Pump vs$.$ System matching A water pump is used to transport water from a large reservoir $(1)$ to another at a higher elevation $(2).$ Both are open to the atmosphere. The manufacturer's pump curve is approximated as $H_{\text{a}\text{v}\text{a}\text{i}\text{l}}=H_0-c{Q}^2$ where $H_0=24\mathrm{.}4m$ and $a=2\mathrm{.}441{10}^8$ with $Q$ in units of $\frac{m^3}{s}.$ Given the system and values shown below, determine the duty point of the pump.

$z_2-z_1=7\mathrm{.}85m$

$D=2\mathrm{.}03cm$

$K_{L_1\text{e}\text{n}\text{t}\text{r}\text{a}\text{n}\text{c}\text{e}}=0\mathrm{.}5$

$K_{L\text{v}\text{a}\text{l}\text{v}\text{e}}=17\mathrm{.}5$

$K_{L\text{e}\text{l}\text{b}\text{o}\text{w}\text{s}}=0\mathrm{.}92$

$K_{L\text{e}\text{x}\text{i}\text{t}}=1\mathrm{.}05$

$L=176\mathrm{.}5m$

$f=0\mathrm{.}02$