Orifice Flow - Foundations of Spiritual and Physical Safety with Chemical Processes

Mechanical Energy Balance¶

If there is no friction and constant density and no shaft work and no change in elevation, the mechanical energy balance simplifies to:

Incompressible Flow through an Orifice¶

Thus for incompressible flow, the velocity of the fluid can be determined from the pressure drop across an orifice. Although the flow is usually not frictionless, the velocity can be estimated from the pressure drop and the density of the fluid according to:

\dot{m} = A C_d \sqrt{2\rho \Delta P}

(5)

or

\dot{n} = \frac{A C_d }{M_w} \sqrt{2 \rho \Delta P}

(6)

where $\dot{m}$ is the mass flow rate, $\dot{n}$ is the molar flow rate, $M_w$ is the molecular weight, $A$ is the area of the orifice, $\rho$ is the density of the fluid, $C_d$ is the discharge coefficient, and $\Delta P$ is the pressure drop across the orifice. For liquid flow, and estimate of the discharge coefficient is 0.65.

Gaseous flow through an Orifice: General Case¶

The flow of a gas through an orifice is also derived from an energy balance assuming adiabatic and frictionless flow. The flow is characterized by the pressure ratio, $P_o/P_{atm}$ and associated Mach number, $Ma$ . $P_o$ is the absolute pressure of the volume. The Mach number can be found from:

Ma = \text{min} \left(1, \sqrt{\frac{2}{\gamma - 1}\left(\left(\frac{P_o}{P_{atm}}\right)^{\frac{\gamma - 1}{\gamma}} - 1\right)}\right)

(7)

where $\gamma$ is the ratio of specific heats (1.4 for air). The flow is choked when $Ma = 1$ and unchoked when $Ma < 1$ .

The molar flow rate is given by:

\dot{n} = P_o A C_d \sqrt{\frac{\gamma}{R T M_w}} Ma \left[ 1+\frac{(\gamma-1)}{2}Ma^2\right]^{\frac{\gamma+1}{2-2\gamma}}

(8)

where $R$ is the gas constant and $T$ is the temperature, $P_o$ is the absolute pressure inside the vessel, $C_d$ is the discharge coefficient (typically about 0.9). If you use all SI units (Pascal, meters, kilograms, mole: so pressure in pascal, area in meters, R in J/mol/K, and Mw in kg/mol), you’ll get mole per second for $\dot{n}$ . See Perry’s Handbook for Chemical Engineers Equation 6-118 for more information.

Flashing Two Phase Flow Estimate¶

\dot{m} = f \frac{\Delta H_v A}{\nu_{fg}}\sqrt{\frac{1}{C_p T_s}}

(9)

where $\dot{m}$ is the mass flow rate, $f$ is a factor depending on the scenario (less than 1), $\Delta H_v$ is the heat of vaporization, $A$ is the area of the orifice, $\nu$ is the specific volume change of the liquid as it flashes ( $\nu_g$ - $\nu_l$ ), $C_p$ is the heat capacity of the fluid, and $T_s$ is the saturation temperature of the fluid at the set pressure. See Chapter 4 of Crowl and Louvar for more information.