Page 357 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 343

is usually adopted in solving the differential equations. By this technique, the
differential equations are substituted by the finite-difference equations.
The first step of this method is to discretize the solution space, that is, to
divide the spatial and time spaces into M and N intervals, x 0 <x 1 <…<x M ,
τ 0 <τ 1 <…<τ N , with the grid steps Δx i ¼x i x i 1 and Δτ j ¼τ j τ j 1 .
According to the finite-difference forms of the spatial derivatives, the
finite-difference method is further divided into the explicit and implicit rep-
resentations and other combinations of these two preliminary
representations.
For a parallel-flow heat exchanger, if a forward finite-difference scheme
is applied to the time derivative, the differential equations (7.21)–(7.23) can
be discretized into following explicit finite-difference equations:

Δτ Δτ
n n 1 n 1 n 1
t ¼ t +1 NTU 1 Δτ t + NTU 1 Δτt (7.99)
1,i 1,i 1 1,i w,i
Δx Δx
Δτ Δτ NTU 2 Δτ NTU 2 Δτ
t n ¼ t n 1 +1 t n 1 + t n 1 (7.100)
2,i 2,i 1 2,i w,i
R τ Δx R τ Δx R τ R τ

n NTU 1 Δτ R 2 NTU 2 Δτ n 1 NTU 1 Δτ n 1 R 2 NTU 2 Δτ n 1
t ¼ 1 t + t + t (7.101)
w,i w,i 1,i 2,i
R w R w R w R w
in which the dimensionless variables and parameters are defined as follows:
_
_
_
x ¼ x=L, τ ¼ τC 1 =C 1 , NTU 1 ¼ αAÞ =C 1 , NTU 2 ¼ αAÞ =C 2 ,
ð
ð
1 2
_ _ _ _
ð
R 2 ¼ C 2 =C 1 , R τ ¼ C 1 =C 2 = C 1 =C 2 Þ, R w ¼ C w =C 1
To maintain numerical stability, Δx and Δτ should be chosen according to
the following constraints:
Δτ Δx
1 NTU 1 Δτ 0or Δτ (7.102)
Δx 1 + NTU 1 Δx
Δτ NTU 2 Δτ R τ Δx
1 0or Δτ (7.103)
R τ Δx R τ 1 + NTU 2 Δx
NTU 1 Δτ R 2 NTU 2 Δτ R w
1 0or Δτ (7.104)
R w R w NTU 1 + R 2 NTU 2

With the given initial and boundary conditions, Eqs. (7.99)–(7.101)
can be used to calculate and analyze the dynamic temperature responses
of the exchanger subject to arbitrary inlet temperature and/or flow
disturbances.