water at $20$C in a horizontal pipe of $10$cm diameter under fully developed conditions with a flow rate of Q$=4*20^-2$Question # $3($Competencies a$1,$ a$2,$ b$1,$ b$4,$ c$3)$

$(20$ Marks$)$

$(3-$a$)$ Find, from velocity distribution for laminar flow in pipe given below.

$u(r)=-\frac{R^2}{4}(\frac{dp}{dx})(1-\frac{r^2}{R^2})$

a$)$ The centerline $($maximum$)$ velocity, $u_{\text{m}\text{a}\text{x}}$

b$)$ The average velocity, $U_{\text{a}\text{v}\text{g}}$

c$)$ The volume flow rate $($discharge$),$ in terms of head loss and pipe

diameter;

d$)$ The friction factor $f$ as function of Reynolds number.