Peclet numbers small, nonzero

C. HEAT TRANSFER FROM A SOLID SPHERE IN A UNIFORM STREAMING FLOW AT SMALL, BUT NONZERO, PECLET NUMBERS... 

A natural question is to what extent the overall rate of heat transfer is modified by convection for small, but nonzero, values of the Peclet number. To obtain a more accurate estimate of Nu for this case, it would appear, from what has been stated thus far, that we must calculate added terms in the regular asymptotic expansion (9-15) for 0. To attempt this, we must substitute (9-15) into (9-7) and (9-8) to obtain governing equations and boundary conditions for the subsequent terms 6 . In this section, we consider only the second approximation, 9. The governing equation and boundary conditions derived from (9-7), (9-8), and (9-15) are... 

To obtain a valid approximate solution for heat transfer from a sphere in a uniform streaming flow at small, but nonzero, Peclet numbers, we must resort to the method of matched (or singular) asymptotic expansions.4 In this method, as we have already seen in Chap. 4, two (or more) asymptotic approximations are proposed for the temperature field at Pe 1, each valid in different portions of the domain but linked in a so-called overlap or matching region where it is required that the two approximations reduce to the same functional form. The approximate forms of (9-1), from which these matched expansions are derived, can be obtained by nondimensionalization by use of characteristic length scales that are appropriate to each subdomain. 

Numerical solution of the mass transfer equation begins at a small nonzero value of z = Zstart, uot at the inlet where Cp, x, y,z = 0) = Ca, miet for all values of x and y. This is achieved by invoking an asymptotically exact analytical solution for the molar density of reactant A from laminar mass transfer boundary layer theory in the limit of very large Schmidt and Peclet numbers. The boundary layer starting profile is valid under the following condition ... 

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