Engineering Transactions, 14, 3, pp. 329-365, 1966

### Wymiana Ciepła przy Burzliwym Przepływie Cieczy Lepkiej w Okrąglej Rurze

T. Drozd
Instytut Lotnictwa
Poland

The paper presents is an analysis of axially symmetric heat exchange between the wall and the agent in turbulent flow through a circular tube, under steady conditions, after a jump-like variability of the wall temperature. The mathematical description of the phenomenon presented here is based on the analogy between the distribution of the coefficients of eddy diffusivity of momentum and those of eddy diffusivity of heat, with a power profile of velocity selected in a. suitable manner and satisfying, together with its derivatives, the boundary conditions at the wall and at the axis, Solution is obtained by separating variables in a parabolic equation with variable coefficients.
The problem is reduced to the Sturm-Liouville problem. The general solution is obtained by reducing the differential equation to a Fredholm-Volterra integral equation and then to a set of algebraic equations. For large n, an approximate solution is obtained. The variability of the boundary condition at the wall has been taken into consideration by means of the Duhamel integral. The characteristic quantities of heat exchange are determined by solving the energy equation. By way of example, the case of Re = 105 and Pr = 1 is solved numerically by expanding the solution in power series; the computation results are confronted with empirical relations. A method is given enabling the known empirical equations for fixed Nu numbers in the case of constant wall temperature to be applied to problems in which the wall temperature varies in axially symmetric. manner. For the sake of comparison functional relations are given for some hydraulic quantities
concerning turbulence, on the basis of the power profile assumed. The paper contains also an analysis of contemporary views of the phenomenon of heat exchange, in particular the papers of Sellars [18] and Sleicher [25] concerning heat transfer in laminar and turbulent flow through a tube and the equations of thermohydrodynamic analogy of various authors.

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