Elaborated method for calculating U values
A general expression for U can be easily obtained as follows. Consider a double pipe heat exchanger in which one fluid through the inner pipe and the other fluid through the annular space between space between the two pipes.
Let L = length of heat exchanger, m
ri = inside radius of inner pipe, m
ro = outside radius of inner pipe, m
Ai = inside surface area of inner pipe (2πriL), m2.
Ao = outside surface area of inner pipe (2πroL), m2.
hi = film coefficient of heat transfer at inside surface of inner pipe, W/m2 C
ho = film coefficient of heat transfer at outside surface of inner pipe, W/m2 C
kw = thermal conductivity of inner pipe wall, W/m C.
ti = temperature of fluid flowing through the inner pipe, C
to = temperature of fluid flowing through the annular space between the two pipes, C
Ri = thermal resistance of fluid film at the inside surface of inner pipe, m2 C/W
Ro = thermal resistance of fluid film at the outside surface of inner pipe, m2 C/W
Rw = thermal resistance of inner pipe, m2 C/W
(i) The rate of heat transfer between the two fluids is given by:
q = ti – to (eq 1.15)
ΣR
Where ΣR = Ri + Ro +Rw (eq. 1.16)
Since Ri = 1/Aihi (eq. 1.17)
Rw = ln (ro/ri) (eq. 1.19)
2πLKw
Ro = 1/Aoho (eq. 1.20)
Hence
q = . (ti – to) (eq. 1.21)
1/Aihi + ln (ro/ri) + 1/Aoho
2πLKw
(ii) If Ui and Uo denote respectively the overall heat transfer coefficient based on unit area of the inside and outside surfaces of the inner pipe, then
q = AiUi (ti – to) = AoUo (ti – to) (eq. 1.22)
from eq. 1.21 and 1.22 Ui = . 1 (eq. 1.23)
1/hi + Ai ln (ro/ri) + [Ai/Ao]. 1/ho
2πLKw
Uo = . 1 (eq. 1.24)
[Ao/Ai].1/hi + [Ao/2πL].ln (ro/ri) + 1/hoKw
(iii) Since Ai = 2πriL and Ao = 2πroL, eq. c and d can also be written as:
Ui = . 1 (eq. 1.25)
1/hi + [ri/kw] ln (ro/ri) + [ri/ro]. 1/ho
Uo = . 1 (eq. 1.26)
[ro/ri].1/hi + [ro/ Kw].ln (ro/ri) + 1/ho