Problem 5-1 should now be solved with the method outlined in this chapter. The boundary condition for x = 0 : T = T 0 is replaced by   (a) Introduce the dimensionless quantities given in Problem 5.1....

Problem 5-1 should now be solved with the method outlined in this chapter. The boundary condition for x = 0 : T = T₀
is replaced by

(a) Introduce the dimensionless quantities given in Problem 5.1. What is the resulting dimensionless differential equation and what are the boundary conditions?

(b) Split the problem into two first order partial differential equations by introducing a solution vector ~S consisting of the temperature and the axial energy flow. What system of differential equations is obtained? Show the associated eigenvalue problem and obtain its solution.

(c) Determine the complete solution of the problem.

(d) Compare your solution to the one of Problem 5.1 for x/h = 1 for different Peclet numbers. Explain your observations.

Problem 5-1

Consider the heat transfer in a parallel plate channel with distance 2 h between the two plates. The flow is laminar and hydrodynamic ally fully developed. The velocity profile is given by

Use the FORTRAN program reported in Appendix C to calculate the eigenvalues for this problem. Plot the first 100 eigenvalues k²
j as a function of j. Provide an equation for large eigenvalues from the plotted results.