Page 137 - Mechanical Engineers' Handbook (Volume 4)

126 Exergy Analysis, Entropy Generation Minimization, and Constructal Theory

straints on the optimization of the smaller system. The principle of thermodynamic isolation
(Ref. 4, p. 125) must be kept in mind during the later stages of the optimization procedure,
when the optimized elements and components are integrated into the total system, which
itself is optimized for minimum cost in the final stage. 2

4 CRYOGENICS
The field of low-temperature refrigeration was the first where EGM became an established
˙
Q
method of modeling and optimization. Consider a path for heat leak ( ) from room tem-
perature (T ) to the cold end (T ) of a low-temperature refrigerator or liquefier. Examples
H
L
of such paths are mechanical supports, insulation layers without or with radiation shields,
counterflow heat exchangers, and electrical cables. The total rate of entropy generation as-
sociated with the heat leak path is
T H ˙
Q
S ˙ gen dT
T T 2
L
˙
where Q is in general a function of the local temperature T. The proportionality between the
˙
heat leak and the local temperature gradient along its path, Q kA(dT/dx), and the finite
size of the path [length L, cross section A, material thermal conductivity k(T)] are accounted
for by the integral constraint
T H k(T) L
˙
T Q(T) dT A (constant)
L
The optimal heat leak distribution that minimizes S ˙ gen subject to the finite-size constraint is 3,4
Q (T) A T H k 1/2 dT k 1/2 T
˙
opt
L
T
T L T H k 1/2 dT 2
A
S ˙
gen,min
L T L T
The technological applications of the variable heat leak optimization principle are nu-
merous and important. In the case of a mechanical support, the optimal design is approxi-
mated in practice by placing a stream of cold helium gas in counterflow (and in thermal
˙
contact) with the conduction path. The heat leak varies as dQ /dT ˙mc P , where ˙mc P is the
capacity flow rate of the stream. The practical value of the EGM method is that it pinpoints
the optimal flow rate for minimum entropy generation. To illustrate, if the support conduc-
tivity is temperature-independent, then the optimal flow rate is ˙m opt (Ak/Lc )ln (T /T ).
P
H
L
In reality, the conductivity of cryogenic structural materials varies strongly with the temper-
ature, and the single-stream intermediate cooling technique can approach S ˙ gen,min only ap-
proximately.
Other applications include the optimal cooling (e.g., optimal flow rate of boil-off helium)
for cryogenic current leads, and the optimal temperatures of cryogenic radiation shields. The
main counterflow heat exchanger of a low-temperature refrigeration machine is another im-
portant path for heat leak in the end-to-end direction (T → T ). In this case, the optimal
L
H
variable heat leak principle translates into 3,4
˙ mc P T H
T
T UA ln T
opt L