Page 365 - Design and Operation of Heat Exchangers and their Networks
P. 365

Dynamic analysis of heat exchangers and their networks  351


                                M w                       M w
                   ∂θ i  _  ∂θ i  X                _  dt i  X
                                                                 ð
                                      ð
                 C i  + C i   +    U ik θ i  θ w,k Þ + ΔC i  +  ΔU ik t i  t w,k Þ
                    ∂τ     ∂x                        dx
                                k¼1                       k¼1
                                M w
                               X
                         _  d^ t i
                                      ^ ð
                                                  ð
                      + C i  +    U ik t i  ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, MÞ  (7.139)
                           dx
                               k¼1
                 Eliminating the first derivatives of t i and ^ t i in Eq. (7.139) with
              Eqs. (7.119), (7.147), we can express Eq. (7.139) as
                                                                  !
                               M w              M w           _
                              X                 X
                  ∂θ i  _  ∂θ i                            ΔC i
                                     ð
                C i  + C i   +    U ik θ i  θ w,k Þ +  ΔU ik    U ik ð t i  t w,k Þ
                  ∂τ      ∂x                                 _
                               k¼1              k¼1         C i
                                      !
                       M w        _ ^
                       X
                            U ik  C i U ik
                     +                  ð ^ t i  ^ t w,k Þ ¼ 0 i ¼ 1, 2, …, Mð  Þ  (7.140)
                                ^ _
                       k¼1      C i
                 With the same method, Eqs. (7.114)–(7.116) can be linearized as follows:
                                   M                 M
                           ∂θ w,k  X                 X
                       C w,k    +     U jk θ w,k  θ j +  ΔU jk t w,k  t j
                             ∂τ
                                   j¼1               j¼1
                              M
                             X
                                     ^
                           +     U jk t w,k  ^ t j ¼ 0 k ¼ 1, 2, …, M w Þ  (7.141)
                                                 ð
                              j¼1
                          N  0               M                   N  0
                          X                 X                    X

                      0      g θ τ  Δτ  0                   00        0    0
                                0
                              0
                θ i τ, x                       g θ j τ  Δτ ij , x     Δg τðÞt
                      i       ik k     ik       ij          j         ik   k
                          k¼1               j¼1                  k¼1
                          M                      N  0
                         X                       X
                                             0
                                                     0
                            Δg ij τðÞt j x  00  +^ t i x    g ^ t  0
                                      j      i       ik k
                          j¼1                    k¼1
                          M
                         X
                            g ^ t j x 00  ¼ 0 i ¼ 1, 2, …, MÞ           (7.142)
                                         ð
                             ij   j
                          j¼1
                         N  0               M                   N  0
                        X                  X                   X
                   00      g θ τ  Δτ  000      00       00  00       000  τ  0
                            000 0
                 θ τðÞ                        g θ i τ  Δτ , x     Δg ðÞt
                   l        lk k      lk       li       li  i        lk  k
                         k¼1               i¼1                  k¼1
                         M                   N  0
                        X                    X
                                                 000 0
                                          00
                              00
                           Δg τðÞt i x 00  + ^ t    g ^ t
                              li     i    l      lk k
                         i¼1                 k¼1
                         M
                        X
                            00  00                  00
                                       ð
                           g ^ t i xðÞ ¼ 0 l ¼ 1, 2, …, N Þ
                            li
                         i¼1
                                                                        (7.143)
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