Showing papers in "Aiche Journal in 1966"

Shape of liquid drops moving in liquid media

Abstract: An investigation of the effects of various physical properties, drop size, and drop velocity on drop shape was carried out for nonoscillating liquid drops falling through stationary liquid continuous phases. The data of forty-five dispersed-continuous phase systems were studied with continuous phase viscosities varying from 0.3 to 46 centipoise and interfacial tensions varying from 0.3 to 42 dyne/cm. A theoretical relation was obtained from the Taylor and Acrivos analysis which quite accurately predicts drop eccentricities for drop Reynolds numbers less than about 20, but is highly inaccurate at higher Reynolds numbers. Relatively simple empirical relations involving the Weber number, Eotvos number, and viscosity ratio were obtained which enable the prediction of the eccentricity of nonoscillating drops over a wide range of Reynolds numbers (6.0 to 1,354) with average deviations of 6 to 8%. These relations may be useful in the estimation of the interfacial area, velocity, and continuous phase mass transfer coefficient of drops distorted from spherical shape.

Newtonian jet stability

Abstract: Although Newtonian jet stability has been the object of numerous experimental and theoretical studies, the total problem of jet disintegration is by no means solved. Theories available in the literature are only applicable to low-speed laminar jets in stagnant air. In practice, the stability of a liquid jet may be influenced by the ambient medium, turbulence in the nozzle, and the extent of development of the velocity profile. None of these factors has received adequate study. This work presents the beginning of a systematic evaluation of the role played by these factors in the destabilization of a liquid jet. Correlations are presented for predicting the stability of both turbulent and high-speed laminar jets in stagnant air.

The incipience of nucleate boiling in forced convection flow

Abstract: An analysis of the incipience of nucleate boiling is developed as a modification and extension of previous analyses. The results are compared with data for subcooled boiling of water in forced convection flow. Apparent anomalies in the data for the onset of boiling in two-phase gas-liquid flow are shown to be a result of a limited size range of cavities on the heater surfaces. Measurements of the surface characteristics of a copper heating tube were made and compared with incipient boiling heat transfer data taken in that tube. The results predicted from the theoretical analysis are consistent with the experimental heat transfer data.

Flow and turbulence in a stirred tank

Abstract: Photographic measurements have been made of the mean and fluctuating components of velocity of water in a fully baffled stirred tank. Confirmation of much of the photographic data was obtained with a Kiel impact tube. Eulerian correlation coefficients and also Eulerian scales of turbulence were calculated from the photographic data. The Eulerian scale was of the same order as the blade dimensions, a result consistent with earlier measurements on the turbulence behind grids. Equations have been developed to describe the flow of energy and the conservation of angular momentum in the impeller stream of a stirred tank with a radial flow impeller and vertical baffles. These are simplifications of the Navier-Stokes equations and the energy equation. They relate energy, angular momentum, and pressure to the mean and fluctuating components of velocity in the impeller stream. The equations derived are used with the photographic data on mean and fluctuating velocities to estimate the angular momentum at different radial sections of the tank and to calculate the flow of energy through these sections. The estimates are compared with more accurate values of the total torque and energy determined with a torque table.

The finite-difference computation of natural convection in a rectangular enclosure

Abstract: A study is made of the natural convection of a fluid contained in a long horizontal enclosure of rectangular cross section with one vertical wall heated and the other cooled. Two-dimensional motion is assumed. The governing vorticity and energy transport equations are solved by an implicit alternating direction finite-difference method. Transient and steady state isothermals and streamlines are obtained for Grashof numbers up to 100,000 and for height-to-width ratios of 1, 2, and 3.

A generalized equation for computer calculation of liquid densities

Abstract: A generalized equation explicitly relating reduced density to reduced temperature and reduced pressure has been developed for calculating liquid densities of pure compounds on a digital computer. The analytical formulation is based mainly on a modified corresponding states principle and the graphical correlation of Lydersen, Greenkorn, and Hougen. The calculated densities from the equation reproduce the literature data within 2% for sixty-two saturated liquids and nineteen compressed liquids. With the aid of a pseudocritical method and a generalized equation of vapor pressure, the same equation is readily applied to the estimation of liquid mixture densities. The method of Prausnitz and Gunn is chosen for the evaluation of pseudocritical constants. For fifteen binary systems, one ternary system, and one quinary system, the one hundred fifty-nine calculated densities agree with the literature data to within 3%. This method is limited to pseudoreduced temperatures less than 1.0.

