Orifices and Nozzles on a Vertical Plane

Written by Jerry Ratzlaff on . Posted in Flow Instrument

When orifices and nozzles are installed having the piping vertically and assuming that there is an elevation change, the following equations can be used.

 

Orifices and Nozzles on a vertical Plane formulas

\(\large{ Q =  C_d \; A_o \; Y \; \sqrt {  \frac{ 2 \; \left( \Delta p \; +\; \rho \; g \; \Delta y  \right) }{ \rho \; \left( 1\; -\; \beta^4  \right)  } } }\)   
\(\large{ Q =  C_d \; A_o \; Y \; \sqrt {  \frac{ 2g \; \left( \Delta h \; +\; \Delta y  \right) }{ \rho \; \left( 1\; -\; \beta^4  \right)  } } }\)   
\(\large{ Q =  C_d \; A_o \; Y \; \sqrt {  \frac{ 2g \; \left( \Delta h \; +\; \Delta y  \right) }{ \rho \; \left( 1\; -\; \beta^4  \right)  } } }\)   
\(\large{ \Delta h = \frac{1}{2\;g} \;  \left( 1 - \beta^4 \right)  \;  \left(  \frac{ Q }{  C_d \; A_o \; Y  }  \right)^2  - \Delta y  }\)  

Where:

\(\large{ Q }\) = flow rate

\(\large{ \rho }\)  (Greek symbol rho) = density

\(\large{ \Delta y }\) = elevation change ( \(\Delta y = y_1 - y_2\) )

\(\large{ Y }\) = expansion coefficient (Y = 1 for incompressible flow)

\(\large{ g }\) = gravitational acceleration

\(\large{ A_o }\) = orifice area

\(\large{ C_d }\) = orifice discharge coefficient

\(\large{ G }\) = orifice gravitational constant

\(\large{ \Delta h }\) = orifice head loss

\(\large{ p }\) = pressure

\(\large{ \Delta p }\) = pressure differential ( \(\Delta p = p_2 - p_1\) )

\(\large{ \beta }\)  (Greek symbol beta) = ratio of pipe inside diameter to orifice diameter

Solve for:

\(\large{ Y =  \frac{ C_{d,c} }{ C_{d,i} }  }\)

\(\large{ C_{d,c}  }\) = discharge coefficient compressible fluid

\(\large{ C_{d,i}  }\) = discharge coefficient incompressible fluid

\(\large{ \beta }\)  (Greek symbol beta) = \(\frac{d_0}{d_u}\)

\(\large{ d_o }\) = orifice or nozzle diameter

\(\large{ d_u }\) = upstream pipe inside diameter from orifice or nozzle

 

Tags: Equations for Orifice and Nozzle