The energy balance equation in spherical coordinates in the

The energy balance equation in spherical coordinates in the radial direction and with heat generation term equal to phi (T) (= beta 2 T), as a function of temperature can be written (after an energy balance is made) as: 1 / r2 d / dr (r2 dT / dr) + beta 2 T = 0 where beta 2 is a constant equal to (phi / alpha (thermal diffusivity) and phi are constants. Using the transformation: T = f (r) / r Obtain a solvable ODE for f r) as: d2 f / dr 2 = - beta 2 Solve for f(r) and find a general solution for T(r) of the form: T (r) = C1 [ ] + C2 [ ] (SOLUTION should contain: Sine\'S and cosine\'s)or Sinh\'s and Cosh\'s)

Solution

T = f(r)/r

dT/dr ={ rf\'(r) - f(r)}/r^2

Or r^2 dT/dr = rf\'(r) - f(r)

Hence taking derivative again

d/dr (r^2 dT/dr) = f\'(r) +rf\"(r)-f\'(r) = rf\"(r)

Or d^2f/dr^2 = 1/r d/dr (r^2 dT/dr) ...(i)

From the equation given we have

1/r^2 d/dr (r^2 dT/dr) = -beta^2 T = -beta^2 f(r)/r

Cancelling 1 r, we get

1/r d/dr (r^2 dT/dr) = -beta^2 f(r)

Or substituting in 1

d^2f/dr^2 = -beta^2 f

Hence proved.