A new theoretically‐based formula is presented for fully developed steady‐state forced convection heat or mass transfer in turbulent tube flow. This relationship, which is proposed for the frill range range of Prandtl or Schmidt numbers between 0.7 (gases) and 100 000 (liquids), is as follows:
Coniparisons a with widely quotcd sets of data show very good agreement IFriend and metzner (heat transfer in liquids), Dcisser and Eian (heat transfer in air), Allen aid Eckert (heat transfer in water), Malina and Sparrow (heat trniisfcr in liquids), Harrott and Hamilton (mass transfcr in liquids)]. Absolute mean deviations vary from 3.6 to 8.1% over the range 0.73 + Pr or Sr ≤ 97 600.
coalderived liquids involves the intraparticle mass transport of large molecules inside a catalyst of cobalt-molybdate supported on alumina. The sizes of molecules in these heavy feedstocks (with molecular weight between 10' and lo5) range between 25 and 150,i, the largest fraction being around 50A (Tschamler and
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...
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