$\frac{dV}{dt}=w_1+w_2-w$ ove ral $(4-65)$

$V\frac{dx}{dt}=w_1(x_1-x)+w_2(x_2-x)$

Ccourpound

Assume that volume $V$ remains constant $($due to an overflow line that is not shown$)$ and consequently, $w=w_1+w_2.$ Inlet composition $x_1$ and inlet flow rates $w_1$ and $w_2$ can vary, but stream $2$ is pure solute so that $x_2=1.$

The nonlinearities in Eq$.4-66$ are due to the product terms, $w_1{x}_1,$ and so forth. The right side of $(4-66)$ has the functional form $f(x,x_1,w_1,w_2).$ For Step $1$ of Fig. $4\mathrm{.}5,$ find the steady$-$state values of $x$ and $w$ by settings the derivatives of $(4-65)$ and $(4-66)$ equal to zero and substituting the steady$-$state values. For Step $2,$ linearize $(4-66)$ about the nominal steady$-$state values to obtain