(a) Using Bernoulli's equation, show that the measured fluid...

(a) Using Bernoulli's equation, show that the measured fluid speed $v$ for a pitot tube, like the one in Figure 12.7$(\mathrm{b})$ is given by $v=\left(\frac{2 \rho^{\prime} g h}{\rho}\right)^{1 / 2}$ where $h$ is the height of the manometer fluid, $\rho^{\prime}$ 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 $\mathrm{m} .$

Key Concepts

Bernoulli's Equation

Bernoulli's equation is a statement of energy conservation for flowing fluids. It relates the pressure, kinetic energy per unit volume, and potential energy per unit volume along a streamline. In fluid dynamics, it helps in understanding how pressure differences convert into changes in fluid speed, which is the principle used when analyzing devices like pitot tubes.

Pitot Tube

A pitot tube is an instrument used to measure the velocity of a fluid flow. It operates by capturing the fluid and converting its dynamic pressure into a measurable static pressure difference. The device directly utilizes Bernoulli's equation to relate the pressure difference to the speed of the fluid, making it widely applicable in aerodynamics and fluid mechanics.

Manometer

A manometer is a device used to measure pressure differences by the displacement of a liquid column. It typically uses a fluid, such as mercury, whose height difference in the columns is proportional to the pressure difference between two points. Manometers are essential for linking the hydrostatic pressure (from the weight of the liquid) to the dynamic pressure in systems like pitot tubes.

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at rest due to the force of gravity. It is directly proportional to the density of the fluid, the gravitational acceleration, and the height of the fluid column. In the context of manometers, the height difference in the liquid column provides a measure of the pressure difference in a flowing fluid, as formulated through Bernoulli’s principle.

Fluid Density

Fluid density is a measure of mass per unit volume and is a crucial property in fluid dynamics. It appears in Bernoulli’s equation and hydrostatic pressure calculations, affecting how pressure differences translate into fluid speeds. Different densities for the moving fluid and the manometer fluid are considered in practical applications to correctly apply the energy conservation principles.

Transcript

00:03 So for part a, we're going to apply bernoulli's equation, and we can say that applying the bernoulli's equation, in the case of the pilot tube, we can say that p sub 1 plus 1 1⁄2, 1⁄2, 0⁄2, 0⁄2, the would be equalling p sub 2 plus one half times row the density times v sub 2 squared plus row the density times the acceleration due to gravity times h and so of course they're at the same height so we don't have to we can eliminate those terms and then we can say that the pressure difference piece of one minus piece of so for part a we're going to apply brunuli's equation and we can say that applying the bernoulli's equation in the case of the pilot tube we can say that p sub 1 plus 2 would be equalling 1 half times the density v sub 2 squared and that is because in the pilot tube the velocity at the first tube is 0 so we can also eliminate this term and so we know that this is equaling the density prime, which in this case would be the density of the fluid inside the pilot tube.

01:57 And we can say that this would be multiplied by g, multiply 1⁄2, 1⁄2, 2, 1⁄2, plus row g0 g2, plus 1⁄2, the density times v .2 squared, plus row g ,000.

02:19 Row the density times the acceleration due to gravity times h.

02:24 And so of course they're at the same height so we don't have to we can eliminate those terms and then we can say that the pressure difference piece of 1 minus piece of two by h and so we can then solve for v sub 2 and v sub 2 would be equaling 2 times row prime times g times h divided by row all raised to the one half power and this would be our solution or this is what they're giving us in the problem statement us applying bernoulli's equation canceling out all of these terms relate to would be equalling one half times the density v sub two squared and that is because the in the pilot tube the velocity at the first tube is zero so we can also eliminate this term and so we know that this is equaling the density prime, which in this case would be the density of the fluid inside the pilot tube.

03:35 And we can say that this would be multiplied by g, multiplied by adding it to relating it to the pressure difference inside a tube.

03:47 This would essentially be the proof.

03:49 And so this would be your answer for part a.

03:53 For part b, for part b, then they simply want you to solve.

03:59 And so the velocity would be equaling the square root.

04:04 This would be two times the density of mercury, 13 ,600 kilograms per cubic meter, multiply h.

04:13 And so we can then solve for v...

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