The diffuser

There are applications in which it is very desirable to recover kinetic energy as a rise in pressure. Probably the classic example is the draught tube that is used on low-head, water turbines. In those turbines there is an incentive to keep the size of the turbine down and this can only be achieved at the expense of having a higher than desirable velocity at exit. A diffuser that starts off circular in section and changes to square or elliptical and turns through 90° as well can recover the inevitable kinetic energy as a rise in pressure that converts to a reduced pressure at exit from the turbine. Another application is the transition that must act as a diffuser between the square working section of a wind tunnel and the inlet to the fan. It is often a make or break component of the tunnel. Get it wrong and the performance is impaired.

 

We have already met a diffuser in connection with the Venturi meter. There the diffuser is straight and just a cone with an included angle of about 6°. In a diffuser the object is to convert kinetic energy to pressure energy with the minimum loss. If the flow were to be like that in a parallel pipe the loss would be attributable to friction and could be calculated using Darcy and integration. Clearly the longer the length of divergent pipe that contains high speed flow the greater the loss will be over that that which would occur in the same length of parallel pipe carrying the same flow. But the flow in the diffuser is not like that in a parallel pipe. We know that when a stream of water issues from a pipe into a reservoir the stream flows straight on and entrains water from around it. If we fitted a conical divergence to the outlet to the pipe the entrainment would be stopped and what then happens depends on the angle of the cone. Figures 8-7 and 8-8 show in principle what happens. In 8-7 the stream of water retains its basic form but creates eddies around it as shown. These eddies extend into the down-stream pipe where most of the kinetic energy is lost. At a smaller included angle the pattern changes and the stream changes to remain attached to the wall on one side and to create a large eddy on the other[1]. Again the loss of kinetic energy takes place in the large pipe. It is hard to see any reason to regard either of these transitions as diffusers. We need to concentrate on diffusers in which the flow is likely to be attached all round and over the whole length.

 

I started looking at this diffuser several days ago and it seems to get more and more complicated as the days go on. For a start this simple frustum of a cone does not produce a constant retardation of the flow. In the end I chose to avoid the complications and turned to Mathcad to have look at a few significant numbers. The programme and its outcome is shown in calculation 8-1. I chose an included angle of 6°, a flow of 0.022 m³/s because it produced velocities in the range 0 to 3 m/s. This is a frustum and it is necessary to calculate from the notional apex. I worked with lengths from 0.4 m and 1 m from the apex. This gave realistic figures. The first graph is of the diameter and that changes from about 42 mm to about 100 mm in a length of 600 mm.

 

The next graph is of velocity on the basis on one-dimensional flow and the range of velocity is from about 16 m/s to about 2.5 m/s and these are practical figures. Then the third graph gives rate of change of velocity with distance along the diffuser.

 

The graph of rate of change of velocity with distance along the diffuser shows that the velocity is changing most quickly at inlet. For this retardation to exist there must be a pressure gradient acting against the flow and this may not come into existence. However it is clear that diffusers do have attached turbulent flow and we must conclude that if detachment is to occur it will be near the inlet. It may be that a trumpet shaped diffuser would be best. It also tells us that this is a complicated flow pattern and that the divergence is nothing like as predictable as a convergence where the loss is usually very small.

 

From an engineering point of view this diffuser is going to be difficult to design and it would be prudent to look at existing designs if you are ever called upon to design one. Be prepared for rules of thumb like “use sweeping curves that are mathematically continuous”.

 

There have been experiments on the performance of simple conical diffusers and the maximum value for included angle for efficient performance is about 16° and there is a well-defined minimum that stretches from about 4° to about 10° with its turning point at about 5°. However one must remember that the factors that influence engineering design may well make it unnecessary to do anything more than use a diffuser with an effective included angle of 6°.

 

As an aside, if you want to design a transition from square to circular find the diameter of a circle of area equal to that of the square and work out a length having a notional taper of 6°. It can be constructed from triangles and parts of a cone.