pentene, and 1,3-butadicne. This development consistcd of three steps. First, the coefficients of the reduced B\VR equation of state were determined based on I-butcnc and mcthylacetylene, respectively. Second, with the aid of the reduced equations of state, the B\VR constants for cis-2-butene, 1-pentene, and 1,3butadiene were established. Finally, in conjunction with the new constants, a set of temperature-dependent Go's was developed for each substance. This last step assured the accurate prediction of the vapor and liquid fugacities up to the critical point. The new B\\'R constants wcre cvaluated by calculating the equilibrium constants of these components in mixtures. Predicted values \$.ere in excellent agreement with cxpcrimcntal results, indicating the applicability of these constants to phase equilibrium calculations.
T . G. Raufmann, ESSO Research and Engineering Co., Florham Park, -\I. J .
IND.ENC. CHEM.FUSDAMENTALS 7, 113-120 (1968)
Wis.
IND.ENG.CHEM.FUNDAMENTALS 7, 90-95 (1968)
VARIATION OF NONISOTHERMAL DIFFUSION COEFFICIENTS
The diffusion coefficient measured in thermal diffusion is typically 20 to 40% larger than that measured in isothermal diffusion, and increases with temperature gradient. Mass-transfer theory does not account for the magnitude of the effect.
Sergio DiCave and Alden H. Emery, Jr., Purdue University, Lafayette, Ind.
IND.END.CHEM.FUNDAMENTALS 7,95-99 (1968)
ESTIMATION OF RATE CONSTANTS FROM MULTIRESPONSE KINETIC DATA
Alternative procedures for the estimation of several parameters from multivariate observations are evaluated for a complex system. Primary attention is given to methods based on generalization of nonlinear least squares and Bayesian estimation as proposed by Box and Draper. The latter method provides an effective means for estimation even in very complicated cases and demonstrates a parametric sensitivity which allows a precision of estimation not possible with other techniques.
Reiji Mezaki and John B. Butt, Yale University, New Haven, Conn. ADSORPTION ISOTHERMS FOR BENZENE-HEXANE MIXTURES
Equilibrium isotherms were obtained for n-hexane and benzene on silica gel, both separately and from mixtures of the two gases. The measurements were made at low surface coverages and indicated a preferential adsorption of benzene. In the mixture studies at a fixed hexane partial pressure, increasing the benzene in the gas significantly reduced the adsorption of n-hexane. The effect of hexane on benzene was much less. Data were taken from 70" to 130' C . and heats of adsorption calculated from these measurements decreased sharply with increasing surface coverage. Results for the mixture measurements showed that AH for hexane decreased with increasing amounts of benzene. The heat of adsorption of benzene was not affected by hexane in the gas.
John Shen and J . M . Smith, University of California, Davis, Calif.
IND.END.CHEM.FUNDAMENTALS 7, 100-105 (1968)
IND.ENG.CHEM.FUNDAMENTALS 7,120-125 (1968)
KINETICS OF COUPLED FIRST-ORDER REACTIONS WITH TIME-DEPENDENT RATE COEFFICIENTS I N TERNARY SYSTEMS
A method for calculating the kinetic behavior of coupled first-order reactions in a ternary system from any given set of time-dependent rate coefficients is based on Lie algebra. The method reduces the integration of a set of simultaneous differential equations with variable coefficients to successive integrations of ordinary differential equations. The application is illustrated by an example.
Lothar Riekert and James Wei, Mobil Research and Development Corp., Princeton, N . J. 08540
IND.ENG.CHEM.FUNDAMENTALS 7,125-131 (1968) RATES OF ADSORPTION IN THE BENZENE-HEXANE SYSTEM
Breakthrough curves were measured for mixture and single-component adsorption of benzene and n-hexane a t the same conditions as previously reported for equilibrium data. The single-component results showed that for the largest particle size and at the highest temperature, 130' C , the rate was controlled by intraparticle diffusion. At these conditions surface diffusion was estimated to account for more than 90% of the intraparticle transport. For smaller particle sizes axial diffusion affected the breakthrough curves, and at lower temperatures surface adsorption was probably a significant resistance. I n the linear isotherm range (132' C ) the rate of adsorption of benzene from mixtures was the same as in single-component adsorption. At other conditions interaction effects were involved and the mixture results could not be predicted from the single-component measurements.
DISASTER PROPAGATION. PROPAGATION OF EXPLOSIVE-LIKE VIOLENCE
The spread of violence through weakly connected systems by a probabilistic mechanism was studied. Probability theory and simulation are used to develop preliminary criteria for the safe design of storage systems through which disaster may spread by random series of detonations. Parametric results are qualitatively compared to those given by data suggested for the specification of explosives storage. This is an initial report on studies in disaster propagation.
John Shen and J . M. Smith, University of California, Davis, Calzy.
A. H. Masso and D. F. Rudd, University of Wisconsin, Madison, W i s . 53706
IND.ENG.CHEM.FUNDAMENTALS 7, 106-114 (1968)
IND.ENG.CHEM.FUNDAMENTALS 7, 131-141 (1968) V O L . 6 0 NO.
z
FEBRUARY 1968
a3
BRIEFS CONJUGATE GRADIENT METHOD FOR NONLINEAR PROGRAMMING PROBLEMS WITH LINEAR CONSTRAINTS
EXPERIMENTAL TECHNIQUE. APPARATUS FOR DETERMINATION OF LIQUID-LIQUID-GAS EQUILIBRIA AT ADVANCED PRESSURES
A new method for solving nonlinear programming problems in which the constraints appear linearly, starting from an initial feasible point, locates new feasible points in an iterative manner by moving in a direction conjugate to previous directions of search. As such, it may be looked upon as an extension of the work of Davidson-Fletcher-Powell to problems involving linear inequality constraints. As the method proceeds, information about the local curvature of the objective function is incorporated into a matrix which determines the direction in which to move and acts as a variable metric for each maximization problem. Numerical examples illustrate its versatility compared to other procedures.
