Page 354 - Design and Operation of Heat Exchangers and their Networks

340 Design and operation of heat exchangers and their networks

solution for a gas-to-gas exchanger with neither fluid mixed was developed.
Romie (1983) and Gvozdenac (1986, 1991) obtained the transient responses
of crossflow exchangers to inlet temperature step changes. Kabelac (1989)
proposed a lumped parameter cell model to calculate the transient responses
of the liquid and gas outlet temperatures of a finned crossflow heat
exchanger. The model allows for inlet temperature and mass flow rate per-
turbations. Luo (1998) developed a dynamic dispersive model to predict the
transient responses to arbitrary inlet temperature variations.

7.2 Solution methods for dynamic behavior of heat
exchangers

The methods suitable for solving governing equations to obtain transient
responses of heat exchangers can be divided into analytical and numerical
methods. The Laplace-transform method is usually adopted in analytical
solutions if the dynamic problem can be regarded as linear, while the
finite-difference method is frequently applied in numerical solution. Some-
times, Laplace-transform and finite-difference methods can be combined to
obtain solution for transient responses of complex heat exchangers.

7.2.1 Direct solution
For some simple or simplified problems, the easiest method is to obtain the
direct analytical solution. For example, the governing equations of the
lumped parameter model can be linearized so that they can be changed into
the following linear differential equations with constant coefficients:

dT
¼ AT + B τ ðÞ (7.92)
dτ
τ ¼ 0 : T ¼ T 0 (7.93)
where T is the temperature vector of M streams and pieces of solid wall, A is
an M M coefficient matrix, and B is the heterogeneous term. The solution
of Eqs. (7.92), (7.93) has been given in Chapter 2 as follows:
τ
ð
Rτ 1 R τ τ Þ 1 0 0
0
ð
T ¼ He H T 0 + He H B τðÞdτ ð 2:140Þ, (7.94)
0
Rτ r 1 τ r 2 τ r M τ
where e ¼diag{e ,e ,…,e }, r i (i¼1, 2, …, M) are the eigenvalues of
the coefficient matrix A, and H is an M M matrix whose columns are the
eigenvectors of the corresponding eigenvalues.