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.

Formulas that use Orifices and Nozzles on a vertical Plane

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

Where:

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

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

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

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

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

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

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

$$\large{ \Delta h }$$ = head loss

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

$$\large{ p }$$ = 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