Question

Derive an expression for the U factor if the inside and outside surfaces of the tubes in a heat exchanger are finned and have fin efficiencies of ?_i and ?_o on the inside and the outside, and A_f,i and A_f,o as the finned area on the inside and the outside, respectively, and similarly A_u,i and A_u,o as the unfinned area on the inside and the outside, respectively.

Please post all steps taken. Thanks!

Answer 1

In order to solve this problem I have to take some assumption because the data is provided only for heat transfer by convection and nothing has been given for conduction. Hence, I assume that the resistance in heat flow due to conduction is very small and negligible as compared to that for convection. Therefore, Entire heat will be transfered by convection.

Now lets get back to the point,

for both side we have finned area and unfinned area. So this will chagne our effective area for heat transfer for inside and out side

Effective Area for inside:

A_{eff_inside} = A_ui +_i * A_fi   (Equation1)

Similarly effective Area for outside:

A_{eff_outside} = A_uo+_o * A_fo ?(Equation2)

Now. Let us assume the h is the convective heat transfer coefficient for inside as well outside and Q is overall heat flow. So Generaly, Q can be written as

Q = U*A_{eff_outside}* (T_o - T_i ) where T_o and T_i are fluid temperature outside and inside respectively.

So 1/U*A_{eff_outside} is resistance to heat transfer.

It can also be written as

1/U*A_{eff_outside} = 1/h*A_{eff_inside} + 1/h*A_{eff_outside}

by putting the values of A_{eff_inside} and A_{eff_outside} from Equation1 and Equation2 ans simplifying the expression

U = h*(A_ui +_i * A_fi ) / { (A_ui +_i * A_fi ) + (A_uo+_o * A_fo) }