# Orifices and Nozzles on a Vertical Plane

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

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

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

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

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

Symbol English Metric
$$\large{ Q }$$ = flow rate $$\large{\frac{ft^3}{sec}}$$  $$\large{\frac{m^3}{s}}$$
$$\large{ \rho }$$  (Greek symbol rho) = density $$\large{\frac{lbm}{ft^3}}$$ $$\large{\frac{kg}{m^3}}$$
$$\large{ \Delta y }$$ = elevation change ( $$\Delta y = y_1 - y_2$$ )   $$\large{ ft }$$  $$\large{ m }$$
$$\large{ Y }$$ = expansion coefficient (Y = 1 for incompressible flow) $$\large{ dimensionless }$$
$$\large{ g }$$ = gravitational acceleration $$\large{\frac{ft}{sec^2}}$$ $$\large{\frac{m}{s^2}}$$
$$\large{ A_o }$$ = orifice area  $$\large{ in^3 }$$ $$\large{ mm^2 }$$
$$\large{ C_d }$$ = orifice discharge coefficient $$\large{ dimensionless }$$
$$\large{ G }$$ = orifice gravitational constant  $$\large{\frac{lbf-ft^2}{lbm^2}}$$   $$\large{\frac{N - m^2}{kg^2}}$$
$$\large{ \Delta h }$$ = orifice head loss   $$\large{ ft }$$   $$\large{ m }$$
$$\large{ p }$$ = pressure $$\large{\frac{lbf}{in^2}}$$  $$\large{ Pa }$$
$$\large{ \Delta p }$$ = pressure differential ( $$\Delta p = p_2 - p_1$$ ) $$\large{\frac{lbf}{in^2}}$$ $$\large{ Pa }$$
$$\large{ \beta }$$  (Greek symbol beta) = ratio of pipe inside diameter to orifice diameter $$\large{ dimensionless }$$

### 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 