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Orifices and Nozzles on a Vertical Plane

Orifices and Nozzles on a Vertical Plane formulas
$Q \;=\; C_d \cdot A_o \cdot Y \cdot \sqrt { \dfrac{ 2 \cdot ( \Delta p + \rho \cdot g \cdot \Delta y ) }{ \rho \cdot ( 1 - \beta^4 ) } }$ $Q \;=\; C_d \cdot A_o \cdot Y \cdot \sqrt { \dfrac{ 2 \cdot g \cdot ( \Delta h + \Delta y) }{ \rho \cdot (1 - \beta^4 ) } }$ $Q \;=\; C_d \cdot A_o \cdot Y \cdot \sqrt { \dfrac{ 2 \cdot g \cdot ( \Delta h + \Delta y) }{ \rho \cdot ( 1 - \beta^4) } }$ $\Delta h \;=\; \dfrac{1}{2 \cdot g} \cdot ( 1 - \beta^4 ) \cdot \left( \dfrac{ Q }{ C_d \cdot A_o \cdot Y } \right)^2 - \Delta y$
Symbol	English	Metric
$Q$ = Flow Rate	$ft^3\;/\;sec$	$m^3\;/\;s$
$\rho$ (Greek symbol rho) = Density	$lbm\;/\;ft^3$	$kg\;/\;m^3$
$\Delta y$ = Elevation Change ( $\Delta y = y_1 - y_2$ )	$ft$	$m$
$Y$ = Expansion Coefficient (Y = 1 for Incompressible Flow)	$dimensionless$	$dimensionless$
$g$ = Gravitational Acceleration	$ft\;/\;sec^2$	$m\;/\;s^2$
$A_o$ = Orifice Area	$in^3$	$mm^2$
$C_d$ = Orifice Discharge Coefficient	$dimensionless$	$dimensionless$
$G$ = Orifice Gravitational Constant	$lbf-ft^2\;/\;lbm^2$	$N - m^2\;/\;kg^2$
$\Delta h$ = Orifice Head Loss	$ft$	$m$
$p$ = Pressure	$lbf\;/\;in^2$	$Pa$
$\Delta p$ = Pressure Differential ( $\Delta p = p_2 - p_1$ )	$lbf\;/\;in^2$	$Pa$
$\beta$ (Greek symbol beta) = Ratio of Pipe Inside Diameter to Orifice Diameter	$dimensionless$	$dimensionless$
Solve for:
$Y = C_{d,c} \;/\; C_{d,i}$ $C_{d,c}$ = discharge coefficient compressible fluid $C_{d,i}$ = discharge coefficient incompressible fluid $\beta$ (Greek symbol beta) = $d_0\;/\;d_u$ $d_o$ = orifice or nozzle diameter $d_u$ = upstream pipe inside diameter from orifice or nozzle

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

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