Analogy Solutions three-layer

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

The walls of the 50-mm diameter pipe are kept at a uniform temperature of 70°C. Air flows through the pipe at a mean velocity of 40 m/s. At a certain section of the pipe, the mean air temperature is 30°C. Assuming that the flow in the pipe is fully developed, find the heat transfer coefficient and the rate of heat transfer per m length of pipe from the pipe to the air at this section using both the Reynolds analogy and the three-layer solution. 

A three layer teehnique to combine immiscible material combinations was provided by the Billion Corporation of France. Their solution to polymer incompatibility for sandwich injection moulding used the third intermediate polymer layer as a binder adhesive, this is analogous to methods used in extrusion blow moulding. However, there are obvious machine cost disadvantages here, because the runner system is complex and a third injection unit is required. 

The ihennal resistance concept or the electrical analogy can also be used to solve steady heat transfer problems that involve parallel layers or combined series-parallel arrangements. Although such problems are often two- or even three-dimensional, approximate solutions can be obtained by assuming one-dimensional heat transfer and using the Ihennal resistance network. 

Ideal Gases at High Temperature. Three fundamentally different approaches have been applied to the treatment of the turbulent boundary layer with variable fluid properties all are restricted to air behaving as an ideal, calorically perfect gas. First, the Couette flow solutions have been extended to permit variations in viscosity and density. Second, mathematical transformations, analogous to Eq. 6.36 for a laminar boundary layer, have been used to transform the variable-property turbulent boundary layer differential equations into constant-property equations in order to provide a direct link between the low-speed boundary layer and its high-speed counterpart. Third, empirical correlations have been found that directly relate the variable-property results to incompressible skin friction and Stanton number relationships. Examples of the latter are reference temperature or enthalpy methods analogous to those used for the laminar boundary layer, and the method of Spalding and Chi [104]. 

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