Heat and mass transfer analogy to find average Nusselt no.

[milkchocolate]

I am trying to find an expression for the average Nusselt number corresponding to heat transfer from an isothermal disk.

Given:
Sh_D≡h_m(r)D/D_AB=Sh_o[1 + a (r/r_o)ⁿ] (1)

Sh_o=h_m(r=0)D/D_AB=0.814Re_D^1/2Sc^0.36 (2)

Relevant equations:
Average nusselt number is defined as Nu_av=h_avD/k
where k is thermal conductivity, D is diameter of disk and h_av is average convection coefficient.

The heat and mass transfer analogy states that
Nu_av/Prⁿ=Sh_av/Scⁿ (3)

Sh is Sherwood number, Nu is Nusselt number, Pr is Prandtl and Sc is Schmidt. In this case n=0.36, from given data.

h_av is defined as: h_av=(1/A_s)∫ h dA_s (4)

where you integrate h, the convection coefficient, over the surface area, A_s.

Solution??
If I solve for h_m(r) in (1), and integrate over the surface area, I am still stuck with the constant (D_AB/D), but this should NOT be in the final answer. This is obviously the wrong approach, but the rest of the answer is correct, so I am on to something, I am just not sure how to use the analogy correctly.

Somehow I need to use the analogy and combine it with the formula for h_av to obtain the average Nusselt number. Apparently the solution is to integrate Sh_o[1 + a (r/r_o)ⁿ] over the area A_s, and just replace Sh_o with 0.814Re_D^1/2Pr^0.36. But why can I do this? I understand that it has something to do with the analogy, but I don't understand how or why. Can someone help me out here?