Chapter 6, Problem 27RE

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

$\frac{d}{dr}\left(r\frac{dT}{dr}\right)=a^{2}r(T-T_m),$

where a² is a constant and T_m is the constant air temperature. Suppose r₀ = 1, r₁ = 3, and T_m = 70. Use the substitution w(r) = T(r) − 70 to show that the solution of the given differential equation subject to the boundary conditions

$T(1)=160,T^{\prime }(3)=0$

is $T(r)=70+90\frac{K_1(3a)I_0(ar)+I_1(3a)K_0(ar)}{K_1(3a)I_0(a)+I_1(3a)K_0(a)}$,

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.