Small Peclet Numbers Research Articles

Diffusion to a particle in a shear gas flow in the case of arbitrary kinetics of the surface reaction

Diffusion to a particle in a shear flow at small Peclet and Reynolds numbers is considered in the case when a chemical reaction, the rate of which depends on the concentration in an arbitrary manner, takes place at its surface. The heat andmass transfer in a particle in a translational flow of viscous incompressible fluid at small Peclet and Reynolds numbers were studied in /1–7/. The diffusion mode of the reaction at the particle surface was studied in /1–3/, and a heterogeneous reaction of the first, second and arbitrary order were considered in /4–6/, /4/ and /7/ respectively. The papers /8–10/ deal with the case of diffusion mode of reaction at the particle surface freely suspended in a shear flow.

The uniqueness of steady state flows of glaciers and ice sheets

Summary We consider the uniqueness of solutions to a model set of equations describing the temperature and flow of cold (or sub-temperate) glaciers. These equations are based on the model developed by Fowler & Larson (1978), and incorporate the essential features of a free surface boundary, free melting boundary, and a fully non-Newtonian flow law. The only real simplification which is made is that of taking small Peclet number (or Graetz number), which implies that advective heat transport is negligible. Parametric estimates indicate that this is unrealistic, but nevertheless it represents a formally self-consistent way of effectively adopting a ‘slab’ model, while avoiding the inherent disadvantages of such models. Since the possibility of non-uniqueness is associated with the non-linear viscous heating term, we expect that non-zero advection of heat will only affect the results quantitatively, but not qualitatively. We find that for the various kinds of basal boundary condition which can occur, the solutions for the temperature (and hence the flow field) are unique, with the possible exception of regions where the basal ice is temperate and sliding, but the rest of the glacier is cold. In such regions one can have up to three solutions (even with the exponential approximation to the Arrhenius term), and such solutions can exhibit hysteretic instability: however, we then show that such multiple solutions must transgress the condition that the ice be below its freezing point, and hence they are not relevant in the present study. We therefore conclude that the solutions are unique, and so the non-linear heating term is unlikely to cause surge-like instabilities in real glaciers (or ice sheets). This does not preclude the possibility that the unique solution is linearly unstable to infinitesimal perturbations: such an instability might also lead to surging states, but not in the explosive manner that non-uniqueness would suggest.

Unsteady heat transfer from a circular cylinder immersed in a Darcy flow

Analyses are made for unsteady heat transfer from a circular cylinder immersed in a porous medium through which a liquid is flowing according to Darcy's law. Asymptotic solutions for large and small Peclet numbers (Pe) are obtained for the case where the unsteady temperature field is produced by a step change in wall temperature. The former is valid for Pe≳200 and the latter for Pe≲0.1. The series solution for small time, which is valid for all values of Pe, is also obtained. By applying the Euler transformation to the series, its convergence is greatly improved, and it appears that the Eulerized series determines the mean Nusselt number adequately for most values of time.

Heat dispersion effect on thermal convection in anisotropic porous media

The effect of hydrodynamic dispersion on the onset of thermal convection in flows through anisotropic porous media is studied theoretically. The porous layer is homogeneous and bounded by two infinite perfectly-heat-conducting impermeable horizontal planes kept at constant temperatures. Horizontal isotropy with respect to permeability and thermal diffusivity is assumed. A pressure-driven basic flow is considered in the limits of small and large Peclet numbers. The analysis shows that the onset of convection in both cases is independent of longitudinal dispersion, while dispersion in lateral directions has stabilizing effects. The preferred mode of disturbance consists of stationary rolls with axes aligned in the direction of the basic flow.

Macroscopic transport properties of a sheared suspension

The present paper is concerned with recent theoretical developments and a state-of-the-art summary of the framework for prediction of macroscopic transport properties of a sheared suspension. Included is (a) a derivation of the fundamental equation relating bulk conductive heat flux to microscale thermal properties of the suspension; (b) a discussion of the method of evaluation of this expression; (c) a theoretical framework based on the concept of hydrodynamic fluctuations for the inclusion of Brownian motion effects; and finally (d) specific evaluation of the effective thermal conductivity for a dilute suspension of drops in the limit of small local Peclet number, for a dilute suspension of rigid spheres in the large Peclet number limit, and for a dilute suspension of rigid spheroids in the limit of small Peclet number but including Brownian motion.

Continuous Casting of Cylindrical Ingots

The Heat Balance Integral Method is applied to solve for the continuous casting of cylindrical ingots. Unlike previously developed integral solutions the present analysis includes the effect of axial conduction. Comparison with the solution derived by Veynick (which neglects axial conduction) shows that, for small Peclet numbers, the discrepancy between the results can be quite appreciable. In particular the pool depth prediction by the present analysis can be as much as 40 percent shorter. In its present form the method assumes constant temperature in the liquid metal and constant properties in the solidified crust. It is applicable to a variety of surface cooling rates including constant film coefficient, constant heat flux and other more general conditions. It can also take into account superheat of the molten metal.

