Page 364 - Design and Operation of Heat Exchangers and their Networks

350 Design and operation of heat exchangers and their networks

where

^ ^r j x i
^ V xðÞ ¼ ^v ij ¼ h ij e (7.135)
M M M M
Eq. (7.134) is valid only if the eigenvalues differ from each other. To
avoid the multiple eigenvalues, we can add small deviations to the thermal
^
_
flow rates C i . Such small deviations have almost no effect on the results.
The exit temperature vector of the streams is obtained according to the
energy balance at the exits of streams, Eq. (7.122), which can be expressed in
the matrix form as

^ 00
^00 ^ ^ 00
T ¼ G T + G T x (7.136)
^ 000 ^0
Substituting Eqs. (3.289, 7.129) and (7.131) into Eq. (7.136), we can obtain
the outlet temperature vector of the exchanger as

1
^00 ^00 ^0
^0 ^0
^ 00
^ ^00
^ 000
T ¼ G + G V V GV G T (7.137)
7.3.3 Linear model and linearized model
Consider a heat exchanger that runs at first at a steady state, and then, the
exchanger experiences sudden changes in flow rates at τ¼0 and arbitrary
inlet temperature variations with time. We denote the flow rates and their
distributions, flow arrangements, and heat transfer parameters at the new
operating point with “¯”, which will keep constant for τ>0. Under such
changes in operation conditions, Eqs. (7.113)–(7.116) are linear and can
be solved by means of the Laplace transform.
_
Eqs. (7.113)–(7.116) will become nonlinear if the thermal flow rates C,
heat transfer parameters U, or matching matrices—G, G , G ,and G —
00
000
0
vary with time in τ>0. For the nonlinear problem of heat exchanger dynam-
ics, it is difficult to obtain the analytical solution, and numerical methods can
be used for the dynamic responses of heat exchangers. However, if the ther-
_
mal flow rates C, heat transfer parameters U, matching matrices—G, G , G ,
0
00
0
000
and G —and inlet temperatures T have sudden changes at τ¼0 and then
undergo small disturbances around the new operating point in τ>0, we can
apply Eq. (2.174).

f τðÞθτðÞ ¼ f θτðÞ + Δf τðÞ θ + ΔθτðÞ f θτðÞ + Δf τðÞθ ð 2:174Þ, (7.138)
to Eq. (7.113) to linearize the nonlinear energy equation for the fluid
streams, which yields