Question

The following is a correlation for the average Nusselt number for natural convection over spherical surface. As can be seen in the above, the Nusselt number approaches 2 as Rayleigh number approaches zero. Prove that this situation corresponds to conduction heat transfer and in conduction heat transfer over sphere, the Nusselt number becomes 2. Hint: First step: Write an expression for heat transfer between two spherical shells that share the same center. Second step: Assume the outer spherical shell is infinitely large.

Answer 1

Answer:

Explanation:

$r_2=$∞

$q=4\pi kT_1(T_2-T_1)$

$q=2\pi kD.$ΔT--------(1)

$q=hA$ ΔT$=4\pi r_1^2(T_2_s-T_1_s)$

$N_u=\frac{hD}{k} = 2+\frac{0.589 R_a^\frac{1}{4} }{[1+(\frac{0.046}{p_r}\frac{9}{16} )^\frac{4}{9} } ------(3)$

By equation (1) and (2)

$2\pi kD.$ΔT=h.4$\pi r_1^2$ΔT

$2kD=hD^2\\\frac{hD}{k} =2\\N_u=\frac{hD}{k}=2$-------(4)

From equation (3) and (4)

So for sphere $R_a$→0