NEWTONIAN FLUID SPHERE WITH RIGID OR MOBILE INTERFACE IN A SHEAR-THINNING LIQUID: DRAG AND MASS TRANSFER

The drag force and the mass transfer rate of a Newtonian fluid sphere, having mobile or rigid interface, moving in a power law fluid, are obtained by an approximate solution of equations of motion in the creeping flow regime. It is shown that both the drag and mass transfer increase as the flow index of the external fluid decreases.

The increase of drag due to the pseudoplastic anomaly is more significant at large viscosity ratio parameter. The results obtained are in good agreement with available experimental data and with those analyses based on variational principle when the non-Newtonian flow behavior is not very pronounced.

Also, the predicted mass transfer rates are in good agreement with the trends presented in the literature. Unlike in the case of drag force, the effect of the pseudoplastic anomaly on mass transfer rate is more pronounced for low values of the viscosity ratio parameter. The analysis was extended to include the case when the surface of the sphere was immobilized by surface-active contaminants.     

DRAG AND MASS TRANSFER IN SLOW NON-NEWTONIAN FLOWS OVER AN ENSEMBLE OF NEWTONIAN SPHERICAL DROPS OR BUBBLES

An approximate solution for the slow motion of an ensemble of spherical drops through a power law fluid is obtained using Happel's free-surface cell model. It is shown that the drag coefficient decreases with decrease of the flow index and that this reduction is more significant at low voidage and large viscosity ratio parameter. The effect of the pseudoplastic anomaly on the mass transfer rate is more pronounced at low voidage for large values of viscosity ratio parameter, unlike the case of a single spherical drop

The present analysis covers the whole range of values of viscosity ratio parameter from infinity (an assemblage of solid spheres) to zero (a swarm of bubbles) and reduces to the solutions for those cases already known

The results for the motion of an ensemble of spherical drops also provide the basis for proposing a tentative expression for the expansion of liquid-liquid fluidized bed at low Reynolds number.     

MASS TRANSFER THROUGH PERMEABLE WALLS: A NEW INTEGRAL EQUATION APPROACH FOR CYLINDRICAL TUBES WITH LAMINAR FLOW

Mass transfer through cylindrical semipermeable walls is analyzed. The solution is obtained in terms of integral equations. Despite the existence of a non-homogeneous boundary condition on the semipermeable wall, the solution thus obtained is particularly advantageous since the associated eigenvalue problem is independent of the Sherwood number. This parameter takes into account the main conductances at the tube wall.

The approach is applied to the case of mass transfer from the interior of a capillary tube with semipermeable walls to an external fluid. The flow in the tube is laminar, and the external flow is assumed turbulent.

The mathematical methodology employed provides a framework to develop numerical schemes of fast and sure convergence.     

FREE CONVECTIVE DRAFT INDUCED BY THERMAL AND CONCENTRATION GRADIENTS INSIDE AN ISOTHERMAL, VERTICAL CYLINDER†

Laminar, free convective flow through a vertical cylinder induced by the thermal and concentration buoyancy forces is investigated. The numerical studies involve development of a steady-state, two-dimensional heat and mass transfer model for the moist air core of the vertical tube. The stream function-vorticity method is employed to simplify the governing, coupled conservation equations which were then numerically solved by the successive over-relaxation (SOR) and alternating direction implicit ((ADI) methods.

A graphical correlation was found between dimensionless flow rate and dimensionless tube length as a function of the buoyancy force ratio N = Gr/Grc. Excellent agreement was obtained for the dimensionless flow rale results with those of Davis and Perona¹⁶ and Kageyama and Izumi¹³ for the case when only the-thermal buoyancy force is considered.

The combined buoyancy force from thermal and species diffusion provides larger local Nusselt Nu and local Sherwood Sh numbers relative to the case when just one buoyancy force is accounted for. Both local Nu and local Sh are seen to asymptotically approach a constant value as flow develops.     

UNSTEADY, TWO-DIMENSIONAL HEAT AND MASS TRANSFER IN A PACKED BED OF FINE PARTICLES WITH AN ENDOTHERMIC PROCESS

The basic differential equations controlling the temperature and concentration field in a single packed bed of fine particles were derived and solved for the general case in which unsteady, two-dimensional heat and mass transfer lakes place with an endothermic process.

The time-change of particle- and fluid-temperature and concentration of water vapor (humidity) were calculated by a numerical method which assumed that the rate of the endothermic process can be expressed by a first-order rate equation and that the fluid flowing through the bed is of the piston flow type.

