Concept explainers
Cooling Fin A cooling fin is an outward projection from a mechanical or electronic device from which heat can be radiated away from the device into the surrounding medium (such as air). See Figure 6.R.1. An annular, or ring-shaped, cooling fin is normally used on cylindrical surfaces such as a circular heating pipe. See Figure 6.R.2. In the latter case, let r denote the radial distance measured from the center line of the pipe and T(r) the temperature within the fin defined for r0 ≤ r ≤ r1. It can be shown that T(r) satisfies the differential equation
where a2 is a constant and Tm is the constant air temperature. Suppose r0 = 1, r1 = 3, and Tm = 70. Use the substitution w(r) = T(r) − 70 to show that the solution of the given differential equation subject to the boundary conditions
is
where and I0(x) and K0(x) are the modified Bessel functions of the first and second kind. You will also have to use the derivatives given in (25) of Section 6.4.
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