The sudden enlargement

The expression that is commonly used for loss in a fitting that is used as an enlargement is interesting because of the way in which the total head equation and the momentum equation are used to derive it in a form that involves measurable quantities and on which we can base a coefficient.

 

The variety of shapes of the enlargement is too great to justify comprehensive experiment so the sudden enlargement shown in figure 8-1d is taken to be the worst case.  The transition is abrupt from one diameter to the other so that a flat annulus is created. An expression for the loss is required which must involve the two diameters and a coefficient. Figure 8-4 shows an enlargement joining two pipes A and B of substantial length and two sections 1 and 2 in the pipes that are remote from the enlargement. If we regard the flow as being one dimensional we can apply the total head equation to 1 and 2 and write :-

    + the loss between 1 and 2. If we follow our previous decision on how to deal with losses in fittings we can act as if the loss occurred at the enlargement and put :-

 

    the loss between 1 and 2 =  the loss in pipe A + the loss in enlargement + the loss in pipe B and find the pipe losses using Darcy.

 

Then if we put the loss in pipe A =  and the loss in pipe B  =  the total head equation can be rearranged to give:-

the loss in the enlargement = 

Now we can apply the momentum equation to the liquid between 1 and 2 and say that the net force exerted on the liquid as it flows from 1 to 2 equals the increase in momentum per second between 1 and 2. The net force will be made up of the pressure forces acting on areas  and , the force exerted by the annulus and the friction drag exerted by the walls of the two lengths of pipe.

 

The friction drag on the walls has to be overcome by a pressure difference which, in pipe A is equal to , and in B is equal to . Then :

,

 

We have to decide what to do about the force on the annulus. There will be a pressure on the annulus but there is no reason to suppose that the pressure is uniform. Nevertheless the best result comes from supposing the pressure on the annulus to be uniform and equal to   and then the force on the annulus is equal to  and,

            

 

which, on dividing by  and simplifying, gives :

                                               

But  , and  and it follows that

Putting this into the expression for the loss gives :-

Loss at enlargement = , which reduces to :

                            The loss at a sudden enlargement = 

This is an attractively simple expression which can be rewritten in the form :-

                            The loss at a sudden enlargement =  which is, of course, all in measurable quantities.

 

It is worth reflecting on this expression. No one can pretend that it is the result of a rigorous analysis. The decision that was taken about the force on the annulus is obviously questionable but the expression that follows from that decision is simple and turns out to be useful. Had this not been the case the expression would have disappeared and the decision about the annulus along with it. The expression follows from the same decisions as we have already taken for dealing with pipe losses so we can expect it to have the same validity as the rest of our expressions.

 

Then we can put :

                            the loss at a sudden enlargement = , where  is a coefficient. For most purposes it is adequate to put  = 1. On such methods is an empirical science constructed. 

 

Real transition fittings are shaped for ease of manufacture and do not have a sudden change in diameter and the loss in such fittings when used as enlargers is less than that in a sudden enlargement. In order to take account of the better flow pattern in the real fitting, a reduced value of k is used. [1]