Theory for the scaling of metal temperatures in cooled compressible flows

The scaling of metal temperature in conjugate heat transfer problems in compressible environments is argued from first principles in this paper, and a new invariant temperature parameter is proposed. The objective is to provide a framework for comparing metal temperatures between different temperature boundary conditions while appropriately dealing with compressibility effects. The theoretical analysis is based on the principle of superposition, applicable due to the linearity in temperature of the governing differential equation: the time-averaged general viscous energy equation. It is demonstrated that if all non-dimensional parameters that govern the flow field are equal for different temperature boundary conditions, then the overall cooling effectiveness, ϕ, is also equal if defined as ϕ = Tw1 - TR / T02 - TR where Tw1 is the external wall temperature, T02 the coolant inlet total temperature (before interaction with any part of the metal surfaces), and TR a local ‘recovery and redistribution temperature’. As an overall cooling effectiveness thus defined is independent of mainstream and coolant temperatures, it is shown that it is the correct parameter to scale experimental test rig measurements to full engine conditions. A numerical model of a reverse pass cooling system, internally and externally cooled and representative of typical turbine geometries, has been developed to support the theoretical analysis, aid interpretation of the parameters / and TR, and provide a demonstration of the scaling theory for a relatively simple flow field. The arguments are presented for a system in which the physics can be prescribed, and therefore fully controlled, to emphasise the physical reasoning behind the newly proposed parameters.