$(30$ points$)$ Lecture $7$B; Step $2$: Find partial derivatives of $f_i$ with respect to $p,w_1,w_2,w$

Step $3$: Select trial values:

$P=730$;$w_1=2\mathrm{.}8$;$w_2=1\mathrm{.}4$;$w=4\mathrm{.}8$

Step $4$: Calculate the numerical values in the matrix including $f_i$

$\left[\begin{array}{cc}?& ?\\ ?& ?\\ ?& ?\\ ?& ?\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\\ x_4\end{array}\right]=$ Step $5$: Substituting the temporary values of the variables into the equations for the partial

derivatives forms, a set of linear simultaneous equations to be solved for the corrections to $x_i$ :

Step $6$: Find solutions of the simultaneous equations after $3$ iterations.

$P=$;$w_1=$;$w_2=$;$w=$

Example $6\mathrm{.}3$: Water$-$pumping system

$($Use Newton$-$Raphson simulation$)($Please do not use the successive substitution method$)$

A water$-$pumping system consists of two parallel pumps drawing water from a lower reservoir

and delivering it to another that is $50$ m higher.

The pressure drop due to friction in the pipe is $8\mathrm{.}0w^2$kPa,

where $w$ is the combined mass flow rate $k\frac{g}{s}.$

Eq$.1(f_l)$:$p[kPa]=P$ due to friction $+P$ due to elevation

Eq$.2(f_2)$: Pump $1$: $p[kPa]=840-30w_1-4\mathrm{.}0w_1^2$

Eq$.3(f_3)$ Pump $2$: $p[kPa]=950-70w_2-28w_2^2$

Eq$.4(f_4)$ Mass balance of water:

Objective: to determine the values of $p,w_1,w_2,$ and $w$ using Newton$-$Raphson method.

Step $1$: Construct four equations, $f_1,f_2,f_3,$ and $f_4$

$f_1=$

$f_2=$

$f_3=$

$f_4=$