The experiments were conducted for the drying of silica-gel and the two-stage dehydration reaction of natural gypsum to demonstrate the applicability of the present theoretical analysis.

It has been found that the calculated results show satisfactory agreement with the measured data within the range of the experimental conditions employed.     

The effect of surfactant on terminal velocity of and mass transfer from a fluid sphere in a non-newtonian fluid

The effect of interfacial tension gradients on creeping flow past a fluid sphere (bubble or drop) in a non-Newtonian fluid is investigated. The drag force and the Sherwood number for contaminated fluid spheres moving in a non-Newtonian fluid are obtained by an approximate solution of equations of motion in the creeping flow regime. It has been found that surface-active agents decrease both the terminal velocity and the mass transfer rate. The influence of the flow index of the power-law on both the drag coefficient and the Sherwood number is superimposed over the influence of the interfacial tension gradient. Comparisons between the present analysis and the available experimental data in the literature show reasonable agreement for the terminal velocity and the Sherwood number.     

A GENERALIZED BUBBLE RISE VELOCITY CORRELATION

A generalized bubble rise velocity correlation is developed to cover the range of conditions:

liquid-phase density = 45.1 to 74.7 lb/ft³,

liquid-phase viscosity = 0.233 to 59 cP., and

interfacial tension = 15 to 72 dynes/cm

The gas-phase is air and the bubble size ranged from 1.2 to 15 mm. The developed correlation is based upon new dimensionless groups which contain the parameters affecting bubble rise velocity as well as their interaction, The correlation is independent of flow regimes and applicable for Reynolds numbers from 0.1 to 10⁴. It is in good agreement with work appearing in the literature.     

减阻型纳米流体在圆管内的流动和换热特性

实验测定了在Reynolds数4000～16000范围内,质量分数0～0.5%的石墨、多壁碳纳米管、Al₂O₃、Cu、Al、Fe₂O₃、Zn纳米粒子加入到100～400 mg·kg^-1浓度的十六烷基三甲基氯化铵(CTAC)减阻剂中所制备的减阻型纳米流体的摩擦阻力系数和对流传热系数。结果表明:在CTAC中加入水杨酸钠(NaSal)与去离子水所配制的减阻剂具有一定的稳定性和很强的减阻特性,当减阻剂浓度为200 mg·kg^-1时其减阻特性最优。石墨纳米粒子在增强对流换热和减少流动阻力方面具有较佳的综合性能,当石墨纳米颗粒质量分数为0.4%时,其综合性能因子K是去离子水的5倍。最后给出了减阻型石墨纳米流体在圆管内的流动阻力和换热关联式,其计算值和实验值吻合良好。     

HEAT TRANSFER IN TRICKLE-BED REACTORS

The mechanism of radial heat transfer in two-phase flow through packed beds is examined. A model with 2 parameters: an effective radial thermal conductivity in the bed, k_e, and a heat transfer coefficient, h_w, at the wall, give a satisfactory interpretation of the radial temperature profile.

k_e was expressed in terms of a stagnant contribution, due to the heat conduction through the solid and the fluid in the void space, and a radial mixing contribution of the gas and liquid phases, due to the radial component of the velocity of both fluids. The radial mixing contribution of the liquid ( k_e)_L was compared with radial mass dispersion data, and a satisfactory agreement was obtained.

Moreover, ( k_e)was much higher than the gas mixing and the stagnant contributions.

Correlations for hw and k_e)_L have been proposed in accordance with the hydrodynamic regimes of the two-phase flow.     

GENERALIZED FINITE DIFFERENCE SOLUTION FOR HEAT AND MASS TRANSFER FROM A FINITE CYLINDER DURING QUENCH

A generalized nondimensional solution is presented that describes heat or mass transfer from a finite cylinder during quench. The solution is applicable to three important cases:

Conduction with convection heat transfer at the surface during any single step hot or cold quench.

Conduction with radiation heat transfer at the surface during a single step cold quench with negligible background radiation.

Diffusion with surface desorption of a diatomic gas from a metal specimen during a single step quench in a high vacuum with negligible background pressure.

Application of the generalized solution, which utilizes the numerical method of finite differences with forward stepping, is illustrated by determining a cylinder's transient temperature distribution and surface transfer rate (both instantaneous and cumulative) for an example L/D ratio of 2.0. Selected results are graphed and tabulated for the three cases. The results for the conduction/convection case are verified using the familiar analytical product solution as well as the lumped solution. For the conduction/radiation and diffusion/desorption cases, no analytical solutions are available other than the lumped limit which is in agreement.     

