Showing papers in "Aiche Journal in 1969"

Single-phase flow through porous media

Abstract: The local volume average of the equation of motion is taken for an incompressible fluid flowing through a porous structure under conditions such that inertial effects may be neglected. The result has two terms beyond a pressure gradient: g, the force per unit volume which a flowing fluid exerts on a porous structure, and the divergence of the local volume-averaged extra stress tensor (viscous portion of the stress tensor). Constitutive equations for g are examined with the aid of the principle of material indifference. When g is assumed to be a function of the velocity of the fluid relative to the solid as well as various scalars, the usual results for a nonoriented (isotropic) porous structure are obtained. When g is assumed to be a function of the local porosity gradient as well, we derive a new expression for g applicable to oriented (anisotropic) porous structures. For a Newtonian fluid with a constant viscosity, the divergence of the local volume-averaged extra stress tensor is proportional to the Laplacian of the averaged velocity vector. Boundary conditions for the averaged velocity vector are discussed. Three problems are solved for the flow of an incompressible Newtonian fluid in a nonoriented permeable medium. These solutions, as well as an order-of-magnitude analysis, suggest that we may often neglect both the Laplacian of average velocity and the boundary conditions for the tangential components of averaged velocity at an impermeable wall. Two specific constitutive equations for g are proposed for the flow of incompressible Noll simple fluids in nonoriented porous structures. Flow through a porous medium bounded by an impermeable cylindrical surface is solved for these two constitutive equations, and the results are compared with previously available experimental data.

Low Reynolds number developing flows

Abstract: Equations are given that relate the entrance length to Reynolds number for pipe and channel geometries with a flat velocity profile as the initial condition. These equations are linear combinations of the creeping flow and boundary layer solutions. The former is obtained by minimization of the viscous dissipation using the finite element method. The equation for the pipe entrance is shown to be in good agreement with experimental data.

On the particle size distribution function and the attrition of cracking catalysts

Abstract: The rate of attrition of a catalyst sample, of a single particle size, can be expressed by a simple function of initial diameter and time. The initial rate is a function of initial diameter, whereas the decrease in attrition rate of a catalyst of a given size as it ages depends only on time. The attrition equation for a single size of particles is introduced into the particle size distribution function and, through mathematical analyses, yields a relationship for the attrition of a full size-range catalyst. The form of this relationship, which includes terms for attritability and severity of attrition conditions, is verified by laboratory and commercial attrition data on two full-range catalysts. The size distribution of an attrited sample is not expressible in simple analytical form, but is readily obtained by numerical analysis.

Turbulence phenomena in drag reducing systems

Abstract: An analysis based on the Townsend-Bakewell model of the eddies in the wall regions of turbulent shear flows shows that viscoelastic fluid properties must lead to significant reductions in the rate of production of turbulent energy. This analysis in turn leads to the proper form of the similarity laws for drag reducing fluids, heretofore deduced empirically. Measurements of the axial and radial turbulence intensities for flow through smooth round tubes are reported, as are measurements of the time-averaged velocity profiles and the drag coefficients. These indicate that for solutions exhibiting drag reduction at all Reynolds numbers the flow may be transitional to Reynolds numbers of the order of 105. This transitional flow consists of alternating patches of laminar and turbulent fluid, within each of which the flow characteristics are approximately similar to those of Newtonian fluids. At high Reynolds number conditions with the turbulent field fully developed the velocity profile in the core is flatter under drag-reducing conditions than for turbulent Newtonian fluids, a change dependent on the increased isotropy of the turbulent field of the drag-reducing fluid. These effects appear to be a result of increases in the time scales of the radial fluctuations caused by the fluid properties. Design calculations based upon the present results suggest that in large diameter pipelines, or in boundary layers on large objects, drag reduction may not be attainable under conditions of practical interest until fluids having relaxation times an order of magnitude larger than those presently available, but with comparable viscosity levels, are developed or, alternately, until fluids exhibiting Weissenberg numbers which do not change with deformation rate, can be found.

The synthesis of system designs. II. Heuristic structuring

Abstract: The use of heuristic structuring strategies in the development of computer programs for the synthesis of process designs is examined. By the employment of selection weights which are adjusted as experience is gained from past successes and failures, the computer is able to learn the sequence of structuring decisions which leads toward the optimal process design. The computer can develop competence in the synthesis of systems in a limited area of technology.

