Page 358 - Design and Operation of Heat Exchangers and their Networks

344 Design and operation of heat exchangers and their networks

If a backward finite-difference scheme is applied to the time derivative,
the discretized algebraic equations become implicit as follows:
Δτ Δτ

n
n 1
1+ + NTU 1 Δτ t ¼ t n 1,i 1 + NTU 1 Δτt n w,i + t 1,i (7.105)
1,i
Δx Δx

Δτ NTU 2 Δτ Δτ NTU 2 Δτ
n 1
n
1+ + t ¼ t n 2,i 1 + t n w,i + t 2,i (7.106)
2,i
R τ Δx R τ R τ Δx R τ
NTU 1 Δτ R 2 NTU 2 Δτ NTU 1 Δτ

1+ + t n ¼ t n
w,i 1,i
R w R w R w
R 2 NTU 2 Δτ
n
+ t + t n 1 (7.107)
2,i w,i
R w
This backward finite-difference scheme converges unconditionally.
However, it is implicit, and an iteration is needed to solve the discretized
equation system.
An advantage of numerical method such as the finite-difference method
is that some of the aforementioned assumptions can be relaxed. For example,
the assumption of constant thermal properties, flow rates, and heat transfer
coefficients is no longer necessary to determine transient behavior of a heat
exchanger.
Besides the finite-difference method, the other numerical methods such
as the finite element and the method of characteristics have also been
used for dynamic simulation of heat exchangers. Under some assumptions,
Heggs and Render (1983) used the method of characteristic curves for
transient response of heat exchangers with one fluid of infinite thermal
flow rate, which include evaporators, condensers, reboilers, precoolers,
and intercoolers.

7.3 Dynamic behavior of multistream parallel channel
heat exchangers

Multistream heat exchangers are widely used in process industries such as gas
processing and petrochemical industries to exchange heat among more than
two fluids with different supply temperatures owing to their higher effi-
ciency, more compact structure, and lower costs than two-stream heat
exchanger networks. The use of multistream heat exchangers is more
cost-effective and can offer significant advantages over conventional two-
stream heat exchangers in certain applications, especially in cryogenic plants
(Yee et al., 1990; Chen et al., 1998; Wang and Sunden, 1999). The steady-
state behavior of multistream heat exchangers can be found in Chapter 3.