(Solution) - An insulated composite rod is formed of two parts arranged -(2025 Original AI-Free Solution)

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An insulated composite rod is formed of two parts arranged end to end, and both halves are of equal length. Part ? has thermal conductivity k?, for 0 ? x ? 1/2, and part b has thermal conductivity kb, for 1/2 ? x ? 1. The nondimensional transient heat conduction equations that describe the temperature u over the length x of the composite rod are
?2u/?x2 = ?u/?t 0 ? x ? 1/2
r ?2u/?x2 = ?u/?t 1/2 ? x ? 1
Where u = temperature, x = axial coordinate, t = time, and r = k?/kb. The boundary and initial conditions are
Boundary conditions u(0, l) = 1 u(1, l) = 1
(?u/?x)? = (?u/?x)b x = 1/2
Initial conditions u(x, 0) = 0 0 < x < 1
Solve this set of equations for the temperature distribution as a function of time. Use second-order accurate finite-difference analogues for the derivatives with a Crank-Nicolson formulation to integrate in time. Write a computer program for the solution, and select values of ?x and ?t for good accuracy. Plot the temperature u versus length x for various values of time t. Generate a separate curve for the following values of the parameter r = 1, 0.1, 0.01, 0.001, and 0.