Thermal Boundary Layer Solution

Lecture 25, March 8, 2004 • Quiz tomorrow (ch4 – ch7) o Up to Blasius solution o Focus on Understanding Thermal Boundary Layer Solution Begin by defining our non-dimensional temperature,

Starting from the Boundary layer energy equation, and applying the similarity parameter and all the results from the derivation of the Blasius momentum equation, we find a second order ordinary differential equation,

We need to apply two boundary conditions to this second order ODE, and they are that

We can solve this equation using the provided Matlab functions, or any other solver, for Pr = 1, and find

Note that the temperature profile does indeed go from 0 to 1, and that the gradient of temperature starts at a finite value representing the heat transfer from the surface, and decreases to 0 in the free stream, where there is no heat transfer since the temperature is uniform. We can now experiment with different Prandtl numbers, and see what effect this has on the solution.

Note that for Pr =1 the thermal boundary layer is exactly the same size as the momentum boundary layer, as anticipated from the original governing equations (for zero pressure gradient they are identical). As we increase Pr, the thermal boundary layer gets smaller. As we decrease Pr, the thermal boundary layer gets larger. This has an effect on the temperature gradient, and correspondingly, on the heat transfer from the plate. We find by investigating solutions for several Pr, that the size of the thermal boundary layer can be correlated to the size of the momentum boundary layer by,

for Pr > 0.6. We can take this information, as well as the solution for the temperature gradients at the wall to define the Nusselt number for heat transfer from a flat plate subject to a laminar boundary layer,

At some critical value of Re, the flow will transition to turbulent flow, and these equations will no longer be valid. Turbulent flow is far more difficult to solve as it is unsteady and three dimensional. Therefore people have resorted to correlating experimental data, and to fitting curves from experimental data of the form dictated by our dimensional analysis (i.e. Nu = f(Re,Pr)). We can then use these empirical correlations as long as we do not exceed the limits of the data obtained by the experiments. Empirical correlations for Turbulent flow The shear stress for a turbulent flat plate boundary layer flow is found to be

The size of the turbulent boundary layer is found to be

And, the heat transfer is characterized by

For comparison sake, lets plot the boundary layer thickness as a function of distance along the plate as the flow transitions from laminar to turbulent.

Note that the turbulent boundary layer grows at a much faster rate than does a laminar boundary layer. You need to be careful if you have a situation as above, since both the laminar and turbulent portions of the plate will be important. Recall the averaging example we did last week. When using correlations, be very careful of the range of applicability in Re and Pr. Also note carefully the definition of the critical Re for transition to turbulence, Xc. The error in some of these correlations can be as large as 20% Æ using them outside their range of

applicability can result in completely erroneous results as the physics can change completely. The question arises in using the correlations as to what temperature the properties should be evaluated at. In most cases these properties should be evaluated at the film temperature, which is defined as the average temperature in your domain.

It is important to consider this point, because we have assumed that the material properties our constant in order to simplify the governing equations, and to decouple the energy equation from the momentum equation. Keep this in mind as you use these Æ when the temperature differences get large, there will be errors due to this assumption.

See section 7.2.4 for an unheated starting length which will apply to circuit boards etc.