Model calculations for capillary condensation

Abstract: Capillary phenomena arising from vapor phase condensation in porous media are discussed in the light of an exact interface curvature theory and a self-consistent thermodynamic theory. The system studied consists of liquid condensed in the form of pendular rings at the contact points between identical spherical particles. The geometrical parameters—the curvature, the confined volume, and the surface area of the liquid-vapor interface—must be expressed in terms of incomplete elliptic integrals. In addition several corrections are introduced for the classical Kelvin relation for lowering of vapor pressure. One of these is based on the density dependence of the isothermal thermodynamic susceptibility. Since the susceptibility vanishes at large negative pressures, an upper limit to the curvature is established. The balance equation for the extensive free energy is considered from the point of view of hydrostatic principles.

Similarity solutions for non‐Newtonian fluids

Abstract: Similarity transformations are possible for power law fluids in the following systems: momentum transfer in general Falkner-Skan flows and Goldstein flows; and momentum and energy transfer in forced convection about a right angle wedge, natural convection with constant heat flux at the boundary surface, and general Falkner-Skan flows with nonconstant heat conductivity and a restricted boundary temperature distribution. Similarity transformations are also possible for momentum and heat transfer of Eyring viscous fluids about a right angle wedge. A numerical solution is obtained for forced convection of power law fluids about a right angle wedge with an isothermal surface. From the numerical results an approximate expression is obtained for the local Nusselt number.

Diffusion and solution of gases in thermally softened or molten polymers: Part I. Development of technique and determination of data

Abstract: A technique has been developed that determines simultaneously solubilities and diffusivities of gases in molten or thermally softened polymers. Henry's law was found to hold up to 20 atm. In addition it was found that pressure had no appreciable effect on diffusion coefficients up to 20 atm. Solubilities and diffusivities were determined for systems involving nitrogen, helium, carbon dioxide, and argon in polyethylene, polyisobutylene, and polypropylene. In addition, solubilities were also determined for the preceding gases in polystyrene and polymethylmethacrylate. Other data were also obtained for neon, krypton, and monochlorodiflouromethane in various polymers.

Hydrodynamics of rivulet flow

Abstract: A theoretical analysis of the hydrodynamics of liquid rivulets flowing down an inclined surface is presented. Steady state solutions are developed for the laminar flow case which relate the flow rate to the rivulet width, the physical properties of the liquid, and the contact angle. Excellent verification of the theoretical predictions was obtained in a number of experiments with various liquids on an inclined glass plate. All constants in the equations were derived from theory. These results will be useful in obtaining a better understanding and correlation of such phenomena as liquid flow over packings and catalysts and flow along walls and tube surfaces, as in condensation and evaporation.

An analytical study of laminar counterflow double‐pipe heat exchangers

Abstract: An orthogonal expansion technique for solving a new class of counterflow heat transfer problems is developed and applied to the detailed study of laminar flow concentric tube heat exchangers. The exchanger problem is solved for fully developed laminar velocity profiles, negligible longitudinal conduction in the fluid streams and in the exchanger walls, and with fluid properties which are independent of the temperature. A description of the variation of the local Nusselt numbers and the temperature at the wall between the two streams is given. Also reported are bulk temperature changes in the two streams and mean overall Nusselt numbers. It is shown that for long exchangers, which are of some industrial importance, asymptotic Nusselt numbers exist in counterflow as in single-phase and cocurrent systems. Numerical values of asymptotic Nusselt numbers are reported for a wide range of parameters. Comparisons are made with single-stream solutions such as the Graetz problem, with empirical correlations of experimental data, and with cocurrent flow exchangers. To solve this problem it was necessary to derive new orthogonality relations, and also expressions for determining positive and negative sets of eigenvalues and eigenvectors. Satisfaction of inlet boundary conditions at both ends of counterflow exchangers requires a complete set of eigenfunctions and thus one must use both the positive and negative sets.

