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A feed stream with concentration c f , 1 enters Tank 1 . The outlet of Tank 1 is fed to Tank 2 along with

A feed stream with concentration cf,1 enters Tank 1. The outlet of Tank 1 is fed to Tank 2 along with an additional feed stream with concentration cf,2. Mole balances for this system give the following differential equations for the solute concentrations in Tank 1 and Tank 2, respectively:
dc1(d)t=1-1(cf,1-c1)
dc2(d)t=2-1(cf,2+c12-c2),
where 1 and 2 are the residence times for each tank. Both tanks initially contain no solute.
(a) What are the steady-state concentrations c,1 and c,2 in each tank?
(b) Transform this system of equations to use the differences c1 and c2 between the concentrations c1 and c2 and their steady-state values (i.e.,ci=ci-c,i for Tank i) as the dependent variables, rather than c1 and c2 themselves.
(c) Solve for c1(t) and c2(t) by first finding c1(t) then finding c2(t).
(d) If cf,1=1M,cf,2=0.5M,1=2min, and 2=4min, plot c1(t) and c2(t) along with their steady-state values.
(e) Rewrite the system of equations from (b) using matrices and vectors. Determine the eigenvalues and eigenvectors of the matrix for the parameters given in (d). Show that these eigenvalues and eigenvectors lead to the same solution as (c).
(f) Use the result from (c) to derive expressions for c1(t) and c2(t) if 1=2=.
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