Question 9.14: Fluid Flow in a Pipe GOAL Solve a problem combining Bernoull...
Fluid Flow in a Pipe
GOAL Solve a problem combining Bernoulli’s equation and the equation of continuity.
PROBLEM A large pipe with a cross-sectional area of 1.00 m² descends 5.00 m and narrows to 0.500 m², where it terminates in a valve at point ① (Fig. 9.33). If the pressure at point ② is atmospheric pressure, and the valve is opened wide and water allowed to flow freely, find the speed of the water leaving the pipe.
STRATEGY The equation of continuity, together with Bernoulli’s equation, constitute two equations in two unknowns: the speeds v_{1} and v_{2}. Eliminate v_{2} from Bernoulli’s equation with the equation of continuity, and solve for v_{1}.
Write Bernoulli’s equation:
(1)\quad P_{1}+\textstyle{\frac{1}{2}}\rho v_{1}{}^{2}+\rho g y_{1}=P_{2}+\textstyle{\frac{1}{2}}\rho v_{2}{}^{2}+\rho g y_{2}
Solve the equation of continuity for v_{2}:
A_{2}v_{2}=A_{1}v_{1}
{\bf(2)}\quad v_{2}=\frac{A_{1}}{A_{2}}\,v_{1}
In Equation (1), set P_{1}=P_{2}=P_{0}, and substitute the expression for v_{2}. Then solve for v_{1}.
({\bf3})\quad P_{0}+\textstyle{\frac{1}{2}}\rho v_{1}^{\mathrm{\bf2}}+\rho gy_{1}=P_{0}+\textstyle{\frac{1}{2}}\rho\left(\displaystyle{\frac{A_{1}}{A_{\mathrm{\bf4}}}}\,v_{1}\right)^{\textstyle{2}}+\rho gy_{2}
v_{1}^{~2}\left[1-\left(\frac{A_{1}}{A_{2}}\right)^{2}\right]=2g(y_{2}-y_{1}\bigr)=2g h
v_{1}={\frac{\sqrt{2g h}}{\sqrt{1~-~(A_{1}/A_{2})^{2}}}}
Substitute the given values:
v_{1}=\ 11.4\,{\mathrm{m/s}}
REMARKS Calculating actual flow rates of real fluids through pipes is in fact much more complex than presented here, due to viscosity, the possibility of turbulence, and other factors.