Local Analysis of Radiation Cooling
For the following simple analysis it is assumed that the continuum approach is valid (for a general introduction to energy transport by thermal radiation see, e. g., E. R.G. Eckert et al. , R. B. Bird et al. ). Slip effects as well as high-temperature real-gas effects are not regarded. Non-convex effects will be treated later, Sub-Section 3.2.5.
The basic assumption is that of a locally one-dimensional heat-transfer mechanism (see the situation given in Fig. 3.1). This implies the neglect of
changes of the thermal state of the surface in directions tangential to the vehicle surface at the location under consideration. It implies in particular that heat radiation is directed locally normal to and away from the vehicle surface.
The general balance of the heat fluxes vectors is
q — q — q =0. (3.12)
—w —gw —rad, 4 7
Case 1 of the discussion at the end of the preceding section is now the point of departure for our analysis. With the heat flux radiated away from the surface
qrad = £&T4w , (3.13)
where є is the emissivity coefficient (0 А є A 1), and a the Stefan-Boltzmann constant, Appendix B.1, we find
qw qgw V qrad b q |ш V ь(тТ |ш, (3.14)
with kw being the thermal conductivity at the wall, and y the direction normal to the surface.
A finite difference is introduced for the derivative of T, with A being a characteristic length normal to the surface of the boundary layer, i. e., a characteristic boundary-layer thickness , and Tr the recovery temperature of the problem. After re-arrangement we find
k T T
rpA ^ n, w ± r,, _ 1 ra. s
га~єаА[ Tr >’
Since we wish to identify basic properties only in a qualitative way, we introduce for A simple proportionality relations for two-dimensional flow.
(Pr, xRe, XiX)0-5