The pressure rise, ∆p = p2 - p1, across the abrupt expansion of Fig. P7.32 through which a liquid is...

Question

The pressure rise, ∆p = p2 - p1, across the abrupt expansion of Fig. P7.32 through which a liquid is flowing can be expressed as    where A1 and A2 are the upstream and downstream cross-sectional areas, respectively, ρ is the fluid density, and V1 is the upstream velocity. Some experimental data obtained with A2 = 1.25 ft2 , V1 = 5.00 ft/s, and using water with ρ = 1.94 slugs/ft3 are given in the following table:     Plot the results of these tests using suitable dimensionless parameters. With the aid of a standard curve fitting program determine a general equation for ∆p and use this equation to predict ∆p for water flowing through an abrupt expansion with an area ratio A1/A2  = 0.35 at a velocity V1 = 3.75 ft/s.       Q

Robert · Accepted Answer

Solution:- 
As per the given 
Perform dimension analysis for the pressure rise relation: 
1 2 1( , , )p f A A V   
Write the dimension of the variables in terms of MLT system 
1 2
2
1
2
2
3
1
p ML T
A L
A L
ML
V LT

 







  
Use pi theorem to find the number of pi terms required, 
M=n-r=5-3=2 
Calculate first pi term by using the dependent variable and combining it with the repeating variables as 
shown 
1 2 1( ) ( ) ( )
, , tan
a b cp A V
Where a b c arecons t
 
  
1 2
1 2 2 3 1
,
( )( ) ( ) ( )o o o a b c
Substituting ML T for p
M L T ML T L ML LT
 
      
Calculate the value of a, b and c so that the resulting component for each of the basic 
Dimensions. 
For M: 
1+b=0 
For L:

The pressure rise, ∆p = p 2 - p 1 , across the abrupt expansion of Fig. P7.32 through which a liquid is flowing can be expressed as where A 1 and A 2 are the upstream and downstream cross-sectional...

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