Felix Aranovich
Senior Project Engineer
Sterling Hydraulics
Elk Grove Village, Ill.

Precisely calculating flow forces in spool valves lets designers maximize flow and minimize solenoid size. For instance, relatively small coils operate a 16-gpm threeposition, fourway Sterling Hydraulics GS0657 Series valve.

Precisely calculating flow forces in spool valves lets designers maximize flow and minimize solenoid size. For instance, relatively small coils operate a 16-gpm threeposition, fourway Sterling Hydraulics GS0657 Series valve.


Flow forces are the result of fluid mass accelerating through the small opening between spool and sleeve. The axial force component <i />F<sub>x</sub></i>is useful for sizing coils and springs.

Flow forces are the result of fluid mass accelerating through the small opening between spool and sleeve. The axial force component Fxis useful for sizing coils and springs.


Critical point <I />m </i>represents the maximum flow force.

Critical point m represents the maximum flow force.


Test circuit shows the valve in a transient position.

Test circuit shows the valve in a transient position.


This simple test circuit includes a nonpressurecompensated pump, relief valve, and a two-way, two-position solenoid spool valve. Theoretical results for a flow of 10 gpm and relief pressure of 3,000 psi are presented as the 0-m-n line. Test results for pilot-operated and direct-acting valves are also shown, indicating a good correlation between predicted and actual results.

This simple test circuit includes a nonpressurecompensated pump, relief valve, and a two-way, two-position solenoid spool valve. Theoretical results for a flow of 10 gpm and relief pressure of 3,000 psi are presented as the 0-m-n line. Test results for pilot-operated and direct-acting valves are also shown, indicating a good correlation between predicted and actual results.


Designers of spool-type, solenoid-operated hydraulic valves often face the vexing problem of increasing flow while shrinking the size of the solenoid actuator. It's an effort to boost performance while reducing size, power consumption, and costs.

But solving the problem means the inconvenient factor of flow forces must be taken into account. Computing flow forces presents practical engineering problems for two main reasons. First, flow forces are generally not as well understood as, say, pressure forces. Second, the pump and other hydraulic-system components dynamically affect transient flow forces in the valve. So let's see how the system affects internal forces and overall valve performance.

Flow force, sometimes referred to as the Bernoulli force, results from acceleration of the fluid mass in the small opening between the valve's spool and sleeve.

The axial component of that force is:

Fx= 2CdCvAPcosa

where Cd= discharge coefficient, Cv= velocity coefficient, A = throttling area, P = pressure differential over the throttling area, and a = outflow angle.

To reflect real-life conditions, let's assume Cd= 0.7, Cv= 1 (so it can be omitted from future discussions), and a = 69°. This angle depends on the geometry of the throttle and sharpness of the spool cutting edge. Here, it's based on the theoretical number computed by R. von Mises for a square-land spool chamber.

Substituting these values in the above equation,

(1) Fx= 0.5AP

In an actual hydraulic circuit P and A are interrelated, so this equation is not practical without knowing the relationship.

The system's affect on valve performance is often seen when two different test stands operate the same valve under the same test conditions. In one case the spool readily shifts while it "hangs up" in transition on the other. This illustrates that estimating flow forces in spool valves must account for other hydraulic-system components. These components can change how fast oil moves through the small opening when the spool starts to shift, resulting in a "tough" system that resists movement or a "soft" one where the spool valve shifts easily.

Consider a simple circuit consisting of a fixed reciprocating pump that is not pressure compensated, a hard-to-shift relief valve, and a two-way, two-position solenoid spool valve. Further assume the relief valve has unlimited flow capacity and keeps inlet pressure constant at any spool position and any value of A. Then, based on Equation 1, the axial component of the flow force versus throttling-area can be presented as a straight line (shown in the Flow force vs. throttling area graphic as Fxat constant P).

Next assume the relief valve is completely closed but the pump has unlimited pressure capacity and delivers constant flow at any value of A > 0. In this case Equation 1 no longer applies. Instead, substitute for P using the modified Bernoulli equation:

where Q = volumetric flow rate and = mass density.

Substituting for P from Equation 1,

Using the same values for Cdand cosa, and = 8.1 X 105 lb-sec2/in.4 (for typical petroleum oil),

(2) Fx = 6.15 X 10-4 Q2/A.

As this equation shows, the axial component of flow force is inversely proportional to the throttling area when flow through the opening is constant. It is graphically presented as the curve of Fxat constant Q.

Actual processes start with the relief valve open in regulating mode, and

follow the linear constant-pressure pattern until the valve completely closes. This is depicted as point m in the graphic. Beyond this point the system follows a constantflow pattern.

Critical point m is actually the point of interest because it represents the maximum flow-force value. The mathematical coordinates of this point can be found by simultaneously solving Equations 1 and 2.

Fm, the maximum value of Fx, can be used to evaluate the maximum axial component of the flow force which counteracts the force trying to open valve.

Am, the throttling area at Fm, can be used to find the approximate critical position of the spool and help designers determine the best interaction between solenoid and spring forces.

An example of how calculations match up with an actual system is shown in the Comparing theory and reality graphic.

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