Small Peclet Numbers Research Articles

Decompositional burning of a droplet in a small Peclet number flow.

Asymptotic results for chemical-reaction and forced-convection effects on the quasi-steady adiabatic vaporization of a rigid uniform spherical droplet immersed in an unbounded expanse of gas are obtained by inner-and-outer expansions. The zeroth-order approximation is the radially symmetric solution yielding the classical logarithmic mass transfer rate. The perturbation introduces the corrections arising from 1) first-order decompositional burning for reaction rates small relative to flow rates (small first Damkohler similarity parameter); and 2) the possible wake-generating role of a slight relative flow past the droplet (small Peclet number). For tractable closed-form solution the flow is taken as incompressible with constant properties; the Lewis number is restricted to unity in the perturbational analysis. The first perturbation to the Sherwood number (normalized mass transfer rate) is found to be independent of Schmidt number, but dependent on the reaction rate. The first modification to the Stokes drag (owing to mass transfer) is found to be independent of the reaction rate, but dependent on the Schmidt number. For indefinitely large Schmidt number the Stokes drag is always increased because of mass transfer, and streamlines display no wake; for orderunity Schmidt number entirely different results are anticipated.

Longitudinal dispersion at small and large Peclet numbers in chemical flow reactors

In applying the longitudinal dispersion model to describe the behaviour of a continuous flow reactor, in which a single chemical reaction is carried out, one obtains in general an ordinary non-linear differential equation of the second order. In this paper a perturbation method was used to get an approximate solution of this differential equation. The analysis has been carried out for both large and small Peclet numbers assuming a general kinetics of the chemical reaction. In order to extend the range of acceptable accuracy of the derived formulae, two perturbation functions were introduced. The results computed on the basis of the derived equations were compared with the analytical and numerical solutions obtained by other authors indicating satisfactory agreement over a broad range of Peclet numbers.

The dispersivity tensor in isotropic and axisymmetric mediums

The tensor form of the dispersivity coefficient Dij, appearing in the equation of hydrodynamic dispersion within isotropic and axisymmetric porous mediums, is derived. It is shown that in isotropic mediums Dij/k = F1 δij + F2 υf υj l2/k2, where F1 and F2 are even functions of υl/ν and υl/k, the Reynolds and Peclet numbers. The form of Dij in the particular cases of very small and very large velocities is also discussed. Physical and dimensional considerations suggest a quadratic dependence on the velocity components for very small Reynolds and Peclet numbers and a linear dependence on the velocity components for large Reynolds numbers. Similar results are obtained for axisymmetric mediums. The relations between the dispersivity and velocity components obtained are more general than those suggested by earlier investigators.

Hydromagnetic heat transfer in the thermal entrance region of a channel with electrically conducting walls

The temperature distributions and heat transfer in the thermal entrance region, which results when a viscous, incompressible electrically conducting fluid flowing steadily in the presence of a transverse magnetic field between parallel electrically conducting walls, which are kept at a constant temperature up to a certain cross section, enters a region with a different wall temperature, are analyzed. Solutions have been obtained in terms of eigenfunctions and eigenvalues for two cases corresponding to a small value of the Peclet number and an infinite value. It is found that downstream temperature conditions do not have any effect upon the upstream temperatures to the left of the transition cross section when the Peclet number is large, whereas for a small Peclet number, this ceases to be so. Thermal entry lengths are seen to be diminished by the increase in conductivity conditions of the walls.

Heat transfer in laminar flow between parallel plates at small Péclét numbers

The problem of heat transfer for laminar flow between two infinite parallel plates, y=±l, x≤0, kept at a constant temperature T 0, and y=±l, x≥0, kept at a different constant temperature T s is formulated to take into account the effect of heat diffusion on the incident fluid. This has been achieved by obtaining solutions of the energy equation for the regions x≤0 and x≥0 and by imposing continuity conditions on the temperature and its derivative at the junction x=0. It is found that at small Peclet numbers the incident temperature is affected by the diffusion of heat from the right (x>0) to the left (x<0). This effect is negligible for large Peclet numbers (Pe ∼ O(1000)). Further the temperature of the incident fluid at x=0 cannot be taken as constant (=T 0) if the heat generated by viscous dissipation is taken into consideration. Detailed solutions are given for Pe=1. Mean-mixed temperatures and local Nusselt numbers for x>0 and x<0 are tabulated and shown graphically.

Heat Transfer to Fluids With Low Prandtl Numbers for Flow Across Plates and Cylinders of Various Cross Section

Abstract Local and over-all heat-transfer coefficients were determined analytically for one and two-dimensional flow of a fluid with small Prandtl number, such as a liquid metal, across a flat plate and cylinders of various cross section. The possibility of heat generation within the fluid was considered. Certain of the results were simplified for convenient calculation, and the approximations investigated. A comparison of the results with those of Pohlhausen and Squire showed good agreement. The procedure for obtaining these coefficients rested upon a consideration of forced convection in a fluid with small Prandtl number which implied that inviscid flow can be assumed, and eddy transport of heat is negligible in comparison to molecular conduction for flows with small or moderate Peclet numbers. The analysis was simplified greatly by means of a transformation and certain results due to Boussinesq (1). The agreement of the analytical results with those obtained experimentally by Hoe (2) for mercury flowing through tube banks and the method for calculating heat-transfer rates in tube banks will be described in a later paper.

The Decay of Isotropic Temperature Fluctuations in an Isotropic Turbulence

The approach of von Karman and Howarth (to isotropic turbulence) has been applied to the decay of isotropic temperature fluctuations in a decaying isotropic turbulence, under the restriction that the temperature differences are too small to have any effect on the velocity field. From the correlation equation it is found that, if the scalar functions characteristic of the pertinent double and triple correlations are assumed to descend to zero faster than r~, the product of the second moment of the double temperature correlation coefficient and the mean square temperature fluctuation is a constant during decay. This is analogous to Loitsiansky's theorem for the constancy of the product of turbulent energy and fourth moment of the velocity correlation coefficient in the turbulence decay. For extremely small Peclet Number the correlation equation reduces to the three-dimensional heat conduction equation with spherical symmetry. The corresponding equation for the linear viscous decrement of the velocity field is a quasi-five-dimensional equation. Estimates of decay and scale growth of the temperature fluctuations a t both extremely small and extremely large Peclet Numbers are compared with the corresponding results for the turbulence. Perhaps the most practical result is that the temperature spottiness always seems to die off at a slower rate than the velocity spottiness.