Venturi meter. The Venturi meter is a device used to measure...

Venturi meter. The Venturi meter is a device used to measure the speed of a fluid traveling through a pipe. Two cylinders are inserted in small holes in the pipes, as shown in Figure $13.48 . \quad$ since the cross-sectional area is different at the two places, the speed and pressure will be different there also. The difference $(h)$ in the heights of the two columns can easily be measured, as can the cross-sectional areas $A_{1}$ and $A_{2} .$ Notice that points 1 and 2 in the figure are both at the same vertical height. (a) Show that $\Delta p=\rho g h,$ where $\rho$ is the density of the fluid and $\Delta p$ is the pressure difference between points 1 and 2 (b) Apply Bernoulli's equation and the continuity condition to show that the speed at point 1 is given by the equation $$ \begin{array}{l}{v_{1}=\sqrt{\frac{2 g h}{\left(A_{1} / A_{2}\right)^{2}-1}},(\mathrm{c}) \text { How would you find the speed at }} \\ {\text { point } 2 ?}\end{array} $$

Key Concepts

Venturi Meter

A Venturi meter is an instrument that measures the velocity of a fluid by utilizing changes in cross-sectional areas in a flow tube. It operates on the principles of Bernoulli's equation and the continuity equation, allowing the calculation of fluid speed by measuring the pressure difference induced by a constriction in the tube.

Pressure Head

Pressure head refers to the height of a fluid column that produces a certain pressure due to the force of gravity. It is often used in fluid statics to convert between pressure and height, and plays a key role in relating measurable height differences in manometers to pressure differences in flowing fluids.

Bernoulli's Principle

Bernoulli's principle states that for an ideal fluid flowing without friction, the total energy (pressure energy, kinetic energy, and potential energy) along a streamline is constant. This principle is used to relate pressures and velocities at different points in a flowing fluid, helping to explain how pressure differences arise when the flow speed changes.

Continuity Equation

The continuity equation is a statement of the conservation of mass for a fluid in steady flow, indicating that the mass flow rate must remain constant throughout a closed system. This means that any decrease in cross-sectional area must be accompanied by an increase in velocity, and vice versa, which is crucial for understanding flow behavior in a Venturi meter.

Transcript

00:01 Okay, so if we look to the figure of this problem, we have the venturi meter, which is something like this.

00:07 Let's draw here.

00:13 We have the first cross -sectional area and we have another area in here.

00:24 So let's see.

00:26 Here we have the other hole, the venture in the pipe.

00:36 So it's something like this.

00:43 It's not quite good but this is divinity divinity meter okay so here we have the water so here we have water and here we have the difference in height so this distance here is what we call and call okay so here is the point two and here are the point one point two so that's our system now that we have the system well established we can answer what the problem wants and well the first item of this problem we just need to show a relation of the pressure okay so first of all, let's look to the system and we're going to say that the pressure on point one.

02:11 Since the hole is open into the air, we have atmospheric pressure plus h1, because h1 is the distance from the liquid.

02:34 To the point one so this is h1 and the pressure on point two let's put in here the pressure on point two it's going to be the atmospheric pressure plus g age 2 where age 2 is just this heighten here so this is age 2 now to answer to answer the question we have to determine what is the difference in pressure between the two points so we can say that p1 minus p2 is going to be delta p the difference in pressure and that is just we can cross the atmospheric pressure because it's equal on both ends and what is left is just that multiplies h1 minus h2.

03:53 But if we see h1 minus h2 is just equal h.

04:00 Therefore, we show in the first item that the difference in pressure between the two points is just hgh.

04:10 That's the first answer.

04:12 Now let's move on to the second item.

04:15 Okay, so in the second item, we have to use the bernoulli's and the continuity equation to show a relation that the problem wants.

04:26 So first of all, let's remember that the bernoulli's equation is just p1 plus the connecting energy of point 1, which is whole v1 square, that is equal the pressure on point 2 plus the connect the energy of point 2...

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