Engineering

ENGR 9985 Advanced Heat Transfer

Assignment #1 Due in Approximately Two Weeks

1) Given the following partial differential equation which governs the conduction of heat from an isothermal moving square heat source on a semi-infinite domain having a thermal conductivity k:

where z is measured into the semi-infinite region and x and y are in the surface plane. The source moves in the positive x-direction and has a uniform temperature Ts and a side length of a. The semi-infinite region is at a constant temperature of To in the regions far away from the heat source and regions outside of the source area are assued to be adiabatic. Note the heat source is a finite discrete patch of dimensions ‘a by a’ on a semi- infinite surface. Using the method scaling analysis determine the thermal resistance R=(Ts-To)/Q for the limiting cases of very slow (U -> 0) moving heat sources and very fast (U -> oo) moving heat sources. Define a non-dimensional thermal resistance R*= Rka for each limit. What is the new dimensionless group that appears in the analysis? Hint: you must consider what happens to the different terms in the energy equation when heat enters the region for slow and fast source velocity. Make sketches to visualize the process. It may help you to understand that x/U is the resident contact time. 2) Obtain the conduction energy equation in polar co-ordinates and spherical co- ordinates, by means of the scaling factors (metric coefficients, gi) for orthogonal curvilinear co-ordinates using the appropriate expressions for the divergence and Laplacian. Include, conduction, advection, thermal storage, and uniform volumetric heat generation.

∂2T ∂x2

+ ∂2T ∂y2

+ ∂2T ∂z2

= U α ∂T ∂x