Effects of transpiration and changing diameter on heat and mass transfer to spheres

Abstract: Mass transfer to spheres suspended in an agitated liquid has been studied both experimentally and theoretically. Finite-difference solutions are obtained for mass transfer from a sphere to a fluid flowing past it in steady viscous flow. The effects of a transpiration velocity at the surface of the sphere and of a continuously changing sphere diameter are included. A normalized presentation of these effects is quite insensitive to the bulk flow Peclet number. When these theoretical corrections for transpiring and shrinking spheres are applied to the mass transfer data for ice spheres that are melting in an agitated brine bath, the corrected mass transfer coefficients are brought into agreement with a generalized correlation published elsewhere. This agreement suggests that the theoretical results apply, with reasonable accuracy, to a shrinking and transpiring sphere that is suspended in a turbulent liquid.

Mass transfer limitations in a trickle-bed reactor

Abstract: The hydrogenation of alpha-methylstyrene to cumene at 20 to 50°C. was studied experimentally with a trickle-bed reactor comprising a single vertical column of spherical porous palladium-on-alumina catalyst pellets. Other studies with powdered catalyst and studies in which catalyst pellets were swirled in a reactor allowed the intrinsic kinetics and effectiveness factor of the catalyst pellets to be determined. At 50°C. the reaction rate in the trickle-bed was about one-half of that in the absence of mass transfer limitations in the outside liquid film. The effectiveness factor of the pellets alone at 50°C. was 0.0057, and the tortuosity factor was 7.5. Experimental results are compared with four theoretical models for tricklebed reactors and criteria are presented for estimating whether mass transfer through the outside film is a significant resistance in an industrial trickle-bed reactor.

Diffusion and solution of gases into thermally softened or molten polymers: Part II. Relation of diffusivities and solubilities with temperature pressure and structural characteristics

Abstract: Solubilities and diffusivities of various gases (helium, nitrogen, carbon dioxide, argon, neon, krypton, and monochlorodifluoromethane) in molten or thermally softened polymers (polyethylene, polypropylene, polyisobutylene, polystyrene, and polymethylmethacrylate) have been correlated with structural characteristics, temperature, and pressure. Temperature dependence of both Henry's Law constants and diffusivities were of the Arrhenius equation form. No appreciable effect of pressure was found for either Henry's Law constants or diffusivities up to 300 atm. Earlier correlations for Henry's Law constants in solid polymer systems were found to be inapplicable for molten and thermally softened polymers. New correlations were developed individually for the latter systems. The correlating factor used was the gas Lennard-Jones force constant. Existing correlations for diffusivities were also found not to apply to molten and thermally softened systems. New correlations were again developed on an individual polymer basis. These related diffusivity to gas Lennard-Jones collision diameter or molecular diameter. Generalized correlations were also developed that held for a number of polymers. These were for both Henry's Law constants and diffusivities.

Internal flow mechanism in filter cakes

Abstract: The definition of filtration resistance is modified by considering relative solid-liquid velocity. The internal flow mechanism in a filter cake is reexamined in view of the movement of solids during compression. Under conditions of short filtrations involving highly concentrated slurries, the velocity of solids is shown to be comparable to the velocity of the liquid. A differential equation is proposed for the flow through compressible cakes in which the pressure gradient is assumed proportional to the difference in average velocities of the liquid and solid rather than to the average velocity of the liquid alone. An improved definition of the average filtration resistance is developed on the basis of the new flow equation.

The laminar boundary layer on a moving continuous flat sheet immersed in a non-Newtonian fluid

Abstract: The laminar boundary layers on a moving continuous flat surface in non-Newtonian fluids characterized by the power law model are investigated using exact and approximate methods. Both pseudoplastic and dilatant fluids are considered. Numerical solutions of the boundary-layer equations are obtained for values of the parameter n in the power law model ranging from 0.1 to 2.0. An integral solution of the momentum equation, which can be used to obtain values of the dimensionless shearing stress that are in good agreement with the exact values, is developed. An integral solution to the energy equation is also presented.

