$(1)(30$ points$)$ Lecture $9$A; $($b$)$ Construct the matrix of partial derivatives

$\left[\begin{array}{ccc}\frac{del{f}_3}{del{x}_1}& \frac{del{f}_3}{del{x}_2}& \frac{del{f}_3}{del{x}_3}\\ \frac{del{f}_2}{del{x}_1}& \frac{del{f}_2}{del{x}_2}& \frac{del{f}_2}{del{x}_3}\\ \frac{del{f}_1}{del{x}_1}& \frac{del{f}_1}{del{x}_2}& \frac{del{f}_1}{del{x}_3}\end{array}\right]$

$($c$)$ Construct the matrix of partial derivatives with numbers using the following values near the

solutions: $t=40C,w_c=1\mathrm{.}2k\frac{g}{s},$ and $UA=320k\frac{W}{K},$

$($d$-1)$ Multiply the lower$-$triangular terms by the constant $($d$-2)$ Solve the matrix to determine the values of the constant $$

$($d$-3)$ Determine whether the calculation shall converge or diverge.

$($e$)$ If your solution diverges, stop here. If converges, determine the flow rate of condensed

water recovered, $w_c,$ through the system simulation using Lagrange multiplier method.

Problem $14\mathrm{.}2$: Pure water recovery $($distillation$)$ in boiler operation.

$($Use Lagrange multiplier method$)$

One of the methods by which some of the water from the blowdown of a boiler can be recovered is to throttle it to a flash tank, generating vapor and concentrated liquid with unwanted metal ions for discharge. Then, the vapor from the flash tank moves to a condenser, where all of the vapor is condensed to the steam, which is recovered as a condensate from the bottom of the condenser as shown below. The condensate can be reused as a boiler feedwater, if desired.

Given:

The enthalpy of water: $h_f[k\frac{J}{k}g]=4\mathrm{.}19t$

The enthalpy of water vapor: $h_g[k\frac{J}{k}g]=2450+1\mathrm{.}9t$

Eq$.1$: Heat balance at condenser: $w_c(h_g-h_f)=m_c^{}(c_p)(T_{c,o}-T_{c,ii})=m_c^{}(c_p)(T_{c,o}-18)[1-e^{-\frac{UA}{m_c(4\mathrm{.}19)}}]$

Eq$.2$: Universal heat transfer rate at condenser: $UA=\frac{350}{(w_c{)}^{0\mathrm{.}2}}$

Eq$.3$: Heat balance at flash tank: $q,$

Blc

$($a$)$ Construct $f_l,f_2,$ and $f_3$

$f_1=$

$f_2=$

$f_3=$