Page 43 - Design and Operation of Heat Exchangers and their Networks

30 Design and operation of heat exchangers and their networks

For the fin around a cylinder (annular finned tube), the heat conduction
along the fin height r is described in the cylinder coordinates as

d dt f
λ f A c, f rðÞ ¼ α f P f rðÞ t f tð Þ (2.54)
dr dr
(2.55)
r ¼ R 0 : t f ¼ t w
dt f dt f
ð
r ¼ R : λ f ¼ α f t f tð Þor ¼ 0 adiabaticat fin tipÞ (2.56)
dr dr
For example, for an annular fin with constant fin thickness,
A c,f (r)¼2πrδ, and P f (r)¼2πr. Eq. (2.54) then turns into

1 d dt f α f
r ¼ ð t f tÞ (2.57)
r dr dr λ f δ
The analytical solutions for several typical fin profiles are available in the
literature (Kraus et al., 2001). For the fins with constant cross-sectional
area, constant wetted perimeter, and adiabatic boundary condition at the
fin tip, the fin efficiency is given by

ð
tanh mhÞ
η ¼ (2.58)
f
mh
where m is referred to as the fin performance factor,
s ffiffiffiffiffiffiffiffiffiffiffiffi
α f P f
m ¼ (2.59)
λ f A c, f
If the heat convection at the fin tip should also be taken into account, we
can still use Eq. (2.58) by extending the fin height with
(2.60)
Δh ¼ A c, f =P f

2.1.1.10 Overall heat transfer coefficient
The overall heat transfer coefficient k is principally based on the heat transfer
coefficients at both sides of the wall separating the two fluids. It can be
expressed as the sum of a series of thermal resistances: convective heat trans-
fer resistances of the two fluids, conductive heat transfer resistance of the
wall, and possible fouling resistances at hot and cold sides:
1 1 R f,h δ w R f,c 1
¼ + + + + (2.61)
kA α h A h A h λ w A m A c α c A c