The change in pore size distribution from surface reactions in porous media

Abstract: When a porous solid is penetrated by a reactive fluid which changes the pore geometry, the macroscopic properties of that porous material may be greatly changed. A model is proposed in which the matrix is visualized as being a number of short cylindrical pores dispersed randomly throughout the solid. The change in the distribution of these cylindrical pores is then represented by a integrodifferential equation which is solved for two special cases. The evolution of the pore size distribution is determined by the particular way in which the solid-liquid boundary takes place. The case considered here is that of a surface reaction which dissolves the solid thus continuously enlarging the pores. The rate of reaction is calculated theoretically using a laminar flow diffusion model and this growth rate expression is then taken as the basis for numerical calculations relating to the action of dilute hydrochloric acid on limestone. A comparison is made with experimental results and it is found that the model behaves in much the same way as the real system although the observed rate of pore growth was two to three times that predicted by the diffusion model. Several possible explanations for this discrepancy are being tested. An exact solution of the integrodifferential equation for highly retarded reaction rates has been found with the change in permeability being given in terms of the change in porosity. This result will permit a prediction of the stimulation that can be achieved in acidizing oil wells with retarded acids.

The effect of pressure on the permeation of gases and vapors through polyethylene. Usefulness of the corresponding states principle

Abstract: The permeation of carbon dioxide through polyethylene membranes has been studied at pressures up to 54.4 atm. and at temperatures above and below the critical temperature of the gas (31.0°C.). The permeability coefficient is independent of pressure at the highest experimental temperature (61.0°C.), but becomes increasingly pressure-dependent as the temperature is lowered. The principle of corresponding states can be used to correlate the solubility of both gases and vapors in polyethylene over a wide range of temperatures. This principle can also be invoked to obtain an upper limit for the penetrant pressure above which the permeability coefficient becomes pressure-dependent. The effect of pressure on the permeability, solubility, and diffusivity of gases and vapors in polyethylene is discussed in some detail.

Transport of heat and mass between liquids and spherical particles in an agitated tank

Abstract: Data are reported for heat transfer from water to melting ice spheres and for mass transfer in the case of dissolving spheres of pivalic acid suspended in water agitated in a stirred vessel. The transport coefficients are found to depend on agitator power input but not on agitator design, in agreement with the Kolmogoroff theory. These experimental results are used with others in the literature to develop a correlation involving Nusselt and Prandtl or Schmidt numbers together with a dimensionless group involving agitation power. The correlation is essentially independent of solid-liquid density ratio in the range 0.8 to 1.25, and in this range the gravity group also appears to be unimportant.

Prediction of jet length in immiscible liquid systems

Abstract: The stability theory is used to predict jet length from jet inception to disruption for injection of one Newtonian liquid into a second immiscible Newtonian liquid. Knowledge of the length is essential for predicting the size of drops formed from jets. At low velocities jet length is controlled by the amplification of symmetrical waves which travel at the interfacial velocity of the jet. At higher velocities an abrupt lengthening of the jet may occur as a result of drop merging, and the jet length is then controlled by the growth rate of sinuous waves which are strongly velocity dependent. Jet disruption results from a geometrical limitation on the maximum amplitude of the sinuous waves. Predictions show good quantitative agreement with experimental data for thirteen mutually saturated systems over a wide range of variables and qualitative agreement with limited experimental data on the effects of initial disturbance level and mass transfer.

Effects of ultrasonic vibrations on heat transfer to liquids by natural convection and by boiling

Abstract: The effects of ultrasonic vibrations on heat transfer to water and methanol by natural convection and by boiling were measured at three ultrasonic energy levels with frequency ranging from 20.6 to 306 kcycles/sec., using electrically heated platinum wires of diameters 0.007 and 0.010 in. Up to an eight-fold increase in heat transfer coefficient was obtained in natural convection, but the effects diminished with increased temperature difference and became negligible in the well-developed nucleate boiling region. High-speed photographs showed that the increase was due to the motion of cavitation bubbles on the wire surface. The heat transfer results were correlated by local cavitation activity values measured by a technique developed for this work.

Effects of surfactants on mass transfer inside drops

Abstract: A model is presented to account for reduced mass transfer to drops falling through a continuous phase which contains a surface active agent. The fluid flow patterns are essentially laminar. The reduction in mass transfer is said to be due to a reduction in available interfacial transfer area and to changes in both velocity and pattern of internal circulation. These are shown to be functions of contact time and can be characterized. Experimental values agreed with the theoretically predicted ones with a deviation of less than 10%.

