Chapter 12, Problem 24PE

(a) Using Bernoulli's equation, show that the measured fluid speed v for a pitot tube, like the one in Figure 12.7(b), is given by $v={\left(\frac{2\rho \text{'}gh}{\rho }\right)}^{1/2}$ , where h is the height of the manometer fluid, $\rho \text{'}$ is the density of the manometer fluid, $\rho$ is the density of the moving fluid, and g is the acceleration due to gravity. (Note that v is indeed proportional to the square root of h, as stated in the text.) (b) Calculate v for moving air if a mercury manometer's h is 0.200 m.

Figure 12.7 Measurement of fluid speed based on Bernoulli's principle. (a) A manometer is connected to two that are close together and small enough not to disturb the flow. Tube 1 is open at the end facing the flow. A dead spot having zero speed is created there. Tube 2 has an opening on the side, and so the fluid has a speed V across the opening; thus, pressure there drops. The difference in pressure at the manometer is $\frac{1}{2}\rho v_2^2$ , and so h is proportional to $\frac{1}{2}\rho v_2^2$ . (b) This type of velocity measuring device is a Prandtl tube, also known as a pitot tube.