Page 140 - Mechanical Engineers' Handbook (Volume 4)

6 Storage Systems 129

˙
˙
S gen, P ) , which they termed the Bejan number, where S means local (volumetric) entropy
gen
generation rate, and T and P refer to the heat transfer and fluid flow irreversibilities,
respectively. This dimensionless group should not be confused with the dimensionless pres-
sure drop that is useful in the optimization of spacings in heat-transfer assemblies with forced
convection, cf. Ref. 7, pp. 136–141.

6 STORAGE SYSTEMS
In the optimization of time-dependent heating or cooling processes the search is for optimal
histories, that is, optimal ways of executing the processes. Consider as a first example the
sensible heating of an amount of incompressible substance (mass M, specific heat C), by
circulating through it a stream of hot ideal gas ( , c , T ) (Fig. 6). Initially, the storage˙m P
material is at the ambient temperature T . The total thermal conductance of the heat ex-
0
changer placed between the storage material and the gas stream is UA and the pressure drop
is negligible. After flowing through the heat exchanger, the gas stream is discharged into the
atmosphere. The entropy generated from t 0 until a time t reaches a minimum when t is
of the order of MC/( c ). Charts for calculating the optimal heating (storage) time interval˙m P
are available in Refs. 3 and 4. For example, when (T T ), the optimal heating time is

0
given by opt 1.256/[1 exp ( N )], where opt t opt ˙ m c /(MC) and N UA/( c ).
˙
m
tu
P
P
tu
Another example is the optimization of a sensible-heat cooling process subject to an
overall time constraint. Consider the cooling of an amount of incompressible substance (M,
C) from a given initial temperature to a given final temperature, during a prescribed time
interval t . The coolant is a stream of cold ideal gas with flow rate ˙m and specific heat c (T).
c
P
The thermal conductance of the heat exchanger is UA; however, the overall heat transfer
coefficient generally depends on the instantaneous temperature, U(T). The cooling process
requires a minimum amount of coolant m (or minimum refrigerator work for producing the
cryogen m),
t c
m ˙m(t) dt
0
when the gas flow rate has the optimal history 3,4
˙ m (t) U(T)A 1/2
opt
C*c (T)
P

Figure 6 Entropy generation during sensible-heat storage. 3