Page 96 - Design and Operation of Heat Exchangers and their Networks

84 Design and operation of heat exchangers and their networks

For rating problems, we can express Eq. (3.75) as

21 e NTU s S Þ
ð
ε s ¼ (3.76)
ð 1+ R s + SÞ 1+ R s Sð Þe NTU s S
This solution can be further extended for different kA in the two tube
passes by replacing Eq. (3.74) with Eq. (3.77) (Roetzel and Spang, 2010,
2013, Table 4):

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
S ¼ 1+ R +2R s 2γ 1Þ (3.77)
ð
s
where γ is the ratio of kA:
kA parallel flow pass
γ ¼ (3.78)
kA
Eqs. (3.73)–(3.76) are valid for both flow arrangements shown in
Fig. 3.4. However, the intermediate temperature of the tube-side fluid at
the outlet of the first pass (also at the inlet to the second pass), t t,i , is distin-
guishing in these two cases:

1
t t,i t 0 Se 2 NTU s 1+ R s 2γ 1ð½ ފ
CaseI : t ¼ 1 ε s (3.79)
t t 0 1
0
s t 2sinh NTU s S
2
2 3
1
NTU s 1+ R s 2γ 1ފ
t t,i t 0 6 Se 2 ½ ð 7
CaseII : t ¼ 1 ε s 1+ 7 (3.80)
6
0
t t 0 4 1 5
s t
2sinh NTU s S
2
3.2.2.5 1-4 shell-and-tube heat exchangers
For 1-4 shell-and-tube heat exchangers, Underwood (1934) derived a
solution that can be rewritten as
2
ε s ¼ (3.81)
S 4 NTU s R s R s NTU s
1+ R s + S 4 coth + tanh
2 2 2 2
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
where S 4 ¼ 1+ R s =2Þ . Eq. (3.81) is valid for both flow arrangements of
ð
parallel flow and counterflow in the first tube pass.
3.2.2.6 1-2m shell-and-tube heat exchangers
For the effectiveness of a 1-2m shell-and-tube heat exchanger, a simplified
form was suggested by Baclic (1989) as