The hole in a tank

I choose the next example because it shows the energy equation being used to influence design.

 

There are applications of engineering where, in the design stage, it is desirable to be able to predict the rate at which liquid would flow out of a hole in the side or bottom of a tank. Such applications are, in dosing where one flowing liquid is mixed with another, in measurement of rate of flow and in estimating the time taken for a tank to empty through a hole.

 

Holes and tanks can have all sorts of shapes and sizes and the hole can be put anywhere. As we cannot make a rigorous analysis of even the most simple combination we must expect to have to experiment. However the number of possible combinations of hole and tank is much too great to justify gathering the data needed to make an accurate prediction in every case.

 

We have to limit the experimental work to something that is manageable and then devise a way to store the resulting limited data so that it can be retrieved and used to make a sensible prediction.

 

Let us start by considering the shape of the hole. Suppose that it is to be made in the plate from which the tank is constructed. The obvious hole would be made by drilling the plate and chamfering off the ragged edge produced by the drilling. A cross-section of such a hole is shown in figure 5-5. If we attempted to store data about the rate of flow through this hole we would need to be able to describe the hole in geometrical terms. For this we would need to state the thickness of the plate, the diameter of the hole, the size and angle of the chamfer and, presumably make a statement about the surface finish of the plate and the hole. This involves five dimensions, each of which may influence the rate of flow.

 

Clearly it would be the work of a life-time to gather sufficient data to be able to predict accurately the rate of flow for any given combination of these dimensions even if the shape of the hole were to be the only criterion. In practice the rate of flow is also dependent on the way in which the liquid approaches the hole and on the properties of the flowing liquid.

 

So we must reduce the geometry to something with fewer dimensions and think of a way to deal with different shaped holes. Presumably the ideal hole for experiment would have one dimension, the diameter, as shown in cross-section in figure 5-6. Such a hole would be impractical because the sharp edge would wear very quickly. In fact the sharp edge is not essential and the device that has become known as the sharp-edged orifice is made as shown in cross-section in figure 5-7. The short axial length of the orifice strengthens the sharp edge of 5-6 but the liquid, as it forms a jet, flows cleanly from the square edge, as shown in figure 5-8, just as if it is sharp. The jet that emerges from such an orifice looks like a rod of glass.

 

We must now consider how we might experiment to gather data on the rate of flow through a sharp-edged orifice. So far we have said nothing about the shape of the tank and if the orifice were to be fitted as shown in figure 4-9 or in figure 4-10 we would not expect the flow pattern of the liquid approaching the orifice to be the same as that in figure 4-8. In both cases the shape of the jet would be affected by the proximity of the walls of the tank and so presumably the rate of flow affected.

The only thing to do is to test a well-made orifice in a large tank with the orifice mounted flush with the flat, horizontal bottom of the tank so that the distant walls of the tank do not affect the flow pattern. The arrangement is shown in figure 5-11. Then the system has only two dominant dimensions, the depth  and the diameter of the orifice .

 

Others have experimented at great length and we have their data. In order to use it we must see how it was stored.

 

The rates of flow  through an orifice of diameter  were measured under a range of steady heads . Then ,  and  were related by:-

                where  is a coefficient called the coefficient of discharge and is a number. This expression defines  and, as  is the area of the orifice and  is the velocity that the liquid would have if it fell, without resistance, through a distance , the product  has the units of rate of flow. In no sense is the expression above an attempt to predict the flow, it is simply the creation of a dimensionally correct expression that involves only the measurable quantities. It is the maximum conceivable flow through the orifice and it is a comprehensible basis for the definition of a coefficient. It should be noted that if the energy equation were to be used to find the velocity of the jet it would have to be applied to the free surface and to the jet where it first becomes parallel. The position of this latter point is not easily determined and it is thought to be better to define the coefficient of discharge in terms of the measurable quantity  than to undertake to locate the end of the contraction of the jet.

 

How attractive this expression would be if it transpired that  is independent of  and  for all known liquids! Of course this is not the case and values of  are stored in the form of graphs like graph 5-2 for each liquid. For water there is a region above  = about 450 mm for which  for each orifice is independent of . As liquids that are inflammable or toxic would not be allowed to flow with a free surface, the practical range of liquids is relatively small and graph 5-2 could be used where the liquid has a similar mobility to that of water.

 

The testing of an orifice under steady flow conditions involves an inflow and this will prevent the water being free from eddies as it flows to the orifice. No information is given by the experimenters on the state of the water in the test tank but in fact the disturbance of the water as a result of filling has no detectable effect on the flow. Indeed even a fully-developed, air-entraining vortex as shown in picture 5-1 makes little difference to the rate of flow. (In picture 5-1 you can see the vortex at the top of the picture that was taken when the level in the tank was low. The vortex could be established easily at any depth.) It seems that in convergent flow disturbances are quickly damped out.

 

The flow through an orifice is not significantly different to that through the nozzle that has already been considered in this chapter, so the typical value of  of 0.63 is lower than might have been expected. However figure 5-8 shows that the jet is smaller in diameter than the orifice and it is this contraction, which reduces the area of the jet to less than that of the orifice, which accounts for the low value of . A further useful piece of information for the engineer is that the loss in the flow as it converges to form the jet varies much more with diameter than with head. The loss ranges from about 9% for  =20 mm to 1% for  = 60 mm.

 

We can now consider our original problem of the tank with a hole. We now know that the energy loss in the converging flow can be related to hole size and that it is the effective area of the jet which dominates . We also know that the effective area depends on the flow pattern upstream of the hole. We might then expect that the flows through the orifices in figures 5-9 and 5-10 would be greater than for an orifice that is not affected by the walls of the tank. Figure 5-12 shows the flow pattern through a short pipe with a square entry. The flow separates at the sharp edge and then contracts just as it does in the sharp-edged orifice but, in the short pipe, it expands again to fill the pipe. Unfortunately divergent flow usually breaks down into small eddies and this happens in this short pipe with considerable additional loss of energy.  varies with the ratio of l/d and, for l/d = 3,  is approximately 0.83.

 

The contraction and divergence of the short pipe can be eliminated by changing the square edge to a radius as shown in figure 5-13. In this case  varies with three dimensions and typically, for r/d = 0.5 and l/d = 3,  is about 0.95.

 

The data given above would be adequate to make a useful estimate of  in most cases. Only the designer would know whether the application justified any further commitment of time and money to get an accurate value.