Developed turbulent transport in ducts for large prandtl or schmidt numbers

Abstract

The problem of developed turbulent heat or mass transfer in a duct is considered for the limit of large u (Prandtl or Schmidt number). The limiting results depend on the behavior of the eddy diffusivity near the solid surface. Since there is a question about whether this variation begins with e a y + 3 + . . . or e cc y+4 + . . . for y * near zero, both possibilities are considered. In each case the first three terms of the asymptotic expansion for u + co Heat or mass transfer under developed conditions for turbulent flow in ducts has long been of practical interest. This is true since for many transport situations in turbulent flow the temperature or concentration achieves its developed profile over most of the transfer region. The particular case where u -+ m has been of special interest both theoretically and in practice, since for heat or mass transfer in many liquids the u values are large.Early work on this problem was based on the analogies of Prandtl-Taylor, von Khrman, and Martinelli (8). The differences between these analogies are attributable to various simplifying assumptions made by the authors. Subsequently, Lyon (10) and Seban and Shimazaki ( 1 1 ) presented analyses which were devoid of simplifying assumptions, but which could only be implemented numerically. The main purpose of this paper is the development of an asymptotic expansion for the rigorous formulation in the limit of u + m.Deissler (2) used the idea of a continuously varying eddy diffusivity to determine an analytical approximation for the Stanton number in a circular tube for the limit of u -+ 5 c . Later Elrod ( 3 ) showed that the Taylor series expansion for the eddy diffusivity in the neighborhood of the wall must begin with a term proportional to y" where n must be greater than or equal to three. The asymptotic dependence of the Stanton number on u for u + cc depends on this value of n. Empirical correlations based on values of n of three ( 4 , 9) and four (2, 1 2 ) have been proposed, mainly of the form S t = A Rea ub. In addition, a number of large Schmidt number mass transfer experiments have been conducted with the goal of determining n. These investigations are not conclusive, since evidence is found for both la = 3 (6, 7) and n = 4 ( 1 2 ) . Part of the reason for this discrepancy may be due to an inadequate formula with which to compare the experimental results.In this paper we derive an asymptotic expansion of the Lyon equation for the limit of u -+ m. The expansion depends in a fundamental way on the eddy diffusivity variation near the wall. Since this variation is currently in quesAlChE Journal (Vol. 18, No. 3) large Schmidt number tion, both of the likely possibilities are considered. Three terms of the asymptotic expansion are generated for both c(!/+) = K3y+3 + K4y+4 + K5y+5 and ~( y ' ) = M4y+' + kfj!y+5 + A!fG!/+'. The errors due to taking only three terms of € ( ! I + ) are also estimated. The results show the corrections to earlier one-term approximations, and thereby indicate the validity of the corresponding sim...

Heat and mass transfer in turbulent flow under conditions of drag reduction

“…For the case of a uniform wall flux boundary condition, the fully-developed Stanton number for heat or mass transfer is given by the Lyon Equation (Hanna and Sandall, 1972).…”

“…The first integral in Eq. 3, representing the contribution of the wall region, has been evaluated asymptotically for large (T by Hanna and Sandall (1972) for the cases where E+ begins with either y+3 or y+,, as follows.…”

“…3 for the case where E + beings as y+, was found to be: For cr > 50 the Stanton number calculated from Eq. 3 is given to a good approximation [within 8% (Hanna and Sandall, 1972)] by these three term expansions for the first integral, representing the wall region. Since the Prandtl number can be as low as 3 under some conditions of interest for heat transfer with drag reduction, we wish to evaluate the second term in Eq.…”

“…where M , the split point for the integrals was taken to be 45 (Notter and Sleicher, 1971). Previous analyses and comparisons with experimental data for large Schmidt number mass transfer and large Prandtl number heat transfer under conditions of no drag reduction have clearly shown that E+ Q y+3 near the wall best describes the data (Notter and Sleicher, 1971;Hanna and Sandall, 1972;Sandall and Hanna, 1979). Thus for drag reduction we would also expect that E+ Q as y+ -+ 0.…”

“…In the present work; there are two primary objectives. The first is to develop fundamental asymptotic analytical relationships for heat or mass transfer with drag reduction which are natural extensions of those previously developed for zero drag reduction (Hanna and Sandall, 1972). The second objective is to formulate a new eddy difhsivity model leading to these analytical relationships in such a way that the results can be viewed as appropriate asymptotic solutions to the problem for the case of large Prandtl or Schmidt number.…”