Effective thermal conductivity of disperse media at small peclet number

The steady heat conduction of disperse media consisting of identical spherical particles is calculated for small Peclet numbers characterizing the heat transfer at the level of the individual particles; heat transfer by contact conduction over the disperse phase is neglected.

Two-Dimensional Analysis of Conduction-Controlled Rewetting With Precursory Cooling

This paper presents a two-dimensional analysis of the effect of precursory cooling on conduction-controlled rewetting of a vertical surface, whose initial temperature is higher than the sputtering temperature. Precursory cooling refers to the cooling caused by the droplet-vapor mixture in the region immediately ahead of the wet front, and is described mathematically by two dimensionless constants which characterize its magnitude and the region of influence. The physical model developed to account for precursory cooling consists of an infinitely extended vertical surface with the dry region ahead of the wet front characterized by an exponentially decaying heat flux and the wet region behind the moving film-front associated with a constant heat transfer coefficient. Apart from the two dimensionless constants describing the extent of precursory cooling, the physical problem is characterized by three dimensionless groups: the Peclet number or the dimensionless wetting velocity, the Biot number and a dimensionless temperature. Limiting solutions for large and small Peclet numbers have been obtained utilizing the Wiener-Hopf technique coupled with appropriate kernel substitutions. A semiempirical matching relation is then devised for the entire range of Peclet numbers. Existing experimental data with variable flow rates at atmospheric pressure are very closely correlated by the present model. Finally a comparison is drawn between the one-dimensional limit of the present analysis and the corresponding one-dimensional solution obtained by treating the dry region ahead of the wet front characterized by an exponentially decaying heat transfer coefficient.

Regular expansion solutions for heat or mass transfer in concentrated two-phase particulate systems at small peclet and reynolds numbers

General solutions for small Peclet number heat or mass transfer in concentrated two-phase particulate systems submerged in arbitrary Stokesian flows with arbitrary temperature or concentration boundary conditions were obtained using regular perturbation expansions of the dependent variable in powers of the Peclet number. Uniformly valid solutions for all orders of approximation were obtained within the domain of interest, subject to a restriction imposed on the value of the Peclet number by the dimensions of that domain. The solutions were specialized to cases of practical interest. For a particulate system submerged in a uniform flow with a linear temperature gradient in the direction of flow, the results reduced to those of Maxwell for effective electrical conductivity of a composite medium. Specific relations were also obtained for local and overall interfacial fluxes and Nusselt numbers. The results reflect the effects of particle volume fraction and retardation of internal circulation by surfactant impurities.

Heat and mass transfer in a dispersive medium

Macroscopic equations for the conservation of heat (or the mass of a diffusing impurity) in a continuous medium containing distributed particles of a dispersed phase are formulated neglecting the effect of random fluctuations of the medium and particles by the transfer process. The problem of convective heat conduction or diffusion near an isolated particle is also formulated, the solution of which permits calculation of all the parameters entering into the indicated equations. This problem has been solved in the particular case of small Peclet numbers, which characterize heat and mass exchange in the vicinity of a single particle.

The effect of deformation on the effective conductivity of a dilute suspension of drops in the limit of low particle peclet number

The effective thermal conductivity of a dilute suspension of slightly deformed droplets is calculated in the limit of small particle Peclet number for the undisturbed bulk shear, u = γy, and the linear bulk temperature distribution, T = αy. Two distinct cases of small deformation are considered; deformation dominated by interfacial tension forces, and deformation dominated by viscous forces in the drop. The results show that the presence of deformation can cause a fundamental change in form of the dominant, flow-induced contribution to the effective conductivity.

Effects of Variable Thermal Properties on Moving-Band-Source Temperatures

Temperatures calculated by moving-heat-source theory for machining and sliding processes are often sufficiently large that the assumption of temperature-independent thermal properties is invalid. In the present paper results of a numerical analysis are presented that consider the effects of variable thermal properties on the temperatures due to a moving-band source. Compared with the constant-property model, the maximum surface temperatures are found to be significantly higher with small Peclet numbers and strong heat sources, but the average surface temperatures within the band are much less affected by the variations of thermal properties with temperature. The variable-property model also indicates significantly larger transverse temperature gradients, a phenomenon that should cause greater thermal stresses.

The effects of crystallographic anisotropy on the growth kinetics of Widmanstätten precipitates

A theoretical model which emphasizes crystallographic considerations during the growth of Widmanstätten precipitates is developed. The theory predicts that the crystallographic anisotropy of interface kinetics process becomes significant under the growth conditions of small peclet numbers, viz. p < a 1 5.5 , where a 1 is the anisotropy parameter. The results of this model are applied to the experimental data on the growth of Widmanstätten cementite plates in hypereutectoid steels.

