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Orifice Flow, Hydraulic - Equations

An orifice is a sudden flow restriction of short length (zero length for sharp edge orifice). Orifices are treated as either a sharp edge orifice or a short tube orifice. Orifices are primarily used to control flow or create a pressure differential (drop). Orifices may be fixed or variable (valve). Many types of valves and flow devices can essentially be viewed as orifices. In valves, there can be numerous flow passages, but usually somewhere in the flow passage is a restriction that controls flow, which is why a valve often behaves like an orifice.

Flow characterization for orifices is done mathematically. However, by understanding the nature of the mathematical equations the behavior of fluid flow in orifices (as well as pipes and servos) can be understood and intuition developed. Also, the equations can be used to model component and system behavior for enhanced analysis and understanding. The orifice flow equation is a key equation for hydraulic systems.

Like pipe flow, fluid flow in orifices can be either laminar or turbulent (see Figure 1). In laminar flow, each fluid particle follows a well defined trajectory, with velocity only in the direction of flow. In turbulent flow (most common in hydraulic systems due to small line diameters and small orifices) each particle flows in the general direction (velocity) of the flow, but is subjected to fluctuating cross current velocities. Equations for computing orifice flow are different for laminar and turbulent flow.

Figure 1 Flow Streamlines for Laminar and Turbulent Orifice Flow


Determination of laminar or turbulent is determined using the Reynolds number, Re

    (1)

where

hydraulic diameter

A flow section area

S flow section perimeter

v flow velocity

μ dynamic viscosity

ν kinematic viscosity

For low values of Re, flow is laminar. For high values of Re, flow is turbulent.

From Bernoulli’s equation, the total energy loss is the energy converted to heat by friction of particles against the wall and each other

    (2)

Assuming, away from the orifice, that v1 = v2 and A1 = A2, the flow becomes a product of the area and speed

   (3)

where ξ is a dimensionless loss coefficient, representing the energy loss associated with the pressure drop. For hydraulic systems, this equation is normally written as

   (4)

where αd is the discharge coefficient and represents the energy loss in the fluid. Equation (4) is the orifice flow equation. The discharge coefficient is the key element to estimate for laminar and turbulent flow regimes. Inspection of the equation (4) indicates that the flow rate varies proportionally with area if the Δp is held constant, and that the flow rate varies with the square root of Δp if the flow area is held constant. Figure 2 shows notional charts of the flow behavior.



Figure 2 Flow Rate Behavior for a Orifice


Turbulent Orifice Flow

For a sharp edge orifice, with turbulent flow and with orifice flow area, Ao << A (pipe flow area), the theoretical αd is

   (5)

For short tube orifices of length L, pipe diameter d, and orifice diameter do,

Figure 3 graphically shows the variation in αd for the above two equations.

Figure 3 Turbulent Flow Discharge Coefficient for Short Tube Orifice


Laminar Orifice Flow

Equation (4) can be used in the turbulent-laminar (transition) region and the laminar flow region using

   (6)

where

   (Sharp Edged Orifice)

   (Rounded Off Orifice)

δ is called the laminar flow coefficient and depends on orifice geometry. αd,turb comes from equation (5). Recrit is the critical (breakpoint) Reynolds number found during empirical testing for the type of orifice (see Figure 4). These equations are theoretical, but have been validated by experiment.

Equation (6) is valid up to the critical Reynolds number (see Figure 4). Experimental data for αd as a function of Recrit for various orifices are shown in the figure below.


Figure 4 Laminar Flow Discharge Coefficient for Orifices



The sloped lines can be used for αd in the laminar flow region. At the breakpoint, turbulent flow occurs and αd = 0.611 should be used.