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Chapter 4 Orifice Basics

      Flow restrictions such as orifices, flow nozzles, and venturis are known for introducing pressure losses associated with flow rate. In hydraulic control systems, the relationship between pressure drop and flow rate established by these restrictions (generally referred to as orifices) is the basis of the operating principle accomplished by most of the control elements in hydraulic systems, such as hydraulic control valves. Therefore, an entire chapter is dedicated to the orifice equation and its uses in hydraulic systems.

4.1 Orifice Equation

      The orifice equation provides the relationship between the flow rate through a generic restriction and the pressure drop across it.

In this section, the orifice equation is derived for the particular case of a sharp orifice (Figure 4.1). As shown in the figure, the internal flow is characterized by different phases. At first, the fluid stream accelerates approaching the restriction. Afterward, the flow separates from the sharp edge of the orifice, which causes recirculation zones downstream of the restriction. In this phase, the mainstream flow still continues to accelerate from the nozzle throat to form a vena contracta at section 2, where the flow area is minimum. The flow then decelerates to fill the duct section. At vena contracta, the flow streamlines are essentially straight, and the pressure is uniform across the section.

      This flow condition can be well studied using the continuity and Bernoulli's equations. Then, empirical correction factors may be applied to estimate the correct flow rate, or to consider different geometrical conditions.

      Under the assumption of incompressible flow, the mass conservation written between sections 1 and 2 of Figure 4.1 gives

(4.1)

Bernoulli's equation applies to both sections 1 and 2 under the additional assumptions of stationary conditions, frictionless flow, and uniform velocities at each section¹. Moreover, these sections are properly considered where no streamline curvature is present so that the pressure is uniform across the sections:

(4.2)

Figure 4.1 Flow through a sharp orifice.

The elevation term in Eq. (4.2) can be simplified by assuming z₁ = z₂.

From Eqs. (4.1) and (4.2), it is possible to obtain the following expression for the volume flow rate Q = Q₁ = Q₂:

      (4.3)

      The actual flow area Ω₂ in the vena contracta is unknown. Therefore, an empirical coefficient called the coefficient of discharge C_d is introduced in order to write the equation referring to the known value Ω_O:

(4.4)

      The coefficient of discharge not only is a pure geometrical ratio but also accounts for other secondary but non‐negligible aspects that affect the actual flow conditions through the orifice. These are the frictional effects due to fluid viscosity and the approximated flow uniformity. For these reasons, the empirical formulas available for C_d show a primary dependency of the coefficient of discharge with the Reynolds number. Empirical formulas for C_d are available in the literature, such as in the Miller handbook [38] or the ASME standards [39].

The term is also referred to as the velocity of approach factor. Usually, the velocity of approach factor and the coefficient of discharge are combined in a single coefficient, often indicated as the flow coefficient or the orifice coefficient. Equation (4.4) then becomes²

(4.5)

Figure 4.2

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