Thermal convection in laminary boundary layers. I

Abstract

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--

Mass transfer from single solid spheres by free convection

“…The local mass transfer rates for laminar free convection are presented in Figure 3. These are somewhat higher than those predicted for heat transfer ( 9 ) , but over most of the surface of the sphere (up to 19 = 140 deg.) the form of the distribution is in good agreement.…”

“…shown theoretically that the product NBh NfEa-'14 is almost independent of W R a for Na, >> 1 ( 9 ) . When one assumes this product to be constant at a given point on the sphere surface, the mean probable errors of the average local transfer rates at successive angles of 30 deg.…”

“…For this method to be applicabIe all viscous and concentration effects must be restricted to a thin layer close to the surface; that is the ratios of transverse co-ordinates and velocities to longitudinal ones are considered to be negligi-d2 u aZ"u" axp au2 ble ( y << x, o << u , -<< -1. Merk and Prins ( 9 ) . for the caze of heat transfer, show that this applies…”

Ice formation in the subcooled brine environment

“…In most cases, N u is a function of the Rayleigh number, Ra, and the Prandtl number, P r, for natural convection only (Gr/Re , N u is a function of Re, Gr and P r; natural and forced convections are in the same order of magnitude (2c) [24][25][26][27].…”

Free convection from a point source of heat, and heat transfer from spheres at small Grashof numbers

“…The heat transfer in this case is enhanced above the pure heat conduction value by a factor of the order of the ratio of the particle size to the thickness of the boundary layer. Merk and Prins [2] used an integral method to obtain the relation between the Nusselt and Grashof numbers in the large Grashof number limit. Acrivos [3] analysed the case of large values of the Prandtl number.…”