Answer to Question #130129 in Mechanics | Relativity for Ariyo Emmanuel

Question

Consider the fully developed flow of glycerin at 40 °C through a 70-mlong, 4-cm-diameter, horizontal, circular pipe. If the flow velocity at the
centerline is measured to be 6 m/s, determine the velocity profile and the
pressure difference across this 70-m-long section of the pipe, and the
useful pumping power required to maintain this flow.

Answer 1

solution:-

given data

density of glycerin ($\rho$ ) =1252 kg/m^3

viscosity of glycerin($\mu$ )=0.3073 kg/m-s

diameter of pipe (D) =0.04 m

length of pipe (L)=70 m

maximum velocity(u_max)=6 m/s

the velocity profile in fully developed laminar flow is

$u(r)=u_{max}(1-\frac{r^2}{R^2})$

$=6(1-\frac{r^2}{(0.02)^2})$

$\fcolorbox{red}{yellow}{$u(r)=6(1-2500r^2)m/s$}$

average velocity

$V_{a}=\frac{u_{max}}{2}=\frac{6}{2}=3m/s$

flow rate

$Q=V_aA$

$Q=3\times\pi\times\frac{(0.04)^2}{4}$

$Q=3.77\times10^{-3}m^3/s$

renolds number can be given

$Re=\frac{\rho V_aD}{\mu}$

$=\frac{1252\times3\times0.04}{0.3073}$

$Re=488.9$

$Re<2300$

so its laminar flow.

therefore friction factor

$f=\frac{64}{Re}=\frac{64}{488.9}=0.1309$

and head loss

$H=f\frac{LV_a^2}{D2g}$

$H=\frac{0.1309\times70\times3^2}{0.04\times2\times9.8}$

$H=105.1m$

by using the Bernoulli equation pressure difference can be written as

$\Delta P=P_1-P_2 =\rho g(Z_2-Z_1+H)$

pipe is horizontal so Z₁ and Z₂ will be zero.

$\Delta P=1252\times9.8(0+105.1)$

$\fcolorbox{green}{yellow}{$\Delta P=1291\times10^3 Pa$}$

pumping power can be given as

$pumping\space power = Q \times\Delta P$

$=3.77\times10^{-3}\times1291\times10^{3}$

$\fcolorbox{green}{yellow}{$pumping \space power=4.87\times10^3 W$}$