Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures
The two-dimensional convective motion generated by buoyancy forces on the fluid in a long rectangle, of which the two long sides are vertical boundaries held at different temperatures, is considered with a view to the determination of the rate of transfer of heat between the two vertical boundaries. The governing equations are set up; they reveal that the flow is determined uniquely by the Prandtl number <r, the Rayleigh number A = g{Tx — T0)(f /(T0kv), and the ratio of the sides of the rectangle l/d. In the case of cavities used for thermal insulation of buildings, which is kept specially in mind throughout the paper, A is usually about 1000 d3 (where d is in centimeters), and l/d takes values between about 5 and 200. The essence of the problem is to determine which of several different flow regimes occurs at any given values of A and l/d. It appears that with the above practical values the flow is not decisively of one single kind, and the discussion of the heat transfer for each of several ranges of values of A and l/d is necessary. Sections 4, 5 and 6 are concerned with laminar flow regimes characterized by very small values of A, large values of l/d, and large values of A, respectively. In Section 7 approximate criteria for these flows to be stable are established, and in Section 8 the expressions for the heat transfer when the flow is turbulent are considered briefly. The unified picture provided by all these different results is considered in Section 9. A comparison of the theoretical predictions about the heat transfer with the limited experimental data, mostly obtained by Mull and Reiher (1930), is made. Theory and experiment agree in suggesting that, under practical conditions, the effect of convection is negligible for d < 1 cm, and that the heat transfer per unit area of vertical boundary decreases as d increases, provided d < 2.5 cm, and remains approximately constant (at a value proportional to Z-1/4) for further increase of d. 1. The background to the problem. The purpose of this paper is to determine the rate at which heat is transferred across the air space between two plane parallel vertical boundaries which are held at different temperatures. The air space is closed by horizontal boundaries distance I apart (I being large compared with the distance d between the vertical walls, in general), as sketched in Fig. 1. In the remaining direction, at right angles to the plane of the sketch, the air space is regarded as extending to infinity. All boundary conditions will be assumed uniform in this latter direction, and the convective