Page 42 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 29

In the curved tubes, the critical Reynolds number is larger than that in
the straight tubes:

0:45
ð
Re cr ¼ 2300 1 + 8:6 r=r c Þ (2.47)
2.1.1.9 Extended heat transfer surfaces
For the extended heat transfer surfaces (fins), there are two parallel heat
transfer processes. The one is the convective heat transfer from the unfinned
surface to the fluid, and the other is the conductive heat transfer through the
fins and then from the fin surface to the fluid by heat convection. The effect
of the conductive thermal resistance on the heat transfer performance of fins
can be expressed by the fin efficiency, defined as the ratio of the heat trans-
ferred from the fin to the heat that would be transferred by the fin if its ther-
mal conductivity were infinite large (i.e., if the entire fin were at the same
temperature as its base):

Q f ,actual
η ¼ (2.48)
f
α f A f t w tÞ
ð
where t w is the wall temperature at the fin base, t is the fluid temperature, and
α f is the heat transfer coefficient at the fin surface (usually we take α f ¼α).
Then, we can express the total heat transfer Q and the overall fin efficiency
η 0 as

ð
ð
ð
Q ¼ α A A f Þ t w tð Þ + η α f A f t w tÞ ¼ η αAt w tÞ (2.49)
f 0
η ¼ 1 1 η α f =αð f ÞA f =A (2.50)
0
The expression of fin efficiency depends on the fin profile. Some typical
examples of the fin profiles are longitudinal fins of rectangular, trapezoidal,
or parabolic profiles; radial fins of these profiles; and cylindrical, truncated
conical, or truncated parabolic spines. Consider a fin on a plate
wall. By assuming one-dimensional heat conduction along the fin
height in x direction, for a given profile, we have the fin cross-sectional area
A c,f ¼A c,f (x) and fin wetted perimeter P f ¼P f (x) at the position x. The heat
conduction along the fin height x can be expressed as

d dt f
λ f A c, f xðÞ ¼ α f P f xðÞ t f tð Þ (2.51)
dx dx
(2.52)
x ¼ 0 : t f ¼ t w
dt f dt f
ð
x ¼ h : λ f ¼ α f t f tð Þor ¼ 0 adiabatic atfin tipÞ (2.53)
dx dx