The energy balance equation in spherical coordinates in the radial direction and

Question

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)

Explanation / Answer

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.

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