Analysis of Conduction-Controlled Rewetting of a Vertical Surface

The present paper presents a two-dimensional analysis of conduction-controlled rewetting of a vertical surface, whose initial temperature is greater than the rewetting temperature. The physical model consists of an infinitely extended vertical slab with the surface of the dry region adiabatic and the surface of the wet region associated with a constant heat transfer coefficient. The physical problem is characterized by three parameters: the Peclet number or the dimensionless wetting velocity, the Biot number, and a dimensionless temperature. Limiting solutions for large and small Peclet numbers obtained by utilizing the Wiener-Hopf technique and the kernel-substitution method exhibit simple functional relationships among the three dimensionless parameters. A semi- empirical relation has been established for the whole range of Peclet numbers. The solution for large Peclet numbers possesses a functional form different from existing approximate two-dimensional solutions, while the solution for small Peclet numbers reduces to existing one-dimensional solution for small Biot numbers. Discussion of the present findings has been made with respect to previous analyses and experimental observations.

Approximate solutions to single and two phase axial dispersion problems in isothermal tubular flow reactors

Approximate collocation solutions to the governing equations for single and two phase isothermal reactions in axial dispersion model reactors involving power-law type kinetics are presented. The solutions assume that the approximate solutions for the power-law type of kinetics are of the same forms as those for pseudo first order reaction under otherwise similar reaction conditions. When the reaction involves two reactants and the rate of the reaction is first order with respect to each reactant, the approximate analytical solution agrees very well (within 10 per cent) with the exact solution. When the reaction is n th order with respect to a single reactant; the approximate solution agrees well with the exact solution at small and intermediate Peclet numbers. At large Peclet numbers, the analytical and approximate solution deviate considerably and the approximate solution of Burghardt and Zaleski[3] should be preferred over the present one.

Dispersion effect on buoyancy-driven convection in stratified flows through porous media

The effect of hydrodynamic dispersion on the onset of convection in flows through porous media is studied theoretically. The medium is isotropic, and bounded by two horizontal impermeable planes having a constant concentration difference. Pressure-driven as well as thermally-driven basic flows are considered. The investigations are valid in the limit of small and large Peclet numbers. The analysis shows that the onset of convection is independent of the longitudinal dispersion coefficient, while lateral dispersion always has a stabilizing effect. The preferred mode of disturbance is stationary, being rolls with axes aligned in the direction of the basic flow (longitudinal rolls).

Mass transfer between bubbles and the continuous phase in a pseudofluidized layer

The problem of mass transfer between an isolated bubble and the continuous phase in a pseudofluidized layer is considered, when the rising velocity of the bubble exceeds the pseudofluidization rate. In this case the bubble with the surrounding region, a so-called two-phase system, is surrounded by a surface current impermeable to the liquid [1–3], and the problem reduces to determining the concentration field and the total flow on the material surface. The problem is solved for large and small Peclet numbers by a boundary layer diffusion method and by asymptotic expansion matching.

Computation of forced convection in slow flow through ducts and packed beds—I extensions of the graetz problem

A computational study of forced convection processes in ducts and packed beds at low Reynolds numbers has been made. The results give a better understanding of these processes, especially for small Peclet numbers. It is demonstrated that two distinct forms of Nusselt numbers are relevant for low Peclet numbeRs. One, used mostly by theorists, is related to the local driving force; the other, used mostly by experimentalists, is related to easily measurable temperatures and concentrations. As an example the Graetz problem has been solved numerically over a wide range of Peclet numbers, and new asymptotes have been obtained for the region of small Peclet numbers.

A SUFFICIENT CONDITION FOR THE EXISTENCE OF A UNIQUE EQUILIBRIUM STATE IN AN ADIABATIC TUBULAR REACTOR

This paper presents a sufficient condition for the existence of a unique equilibrium state in an adiabatic tubular reactor in which a single exothermic chemical reaction takes place and Peclet number for mass transfer is equal to that for heat transfer. The condition is obtained by analyzing the relations between properties of the heat generation curve for an adiabatic tubular reactor and those of the curve for an adiabatic complete mixing reactor with the same residence time, inlet temperature and inlet concentration as the tubular reactor, and is given in terms of the character of the heat generation curve for the complete mixing reactor, which can be analyzed much more easily than the tubular reactor. Compared with other conditions for a first-order reaction, the condition is shown to be stronger than the condition of Luss and Amundson for equilibrium states with small Peclet numbers and low temperature, but more conservative for equilibrium states with large Peclet numbers.

The drag of a body moving transversely in a confined stratified fluid

The slow motion of a body through a stratified fluid bounded laterally by insulating walls is studied for both large and small Peclet number. The Taylor column and its associated boundary and shear layers are very different from the analogous problem in a rotating fluid. In particular, the large Peclet number problem is non-linear and exhibits mixing of statically unstable fluid layers, and hence the drag is order one; whereas the small Peclet number flow is everywhere stable, and the drag is of the order of the Peclet number.