Page 347 - Design and Operation of Heat Exchangers and their Networks
P. 347

Dynamic analysis of heat exchangers and their networks  333


                                    ^ _  d^ t 1
                                              ^ ð
                                   C 1   + ^ U 1 t 1  ^ t w Þ ¼ 0        (7.50)
                                      dx
                                     n ^ _  d^ t 2
                                 ð  1Þ C 2  + ^ U 2 t 2  ^ t w Þ ¼ 0     (7.51)
                                                 ^ ð
                                         dx
                                  ^ U 1 t 1  ^ t w Þ + ^ U 2 t 2  ^ t w Þ ¼ 0  (7.52)
                                    ^ ð
                                                 ^ ð

                                                 ^
                                      ^
                                            0
                                   ^ t 1 x 0  ¼ ^ t , ^ t 2 x 0  ¼ ^ t 0  (7.53)
                                       1    1     2     2
                 Substitution of Eq. (7.52) into Eqs. (7.50), (7.51) yields
                                   2   3
                                    d^ t 1
                                           "       #"#
                                             ^ 11 ^a 12
                                   6  dx  7  a        ^ t 1
                                   6   7
                                         ¼                               (7.54)
                                   4   5
                                             ^ 21 ^ 22
                                    d^ t 2   a  a     ^ t 2
                                     dx
              or
                                           ^
                                          dT
                                               ^ ^
                                             ¼ AT                        (7.55)
                                          dx
              in which
                               ^                            ^
                                           ^
                                                                      ^
                            ^ U 1 U 2                     ^ U 1 U 2
                  a ^ 11 ¼           ¼ NTU 1 , ^a 12 ¼            ¼ NTU 1
                         ^ _     ^  ^                 ^ _     ^  ^
                        C 1 U 1 + U 2                 C 1 U 1 + U 2
                       n   ^                           n   ^
                   ð  1Þ ^ U 1 U 2  n              ð  1Þ ^ U 1 U 2   n
                                       ^
                                                                        ^
              a ^ 21 ¼         ¼ 1ð  Þ NTU 2 , ^a 22 ¼         ¼   1ð  Þ NTU 2
                   ^ _     ^                       ^ _     ^
                  C 2 U 1 + ^ U 2                 C 2 U 1 + ^ U 2
                 The general solution of Eq. (7.55) can be written as
                                         ^
                                                  ^
                                               ^ Rx
                                         T ¼ ^ He D                      (7.56)
                                                         ^ 1 x
                                                         r
                                                        e    0
                                                  ^ Rx
                             ^ Rx
              Here, we denote e  as a diagonal matrix, e  ¼  ^ 2 x  , in which ^ 1 and
                                                                         r
                                                             r
                                                         0 e
                                                        ^
              r ^ 2 are the eigenvalues of the coefficient matrix A:

                                      a
                                       ^ 11  ^r  a
                                              ^ 12
                                                     ¼ 0                 (7.57)
                                                  r
                                             a
                                        a    ^ 22  ^
                                       ^ 21
                 Then, we have
                            "          #
                                      n      ^
                              1   ð  1Þ   ^ U 1 U 2              n
                                                       ^
                                                                    ^
               r
               ^ 1 ¼ 0, ^r 2 ¼   +              ¼ NTU 1   1Þ NTU 2 (7.58)
                                                             ð
                             ^ _    ^ _  ^    ^
                             C 1   C 2   U 1 + U 2
   342   343   344   345   346   347   348   349   350   351   352