Page 432 - Design and Operation of Heat Exchangers and their Networks
P. 432
Experimental methods for thermal performance of heat exchangers 415
The Laplace transform of the fluid temperature can be easily obtained as:
asinh b 1 xÞ + bcosh b 1 xÞ 0
ð
½
½
ð
e
θ ¼ 2 2 e θ (8.109)
ax e
a + b
sinh bðÞ + bcosh bðÞ
2a
where
Pe
a ¼ (8.110)
2
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2aNTUs
2
b ¼ a + +2aBs (8.111)
s + NTU
The inverse Laplace transform can be obtained by the use of the residuum
theorem (Anon., 1979):
σ + i∞
1 sτ X s j τ
h i ð ∞ h i
θ τðÞ ¼ L 1 e θ sðÞe ds ¼ res fs j e (8.112)
e
e
θ sðÞ ¼
2πi
σ i∞ j¼0
to obtain the analytical expression of the real-time temperature dynamics.
For a unit step change in inlet fluid temperature, the integrand becomes
1 asinh b 1 xÞ + bcosh b 1 xÞ ax sτ
ð
½
½
ð
e e (8.113)
2
s a + b 2
sinh bðÞ + bcosh bðÞ
2a
One of the singularities is s 0 ¼0, which is of the first order. Other singular-
2
ities are zero points of the denominator a + b 2 sinh bðÞ + bcosh bðÞ, which
2a
yield
" 2 2 #
1 a + β j
s j ¼ NTU 1 + BÞ +
ð
2B 2a
v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u 2 2 2 2 ! 2
u a + β a + β
1 t 2 j j
NTU 1+2Bð Þ + 2NTU + NTUB ð j ¼ 12…;∞Þ
2B 2a 2a
(8.114)
in which β j is the jth root of the function
2aβ
j
tanβ ¼ 2 (8.115)
j
β a 2
j
