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4.1 Orifice Equation
ОглавлениеThe 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
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:
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 Cd is introduced in order to write the equation referring to the known value ΩO:
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 Cd show a primary dependency of the coefficient of discharge with the Reynolds number. Empirical formulas for Cd 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
Figure 4.2 Orifice area for a poppet needle valve. (a) Entire popper valve. (b) Eetail on the throat flow area. (c) ISO symbol.
The same abovementioned references collected empirical formulas for the orifice coefficient for various geometries and sizes. In many cases, it can be simply assumed that Ω1 ≫ ΩO: hence, the effects of velocity of approach are negligible and the value of Cf can be approximated with the coefficient of discharge Cd.
The orifice equation has a very broad application in hydraulics because it can be used to describe the flow through any element of the system introducing one or more flow restrictions. Figure 4.2 shows the example of a needle valve based on a poppet design. The orifice area ΩO is represented by the minimum flow area and, in this case, it has an annular shape.
For any specific geometric case, the orifice coefficient should be determined experimentally. Several authors report the theoretical evaluation of such a coefficient for different geometries, under the assumptions of frictionless, incompressible and irrotational flow. For example, Von Mises [40] provided the analytical results for the flow coefficient for several different orifice geometries. A comparison between Von Mises' results and experimental data is also extensively discussed in [36].
As shown in [2, 32, 36], typical values for Cf are in the range of 0.6–0.8. In particular, 0.611 is the von Mises' theoretical value for a circular sharp edge orifice (Figure 4.1).
The value of the Cf coefficient can be considered constant only for high Reynolds numbers (turbulent flow conditions). In case of laminar flow, the flow through an orifice can be solved analytically from Navier‐Stokes equations. In the case of a circular sharp edge orifice, the relationship becomes [41]
Figure 4.3 shows the trend for the Cf coefficient as a function of the Reynolds number, Re. In the figure, both theoretical solutions (Von Mises and Eq. (4.6)) are reported along with experimental results, as described in [36]. The measured trend for Cf shows a smooth transition from the laminar behavior, typical of low fluid velocities, and the turbulent conditions. Qualitatively, the same trend represented in Figure 4.3 applies to the orifice geometries and shape characteristics of common hydraulic control elements.
Figure 4.3 Theoretical and experimental trends of the flow coefficient vs. the Reynolds number.
The general orifice equation (Eq. 4.5) shows how an orifice can be seen as a hydraulic resistance, according to the definition provided in Chapter 3 (Eq. (3.38)):
Therefore,
