Written by Jerry Ratzlaff on . Posted in Flow Instrument

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

### Where:

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

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

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

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

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

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

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

$$\large{ \beta }$$  (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