Page 349 - Design and Operation of Heat Exchangers and their Networks

Dynamic analysis of heat exchangers and their networks 335

yτ f gτ + g Δf τ
can be applied to Eqs. (7.38)–(7.40), which yield the following linear
problem:
!
∂θ 1 _ ∂θ 1 ^ U 2 ^ U 1 _
ð
C 1 + C 1 U 1 θ w θ 1 Þ C 1 U 1 ð ^ t 1 ^ t 2 Þ ¼ 0 (7.65)
^
∂τ ∂x ^ U 1 + ^ U 2 C 1
_
" #
n
∂θ 2 n ∂θ 2 ^ U 1 ð 1Þ ^ U 2
ð
ð
C 2 + 1Þ _ C 2 U 2 θ w θ 2 Þ + _ C 2 U 2 ð ^ t 1 ^ t 2 Þ ¼ 0
∂τ ∂x ^ U 1 + ^ U 2 ð ^ n ^
1Þ _ C 2
(7.66)
∂θ w ^ U 2 U 1 ^ U 1 U 2
C w U 1 θ 1 U 2 θ 2 + U 1 + U 2 θ w ð ^ t 1 ^ t 2 Þ ¼ 0 (7.67)
∂τ ^ U 1 + ^ U 2

0 0 0 0
θ 1 x , τ ¼ θ τðÞ, θ 2 x , τ ¼ θ τðÞ (7.68)
1 1 2 2
τ ¼ 0 : θ 1 ¼ θ 2 ¼ θ w ¼ 0 (7.69)
A special case is that at τ¼0, the flow direction, thermal flow rates, and
heat transfer parameters have a sudden change, and then for τ>0, they do
_
_
not vary with time any more, that is, C ¼ C, U ¼ U, and the inlet fluid tem-
perature variations can be arbitrary. This case belongs to a linear problem,
and Eqs. (7.65)–(7.67) are exact. Otherwise, Eqs. (7.65)–(7.67) are
approximate.

7.1.2.7 Analytical solution in the Laplace domain
The Laplace transform of Eqs. (7.65)–(7.68) yields a nonhomogeneous
ordinary differential equation system:
!
e
_ dθ 1 ^ U 2 ^ U 1 e _
e
e
ð
C 1 + sC 1 + U 1 Þθ 1 U 1 θ w C 1 e U 1 ð ^ t 1 ^ t 2 Þ ¼ 0 (7.70)
^
dx ^ U 1 + ^ U 2 C 1
_
e
n dθ 2
ð 1Þ _ C 2 + sC 2 + U 2 θ 2 U 2 θ w
e
e
dx
" #
n
^ U 1 ð
+ 1Þ ^ U 2 e _ ^ t 1 ^ t 2 Þ ¼ 0 (7.71)
C 2 e U 2 ð
^ n ^
^ U 1 + ^ U 2 ð _
1Þ C 2
1 ^ U 2 U 1 ^ U 1 U 2
e
e
θ w ¼
e
e U 1 θ 1 + U 2 θ 2 + ^ t 1 ^ t 2 Þ (7.72)
ð
e
sC w + U 1 + U 2 ^ U 1 + ^ U 2 sC w + U 1 + U 2
0 0
0 0
θ 1 x , s ¼ θ sðÞ, θ 2 x , s ¼ θ sðÞ
e e e e (7.73)
1 1 2 2