A stainless steel
body of conical section (see Figure 1) is immersed in a fluid at a temperature Ta. The body is of circular cross-section that
varies along its length, L. The large end is located at x=0 and is
held at a temperature Ta=5. The small end is located at x=L=2 and is
held at Tb=4.
A heat balance
equation can be developed at any cross-section of the body using the principles
of conservation of energy. When the body is not insulated along its length and
the system is at a steady-state, its temperature satisfies the following o.d.e.
(ordinary differential equation).
d^2T/dx^2 + a(x)dT/dx+b(x)T=f(x)
Where a(x), b(x) and c(x) are
functions of the cross-section area, heat transfer coefficients and heat inside
the body. In the current example they are given by:
We have the analytical and numerical solution.... click on BUY NOW.
1.- Solve the
equation using the shooting method as follows.
(a) Convert the
second-order o.d.e. to a system of two first-order o.d.e.
(b) Use the
shooting method to numerically solve the system of equations with a step size
of 0.5
(c) Plot the
temperature distribution along the body.
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