Page 40 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 27

2.1.1.7 Heat transfer in concentric annular ducts
For heat transfer in concentric annular ducts, three boundary conditions
have often been met: (1) heat transfer through the inner tube, with the insu-
lated outer tube; (2) heat transfer through the outer tube, with the insulated
inner tube; and (3) heat transfer through both the inner and outer tubes hav-
ing the same wall temperature. A typical example is double-pipe heat
exchangers, which are usually treated as boundary condition (1).
Stephan (1962) developed a set of correlations. For boundary conditions
6
(1) and (2), the Nusselt number for turbulent flow (2300 Re 10 ) can be
evaluated by the following equations, respectively:
h i
0:45 2=3 0:75 0:42 0:14
ð
Nu i ¼ 0:033 d o =d i Þ 1+ d h =LÞ Re 180 Pr ð μ=μ Þ
ð
w
(2.37)
h i
2=3 0:75 0:42 0:14
ð
Nu o ¼ 0:037 1 0:1d i =d o Þ 1+ d h =Lð Þ Re 180 Pr ð μ=μ Þ
w
(2.38)
For boundary condition (3), Stephan suggested the following relation:
Nu i d i =d o +Nu o
Nu ¼ (2.39)
d i =d o +1
For hydrodynamically developed laminar flow in the thermal entrance
region, Stephan expressed the Nusselt number as follows:

0:8
0:19 RePrd h =LÞ
ð
Nu ¼ Nu ∞ + fd i =d o Þ (2.40)
ð
0:467
1+0:117 RePrd h =LÞ
ð
in which Nu ∞ is the Nusselt number for fully developed laminar flow under
the corresponding boundary condition, and the function f (d i /d o ) was given as
8
1=2
ð
1+0:14 d i =d o Þ , heat transfer inner tube
<
1=3
ð
fd i =d o Þ ¼ 1+0:14 d i =d o Þ , heat transfer outer tube (2.41)
ð
: 0:1
1+0:14 d i =d o Þ , heat transfer both tubes
ð
Martin’s expressions (Gnielinski, 2010b, 2013b) can be used for Nu ∞ :
8
0:8
3:66 + 1:2 d i =d o Þ , heat transfer inner tube
ð
>
>
< 0:5
3:66 + 1:2 d i =d o Þ , heat transfer outer tube
ð
Nu ∞ ¼
> 3:66 + 4 0:102= 0:02 + d i =d o Þ½ ð Š heat transfer both tubes
>
: 0:04
ð d i =d o Þ ,
(2.42)