# 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.  The elevation change is  $$\large{ \Delta z = z_1 - z_2 }$$ .

### Orifices and Nozzles on a vertical Plane Formula

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

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

$$\large{ \Delta P = \frac{1}{2} \; \rho \; \left( 1 - \beta^4 \right) \; \left( \frac{ Q }{ C_d \; A_o \; Y } \right)^2 - \rho \; g \; \Delta z }$$

$$\large{ \Delta h = \frac{1}{2\;g} \; \left( 1 - \beta^4 \right) \; \left( \frac{ Q }{ C_d \; A_o \; Y } \right)^2 - \Delta z }$$

Where:

$$\large{ A }$$ = area

$$\large{ h }$$ = fluid head

$$\large{ \Delta h }$$ = change in fluid head

$$\large{ C_d }$$ = discharge coefficient

$$\large{ z }$$ = elevation

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

$$\large{ Q }$$ = flow rate

$$\large{ G }$$ = gravitational acceleration

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

$$\large{ A_o }$$ = orifice area

$$\large{ P }$$ = pressure  $$\large{ \left( P_d - P_u \right) }$$

$$\large{ P_u }$$ = upstream pressure of orifice or nozzle

$$\large{ P_d }$$ = downstream pressure of orifice or nozzle

$$\large{ \Delta P }$$ = pressure differential

$$\large{ \beta }$$  (Greek symbol beta) = ratio of pipe diameter to orifice diameter  $$\large{ \left( \frac{d_o}{d_u} \right) }$$

$$\large{ d_o }$$ = orifice or nozzle diameter

$$\large{ d_u }$$ = upstream pipe diameter from orifice or nozzle

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