Page 340 - Design and Operation of Heat Exchangers and their Networks

326 Design and operation of heat exchangers and their networks

which yields
^ _ ^ _ ^ _ ^
C h C c + ^ U ^ t h,in + C c U^ t c,in
(7.11)
^ t h ¼ ^ ^
_
_
C h + ^ U C c + ^ U ^ U 2
^ _ ^ _ ^ _ ^
C c C h + ^ U ^ t c,in + C h U^ t h,in
(7.12)
^ t c ¼ ^ ^
_
_
C h + ^ U C c + ^ U ^ U 2
With the same method, we can obtain the steady-state solutions of the
fluid temperatures under the new mean operating condition as follows:

_ _ _
C h C c + U t h,in + C c Ut c,in
(7.13)
t h ¼
_
_
C h + U C c + U U 2

_ _ _
C c C h + U t c,in + C h Ut h,in
(7.14)
t c ¼
_
_
C h + U C c + U U 2
For the dynamic analysis, we introduce an excess temperature defined by
θ ¼ t t (7.15)
For small disturbances in the inlet fluid temperatures, heat transfer coef-
ficients, and mass flow rates around the new mean operating condition, the
dynamic response of the excess temperature θ might be small. In such a case,
we can use Eq. (2.13) to linearize the energy equations. Thus, Eqs. (7.6)–
(7.8) can be expressed with the excess temperature approximately as follows:

dθ h _ _
ð
ð
ð
ð
C h ¼ C h θ h,in θ h Þ + U θ c θ h Þ + ΔC h t h,in t h Þ + ΔU t c t h Þ (7.16)
dτ
dθ c _ _
ð
ð
ð
ð
C c ¼ C c θ c,in θ c Þ + U θ h θ c Þ + ΔC c t c,in t c Þ + ΔU t h t c Þ (7.17)
dτ
(7.18)
τ ¼ 0 : θ h ¼ ^ t h t h , θ c ¼ ^ t c t c
_
_
_
_
_
_
where ΔC h ¼ C h C h , ΔC c ¼ C c C c , ΔU ¼ U U, and the symbol
“¯” indicates the mean value under the new steady-state operating condition
or in the new operating period.
Applying the Laplace transform to Eqs. (7.16), (7.17) and solving the
algebraic equations in the Laplace domain, we have