Mathematical analysis of bubble dissolution

Abstract: A perturbation series solution is derived for isothermal bubble dissolution and bubble growth from an initially finite size The new solution is superior in several aspects to the quasi-stationary and quasi-steady state approximations derived previously The accuracy and range of validity of the new results are investigated by comparison with finite-difference solutions of the equations governing bubble growth or dissolution In addition, previous numerical solutions of the problem are compared to the finite-difference results of this study

Thermal and material transport in nonisothermal packed beds

Abstract: The turbulent transport of mass, energy, and momentum was studied in a 4 in. diameter cylindrical column packed with 0.3 in. diameter stoneware spheres. Helium was used as a tracer material, with air as the mainstream fluid. The concentration of tracer present, the temperature, and the velocity of the gas were measured at several axial and radial increments. These data were analyzed numerically to obtain the radial component of the effective thermal conductivity and the radial component of the mass-dispersion coefficient as functions of radial position. Experimental conditions covered isothermal determinations at room temperature and nonisothermal determinations in which a temperature gradient was established in the radial direction. Significant differences were found between the isothermal and strongly nonisothermal results, primarily for the velocity profiles and thermal conductivities. Correlations were developed for the local values of the parameters.

A digital simulation of transient oxygen transport in capillary‐tissue systems (cerebral grey matter). Development of a numerical method for solution of transport equations describing coupled convection‐diffusion systems

Abstract: The mechanism by which oxygen is transported from capillaries into tissue for cell respiration has been a subject of interest since the work of August Krogh in 1919. Mathematical analyses of the diffusion process have been obtained by several investigators (11, 12, 18), but are limited in scope by the complicated nature of the problem. However, the solutions that exist give enough insight to inspire a more rigorous treatment. With the advant of electronic computers more complete analyses are now within reach (16). The human brain accounts for less than 4% of the total body weight, yet it uses 15 to 20% of the total body oxygen consumption. This high metabolic rate makes the brain particularly vulnerable to any change in the normal oxygen supply. The fact that following a cessation of cerebral blood flow, unconsciousness develops within 10 sec., and irreversible damage within 10 min., demonstrates the importance of understanding the dynamics of oxygen supply to the brain.

Flow properties of suspensions with high solids concentration

Abstract: Equations were developed for evaluating the laminar flow behavior of high-solids suspensions from the physical properties of the liquid and solid components. A technique was developed for calculating suspension flow rates as a function of pressure drop. The technique is applicable to the design of pipe lines. Flow measurements were made in pipe-line viscometers of a unique design that minimized entrance and exit effects. Experimental flow data were obtained for suspensions consisting of nickel, alumina, copper, or glass solids in sodium, xylene, or glycerine vehicles with solids concentrations of 28 to 55 vol. %. The basis for the correlation of the data was an analytical investigation of the flow behavior that considered the particle-particle interaction that takes place in a settled suspension. The correlation equations fit all systems investigated. They take into account the effects of liquid viscosity, liquid and solid densities, particle size, size distribution, particle surface area, volume fraction of solids in the suspension, and volume fraction of solids at maximum settled conditions.

Converging flows of viscoelastic materials

Abstract: The kinematics of converging velocity fields such as those found in flows from a large duct or reservoir into a small tube are especially simple in the case of viscoelastic materials as they may be approximated by means of a diagonal deformation rate tensor. This result is shown to be valid in the present study in which aqueous polymeric solutions were utilized and is inferred as having been valid under the experimental conditions employed by Bagley in studies of molten polymers. It is suggested that asymptotic approximations based upon the diagonality of the deformation rate tensor may be of general use in analysis of flows of viscoelastic materials, that is, they could represent, potentially, simplifying approximations comparable in utility to the boundary layer approximations employed commonly in the analysis of flows of Newtonian fluids. An interesting prediction of the present analysis is that for flows from a large duct into a small one a plot of isotropic pressure vs. axial position exhibits a minimum near the inlet to the smaller duct. Experimental results are presented in partial support of this unusual behavior. The analysis also suggests that an orfice jet thrust technique for measurement of normal stresses, closely related to recent independent studies by Middleman and by Fabula, may be an indirect but especially simple and sensitive tool for measurement of material properties.

