Problem 4.2.2

 

Air is flowing through a pipe at 70 degrees Fahrenheit and 1.5 bar pressure.  If pressure drop is 0.1 bar, estimate the flow rate for the following cases:

 

  1. Square-edged orifice
  2. Nozzle
  3. Venturi.

Also, estimate flow rate through a nozzle under critical conditions.

 

Flow meter coefficient, C, is dependent upon Reynolds number.  For orifices and nozzles, Reynolds number is based on larger diameter (pipe diameter).  For venturis, Reynolds number is calculated using smaller diameter.

 

Nozzles:  As we do not know the value of nozzle coefficient, one can assume turbulent flow condition.  Nozzle coefficient is plotted as function of Reynolds number and beta, where beta is ratio of nozzle diameter to pipe diameter.  The following procedure is recommended.

 

  1. Assume turbulent flow and read the value of C for corresponding beta value.
  2. Obtain expansion factor, Y from Crane’s chart A-20, where Y is plotted as a function of pressure ratio and beta.  Use appropriate scale along x-axis that depends upon the value of k.
  3. Calculate mass flow rate, w, using the formula: C Y A1 square root of 2 gc delta P rho.  A1 is the cross-sectional area of the nozzle.  Verify if the assumed value of flow coefficient, C, is correct.
  4. Calculate volumetric flow rate, q using the density of fluid.
  5. Use, pipe diameter, d2 to calculate Reynolds number.  
  6. Calculate cross-section area of the pipe, A2.
  7. Divide q by A2 to obtain V2.
  8. Calculate Reynolds number, V2 d2 rho by mu.
  9. Read the value of C using beta and Reynolds number.
  10. Repeat these calculations with revised value of C, till the solution converges.

 

For the given situation, mass flow rate through nozzle is found to be 0.093 kilograms per second.  Similar approach is used for orifice.  For orifice mass flow rate is found to be 0.066 pounds per second. 

 

For venturis, Reynolds number is based upon d1.  Assume turbulent flow.  Mass flow rate is found to be 0.092 kilograms per second. 

 

For flow under critical condition, obtain the value of critical pressure ratio for the given value of beta and k.  For beta of 0.484, and k of 1.4, rc is read as 0.535.  This gives delta P over Pi as 0.465.  Expansion factor can read from Figure A-20 as 0.62.  Mass flow rate is 0.159 kilogram per second.