A Venturi meter with a minimum diameter of 3 in. is to be...

A Venturi meter with a minimum diameter of 3 in. is to be used to measure the flowrate of water through a 4 -in.-diameter pipe. Determine the pressure difference indicated by the pressure gage attached to the flowmeter if the flowrate is $0.5 \mathrm{ft}^{3} / \mathrm{s}$ and viscous effects are negligible.

Key Concepts

Continuity Equation

The continuity equation is a fundamental principle in fluid mechanics that states the mass flow rate of an incompressible fluid remains constant along a streamline. In practice, this means that as the cross-sectional area of a pipe decreases, the velocity of the fluid must increase to maintain the same flow rate. This concept is essential for calculating the changes in velocity between the larger pipe and the constricted throat of the meter.

Bernoulli's Equation

Bernoulli's equation relates the pressure, velocity, and elevation of a fluid in steady, incompressible, and frictionless flow. It is used to understand how the increase in velocity in a constricted area of the flow leads to a decrease in pressure. This pressure difference is the working principle behind devices like the Venturi meter for measuring flow rates.

Venturi Meter

A Venturi meter is an instrument used for measuring the flow rate of a fluid by constricting the flow in a pipe and using the pressure difference between the wide and narrow sections. The meter harnesses the concepts of the continuity equation and Bernoulli's equation by creating a high-velocity, low-pressure region at the throat of the meter, which can be correlated to the volumetric flow rate.

Transcript

00:01 So in this question, what we have been given is the, there is a venturi meter, right, and its minimum diameter is equal to three inches, right? now, it says that it is used to measure the flow rate of water.

00:21 Now, the diameter of the pipe through which it can measure the flow of the water, it's equal to four inches.

00:28 Now, we are required to determine the pressure difference, right? the pressure difference that is indicated by the pressure gate attached to the flow meter, right? and it says that the flow rate, the flow rate is equal to 0 .5 feet cube per second, right? and say that the viscous effects, they are negligible.

00:56 So let's start with the solution.

00:59 Of this question is very simple we are required to find out the pressure difference right so the generic the sorry the general volumetric flow rate right for the venture meter sorry the venture meter is given by this expression right that is the volumetric flow rate that is equal to a sub 2 under root 2 times the pressure difference divided by the density of the water times one negative the ratio of the area of the surfaces right and then what we are going to do is that we are going to plug in the value for this area right and for this area it's equal to pi divided by four d sub two square right and this fraction it's given by the expression d that is a diameter right in terms of the diameter square right so let's plug them in over here right so that we have the volumatic 4x2 rate is equal to pi divided by 4 d sub 2 square then we have the under root in the under root there is a numerator so in that numerator we have two times the pressure difference and it's divided by the density of the water and then again we have in close parenthesis so over there we have one negative the sub 2 the sub 1 and the squares are involved right so what we're going to do is that we are going to refer to the table 1 .5 right and we are going to substitute the value of the density from there right so the value of density for water in the british carbureational system is 1 .94 slugs per feet cube, right? and then we have all the other physical quantities values available, right? so we just need to plug them in.

03:21 And we can find all the pressure difference, right? so let's take a look at that...

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