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?

 

[mark726]

Thank you for your question. Let me try to explain the solution in a step-by-step manner.

Step 1: Solving for hm(r)
As you correctly mentioned, we need to solve for hm(r) in equation (1) in order to obtain the expression for the average Nusselt number. To do this, we can rearrange equation (1) as follows:

ShD/Sho = 1 + a(r/ro)n
hm(r)D/DAB = 1 + a(r/ro)n

Now, we know that ShD/Sho = 1 at the center of the disk (r=0), so we can substitute this into the above equation to get:

hm(r=0)D/DAB = 1 + a(0/ro)n
hm(r=0)D/DAB = 1

Therefore, we can rewrite equation (1) as:

hm(r)D/DAB = 1 + a(r/ro)n

Step 2: Using the heat and mass transfer analogy
Next, we can use the heat and mass transfer analogy (equation 3) to relate the Sherwood and Nusselt numbers. Substituting n=0.36 (from given data) into equation (3), we get:

Nuav/Pr^0.36 = Shav/Sc^0.36

Step 3: Solving for Shav
To solve for Shav, we can use the definition of average Nusselt number (Nuav = havD/k) and the definition of average convection coefficient (equation 4). Substituting these into equation (3), we get:

havD/kPr^0.36 = Shav/Sc^0.36

Solving for Shav, we get:

Shav = havDkPr^0.36Sc^-0.36

Step 4: Substituting for Shav in equation (1)
Now, we can substitute the expression for Shav into equation (1) to get:

hm(r)D/DAB = havDkPr^0.36Sc^-0.36[1 + a(r/ro)^0.36]

Step 5: Integrating over the surface area
Finally, we can integrate this equation over the surface area (As) to get the average Nusselt number (Nuav). Using equation (4) and the fact