upper O Baseline StartRoot StartFraction 2 normal upper Delta p Over rho EndFraction EndRoot"/>
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,
4.2 Fixed and Variable Orifices
A fixed orifice is a hydraulic element characterized by a specific throat area ΩO; in a variable orifice, the area ΩO can vary according to the instantaneous geometric configuration.
Figure 4.2 gives an example of a needle valve, which is an adjustable orifice. The characteristics of both fixed and variable orifices are represented in Figure 4.4. The relationship between pressure drop and flow rate for turbulent flow conditions is parabolic, and the curves show the trend for different orifice areas. The component working conditions pertain only to the first (positive flow) and the third quadrant (negative flow). The figure also highlights the symbols used to indicate both the fixed and the variable orifice cases.
It is important to remark the square root dependence between Δp and Q highlighted in Figure 4.4. Because of this dependence, in order to double the flow across the orifice, it is necessary to increase the pressure by four times.
In hydraulic control valves, variable orifices are often used to represent the positions and the connections implemented by the valve. Figure 4.5 shows the example of two proportional valves, one is a two‐position two‐way (a) and the other one is a three‐position four‐way (a). In both cases, each square represents the possible configurations of the port connections obtained by the valve, and the continuous lines above and below the symbols indicate that the orifice area of every connection can be continuously varied.
The valve symbol includes different details when compared to the representation with basic orifices. For instance, in Figure 4.5, the information about the closed configuration (all ports completely closed ΩO = 0) is not provided by the basic orifice symbols.
Figure 4.4 Orifice equation plotted in a (p, Q) layer for different area openings.
4.3 Power Loss in Orifices
As explained in Chapter 3, the product Q · Δp represents hydraulic power. For the case of an orifice, this product represents the power dissipation introduced by the orifice itself:
