Page 93 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 81

the maximum temperature of the heat transfer surface or to controlling
either the temperature rise of cold fluid being heated or the drop in temper-
ature of the hot fluid being cooled.
The effectiveness is determined by the number of transfer units (NTU),
the thermal capacity rate ratio R, and the flow arrangement. For a particular
two-stream heat exchanger, the heat transfer between the two fluid streams
can be modeled with the energy balance of a small control volume in the
exchanger, which yields a set of governing differential equations and bound-
ary conditions. The solutions of such mathematical models can be obtained
analytically or numerically. If an explicit analytical solution of the fluid
temperature distribution for a particular flow arrangement is available, the
effectiveness can be expressed explicitly according to its definition
(Eq. 3.55). We have already discussed the temperature distributions in
parallel-flow and counterflow heat exchangers. By the substitution of
Eq. (3.26) for the parallel-flow arrangement and Eq. (3.38) for the counter-
flow arrangement into Eq. (3.55), the effectiveness can be easily obtained.

3.2.2.1 Parallel-flow heat exchangers
We first apply the ε-NTU analysis to a parallel-flow heat exchanger. If
_
_
_
_
C c C h (i.e., R ¼ C c =C h ), the relationship between ε and NTU for
the parallel-flow heat exchanger has already been given by Eq. (3.25) as
ð
ð
ð
t t 0 1 e kA= _ C cÞ 1+ _ C c = _ C hÞ 1 e NTU 1 + RÞ
00
ε ¼ c c ¼ ¼ (3.58)
t t 0 _ _ 1+ R
0
h c 1+ C c =C h
_
_
_
_
If C c > C h (i.e., R ¼ C h =C c ), according to the definition of ε, we have
ð
ð
t t 00 t t 1 e kA= _ C hÞ 1+ _ C h = _ C cÞ 1 e NTU 1 + RÞ
ð
0
0
00
_
_
ε ¼ h h ¼ c c C c =C h ¼ ¼
t t 0 t t 0 _ _ 1+ R
0
0
h c h c 1+ C h =C c
(3.59)
In practice, it is more convenient to use a similar definition, the dimen-
sionless temperature change of each fluid:
0
t t 00
h
h
ε h ¼ (3.60)
t t 0
0
h c
t t 0 c
00
c
ε c ¼ (3.61)
t t 0
0
h c