Question

What is the dimensionless heat conduction rate for a sphere at surface temperature T1 buried in an infinite medium of temperature T2?a. 2*D.b. 2.34*D.c. Pi*D.d. 5.93*D.

Answer 1

Solution :

The dimensionless conduction heat rate, $q_{ss}^*$

$q_{ss}^*=\frac{q\times L_c}{K A_s(T_1-T_2)}$   ...........(1)

where $L_c$  = characteristic length

                $=\left(\frac{A}{4\pi} \right)^{1/2}. \sqrt{\frac{D^2}{4}}$

              A is surface area

          q = heat transfer

$q=Sk(T_1-T_2)$   ..................(2)

where, S = conductor shape factor

Now substituting (2) in (1),

$q_{ss}^* = \frac{Sk(T_1-T_2)L_c}{kA(T_1-T_2)}$

$q_{ss}^* = \frac{S \times L_c}{A}$

$q_{ss}^* = \frac{S \times D/2}{\pi D^2}$

$q_{ss}^* = \frac{S \times D}{2\pi D^2}$   ...................(3)

For a sphere, we know S = 2πD

Putting this in (3),

$q_{ss}^* = \frac{2 \pi D \times D}{2\pi D^2}$

$q_{ss}^* = \frac{2 \pi D^2}{2\pi D^2}$

$q_{ss}^* = 1$

Therefore, the dimensionless heat conduction rate for a sphere is 1.