A NUMERICAL STUDY OF FLUID FLOW AND HEAT/MASS TRANSFER OVER A STACK OF PLANKS

Heat/mass transfer by air flow over a sample stack of planks is studied numerically. For the simulations, the low Re k-epsilon turbulence model and bounded QUICK scheme are used. The calculated Nusselt numbers are in good agreement with the experimental data

The results of our study show that the low Re turbulence models have advantages over the conventional high Re models for this type of industrial application. This is mainly due to the small height of separation bubbles resulting from the selected large blockage ratios (more than 50 percent) occurring in such flows

Numerical simulations were carried out to study the effect of the vertical air gap due to shrinkage and non uniform sawing as well as the non uniformity in the height of boards on the flow field and heat/mass transfer characteristics. The results show that the selected gap size significantly affects the local and average Nu numbers across the stack. We have suggested optimum gap sizes for maximum heat/mass for different flow velocities (Re numbers).     

AIR FLOW, TEMPERATURE AND HUMIDITY PATTERNS IN A CO-CURRENT SPRAY DRYER: MODELLING AND MEASUREMENTS

In a co-current pilot plant spray dryer measurements were done of the airflow pattern (no spray) and the temperature and humidity pattern (water spray). These patterns were simulated with a computational fluid dynamics package (FLOW3D)

The measured air velocities showed large fluctuations. The measured and predicted flow pattern showed good agreement qualitatively, but the measured profiles showed less variation than the predicted ones

The measured temperatures and humidities showed good agreement in large areas of the dryer, but the agreement in the zone near the central axis leaves room for improvement.     

MASS TRANSFER WITH A FIRST ORDER CATALYTIC REACTION IN A THREE-PHASE SYSTEM. TRICKLE FLOW ANALYSIS IN A PERIODICALLY CURVED CATALYTIC SOLID

The present work analyzes the process of mass transfer with chemical reaction in a system formed by a periodically curved catalytic wall, which is wetted by a descending film. Through the film a limiting gaseous reactant is transferred from the stagnant gas phase to the catalyst where the chemical reaction takes place.

The film hydrodynamics is first solved with the unknown free surface through a regular perturbation technique, by expanding the resulting equations in terms of a small parameter: the ratio between the film average thickness and the wave length of the curved solid wall. Assuming that the system is isothermal and at steady state, the mass transfer of the gas is afterwards incorporated. A first order kinetics whose limiting reactant is in the gas phase occurs in the solid phase.

Once the model is established and solved, the influence of the dimensionless parameters upon the effectiveness factor and the solid-liquid Biot number is then studied; important effects are found by changing the solid surface curvature at constant flow rate and catalyst volume. Besides, changes in the flow rate, the Peclet number and the ratio between the solid average width and the film average thickness, show significant effects on the net mass transfer process.     

CONTRIBUTION TO THE ANALYSIS OF THE SELECTIVITY OF GAS-LIQUID REACTIONS PART II: COMPARISON OF EXPERIMENTAL RESULTS WITH MODEL PREDICTIONS

The chlorination of paracresol is used in an experimental study of selectivity in gas-liquid contactors.

Experiments in a batch reactor show the influence on selectivity of the dimensionless numbers presented in Part I and involving competition between mass transfer and chemical reaction together with the hydrodynamics.

The extension of open reactor model presented in Part 1 to the batch reactor permits a comparison between theory and experiments and shows a good agreement     

MOTION OF A CIRCULATING POWER-LAW DROP TRANSLATING THROUGH NEWTONIAN FLUIDS AT INTERMEDIATE REYNOLDS NUMBERS

The equations of motion for the flow of a power-law fluid sphere in a Newtonian continuous phase have been approximately satisfied by Galerkin's method: External and internal stream functions have been obtained for the following ranges of variables: 10-50 for Re₀, 0.1-1000 for the viscosity ratio parameters, X, and 0.6-1.0 for the dispersed phase power-law index, n_i,. It is predicted that pseudoplasticity in the dispersed phase causes a reduction in the circulation within the drop and that it has a minor effect on the drag coefficients and the rates of mass transfer in the continuous phase     

EFFECT OF MASS FLUX AND SYSTEM PRESSURE ON TWO-PHASE FRICTION MULTIPLIER

A horizontal boiling water loop was used to obtain pressure drop and heat transfer data for two-phase steam-water flow for pressures of up to 825 kPa. The data were used to examine the predictions of the separated flow model using the Lockhart-Martinelli method of estimating the two-phase friction multiplier.