Thermally induced solid state polycondensation of nylon 66, nylon 6‐10 and polyethylene terephthalate

Abstract: The mechanism of solid state polycondensation has been subjected to a fundamental analysis. Equations were formulated for combined diffusion and chemical reaction for two separate situations. One was for solid state polycondensation in polymer flakes or chips. The other dealt with polymer powders. The resultant solutions related molecular weight changes to rate functions. A technique for deriving the rate functions from experimental data is described. Solid state polycondensations were then studied for nylon 66, nylon 6-10, and polyethylene terephthalate. These data which ranged from 120 to 200°C. were tested with various mechanisms. The most appropriate one was found to be that developed in the present work. Chemical reaction was found to be the rate controlling step in solid state polycondensation in nylon 66, polyethylene terephthalate, powders of nylon 6-10 and larger particles of nylon 6-10 at and above 160°C. Diffusion of byproduct through the solid was the rate controlling step for larger particles of nylon 6-10 at temperatures below 160°C. Thermograms of nylon 6-10 indicated morphological changes which possibly influenced the behavior of the larger nylon 6-10 particles. The Arrhenius relation was fitted to the situations where chemical reaction controlled.

Physical effects in red blood cell trauma

Abstract: Red blood cell damage and destruction are important problems in the use of artificial valves, heart-lung machines, and other devices which pump or process blood. An experimental study has been made on the mechanism of cell damage. Damage was defined by three types of observations on blood which had been subjected to trauma: (a) release of hemoglobin from cells (hemolysis), (b) morphological changes observed microscopically, and (c) red cell life span studies in rabbits using a Cr51 tagging technique. Three types of physical forces which might be injurious to red cells were studied; shearing stress (of known, constant magnitudes from a concentric cylinder viscometer), pressure variations (from studies in a static pressure cell), and direct impact of solid surfaces (from studies in a device which simulates the seating action of artificial heart valves). The study shows that high shearing stress may be primarily responsible for mechanical cell damage under certain important circumstances. There is a critical shearing stress above which cell damage increases markedly. Much of the cell damage does not appear as an immediate release of hemoglobin. Many cells undergo morphological changes and exhibit shortened average life span in vivo. The morphological changes due to shearing stress are very similar to the changes observed in patients who have hemolytic anemia associated with artificial valves.

Adsorption of carbon monoxide-nitrogen, carbon monoxide-oxygen, and oxygen-nitrogen mixtures on synthetic zeolites

Abstract: Experimental results for the adsorption of the binary gas mixtures oxygen-nitrogen, oxygen-carbon monoxide and nitrogen-carbon monoxide on two synthetic zeolites are reported. In all of these experiments the temperature was −200°F. and the total pressure was 1 atm. Also reported are the isotherms for the three pure gases on the two zeolites at −200°F. The results indicate that these zeolites have a surface selectivity which is independent of any sieving effect based on the size of the adsorbed molecules. It does not appear that the strong separations obtained can be explained in terms of the van der Waals forces which are generally believed to be dominant in physical adsorption. The available methods of predicting binary adsorption data from the pure gas isotherms have been examined.

Scale‐up criteria for stirred tank reactors

Abstract: A method is derived for the design of stirred tank reactors for homogeneous reactions. A simple mixing model proposed previously by Curl (4) is used to compute the effects of finite mixing time on complex chemical reactions. It is also shown how the parameters of the model can be obtained by tracer experiments, or estimated theoretically by the assumption of isotropic turbulence. It is shown that in many practical cases the assumption of ideal mixing is a good approximation. However, the effects of imperfect mixing are more likely to be felt in a large reactor than in a pilot plant. Some quantitative examples are discussed. Methods are derived to compute the average outlet concentration for complex systems such as autothermic reactions, polymerization, crystallization, etc.

Viscoelastic jet stability

Abstract: The breakup of a low speed horizontal jet is investigated. Weber's theory for the Newtonian jet is extended to a linear viscoelastic fluid. The theory predicts a dependence of breakup length on the elasticity number. Breakup lengths are measured for low concentration solutions of polyisobutylene in tetralin. Two molecular weights, several concentrations, and five capillary diameters were studied. A single correlation is obtained for all data which gives the breakup length as a function of the elasticity number, and the parameters of Weber's theory. At constant values of the Ohnesorge number and Weber number, the breakup length decreases with increasing elasticity number. The effect of the length of the capillary is studied. At large elasticity numbers short tubes give rise to slightly shorter breakup lengths than long tubes under identical flow conditions.