6.3. Example Problem 5: Venturimeter

The example 5, which is a very simple problems on venturimeter as given in this figure. That there is a venturimeter with having the diameter of 15 centimeters is reduced to the 7.5 centimeters, which is installed on a horizontal pipeline. And the rate of the fluid passing through this, the Q is equal to 30 liters per seconds, Q equal to 30 liter per second.

In this example, we are discussing a device called a venturimeter, which is used to measure the flow rate of fluids in pipes. The venturimeter works by having a pipe with varying diameters; in this case, the diameter decreases from 15 centimeters to 7.5 centimeters. When fluid flows through this section of the pipe, it speeds up due to the decrease in diameter, which leads to a change in pressure. We know that the fluid passes through this venturimeter at a rate of 30 liters per second.

Think of a garden hose. When you put your thumb over the end of it slightly, the water speeds up and shoots out further. This is similar to how a venturimeter functions—when the diameter of the pipe narrows, the fluid speeds up resulting in a measurable decrease in pressure.

And the pressure difference between this throat and enlarged part. That is the pressure difference between P 1 and P 2 as we can find out the pressure difference as given here the pressure difference is equal to 2.45 meters in terms of the water head, in terms of the water head.

The pressure difference is crucial for determining the discharge coefficient of the venturimeter. Here, P1 refers to the pressure at the wider section of the pipe before the fluid passes through the narrower throat, while P2 is the pressure at the throat itself. In our example, this pressure difference is provided as equivalent to a height of 2.45 meters of water.

Imagine filling a U-shaped tube with water and noticing the difference in water levels when you push water through one side. The difference in height represents the pressure difference and is similar to what happens in a venturimeter.

This case, we can take a the centrally flow part which will be a streamlined part this. And along the streamlines, we can apply the Bernoulli’s equations to find out what could be the Q theoretical value.

In fluid dynamics, Bernoulli’s equation helps us relate the pressure, velocity, and height of a flowing fluid. In our case, we apply it along the streamline of the venturimeter. By doing so, we can calculate the theoretical flow rate (Q) based on the cross-sectional areas of the pipe before and after the throat and the measured pressure difference.

Consider driving a car: the faster you go (higher velocity), the more you feel the wind resistance (lower pressure inside the car). Similarly, in a venturimeter, as fluid speeds up in the narrow section, it creates a pressure drop that can be calculated using Bernoulli’s equation.

If I apply this Bernoulli’s equations, Bernoulli’s equations as we apply on along this the streamlines, it can easily we can write it the same form. Since in this case, whatever the Q amount of the flow comes it, the Q will be the A V, A V. And since it is, we know the pressure difference between 1 and 2 points, that is the reasons we can easily compute this pressure difference between two point as it is measure it.

By applying Bernoulli’s equation and knowing the cross-sectional areas at points 1 and 2, we can derive a formula for the flow rate. Given the known values for pressure difference and pipe dimensions, we can determine the theoretical discharge (Q theoretical) of the venturimeter. The actual discharge can then be used to compute the discharge coefficient (C_d).

Think of it this way: if you're trying to optimize the output of a juice extractor, you would measure how much juice it should ideally produce (theoretical) based on the fruit size and type, then compare it to what you actually get (actual) to understand its efficiency (discharge coefficient).

So the basic idea is here to apply the Bernoulli’s equations along this streamlines. If I apply this Bernoulli’s equations, you know this value of A A value as the diameter is given to us 2.45, 2g all the value is 1 2 given to us. So we can compute the C d value which comes out to be 0.95 for this case.

After substituting all known values into our equation derived from Bernoulli’s, we can isolate the variable for C_d (discharge coefficient) to find that it equals 0.95. This number helps us understand how efficient this venturimeter is in moving fluid through the system compared to its theoretical flow rate.

If we think of the discharge coefficient as a performance rating, a C_d of 0.95 means this venturimeter is quite effective, like having a highly efficient appliance that uses almost all of its input energy to produce output.

--