The Venturi meter

The best-known application of the energy equation must be in the flow measurement device called the Venturi meter. It was invented by Herschel but named after Venturi. It exploits the drop in pressure that occurs when a liquid or a gas that flows along a pipe passes through a smooth contraction to make it into a flow meter. This is simple in principle but has to be applied in practice. All the work needed to make a practical flow meter was done at least 100 years ago and nothing has changed. You can find it all in Gibson’s book of 1912 called “Hydraulics and its applications”.

 

The success of the simple contraction shown in figure 5-14 depends on the existence of an expression on which a coefficient can be defined so that the volume flow can be related to the dimensions of the contraction and the pressure drop between sections 1 and 2. Then the coefficient can be found by experiment.

For the contraction we can write using the energy equation that :-

        .

Then:-

             .

 

 We know that :-

 and substitution gives:-

       . However, if we let  this reduces to :-

                                              or:-

                                                              

This expression is very simple. It says that the difference in head between inlet and throat is equal to the kinetic energy at inlet multiplied by a number that depends on the ratio of the diameters at inlet and throat. This makes sense.

 

We can explore this expression using say Mathcad because the range of  is not great in practical pipes and will not normally exceed say 3 m/s.

 

I have run this out in graph 5-1 and it is evident that it will be necessary to choose a diameter ratio to match the instrument used to make the measurement to the manometric head developed between the inlet and throat. Further the Venturi meter is not suitable for measuring flows at low velocity e.g. 0.3 m/s.

 

Generally a Venturi meter is used to measure flow and the expression can be rearranged to give volume flow for a given difference in head. It becomes:-

                         

 

This is an expression for flow in terms of the measurable quantities and the real flow will be given by multiplying by a coefficient  to be found by experiment.

                                                                 

 

The application of the energy equation and continuity has yielded an expression that tells us that this contraction can be used as a flow meter. In order to get it we did not analyse the behaviour of liquid as it flows though the contraction, we simply applied the equations in a mechanistic way to get an expression that is suitable for gathering and storing data. Now we have to look at the engineering design that will make it into a flow-meter.

 

Others, and Herschel was the first, have found out what we need to do. Normally the meter is made with a conical contraction and a conical divergence and the two are joined with a profile that is an easy curve to produce a throat. Clearly an essential feature of the design is the method of detecting the pressures at the inlet to the contraction and at the throat.

 

I have drawn just the contraction in figure 5-15. The proportions come from Gibson who suggests a length equal to 2.5 times the pipe diameter and a throat diameter equal to 1/3 of the pipe diameter. The manometric connections are made to two piezometer rings that are annular rings like an egg band on an earthworm. The rings are cast as part of the instrument and at least four holes are drilled through the wall of the contraction so that liquid fills the rings and any pipes connected to the rings. The holes must not be so small that they clog with dirt or so big that they disrupt the flow over them. They must be free from any rag or burr. Then, if the contraction follows a straight run of pipe at least 15 pipe diameters in length and the diameter of the pipe is greater than 50 mm the value of the coefficient lies between 0.97 and 1. In effect, for most purposes the Venturi meter can be used without calibration.

 

This result means that the flow in the contraction has not caused any significant loss of energy to internal friction. However the water is moving at high speed in the throat and if we just have an abrupt change in diameter back to that of the pipe all that kinetic energy will be lost. This may not matter but, if the water is being pumped along the pipe there will be financial cost to be met and it may be significant. Certainly its importance is set to grow. So a transition must designed to reduce this potential loss if not to minimise it.

 

If the transition is conical, and this is certainly the easiest shape to make, it turns out that the included angle of the cone for minimum loss is 5.1°. I have drawn the profile of such a Venturi meter in figure 5-16 and this is a very slow taper and I find it to be counter intuitive. I say this because one would think that, as the loss is caused by the high velocity, it would be best to reduce the velocity as soon after the throat as possible. Massey gives a graph of loss versus included angle of the cone and it is clear that the loss is small over a range of angle from 3° to about 11° with the lowest at this 5.1°. Somewhere about 20° the flow breaks down and the losses increase rapidly with increase in included angle. The consequence is that, if lowest running cost is an important factor, the Venturi meter will be about 10 diameters long.

 

There is nothing in the application of the energy equation that would be affected if the convergence were to be much shorter and the high value of the coefficient means that the convergent flow is almost free from losses so the convergence could be shortened without affecting the calibration very much. The divergence could be shortened as well with a consequent increase in running cost.[1] Balancing all these factors is what engineers are paid to do.

 

Taken all round the Venturi meter is a versatile device capable of being used for pipes of any size from 50 mm and upwards. It can often be used without calibration and be physically adaptable to circumstances.

 

It depends on a simple expression derived from the energy equation. I like to call such expressions rational expressions because they are not the outcome of some mathematical analysis but of a search for a simple expression that contains only measurable quantities, has the correct dimensions, and can be used with a coefficient to create a storage and retrieval system for data.