The exact value of the heat loss in laminar thermal convection at the surface of horizontal cylinders and spheres is as yet difficult to calculate explicitly with the aid of the so-called exact boundary layer theory. In this paper we shall therefore calculate the local heat transfer by an approximation method first introduced by S qui r e for a flat plate. The calculations are performed for horizontal cylinders and spheres in first approximation for some values of the Prandtl number between 0.7 and ex» and in second approximation for Pr = co. The results look reasonable in themselves.while the total heat transfer is in rather good agreement with experiment. especially for a horizontal cylinder in air. This more or less justifies the approximations introduced. In the last paragraph we consider some details of the second approximation for large values of the Prandtl number and compare the theoretical results for different shapes of the body. § 1. Two-dimensional case. In our first article, to which we refer as part 1 1 ) , we showed that in the neighbourhood of a stagnation point the solution of the boundary layer equations is "similar". If the stagnation point is a regular point of the surface, the local Nusselt number is a constant, and the maximum tangential velocity component increases linearly as a function of the distance from the stagnation point. In general the solution loses its similarity at some distance from the stagnation point, so that it is then no longer possible to describe the heat transfer with two ordinary differential equations. Expanding sin e in terms of~, we may calculate the heat transfer by means of series in the same way as has been done by B I a s ius and Hie men z for isothermal flow in the boundary layer and by F r 6 s s lin g 7) for forced convection. In that case we have to solve many differential equations. To avoid this we shall use -207-
The heat loss of a hot body with constant surface temperature by thermal convection in a laminary boundary layer is described by partial differential equations. These can only be reduced to ordinary differential equations if the temperature and velocity profiles at any two points are similar. This leads to a geometrical condition that is given here for cases of twodimensional and of rotational symmetry. Some examples of the resulting equations are worked out in first approximation. The horizontal cylinder and the sphere do not satisfy the condition, but the flow in the neighbourhood of their lower stagnation points may be found approximately in just the same way. Tile results of other authors are discussed and extended. Other cases will be treated in a second paper. § I. Introduction. In thermal convection the upward volume forces are due to the thermal expansion of the fluid in the neighbourhood of a hot solid wall. Inserting this force into the well known equations of Navier-Stokes (2), (3) for laminary viscous flow and of Fourier-Poisson (4) for heat transfer and making these equations and their boundary conditions (5) dimensionless by expressing the coordinates in a fundamental Unit of length L and the velocities in v/L (v being the kinematica[viscosity#/o), one finds that only two dimensionless parameters are left in the system (2), (3), (4), (5). These parameters, governing the solution, are the numbers of Grashof and Prandtl Gr = g~O,La/v 2, Pr = v/a,g being the gravitational acceleration, ~ tile coefficient of thermal expansion, a = 2/oc the thermal diffusivity of the fluid, and 0, the temperature difference between the hot wall and the fluid at infinity.For extreme cases like glycerol and mercury Pr mav vary from --11--
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