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

Step-by-step 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=(hD)/(k) = 2+(0.589 R_a^(1)/(4) )/([1+((0.046)/(p_r)(9)/(16) )^(4)/(9) ) ------(3)`

By equation (1) and (2)

`2\pi kD.`ΔT=h.4
`\pi r_1^2`ΔT

`2kD=hD^2\\(hD)/(k) =2\\N_u=(hD)/(k)=2\\`-------(4)

From equation (3) and (4)

So for sphere
`R_a`→0