GENERAL PRINCIPLES OF HYDRAULICS 457 
A’B’D’C’ in ¢ seconds. Let the pressure and velocity at AB be P, and »,, 
ind let the pressure and velocity at CD be P and v, Experiment has 
that in the enlarged part of the pipe where it joins the smaller 
part the pressure is the same as in the smaller part, namely, P,. 
The forces urging the mass of water ABDC forward are, P,a, at AB, 
and P,(a—a,) on the annular surface between A’B’ and EF. The force 
‘retarding the forward motion is Pa at CD. Hence the resultant force 
on ABDC in the direction of motion is P,a, + P,\(a-a,) - Pa=(P, - P)a. 
The impulse of this force is (P, — P)at, which must be equal to the change 
in the momentum of the mass of water ABDC in the time ¢ seconds. 
But since there is no change in the momentum of the mass of water 
between the sections A’B’ and CD, the change in the momentum of 
ABDC must be equal to the difference between the momenta of the 
masses AA’B’B and CC’D’D, that is, the change in the momentum of 
_ @ mass equal to v,a,t=vat. Hence 
(P, - Pat = mala ~ v,), therefore ie fe °(0— %) 
Php mi P(e) 
w 2g w 229g 29 
) from which it follows that 
Rut if there had been no loss of energy in passing from the smaller to 
2 
the larger part of the pipe, Bernoulli’s theorem shows that P, + = would 
: w 2g 
P 
have been equal to = =, therefore the loss of energy due to the sudden 
_v/? 
enlargement of the pipe is oe per lb. of water passing. 
In the foregoing discussion no account has been taken of the effect of 
friction, but in the short length of pipe considered the effect of friction 
would be very small. 
396. Loss of Energy or Head due to Sudden Contraction of Pipe.— 
_ In passing from the larger to the smaller part of the pipe (Fig. 750) the 
) stream follows the contour of the larger part 
_ almost right up to the smaller part, and then 
contracts to a cross section of area a, at a 
section AB within the smaller part, a, being ; 
less than a, the area of the section of the : 
smaller part of the pipe. The only loss up 
to AB is due to friction, and may here 
be neglected. After passing AB the stream 
expands and fills the remainder of the pipe, as shown. In passing 
from AB to CD there is a loss due to the sudden enlargement, as 
in the case considered in the preceding Article. Using the formula of 
the preceding Article, the loss of energy or head between AB and CD is 
@,-) - e( = 1) a ms . If kis the coefficient of contraction at AB, 
¥Fia. 750. 
then 4% — iz The coefficient & varies with the ratio of a to a), where ay is 
ba | 
the area of the section of the larger part of the pipe. If ay is very large 
’ 
