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

Question #130129
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.
1
Expert's answer
2020-09-03T14:10:58-0400

solution:-

given data

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

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

diameter of pipe (D) =0.04 m

length of pipe (L)=70 m

maximum velocity(umax)=6 m/s

the velocity profile in fully developed laminar flow is


u(r)=umax(1r2R2)u(r)=u_{max}(1-\frac{r^2}{R^2})


=6(1r2(0.02)2)=6(1-\frac{r^2}{(0.02)^2})


u(r)=6(12500r2)m/s\fcolorbox{red}{yellow}{$u(r)=6(1-2500r^2)m/s$}


average velocity

Va=umax2=62=3m/sV_{a}=\frac{u_{max}}{2}=\frac{6}{2}=3m/s


flow rate

Q=VaAQ=V_aA


Q=3×π×(0.04)24Q=3\times\pi\times\frac{(0.04)^2}{4}

Q=3.77×103m3/sQ=3.77\times10^{-3}m^3/s

renolds number can be given


Re=ρVaDμRe=\frac{\rho V_aD}{\mu}


=1252×3×0.040.3073=\frac{1252\times3\times0.04}{0.3073}


Re=488.9Re=488.9


Re<2300Re<2300

so its laminar flow.

therefore friction factor

f=64Re=64488.9=0.1309f=\frac{64}{Re}=\frac{64}{488.9}=0.1309


and head loss

H=fLVa2D2gH=f\frac{LV_a^2}{D2g}


H=0.1309×70×320.04×2×9.8H=\frac{0.1309\times70\times3^2}{0.04\times2\times9.8}

H=105.1mH=105.1m

by using the Bernoulli equation pressure difference can be written as


ΔP=P1P2=ρg(Z2Z1+H)\Delta P=P_1-P_2 =\rho g(Z_2-Z_1+H)


pipe is horizontal so Z1 and Z2 will be zero.


ΔP=1252×9.8(0+105.1)\Delta P=1252\times9.8(0+105.1)


ΔP=1291×103Pa\fcolorbox{green}{yellow}{$\Delta P=1291\times10^3 Pa$}


pumping power can be given as


pumping power=Q×ΔPpumping\space power = Q \times\Delta P

=3.77×103×1291×103=3.77\times10^{-3}\times1291\times10^{3}


pumping power=4.87×103W\fcolorbox{green}{yellow}{$pumping \space power=4.87\times10^3 W$}



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