where M is the vector (Mi,M2) describing the contents of the two reservoirs and k the matrix:
The eigenvalues of k are the solutions to the equation
where Ei is the nonzero eigenvalue with smallest absolute value (Lasaga, 1980). The treatment a\ = 0 and k2 - - ~(k\2 + kr\). The general solution to Equation (17) can be written as
where \j/i and ij/2 are the eigenvectors of the matrix k. In our case, we have
or, in component form and in terms of the initial conditions:
It is seen that in the steady state the total mass is distributed between the two reservoirs in proportion to the sink coefficients (in reverse proportion to the turnover times), independent of the initial distribution.
In this simple case there is only one time scale characterizing the adjustment process, that is (k]2 + k2\ ) This is also the response time, Tcycie/ as defined by Equation (13).
or, if expressed in terms of the turnover times of the two reservoirs:
The response time in this simple model will depend on the turnover times of both reservoirs and will always be shorter than the shortest of the two turnover times. If r0i is equal to t02/ then icycie will be equal to half of this value.
An investigation of the dynamic behavior of a coupled three-reservoir system using the techniques described above is included in the problems listed at the end of the chapter.
It should be noted that the steady-state solution of Equation (12) is not necessarily unique. This can easily be seen in the case of the four-reservoir system shown in Fig. 4-7. In the steady state all material will end up in the two accumulating reservoirs at the bottom. However, the distribution between these two reservoirs will
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