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In some hydraulic circuits, orifices appear in series or in parallel configuration (Figure 4.6). In these cases, it can be useful to evaluate the overall behavior of the system of orifices.

One way to address this problem is to consider the definition of hydraulic resistance for an orifice. Then, consider that

(4.10)

and

(4.11)

where the hydraulic resistance of the orifice has the expression of Eq. (4.7).

From Eqs. (4.10) and (4.11), the area of the equivalent orifice can be calculated. Another way for determining the area of the equivalent orifice is to directly derive the relationship between Q and Δp of the set of orifices, as shown in the figure.

Figure 4.6 Orifice in parallel and in series.

For orifices connected in parallel, the overall flow is the sum of the individual flows, while the upstream and downstream pressures are equal for all:

(4.12)

Therefore, if the same flow coefficient is assumed for all orifices, the area of the equivalent orifice is represented by the sum of the individual areas, and the equivalent diameter is square root of the sum of the square of the individual diameters:

(4.13)

For orifices in series, the flow across the orifices is constant, while the Δp at each orifice is different:

(4.14)

By definition, the equivalent orifice satisfies the equation:

(4.15)

By equating the last two equations, it is possible to provide the expression for Ω_{eq, ser}. This can be shown for the case of two orifices below, assuming again the same flow coefficient for all orifices:

(4.16)

Therefore, the equivalent orifice area is

(4.17)

This equation can easily be generalized for more orifices:

(4.18)