Answer to Question #234071 in Mechanical Engineering for Randal Rodriguez

Question #234071

6. A tank is in the form of a right circular cylinder with hemispherical ends. The overall length of the tank is

4 meters and the diameter of the hemisphere is 1 meter. If a pump discharges a fluid whose density is

1.2 kg m /Liter in this tank at a rate of225 liters per minute determine a.) Weight of liquid inside the tank if

it is half full. b) total time to fill the tank assuming it is initially empty.

1
2021-09-09T00:33:20-0400

Let "d=" the diameter of the hemisphere, "l="the overall length of the tank

Find the volume of the hemisphere

"V_1=\\dfrac{1}{2}\\big(\\dfrac{4}{3}\\pi(\\dfrac{d}{2})^3\\big)=\\dfrac{\\pi d^3}{12}"

Find the volume of the right circular cylinder

"V_2=\\pi\\big(\\dfrac{d}{2}\\big)^2(l-2(\\dfrac{d}{2}))=\\dfrac{\\pi d^2(l-d)}{8}"

Find the mass of liquid inside the tank if it is half full

"m=\\rho(V_1+V_2)=\\rho\\dfrac{\\pi d^2(2d+3l-3d)}{24}"

"=\\rho\\dfrac{\\pi d^2(3l-d)}{24}"

"m=1.2\\ \\dfrac{kgm}{10^{-3}\\ m^3}\\cdot\\dfrac{\\pi (1\\ m)^2(3(4\\ m)-1\\ m)}{24}"

"=550\\pi\\ kgm\\approx1728\\ kg"

Find the mass of liquid inside the tank if it is half full

"W=mg"

"W=550\\pi\\ kgm(9.81\\ m\/s^2)\\approx16950\\ N"

"V=2(\\dfrac{1}{2})\\big(\\dfrac{4}{3}\\pi(\\dfrac{d}{2})^3\\big)=\\dfrac{\\pi d^3}{6}"

b) Find the volume of the tank

"V=2V_1+V_2"

"=2(\\dfrac{\\pi d^3}{12})+\\dfrac{\\pi d^2(l-d)}{8}"

"=\\dfrac{\\pi d^2(4d+3l-3d)}{24}"

"=\\dfrac{\\pi d^2(3l+d)}{24}"

"V=\\dfrac{\\pi (1\\ m)^2(3(4\\ m)+1\\ m)}{24}=\\dfrac{13\\pi}{24}\\ m^3"

Find the total time to fill the tank assuming it is initially empty

"t=\\dfrac{\\dfrac{13\\pi}{24}\\ m^3}{0.225 \\ m^3\/min}\\approx7.6\\ min"

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