Problem $2$:

An inlet flow $q_i(t)$ flows to a water tank with uniform cross sectional area of $1m^2.$ The tank is connected

through a pipe to the atmosphere. At steady state $($nominal condition$),$ the inlet flow is $1000m\frac{l}{sec},$ and equals

the outflow $q_o(t),$ the pressure of the water tank is $250000$ Pa $.$

For a condition that the flow through the pipe is turbulent flow, as Reynolds number is larger than $2100$ and

the inlet flow varies from the steady state value by a function $qi(t)=qi\_$nom $+200sin10tm\frac{l}{sec}.$

If for turbulent flow in pipe, the pressure drop of the pipe is related to the volume flow in the following

equation:

$P(t)=k**q(t{)}^2$ where

$k=0\mathrm{.}25P\frac{a}{(m\frac{l}{sec}{)}^2}$

$($a$)$ Draw the equivalent circuit system

$($b$)$ Find $R_{\text{l}\text{i}\text{n}\text{e}\text{a}\text{r}\text{i}\text{z}\text{e}\text{d}}$ at the nominal flow rate.

Ans. $500P\frac{a}{m\frac{l}{sec}}$

$($c$)$ Find the capacitance of the tank

Ans. $102m\frac{l}{P}a$

$($d$)$ Find the transfer function from $TF=Q_0\frac{S}{Q_i}(S)$

Ans. $TF=\frac{1}{5100s+1}$