Axial dispersion of spheres fluidized with liquids

Abstract: Solids dispersion due to the simultaneous diffusion (random particle motion) and classification (segregation) of fluidized spheres has been investigated. The model developed permits calculation of the mean concentration of spheres of both sizes in a mixture of two sizes of fluidized spheres as a function of bed length. Fick's law has been applied to the diffusional phenomenon. A hypothesis has been advanced and confirmed which permits calculation [Equation (5)] of the classification velocity for each size of sphere in a fluidized mixture. Calculation of the classification velocities in a mixture of spheres is based on the relation of void fraction to superficial velocity for the individual sizes of spheres. To facilitate carrying out these calculations, a generalized equation has been developed for the relation of void fraction to superficial velocity in terms of the Galileo number (d3gΔρρƒ/μ2) and sphere-to-column diameter ratio. Measurement of the bead size gradients (change in bead size with bed length) at steady state fluidization was used to investigate dispersion. Closely sized glass beads of 0.1 and 0.2 cm. diameters were fluidized with liquids of 1 and 15 cps. at void fractions of 0.5 to 0.8 in 2.5 and 5.0 cm. columns. Lead beads of 0.12 cm. diameter were also used. Further, several experiments were conducted by a method which is analogous to that used in molecular diffusion cells.

A correlation of the frictional characteristics for turbulent flow of dilute viscoelastic non‐Newtonian fluids in pipes

Abstract: Turbulent flow of dilute viscoelastic non- Newtonian fluids in pipes, noting frictional characteristics, elastic fluid parameters, etc

Coalescence in fibrous beds

Abstract: The factors affecting the efficacy of a closely packed bed of mixed cotton and a supporting fiber (Teflon, glass, Dynel) were evaluated for oil-in-water dispersions. Several other water-organic systems were also tested. Superficial velocities ranged from 0.2 to 3.5 ft./min. Successful coalescence was attained at interfacial tensions as low as 3.5 dynes/cm. Dispersed phase viscosity was varied form 1.4 to 137 centipoise. For a mixed-fiber bed with a specific ratio of fiber species, there is an optimum bed depth for best performance. High-speed cinephotomicrographic observations at 100 × and up to 4,000 frames/sec. indicated that fiber wettability is not the most important factor for successful operation.

Gas bubble entrainment by plunging laminar liquid jets

Abstract: A free, vertical jet of liquid plunging into a uiescent surface of the same liquid entrains the surrounding gas into the receiving liquid to form bubbles. The entrainment characteristics of such jets of Newtonian liquids of varying physical properties have been investigated by means of high-speed photography. Although both laminar and turbulent jets entrain gas bubbles, the mechanisms governing the entrainment process of the two types of jets are clearly different. Entrainment by turbulent jets results from the disturbances on the free surface caused by the jet instability; entrainment by laminar jets is accomplished by the formation of a thin shell of gas around the jet at the point of entrance, by the development of oscillations in the shell, and by its subsequent breakup into bubbles. Entrainment occurs only when the average jet velocity exceeds a certain critical value termed minimum entrainment velocity. For a laminar jet having a flat velocity profile at the point of entrance, the following correlation permits prediction of the minimum entrainment velocity: where the dimensionless numbers are based on the liquid properties and the jet diameter at the point where the jet meets the surface of the receiving liquid.

Vapor-Liquid equilibria for five cryogenic mixtures

Abstract: Phase equilibrium data have been obtained for the binary systems nitrogen-argon and nitrogen-carbon monoxide at 83.82°K. and for the systems nitrogen-methane, argon-methane, and carbon monoxide-methane at 9.67°K. Total pressures and compositions of both phases were measured. Parameters have been calculated for the Redlich-Kister and Wilson equations for the excess Gibbs energies of the mixtures. The data are thermodynamically consistent within experimental error.

