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A body of conical section fabricated from stainless steel is immersed in air at a temperature Ta = 0. It has circular cross section that varies with x. The large end is located at x = 0 and is held at temperature TA = 5 as shown in the figure. The small end is located at x = L = 2 and is held at TB = 4. To TA T(x) TB Ta x x = 0 x = L Conservation of energy can be used to develop a heat balance equation at any cross section of the body. When the body is not insulated along its length and the system is at a steady state, its temperature satisfies the following ODE: d-T 2 a = where a(x), b(x), and f(x) are functions of the cross-sectional area, heat transfer coefficients, and the heat sinks inside the body. In the present example, they are given by + x+3 + and Write a function that uses finite differencing to solve the problem. Discretize the interval from X = 0 to x = L using N + 1 points (including the boundary points): The temperature at point j is denoted by Tj (i) Discretize the differential equation using the central difference formulas for the second and first derivatives. The discretized equation is valid for j = 1,2, N and therefore yields N- - 1 equations for the unknowns To T1, TN+1. (ii) Obtain two additional equations from the boundary conditions (TA = 5 and TB = 4) and write the system of equations in matrix form A T = f. Solve this system with N = 20. Plot the temperature. Challenge: Reformulate the problem to implement the insulated boundary condition at x = L. If the body is insulated at x = L, the boundary condition becomes dT/dx = 0. You need to write a difference equation for the boundary condition. Set up the new system of equations and solve the problem and plot the temperature.

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