Page 125 - Design and Operation of Heat Exchangers and their Networks

Steady-state characteristics of heat exchangers 113

The middle region,2π <θ<2(n 1)π:
The energy equation for the cold fluid is the same as Eq. (3.209). For the
hot fluid, it is given as

_ dt h θðÞ
½
ð
ð
ð
C h + khrφ r ðÞ t h θ ðÞ t c θ ðÞ½ Š + kh r s h Þφ r s h Þ t h θ ðÞ t c θ 2πފ
dθ
¼ 0
(3.210)
The outmost region,2(n 1)π <θ<2nπ:
The energy equation for the hot fluid is the same as Eq. (3.210). For the
cold fluid, we have
_ dt c θðÞ
C c + khrφ rðÞ t h θðÞ t c θðÞ½ Š ¼ 0 (3.211)
dθ
In the previous equations, n is the number of channel turns, r 0 the initial
radius, h the channel height, s h the hot fluid channel pitch, and s c the cold
fluid channel pitch. The channel spacing is equal to the channel pitch minus
plate thickness. The radius can be expressed as
θ

ð
r ¼ r 0 + s h + s h + s c Þ (3.212)
2π
Because the two plates are rolled into an Archimedes spiral with a pitch
of s h +s c , the length enlargement ratio φ(r) can be calculated with
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
s h + s c
φ r ðÞ ¼ 1+ (3.213)
2πr
The overall heat transfer coefficient k varies with the channel curvature;
therefore, it is a function of r. With the boundary conditions at the fluid
inlets given by

t h 0ðÞ ¼ t h,in (3.214)

ð
t c 2nπÞ ¼ t c,in (3.215)

the previous ordinary differential equation system can be solved numerically.
To simplify the problem, Bes and Roetzel (1992a) assumed φ 1;
_
_
s h ¼s c ¼s; and k, C h , and C c are constant. They further assumed a high
number of turns n so that the special situation in the first and last turn does
not have to be taken into account. Then, they obtained an approximate