H Heat Transfer in Unidirectional Flows

With the material properties, the latent heat, and the freezing temperature specified, we could solve this equation for ft. Rather than provide numerical results, we go one step further in simplifying, so that we can highlight certain important qualitative features without the need for a lot of tedious numerical work. To do this, we assume that all of the material properties are equal in the solid and liquid phases. In this case, (3-193) reduces to the form 

One thing that we can see from this expression is that ft is reduced as St (or L) is increased. The more latent heat that is released in the transformation from liquid to solid, the slower the interface moves. This is because the process becomes limited by the rate at which heat can be removed from the interface. As more heat is released, the process is slowed down. In contrast, as the specific heat of the solid increases, the rate of freezing increases because more of the heat can be adsorbed with a decreased need to transport the heat away from the interface. 

We have seen several examples of unidirectional and ID flows for which the Navier-Stokes equations simplify to a linear form so that exact analytical solutions can be obtained. The closest analogy would be problems for which u V0 = 0. Of course, this is just the limit of pure conduction (or pure diffusion) such as the problem considered in the previous 

In this section, we instead consider two well-known examples of heat transfer in the fully developed, laminar, and unidirectional flow of a Newtonian fluid in a straight circular tube. We begin with a problem in which there is a prescribed heat flux into the fluid at the walls of the tube, so that there is a steady-state temperature distribution in the tube. At the end of the section, we consider the transient evolution of the temperature distribution beginning with an initially sharp temperature jump within the fluid at a fixed position (say z = 0), which illustrates an important phenomenon that is known as Taylor dispersion. 

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