# Heat and mass transfer analogy to find average Nusselt no.

• milkchocolate

#### milkchocolate

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

Given:

ShD≡hm(r)D/DAB=Sho[1 + a (r/ro)n] (1)

Sho=hm(r=0)D/DAB=0.814ReD1/2Sc0.36 (2)

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

The heat and mass transfer analogy states that
Nuav/Prn=Shav/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.

hav is defined as: hav=(1/As)∫ h dAs (4)

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

Solution??
If I solve for hm(r) in (1), and integrate over the surface area, I am still stuck with the constant (DAB/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 hav to obtain the average Nusselt number. Apparently the solution is to integrate Sho[1 + a (r/ro)n] over the area As, and just replace Sho with 0.814ReD1/2Pr0.36. 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?