The influence of mass flux on the two-phase friction multiplier has been reported for high pressure systems by many workers. The present work confirms the existence of this influence at low pressures as well. System pressure is also found to be a parameter. A correlation for the two-phase friction multiplier, incorporating the effects of mass flux and pressure, is presented here. The correlation has been tested against data from two independent sources. The predictions have been found to be in very good agreement with the data.     

GAS-LIQUID DISPERSION WITH PITCHED BLADE TURBINES

The performance of pitched blade turbines in a gas-liquid dispersion has been studied. The two-phase hydrodynamics, gassed power consumption and mass transfer properties have been examined using six blade open turbines with blade angles from 30 to 60 degrees to the horizontal, mounted for down flow.

There are two distinct regimes by which gas leaving the sparger reaches the impeller: at low gas rates this is indirect via the recirculation loops, while at higher gas flow rates the flow is direct.

The transition between these regimes is reflected in power consumption and mass transfer characteristics and is related to the formation of large cavities behind the blades. It was also concluded that, with respect to mass transfer efficiency, a pitched blade turbine is at least as good as a Rushton turbine.     

SOLID-LIQUID MASS TRANSFER AT THE WALLS OF A RECTANGULAR AGITATED VESSEL

Rates of mass transfer between the walls of a rectangular agitated vessel and solution were studied by measuring the limiting current of the cathodic reduction of ferricyanide ion at the vessel wall. Variables studied were rotation speed of 45° pitched blade turbine impeller, physical properties of the solution and the presence of drag reducing polymer in lhe solution (Polyox WSR-301). The data were correlated for polymer free solution by the equation:

Sh= 1.925 Sc^0.33 Re^0.5

Comparison of the present data with previous heat and mass transfer studies conducted in cylindrical vessels has shown that the mass transfer behaviour of agitated rectangular vessels lie between baffled and unbaffled agitated cylindrical vessels. Polyox addition was found to reduce the mass transfer coefficient by an amount ranging from 5.6 to 37% depending on polymer concentration. Practical implications of using drag reducing polymers in agitated vessels were discussed.     

THE LOCAL VOLUME-AVERAGED EQUATIONS OF MOTION FOR A SUSPENSION OF NON-NEUTRALLY BUOYANT SPHERES

The local volume averages of the equations of motion as well as the appropriate boundary conditions are developed for a flowing suspension of non-neutrally buoyant, uniform spheres in an incompressible Newtonian fluid under conditions such that inertial effects can be neglected. These equations do not represent an asymptotic theory with respect to the volume fraction of solids. Higher order terms have been retained everywhere, except where it has been necessary to estimate the velocity distribution within the immediate neighborhood of each sphere by neglecting the effects of the other spheres present. The resulting local volume-averaged equations of motion and boundary conditions involve no free or undetermined parameters.

For the special case of a very dilute suspension of neutrally buoyant spheres, the total local volume average of Cauchy's first law reduces to the form of the Navier-Stokes equation with the effective viscosity computed by Einstein (1906, 1956).

In two succeeding papers, we demonstrate for several flows [in vertical tubes, in a cone-plate viscometer, between rotating concentric cylinders (Couette flow), and between rotating parallel plates] that our general theory describes more concentrated neutrally buoyant suspensions than does its limiting case of very dilute suspensions.     

Effect of the viscosity ratio on mass transfer from a fluid sphere at low to very high Peclet numbers

The effect of the viscosity ratio on mass transfer from a fluid sphere is examined in this paper. Numerical solutions of the Navier-Stokes equations off motion and the equations of mass transfer have been obtained for the unsteady state transfer from a fluid sphere moving in an unbounded fluid medium of different viscosity. The effects of the viscosity ratio and the flow on the concentration profiles were investigated for Reynolds number, viscosity ratio and Péclet number ranges of 0?Re?400, 0?κ?1000 and , respectively. The local and average Sherwood numbers are also presented graphically. The steady state results show that the average Sherwood number is increasing as Peclet number increases for a fixed viscosity ratio. However, for a fixed Peclet number, the average Sherwood number is decreasing as the viscosity ratio increases and reaches a limit value corresponding to the average Sherwood number for a solid spherical particle. From the numerical results, a predictive equation for the Sherwood number in terms of the Peclet number, the Reynolds number and the viscosity ratio is derived.