Orifices and Nozzles on a Vertical Plane
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 |
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\(\large{ Q = C_d \; A_o \; Y \; \sqrt { \frac{ 2 \; \left( \Delta p \; +\; \rho \; g \; \Delta y \right) }{ \rho \; \left( 1\; -\; \beta^4 \right) } } }\) \(\large{ Q = C_d \; A_o \; Y \; \sqrt { \frac{ 2g \; \left( \Delta h \; +\; \Delta y \right) }{ \rho \; \left( 1\; -\; \beta^4 \right) } } }\) \(\large{ Q = C_d \; A_o \; Y \; \sqrt { \frac{ 2g \; \left( \Delta h \; +\; \Delta y \right) }{ \rho \; \left( 1\; -\; \beta^4 \right) } } }\) \(\large{ \Delta h = \frac{1}{2\;g} \; \left( 1 - \beta^4 \right) \; \left( \frac{ Q }{ C_d \; A_o \; Y } \right)^2 - \Delta y }\) |
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| 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: |
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\(\large{ Y = \frac{ C_{d,c} }{ C_{d,i} } }\) \(\large{ C_{d,c} }\) = 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 |
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