Page 438 - Design and Operation of Heat Exchangers and their Networks

Experimental methods for thermal performance of heat exchangers 421

temperature varies, the temperature of the fin at the half fin height varies
more rapidly than that of the fin root. Therefore, transient heat conduction
occurs along the fin height. Luo and Roetzel (2000, 2001) proposed a new
mathematical model for plate-fin heat exchangers to take the lateral heat
conduction resistance along the fin height into account. Their research
shows that for plate-fin heat exchangers with low fin efficiency, the tradi-
tional model would yield an error as high as 5%.
Since the primary wall is usually thicker than the fins, the temperature
subsidence near the fin root is negligible; therefore, the temperature distri-
bution in the direction of the heat exchanger width can be considered uni-
form. The temperature across the fin thickness is assumed to be constant
since the fins are usually very thin. We further assume that the axial thermal
conduction in the plates and the axial dispersion in the fluid are negligible.
The energy equation for the fluid can be represented as
h
ð
C ∂t ∂t A p α A f
+ C _ ¼ t p t + α t f tÞdy (8.126)
ð
L ∂τ ∂x L hL
0
where A p and A f are the primary surface area (plate surface) and the second-
ary surface area (fin surface), respectively. For the plates, the heat transferred
from the fins to the plates by heat conduction should be included. For the
one-dimensional problem (not the crossflow), the energy equation can be
expressed as

∂t p ∂t f
C p ¼ αA p t t p +2A c, f λ f (8.127)
∂τ ∂y
y¼0
The initial and boundary conditions are the same as Eqs. (8.81)–(8.85).
In Eqs. (8.126), (8.127), there is an unknown fin temperature; therefore,
the partial differential equation for transient heat conduction in fins together
with the initial and boundary conditions is supplemented as follows:
2
∂t f ∂ t f
ð
C f ¼ λ f A c, f h + α f A f t t f Þ (8.128)
∂τ ∂y 2
(8.129)
τ ¼ 0 : t f ¼ t 0
y ¼ 0 andy ¼ h : t f ¼ t p (8.130)
in which A c,f is the cross-sectional area of fins perpendicular to the fin height,
and the heat conduction in fins in the flow direction is neglected.