The Spool with Annular Orifice (IL) represents a one-dimensional movement of the spool in a sleeve with annular holes. Depending on the value of the Edges geometry parameters, the edges of the spool can be sharp or rounded. The equations for calculating the annular orifice area, flow rate and hydrodynamic force are different for sharp and rounded edges.
The pressure at the isothermal liquid ports is entered in bar, and the flow rate in l/min and volume in cm3 are calculated and output at both of these ports.
Velocity in m/s and rod displacement in m are entered at port R_A and transferred unchanged to port R_B. The force in N is input to port R_B and the output force is calculated and transferred to port R_A.
The resultant force acting on the spool is due to the pressure force and external forces. It is assumed that the pressure in port B acts on the active region adjacent to the orifice and tends to open the orifice. The pressure in port A does not act directly on the spool. These assumptions give the pressure force acting on the spool. This force can be corrected by the hydrodynamic force. For a spool with sharp edges, it is assumed that the jet angle is constant. If the spool with rounded edges is modelled, the jet inclination angle is determined by interpolation of experimental results.
The area of the open orifice is a variable related to the spool movement.
Sometimes it is useful to limit the orifice area to a minimum and/or maximum value. The minimum area can be used to model a leak or a special orifice that allows flow even when the spool is in the overlap position. The maximum area can be used to model the flow area adjacent to the orifice when the valve is wide open.
The minimum and maximum orifice areas are determined by the corresponding overlap values (Underlap corresponding to maximum area, Underlap corresponding to minimum area). By default, there are no area limits. The lower limit value must be greater than zero. Regardless of the upper limit , the flow area must not exceed the annular area defined by the spool and rod diameter.
The flow rate is calculated taking into account the movement of the spool valve.
Equations
The overlap value is defined as
where
- overlap corresponding to zero displacement, the value of the parameter Underlap corresponding to zero displacement;
- spool movement, which is output to the R_A port.
The chamber length is defined as
where is the chamber length at zero displacement, the value of the parameter Chamber length at zero displacement.
The volume of the chamber is
where
- spool diameter;
- stem diameter.
The flow coefficient is calculated as
where
- is the pressure drop between the ports;
- hydraulic diameter;
- kinematic viscosity;
- average density of the fluid.
The flow coefficient is calculated as
where
- is the maximum flow coefficient, the value of the parameter Maximum fl ow coeffi cient;
- critical flow coefficient, the value of the Critical fl ow number parameter.
For , the value of does not change much. For low , the value of varies linearly with the change of .
A reasonable value of by default is 1000. However, for holes with complex (rough) geometry it may be less than 50. For very smooth geometry it can be set to 50000.
The average fluid velocity is:
The volume flow rate is:
where
- is the area of the annular hole;
- is the density of the liquid at atmospheric pressure.
Sharp-edged annular bore spool valve
The overlap value is limited between and the smaller of and , where is the value at which the annular area is equal to the throat area:
The value of the minimum overlap is typically zero, but may be larger to model the leakage flow rate. The maximum overlap value is normally very large (Inf), but can be set much smaller to model an additional orifice.
The area of the annular orifice is:
and the hydraulic diameter is:
The contribution to the flow rate due to the spool valve movement is calculated as
where is the speed of the spool movement.
The hydrodynamic force acting on the spool is determined by evaluating the change in momentum. This force tends to close the valve. For steady-state fluid flow, the hydrodynamic force is equal to:
where is the jet angle, which is considered constant for the sharp-edged spool valve and is set in the Jet angle parameters.
Dependence of hydrodynamic force on overlap is defined as follows:
In this case, there is no reactive force in the slab. The port force R_A is calculated with the port force R_B, the pressure force and the hydrodynamic force as follows
Slide spool with rounded-edged annular port
If the Edges geometry parameters are set to Rounded, it is assumed that the spool edges are rounded and there is a diameter gap between the spool and the sleeve, which is a more realistic geometric model.
The roundedness is defined by the following values:
rounding radius , the value of the Rounded corner radius parameters;
diameter gap , value of the parameter Clearance on diameter, note that for diameter gaps greater than 60 µm the leakage should be overestimated;
relative eccentricity , the value of the Eccentricity ratio parameter, which is defined as the ratio of eccentricity and half of the diameter gap:
The value of relative eccentricity should be between 0 and 1.
*♪ Positive overlap
If the overlap value is positive , the flow is calculated as the flow through the orifice
The area of the annular orifice is:
and the hydraulic diameter is:
Note that for sufficiently large sizes, the area calculated by the above formula will exceed the throat diameter:
The overlap value is limited between 0 and the smaller of and , where is the value at which the annular area is equal to the neck area. The maximum overlap value is usually very large (Inf), but can be set much smaller to simulate an additional hole.
*Negative overlap
The flow at negative overlap is the leakage flow through the annular orifice between the spool and its housing, it is assumed to remain laminar.
The area of the annular orifice is:
and the hydraulic diameter is:
The leakage flow rate between spool and sleeve is determined by the standard leakage flow equation:
where
- is the absolute average viscosity of the liquid;
- continuity coefficient, which ensures the continuity of the flow.
When the overlap is negative , it is assumed that the flow is laminar. The transition from laminar to turbulent flow is made using the continuity coefficient to avoid discontinuities between regions.
The contribution to the flow rate due to the spool movement is calculated as
The hydrodynamic force acting on the spool is determined by evaluating the change in momentum. This force tends to close the valve. For steady fluid flow, the hydrodynamic force is equal to:
where is the angle of inclination of the jet.
The cosine of the jet inclination angle is found by interpolating the experimental results shown in the figure below. Linear spline interpolation is used for this purpose.
The dependence of the hydrodynamic force on the overlap x is defined as follows:
Checking the Couette effect box allows to take into account the Couette effect, i.e. to take into account the contribution of the Couette flow to the viscous friction force.
The viscous friction force is defined only in the leakage (negative overlap ) and is calculated as:
The force at port R_A is calculated considering the force at port R_B, the pressure force, the viscous friction force and the hydrodynamic force as
In this simple model, the jet angle is assumed to be constant. For most applications this value can be left by default. The jet angle is set relative to the valve axis.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp'.
Values
deg | rad | rev | mrad
Default value
69.0 deg
Program usage name
jet_angle
Evaluatable
Yes
#Jet force coefficient —
hydrodynamic force coefficient
Details
A hydrodynamic force coefficient which, at a value of 0 (by default) turns off the hydrodynamic force and at a value of 1 turns it on. If experimental data for this coefficient is available, you can adjust the model to this data.