Page 138 - Mechanical Engineers' Handbook (Volume 4)

5 Heat Transfer 127

where T is the local stream-to-stream temperature difference of the counterflow, ˙mc P is the
capacity flow rate through one branch of the counterflow, and UA is the fixed size (total
thermal conductance) of the heat exchanger.

5 HEAT TRANSFER
The field of heat transfer adopted the techniques developed in cryogenic engineering and
applied them to a vast selection of devices for promoting heat transfer. The EGM method
was applied to complete components (e.g., heat exchangers) and elemental features (e.g.,
ducts, fins). For example, consider the flow of a single-phase stream ( ˙m) through a heat
exchanger tube of internal diameter D. The heat transfer rate per unit of tube length q is
given. The entropy generation rate per unit of tube length is

q 2 32 ˙m ƒ
3
˙
S
gen
kT Nu TD 5
2 2
2
where Nu and ƒ are the Nusselt number and the friction factor, Nu hD/k and ƒ ( dP/
˙
2
2
dx) D/(2G ) with G ˙m /( D /4). The S expression has two terms: the irreversibility
gen
contributions made by heat transfer and fluid friction. These terms compete against one
another such that there is an optimal tube diameter for minimum entropy generation rate, 3,4
Re D,opt 2B 0.36 Pr 0.07
0
q ˙m
B
0
(kT) 1/2 5 /2
2
where Re VD/ and V ˙m /( D /4). This result is valid in the range 2500 Re
D
D
6
10 and Pr 0.5. The corresponding entropy generation number is
˙
S 0.856 0.8 0.144 4.8
Re
Re
N gen D D
˙
S
S gen,min Re D,opt Re D,opt
where Re /Re D,opt D /D because the mass flow rate is fixed. The N criterion was used
D
S
opt
extensively in the literature to monitor the approach of actual designs to the optimal irre-
versible designs conceived subject to the same constraints. This work is reviewed in Refs.
3 and 4.
The EGM of elemental features was extended to the optimization of augmentation tech-
niques such as extended surfaces (fins), roughened walls, spiral tubes, twisted tape inserts,
and full-size heat exchangers that have such features. For example, the entropy generation
rate of a body with heat transfer and drag in an external stream (U , T )is

˙
Q (T T ) FU

S ˙ gen B B D
TT T
B
˙
where Q B , T , and F are the heat transfer rate, body temperature, and drag force. The relation
D
B
˙
between Q B and temperature difference (T T ) depends on body shape and external fluid
B

7
and flow, and is provided by the field of convective heat transfer. The relation between F ,
D
7
U , geometry and fluid type comes from fluid mechanics. The S ˙ gen expression has the

expected two-term structure, which leads to an optimal body size for minimum entropy
generation rate.
The simplest example is the selection of the swept length L of a plate immersed in a
parallel stream (Fig. 5 inset). The results for Re L,opt U L / are shown in Fig. 5, where
opt
B is a constraint (duty parameter)