Mass transfer from large oscillating drops

Abstract: A mass transfer model for vigorously oscillating single liquid drops moving in a liquid field has been developed with the concepts of interfacial stretch and internal droplet mixing. The model takes into account both amplitude and frequency of drop oscillations. Experimental values of fraction extracted were predicted with an average deviation of 15%. Oscillations break up internal circulation streamlines and a type of turbulent internal mixing is achieved.

Constitutive equations for viscoelastic fluids for short deformation periods and for rapidly changing flows: Significance of the deborah number

Abstract: The proper forms and asymptotic characteristics of constitutive equations which may be useful for the description of viscoelastic fluids in flow fields of engineering interest are considered. It is seen that the Deborah number emerges as a natural ordering parameter which determines, on the one hand, whether simple approximations explicit in stress may suffice to describe the fluid properties or, on the other hand, whether implicit or integral equations are required. Methods of scale-up are discussed.

Laminar converging flow of dilute polymer solutions in conical sections: Part I. Viscosity data, new viscosity model, tube flow solution

Abstract: This investigation is reported in two parts. Necessary background information is introduced in Part I. The converging flow investigation proper is described in Part II (18). In Part I, viscosity data are presented for polymer solutions used in the converging flow experiment. These data are fitted with a new three-parameter viscosity model which fits the data better than previous three-parameter models. (The viscosity model parameters are used in Part II to characterize rheological behavior of the polymer solutions in the converging flow experiment.) The corresponding relationship between flow rate and pressure drop for laminar flow in cylindrical tubes is derived. (In Part II this relationship is used in deriving an analogous relationship for slow non-Newtonian flow in conical sections.) The primary purpose of Part I is to provide background information for Part II. However, the new viscosity model and the tube flow relationship are of some interest in themselves. The new viscosity model should prove useful for describing viscosity data of a variety of polymer solutions and polymer melts. A simple procedure for fitting the model to viscosity data is described. The tube flow relationship can be used for predicting pressure losses once the viscosity model parameters have been determined. Conversely, it can be used for determining the viscosity model parameters from tube flow data.

An experimental study of falling liquid films

Abstract: A new method for measuring wavelength and wave velocity is described, and experimental values for water flowing down a vertical plane are compared with a numerical solution of the Orr-Sommerfeld equation Good agreement is obtained in the region near the top of the film where small disturbance theory is expected to be valid Experimental Reynolds numbers ranged from 8 to 120

Velocity and pressure profiles for Newtonian creeping flow in regular packed beds of spheres

Abstract: A theoretical analysis is made of Newtonian creeping flow through dense cubic and simple cubic beds of spheres. Velocity and pressure profiles are obtained in equation form by Galerkin's error-distribution method. The profiles satisfy the conservation equations approximately, and the boundary conditions and symmetry conditions exactly. The calculated friction factors are within 5% of experimental values for both packing arrangements.

A theory of withdrawal of cylinders from liquid baths

Abstract: A theory of the amount of liquid entrained by cylinders upon withdrawal from liquid baths is derived for a wide range of cylinder radii. The theory is based on matching curvatures for static and dynamic menisci. Predicted values are expressed as the effect of the dimensionless wire radius (Goucher number) and dimensionless withdrawal speed (capillary number) on the dimensionless flux. The theory was verified experimentally for all wire radii by removing short cylinders from oily fluids and with other information. The fluids used included kerosene, mineral oil, motor oil, and glycerine, with viscosities from 2 to 500 centipoise; Goucher numbers ranged from 0.05 to 1.2. Deviations, which were noted at high capillary numbers where velocity gradients become appreciable, indicated that the theory is a plug flow or low speed theory. Also discussed are differences found with water and the conditions under which films coalesced into droplets.

Effect of axial diffusion of vorticity on flow development in circular conduits: Part I. Numerical solutions

Abstract: An entrance model is presented which extends the analysis of the flow development in the entrance region to flow regimes which are inadequately described by a boundary-layer analysis. Results are presented for the numerical solutions of the boundary-layer equations and the complete equations of motion at five different Reynolds numbers. The effect of the axial diffusion of vorticity on the pressure drop, entrance length, and development of the vorticity and velocity fields is demonstrated.

