A Simplified Theory of Porous Wall Cooling

The rate of heat transfer between a fluid stream in turbulent flow and a smooth, solid wall is largely controlled by the relatively high resistance of the laminar sublayer next to the wall. Although this laminar layer ii extremely thin, heat can be transferred through it only by molecular diffusion. Hence the resistance of this layer is very much greater than for a layer the same thickness farther out in the stream where turbulent exchange is the controlling factor. The thickness of the laminar layer is difficult to define precisely, since there is a gradual transition to the turbulent flow outside, but for the usual scale of many engineering applications almost half the temperature difference between the fluid and the wall occurs in a layer of a few thousands of an inch in thickness. When the wall is made of porous material and a coolant gas is forced through the wall into the stream, it has been found that a very small flow rate of the coolant is remarkably effective in keeping the wall at a low temperature. The coolant flow rate required is such as to give an average velocity normal cooling wall of the order of 1 per cent of the main stream velocity. This flow rate is so low that clearly the injected gas must act as an insulator rather than as a normal coolant. Because of its relatively low velocity, the injected gas can have very little influence on heat convection or momentum transfer in the turbulent stream, and its effect must be confined to the laminar sublayer. The possible influence of the coolant flow on the thickness of the laminar layer will be discussed in Section V.