A novel apparatus has been designed and built for studying liquidliquid-gas equilibria. The experimental equipment features a new magnetic pump to recirculate all three phases, sampling from the recirculating streams, minimization of dead space, and analysis of all phases by gas chromatography. Typical operating pressures are up to about 1000 p.s.i.a. Special methods for sampling gases and liquids under pressure were developed.
R. N . Fleck and J. M . Prausnitr, Department of Chemical Engineering, Universib of California, Berkelgy, Calt’f. IND.
ENG.CHEM. F U N D A M E N T A L S 7, 174-176 (1968)
Donald Goldfarh and Leon Lapidus, Princeton University, Princeton, N . J . IND.END.CHEM.FUNDAMENTALS 7,142-151 (1968)
QUASILINEARIZATION AND THE ESTIMATION OF PARAMETERS IN DIFFERENTIAL EQUATIONS
The estimation problem is formulated as a two-point or multipoint boundary-value problem. The classical least squares criterion is used. T o illustrate the approach, the Peclet group and the reaction rate group in a tubular reactor with axial mixing are obtained.
E. Stanley Lee, Phillips Petroleum Go., Bartlesville, Okla. IND.ENC.CHEM.FUNDAMENTALS 7, 152-158 (1968)
COMMUNICATION. COMPOSITION DEPENDENCE OF THE VISCOSITY OF BINARY LIQUID SYSTEMS
The statistical mechanical theory of transport in liquid mixtures is used to develop a predictive relationship for the variation of the shear viscosity of binary liquid systems based on previous work on the binary diffusion coefficient. The relationship is expected to be valid for systems which are approximately regular in the thermodynamic sense. Comparison with experimental data indicates excellent agreement for a variety of binary systems.
Harry T . Cullinan, Jr., State University of New York at Buffalo, Buffalo, N . Y .
IND.ENC.CHEM.FUNDAMENTALS 7, 177-180 (1968) LINEAR PROGRAMMING. Z-TRANSFORM DESIGN OF DIGITAL CONTROLLERS FOR REGULATOR SYSTEMS
Digital controllers for linear regulator control systems can readily be designed by linear programming coupled with the Z-transform method. This method permits designs based on performance criteria and classes of deterministic input functions which are not suitable for conventional design techniques. The method is described in detail and illustrated by an example.
C. I . Huber and R. I . Kermode, Carnegie Institute of Technology, Pittsburgh, Pa. IND.ENC.CHEM.FUNDAMENTALS 7, 158-164 (1968)
INVARIANT IMBEDDING, NONLINEAR FILTERING, AND PARAMETER ESTIMATION
The invariant imbedding concept is used to derive sequential estimator equations in the theory of nonlinear filtering and estimation. The reaction rate constant and the concentration of the reactant are estimated by these estimator equations. The least squares criterion is used.
E. Stanley Lee, Phillips Petroleum Co., Bartlesville, Okla. IND.ENC.CHEM.FUNDAMENTALS 7,164-171 (1968)
EXPERIMENTAL TECHNIQUE. MEASUREMENT OF TENSILE STRESSES IN RAPIDLY DEFORMING VISCOELASTIC MATERIALS
The geometry of a jet of viscoelastic material issuing abruptly from a nozzle under the influence of an impact is measured by a frameby-frame analysis of high-speed film. Application of the momentum balance equation, and a few simplifying assumptions, allow evaluation of the tensile stresses within the jet during the rapid deceleration phenomenon following the end of flow through the nozzle.
Gianni Astarita, Giusepfie Marrucci, and Domenico Acierno, Istituto di Elettrochimica, Universitd di iVapoli, Piarrale Tecchio, Naples, Italy IND.ENG.CHEM.FUNDAMENTALS 7, 171-173 (1968) 84
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
COMMUNICATION. RE-EXAMINATION OF HOUGEN-WATSON RATE MODELS INCLUDING A WEIGHTED LEAST SQUARES APPROACH
A chemical example of fitting observations to two Hougen-Watson reaction rate models which are nonlinear in the parameters is discussed. The data have been obtained from a study of catalytic isomerization of n-pentane to isopentane, and the experiment and its analysis have been reported. R. A . Johnson and N . A . Standal, C’niversity of Wisconsin, Madison, WZJ., and Reiji Metaki, Yale Unioersity, New Haven, Conn. Norman L. Carr, Gulf Research C3 Development Co., Pittsburgh, Pa. IND.ENG.CHEM.FUNDAMENTALS 7,181-183 (1968)
CORRESPONDENCE
Comments on a published article on numerical differentiation of equally spaced and not equally spaced experimental data and a reply.
Donald T. Winski, International Flavors f3 Fragrances, Union Beach, N .J . Harry C. Hershey, Ohio State Universib, Columbus, Ohio Jacques L. Zakin, The Universiv of Missouri at Rolla, Rolla, Mo. IND.ENG.CHEM.FUXDAMENTALS 7,183-184 (1968)
CORRESPONDENCE
Comments on an article on mechanism in the oxidation of partsper-million quantities of nitric oxide. W . H . Corcoran, California Institute of Technology, Pasadena, Calif. IND. END.CHEM. F U N D A M E N T A L S 7,184 (1968)
