Question 14.8: The Venturi Tube The horizontal constricted pipe illustrated...
The Venturi Tube
The horizontal constricted pipe illustrated in Figure 14.19, known as a Venturi tube, can be used to measure the flow speed of an incompressible fluid. Determine the flow speed at point 2 of Figure 14.19a if the pressure difference P_1 – P_2 is known.
Conceptualize Bernoulli’s equation shows how the pressure of an ideal fluid decreases as its speed increases. Therefore, we should be able to calibrate a device to give us the fluid speed if we can measure pressure.
Categorize Because the problem states that the fluid is incompressible, we can categorize it as one in which we can use the equation of continuity for fluids and Bernoulli’s equation.
Analyze Apply Equation 14.12 to points 1 and 2, noting that y_1=y_2 because the pipe is horizontal:
P_1+\frac{1}{2} \rho v_1^2+\rho g y_1=P_2+\frac{1}{2} \rho v_2^2+\rho g y_2 (14.12)
(1) P_1+\frac{1}{2} \rho v_1{}^2=P_2+\frac{1}{2} \rho v_2{}^2
Solve the equation of continuity for v_1 :
v_1=\frac{A_2}{A_1} v_2Substitute this expression into Equation (1):
P_1+\frac{1}{2} \rho\left(\frac{A_2}{A_1}\right)^2 v_2{}^2=P_2+\frac{1}{2} \rho v_2{}^2Solve for v_2 :
v_2=A_1 \sqrt{\frac{2\left(P_1-P_2\right)}{\rho\left(A_1{}^2-A_2{}^2\right)}}Finalize From the design of the tube (areas A_1 and A_2) and measurements of the pressure difference P_1 – P_2, we can calculate the speed of the fluid with this equation. To see the relationship between fluid speed and pressure difference, place two empty soda cans on their sides about 2 cm apart on a table. Gently blow a stream of air horizontally between the cans and watch them roll together slowly due to a modest pressure difference between the stagnant air on their outside edges and the moving air between them. Now blow more strongly and watch the increased pressure difference move the cans together more rapidly.