Page 243 - Design and Operation of Heat Exchangers and their Networks

Optimal design of heat exchanger networks 233

6.1 Mathematical model and its general solution for
rating heat exchanger networks

Thermal analysis of a heat exchanger network is a basis for its optimal design,
synthesis, regulation, and online control. Usually, the calculation is difficult
because the inlet fluid temperatures of some heat exchangers in the network
are unknown. For a heat exchanger network with a simple sequential
arrangement of heat exchangers, the exit stream temperatures of the net-
work can be obtained by calculating the outlet fluid temperatures of each
heat exchanger sequentially. However, a practical network might have loops
and branches; therefore, the unknown inlet fluid temperatures of some heat
exchangers have to be assumed. To avoid an arduous iterative calculation,
we introduce an explicit analytical solution for thermal calculation of heat
exchanger networks (Roetzel and Luo, 2005; Chen et al., 2007).
Consider a heat exchanger network having N E heat exchangers, N M
0
00
mixers, N stream entrances, and N stream exits. In each heat exchanger,
there are two fluid channels for hot and cold streams, respectively. A mixer is
used to express a node at which two or more streams are mixed together and
splitted again. One mixer is regarded as one channel. Therefore, the total
number of channels N¼2N E +N M . The channel indexes are related to
the exchanger indexes, that is, the index of the hot stream in the jth
exchanger is 2j 1 and that of the cold stream is 2j. The index of the mth
mixer is 2N E +m. The indexes of the network entrances and network exits
can be arbitrarily labeled. The outlet stream temperatures of the jth heat
exchanger can be expressed as

t 00 t 0
E,h, j v hh v hc E,h, j
t 00 E,c, j ¼ v ch v cc t 0 E,c, j ð j ¼ 1, 2, …, N E Þ (6.1)
or in the matrix form

T 00 E, j ¼ V j T 0 E, j ð j ¼ 1, 2, …, N E Þ (6.2)
00
in which T E, j and T E, j are the inlet and outlet temperature vectors of the
0
jth heat exchanger, respectively. The coefficient matrix V j can be calculated
with Eq. (6.3):

v hh, j v hc, j 1 ε j ε j
V j ¼ ¼ (6.3)
v ch, j v cc, j R j ε j 1 R j ε j