Page 59 - Design and Operation of Heat Exchangers and their Networks

46 Design and operation of heat exchangers and their networks

This expression can be applied to multichannel heat exchangers.
For plate-fin heat exchangers, however, the temperature along the fin height
is not constant, and therefore, Eq. (2.93) should be used.
For the solid wall of the heat exchanger, we will assume that the heat
conduction resistance in the direction perpendicular to its heat transfer sur-
face is negligible, that is, the wall temperature across the wall thickness
is uniform. Then, the partial differential equation for heat conduction in
the wall can be written as

∂t w ∂ ∂t w ∂ ∂t w
δλ
δλ
ð δρcÞ ¼ ðÞ + ðÞ
w ∂τ ∂x w ∂x ∂y w ∂y
Z
N f
X
A w,i
α w,i ð t w t i Þ + s w dz (2.95)
A xy
i¼1 δ w
where N f is the number of the fluid streams, δ is the wall thickness, and A w,i /
A xy is the heat transfer area between the wall and the ith stream per unit area
on the x-y plane.
Eqs. (2.92), (2.95) are the fundamental equations of the distributed
parameter model, which can well describe the steady-state or transient ther-
mal performance of a heat exchanger if the flow in the exchanger is a plug
flow. In practice, the heat conduction in the two fluids and solid wall
is much smaller than the heat transfer between the two fluids through
the wall, and therefore, the heat conduction terms in Eqs. (2.92), (2.95)
can usually be neglected. For a two-stream heat exchanger without heat
source in the fluids, they can be expressed as follows:
∂t h _ ∂t h 0
ð
ð
A c ρc p + C h ¼ αAÞ t w t h Þ, t h j x¼0 ¼ t , t h j τ¼0 ¼ t h,0 (2.96)
h
h ∂τ ∂x h
∂t w
ð
ð δρcÞ w ∂τ ¼ αAÞ t h t w Þ + αAð Þ t c t w Þ, t w j τ¼0 ¼ t w,0 (2.97)
ð
ð
c
h
∂t c ∂t c
_
For parallel flow, A c ρc p + C c ¼ αAÞ t w t c Þ,
ð
ð
c ∂τ ∂x c
0
t c j ¼ t , t c j ¼ t c,0 (2.98)
x¼0 c τ¼0
∂t c ∂t c
_
For counterflow, A c ρc p C c ¼ αAÞ t w t c Þ,
ð
ð
c ∂τ ∂x c
0
t c j ¼ t , t c j ¼ t c,0 (2.99)
x¼L c τ¼0