Page 60 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 47

∂t c ∂t c
_
0
For crossflow, A c ρc p + C c ¼ αAÞ t w t c Þ, t c j ¼ t ,
ð
ð
c ∂τ ∂y c y¼0 c
t c j ¼ t c,0 (2.100)
τ¼0
If the heat exchanger operates at a steady state, Eqs. (2.96)–(2.100)
further reduce to
_ dt h 0
ð
C h ¼ kA t c t h Þ, t h j x¼0 ¼ t h (2.101)
dx
_ dt c 0
ð
For parallel flow, C c ¼ kA t h t c Þ, t c j x¼0 ¼ t c (2.102)
dx
_ dt c 0
ð
For counterflow, C c ¼ kA t h t c Þ, t c j x¼L ¼ t c (2.103)
dx
_ dt c 0
ð
For crossflow, C c ¼ kA t h t c Þ, t c j y¼0 ¼ t c (2.104)
dy
In the dynamic analysis of heat exchangers, Eqs. (2.92), (2.95) have
often been simplified by the lumped parameter model. In the lumped
parameter model, it is assumed that each fluid in the whole heat exchanger
has the same uniform temperature. The temperature is only a function of
time. The same is for the solid wall. By integrating Eqs. (2.92), (2.95) over
the whole volumes of the fluid and the solid wall, respectively, and expres-
sing the temperatures and parameters with their volumetric average
values, we obtain the energy equations for the hot and cold fluids and solid
wall as

N w
dt i _ 0 00 X
ð
ð
C i ¼ C i t t i ð αAÞ Δt m, ij + sVÞ i ¼ 1, 2, …, N f Þ (2.105)
ij
i
i
dτ
j¼1
N f
dt w, j X
ð
C w, j ¼ ð αAÞ Δt m, ij + sVÞ w, j ð j ¼ 1, 2, …, N w Þ (2.106)
ij
dτ
i¼1
in which N f is the number of fluid streams and N w the number of walls. The
mean temperature difference between the ith fluid and jth wall is calculated
by integration over the heat exchanger area:
Z
1
Δt m, ij ¼ t i t w, j dA (2.107)
A ij
A ij
Different definitions of the volumetric mean temperatures and mean
temperature difference will yield different lumped parameter models.