Similarity solutions thermal boundary-layer

In the case of a solid sphere, considered in the preceding section, we solved the thermal boundary-layer equation analytically by using a similarity transformation. An obvious question is whether we may also solve (9-257) by means of the same approach. To see whether a similarity solution exists, we apply a similarity transformation of the form... 

Problem 11-1. Similarity Solutions. If ue=xm, find the most general functional form for the surface temperature, 9S = 6S (x), that allows a similarity solution of the thermal boundary-layer equation in two dimensions for large Re )S> 1 and Pr = 0(1). 

We have considered the thermal boundary-layer problem in this chapter for an arbitrary 2D body with no-slip boundary conditions for Re 1 and Pr (or Sc) either arbitrarily large or small. If we assume that we have a body of the exact same shape, but the surface of which is a slip surface (e.g., it is an interface, so that the surface tangential velocity is not zero), the form of the correlation between Nusselt number and Pr will change for Pr 1. Solve this problem, i.e., derive the governing boundary-layer equation, and show that it has a similarity solution. What is the resulting form of the heat transfer correlation among Nu, Re, and Pr ... 

Solutions for Pr = 0.5 and 1.0 are shown in Fig. 6.2 as solid curves. The abscissa of this figure is the thermal boundary layer thickness parameter t H, consisting of the Blasius boundary layer similarity parameter multiplied by Pr1/3. The close agreement of the two solid curves suggests for Pr near unity that the thermal boundary layer thickness where Y0 = 0.01 is inversely proportional to approximately Prl/3 or... 

Figure 6.5 shows the velocity distributions in a boundary layer of a liquid with Pr , = 100 (e.g., sulfuric acid at room temperature). For this Prandtl number, the thermal boundary layer penetration into the liquid is much less than the flow boundary layer, and the regions where viscosity variations occur are confined close to the surface. The curve corresponding to p /pe = 1 is the Blasius solution (see Fig. 6.1). The curve labeled p ,/pe = 0.23 corresponds to a heated surface where the low viscosity near the surface requires steeper velocity gradients to maintain a continuity of shear with the outer portion of the boundary layer. The heated free-stream cases reveal the opposite effects. In general, the outer portions of the flow boundary layers act similarly to the velocity distribution of the Blasius case except for their being displaced in or out by the effects that have taken place in the thermal boundary layer.

Similar to the momentum boundary layer entrance length in a closed channel, there is a thermal boundary layer development region. For laminar fully developed flow (Re < 3000), an exact solution can be found for different boundary conditions. For fully developed and laminar heat transfer in a rectangular channel with equal depth and width, the heat transfer coefficient is constant and can be determined as follows... 

Material transport is usually associated with thermal transport except in situations involving homogeneous phases which can be treated as ideal solutions (L4). For this reason it is necessary to consider the behavior of combined thermal and material transport in turbulent flow. The evaporation of liquids under macroscopic adiabatic conditions is a typical example of such a phenomenon. Under such circumstances the behavior in the boundary layer is similar to that found in the field of aerodynamics in a blowing boundary layer (S4). However, it is not... 

Similarity solutions to the boundary layer equations for certain other thermal boundary conditions at the surface of the plate can be obtained, e.g., such a solution can be obtained for a plate with a uniform heat flux at the surface. 

Temperature profiles for flow over an isothermal flat plate are similar, just like the velocity profiles, and thus we expect a similarity solution for temperature to exist. Further, the thickness of the thermal boundar y layer is proportional to /i. T/V,just like the thickness of the velocity boundary layer, and thus the similarity variable is also t), and 0 = 6(ri). Using thechain rule and substituting the It and tt e.xpres ions from Eqs. 6-46 and 6—47 into the energy equation gives... 

Summarizing, we should note that the methods presented in the present section can be applied without any modifications to heat exchange problems, because temperature distribution is described by an equation similar to the diffusion equation. The boundary conditions are also formulated in a similar way. One only has to replace D by the coefficient of thermal diffusivity, and the number Peo - by Pej-. The corresponding boundary layer is known as the thermal layer. Detailed solutions of heat conductivity problems can be found in [6]. 

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