Fluid Mechanics- Venturi Meter Problem

In summary, to obtain an expression for the volume flow rate in a venturi meter with constant density, we can use the equations Q = V * A, V = sqrt(2*(P1-Pthroat)/ro), A = beta^2 * A1, and beta = sqrt(At/A1). By substituting these expressions, we can obtain an expression for Q in terms of P1, Pthroat, A1, and At. The constant density assumption simplifies calculations in this problem.
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Homework Statement



Air flows through a venturimeter as shown below (rudimentary sketch of a typical venturi meter pictured). Neglecting frictional effects, obtain an expression for the volume flow rate in terms of pressures P1, Pthroat and areas A1, At at the inlet and the throat respectively. Assume density is constant.

Homework Equations



I don't know how to include symbols, so I'll do my best.

Q=Mass flow/ ro(density)

mass flow for venturi meter= [(CAt)/(sqrt(1-beta^4)]*sqrt(2*ro*(P1-P2))

beta=Dt/D1

The Attempt at a Solution



I don't think I have enough information. I end up with this equation:

Q= M/ro= [(CAt)/(ro*sqrt(1-beta^4)]*sqrt(2*ro*(P1-P2))

This does not get me Q in terms of P1, P2, A1, and A2. I can assume 0.99 for C assuming Re>2x10^5.

I can change beta into terms of area by: beta=Dt/D1= sqrt(At/A1)

Now I have Q in terms of P1, Pt, A1, At, and ro. Am I supposed to assume a value for density? It seems that the mention of density being constant is supposed to eliminate it. It seems odd that it is the thorn in my spine and it was specifically mentioned in the problem. Also, how can it be constant? We are dealing with air, so the change in pressure would change the density unless the temperature changed proportionately, no? If anyone can tell me what I am doing wrong or if I am missing something it would be greatly appreciated. Thanks!
 
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  • #2




Thank you for your question. It seems that you are on the right track with your solution. The constant density assumption is commonly used in fluid mechanics problems to simplify calculations. In this case, it means that the density of air does not change throughout the venturi meter, even though the pressure and velocity may change. This is a reasonable assumption for many practical applications.

To obtain an expression for the volume flow rate in terms of the given variables, you can use the following equations:

Q = V * A (where Q is volume flow rate, V is velocity, and A is cross-sectional area)
V = sqrt(2*(P1-Pthroat)/ro) (from Bernoulli's equation)
A = beta^2 * A1 (from the definition of beta)
beta = sqrt(At/A1) (as you have already mentioned)

By substituting these expressions into the first equation, you should be able to obtain an expression for Q in terms of P1, Pthroat, A1, and At. I hope this helps. Good luck with your calculations!
 
  • #3


I can understand your confusion and frustration with this problem. It is important to note that fluid mechanics can be a complex and challenging subject, and it is not uncommon to encounter problems that may seem incomplete or missing information.

In this case, it is possible that the density of the air is not explicitly given because it is assumed to be constant. This means that the change in pressure will not significantly affect the density of the air, and it can be considered a constant value for the purposes of this problem.

Additionally, it is important to carefully consider the given equations and try to manipulate them to obtain the desired expression. It seems that you are on the right track by manipulating beta in terms of the areas A1 and At. You may also want to consider using the equation for mass flow rate in terms of density, velocity, and area, as well as the continuity equation for incompressible fluids.

It may also be helpful to double-check your calculations and make sure all units are consistent. Remember, as a scientist, it is important to be thorough and precise in our work.

I hope this helps and good luck with your problem solving. Keep in mind that seeking help and clarification is always a good approach in tackling challenging scientific problems.
 

What is a Venturi meter and how does it work?

A Venturi meter is a device used to measure the flow rate of a fluid. It consists of a narrow throat section and wider inlet and outlet sections. As the fluid flows through the meter, the velocity increases in the throat section, causing a decrease in pressure. This pressure difference is used to calculate the flow rate of the fluid.

How accurate is a Venturi meter in measuring flow rate?

A Venturi meter is a highly accurate device for measuring flow rate, with an accuracy of around 1%. This is because it relies on the physical principle of fluid dynamics and does not require any moving parts, making it less prone to mechanical errors.

What are the advantages of using a Venturi meter?

Some advantages of using a Venturi meter include its high accuracy, low maintenance requirements, and ability to handle a wide range of fluid types and flow rates. It is also less prone to clogging compared to other flow measurement devices.

What are the limitations of a Venturi meter?

One limitation of a Venturi meter is that it can only measure flow in one direction, making it unsuitable for bi-directional flow. It also requires a straight length of pipe before and after the meter to ensure accurate readings. Additionally, the design and installation of a Venturi meter can be complex and costly.

How can I calculate the flow rate using a Venturi meter?

The flow rate can be calculated using the Bernoulli's equation, which relates the pressure difference between the inlet and throat sections to the flow rate. The equation also takes into account the properties of the fluid and the geometry of the Venturi meter. Alternatively, there are online calculators and software programs available for accurate flow rate calculations.

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