# 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

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

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

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

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

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) $$dimensionless$$
$$g$$ = gravitational acceleration $$ft\;/\;sec^2$$ $$m\;/\;s^2$$
$$A_o$$ = orifice area  $$in^3$$ $$mm^2$$
$$C_d$$ = orifice discharge coefficient $$dimensionless$$
$$G$$ = 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 $$dimensionless$$

### 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

Tags: Orifice and Nozzle