Page 463 - Design and Operation of Heat Exchangers and their Networks
P. 463
446 Design and operation of heat exchangers and their networks
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_
(or cold) stream entering and exiting the heat exchanger, C h and C c the
thermal capacity rates of the hot and cold streams, and c h and c c the bypass
fractions for hot and cold streams, respectively. For the steady-state opera-
0
0
tion condition, if the heat loss is negligible, we have t E,h ¼t h and t E,c ¼t c .
0
0
To present whether a bypass control exists, we can further introduce a binary
variable y associated with the bypass fraction c. If there exists a bypass control,
y¼1, otherwise y¼0.
The exit stream temperatures of the ith heat exchanger can then be
obtained from Eqs. (9.13), (9.14) as
00
0
t ¼ 1 1 y h,i c h,i Þε i t +1 y h,i c h,i Þε i t 0 (9.35)
½
ð
ð
h,i h,i c,i
00
0
t ¼ 1 y c,i c c,i ÞR i ε i t +1 1 y c,i c c,i ÞR i ε i t 0 (9.36)
ð
½
ð
c,i h,i c,i
where NTU i and R i are calculated with thermal capacity rates through the
exchanger:
k i A i
NTU i ¼ (9.37)
_
ð 1 y h,i c h,i ÞC h,i
_
ð 1 y h,i c h,i ÞC h,i
R i ¼ _ (9.38)
ð 1 y c,i c c,i ÞC c,i
The dimensionless temperature change of the hot stream is calculated
with Eqs. (9.17)–(9.22).
For a general HEN composed of N E heat exchangers including heaters
and coolers for heating N c cold process streams and cooling N h hot process
streams, we use the following matrix relationship derived from Eqs. (9.35),
(9.36) to obtain the outlet temperatures of the heat exchangers:
00
AT ¼ T 0 (9.39)
where T contains the nodal temperatures of the HEN and A is a coefficient
matrix obtained according to Eqs. (9.35), (9.36) for each heat exchanger of
the network. For a HEN running at the nominal operation point, Eq. (9.39)
is expressed as
A N T ¼ T 0 (9.40)
00
N N
If there exist parameter deviations of the process streams from their nom-
inal values, the deviations of all nodal temperatures θ of the HEN can be
evaluated by
00 00 1 0 0 00
θ ¼ T T ¼ A N + ΔAð Þ T + ΔT T (9.41)
N N N
