Page 30 - Design and Operation of Heat Exchangers and their Networks

Basic thermal design theory for heat exchangers 17

Nu H λ 4:36 0:0281
2
α H ¼ ¼ ¼ 5:833 W=m K
d i 0:021
The overall heat transfer coefficient between the tube outside surface
and the air flow can be expressed as
1 1
1 ln d o =d i Þ 1 ln 0:025=0:021Þ
ð
ð
k ¼ + d i ¼ +0:021
α H 2λ w 5:833 2 15
2
¼ 5:829 W=m K
If we assume that the air properties are constant and the heat conduction
in the tube wall and air flow along the tube length is negligible, then for the
constant heat flux boundary condition and constant heat transfer coefficient,
the temperature difference between the tube wall and the fluid is a constant,
and the temperature distributions in the tube wall and air flow along the
tube length are two parallel straight lines. Therefore, the maximum wall
temperature happens near the end of the heating section, x¼L:
Δt ¼ t w,o t ¼ t w,o tð Þ ¼ 200 80 ¼ 120 K
x¼L
The required tube length is determined by Q¼kA△t, which yields
Q 26:07
L ¼ ¼ ¼ 0:5649 m
πd i kΔt π 0:021 5:829 120
Finally, we will check the entrance length by calculating the parameter
1330 0:7047 0:021
RePrd i =L ¼ ¼ 34:86
0:5649
which is larger than 10. That means the heating section is still in the entrance
region, and the design with L¼0.57m is a little conservative, but it would
be safe.
The detailed calculation can be found in the MatLab code for Example
2.1 in the appendix.

2.1.1.2 Thermally developing and hydrodynamically developed
laminar flow in straight circular tubes
For the thermally developing and hydrodynamically developed laminar flow
(the Nusselt-Graetz problem), Gnielinski (1989) suggested an asymptotic
equation of the mean Nusselt number for constant wall temperature as
h i 3
3 3 3 1=3
Nu ¼ 3:66 +0:7 +1:615 RePrd=Lð Þ 0:7 (2.9)
T
Compared with the analytical results given by Shah and London (1978,
6
Table 13), the maximum deviation of Eq. (2.9) in 0.1<RePrd/L<10 is