The evaporation of drops of pure liquids at elevated temperatures: Rates of evaporation and wet‐bulb temperatures

Abstract: The rates of evaporation and the wet-bulb temperatures have been correlated for drops of pure liquids evaporating in streams of high-temperature air. The four liquids studied were acetone, benzene, n-hexane, and water. The drops were of a millimeter in diameter and were suspended in a free jet of dry, vapor-free air that ranged in temperature from 27° to 340°C. Reynolds numbers ranged from 24 to 325. Corrections to the Nusselt number to account for the heat lost the outwardly diffusing vapor ranged up to about 35%.

Performance of fouled catalyst pellets

Abstract: Equations are developed for the bulk rate of a gaseous reaction on a porous catalyst whose activity changes with time due to a decrease in active surface. The performance is evaluated in terms of a pellet effectiveness factor which is a function of time and a Thiele (diffusion-reaction) modulus. By a stepwise numerical technique, the equations can be solved without resort to assumptions regarding the distribution of fouled surface within the pellet. The method is applicable at isothermal conditions for any form of the rate equations for the main and fouling reactions and for any diffusivity-concentration relationship. To illustrate the method, results are given for first-order isothermal reactions for three types of fouling processes. For a series form of self-fouling, a catalyst with the lowest intraparticle diffusion resistance gives the maximum activity for any process time. In contrast, for parallel self-fouling a catalyst with an intermediate diffusion resistance is less easily deactivated and can give a higher conversion to desirable product, particularly at long process times. A simpler solution is possible by supposing that the shell model represents the disposition of fouling material in the pellet. It is shown that for parallel self-fouling and independent fouling this model gives reasonably good results, even when the reaction resistance for the main reaction is important. However, the shell concept does not appear suitable over a range of conditions when the fouling is of the series type. The single-pellet effectiveness factors can be used to determine the effect of fouling on the conversion in a fixed-bed reactor. To illustrate the method of approach curves of conversion as a function of time and position in the bed are presented for a parallel, self-fouling reaction system. The results show the influence of intraparticle diffusion on the overall effects of fouling.

Axial dispersion in liquid flow through packed beds

Abstract: Step-function injection and purging of a dilute salt tracer in water was used to measure axial dispersion for low Reynolds number liquid flow through beds of uniform sized, random packed glass spheres. The resultant data and those of several previous studies are coordinated and interpreted in terms of Reynolds, Schmidt, and Peclet numbers.

Viscosities of methane and propane at low temperatures and high pressures

Abstract: The viscosities of methane and propane were determined at pressures to 5,000 lb./sq. in. abs. over a temperature range from −170° to 0°C. and −100° to 0°C., respectively, and the densities of propane were measured at pressures to 5,000 lb./sq. in. abs. over the temperature range from −100° to 0°C. Reproducibility of the viscosity data was about ± 1.2% and for the most part agreement with literature values was within ± 2%. the estimated accuracy of the density measurement was ± 0.5%. The data reported were well correlated in terms of residual viscosity vs. density.

Bubble motion and mass transfer in non‐Newtonian fluids

Abstract: Instantaneous mass transfer coefficients were obtained for the absorption of carbon dioxide bubbles rising in an aqueous solution of sodium carboxymethylcellulose. The rheological character of the solutions was well described by the Ellis model. Mass transfer coefficients were high initially but trailed off rapidly with bubble age. Exceptions were found at specific diameters where the bubble shape went through a transition. At about 0.2 cm. a transition from ellipsoidal to a sphere shape occurred, which has also been observed in Newtonian fluids. At a larger diameter, however, the non-Newtonian fluid showed a shape change from a spherical “cap” to a “top” shape and finally to an ellipsoid. A sudden increase of mass transfer coefficient accompanied each shape transition. Drag coefficient data were correlated successfully with a new Reynolds number. The Newtonian and power law portion of the Ellis model each contributed a component to the Reynolds number, which, when added together, correlated drag data for the bubbles as well as for glass spheres. Attempts to account for transition shape changes and bubble tailing in the non-Newtonian fluid and their effects on bubble mass transfer are included.