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.
x = 0
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
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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