Block Spool with Orifice Hole (IL) represents a one-dimensional movement of the spool in a sleeve with one or more 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 bore area and hydraulic diameter are different for different block parameters.
If `Sharp' is selected for the Edges geometry parameter and `Planar' is selected for the Effective flow area model parameter, it is possible to set the orifice geometry in the Geometry of section orifice parameter:
Circular;
Triangular;
Rectangular;
`Trapezoidal';
`Trapezoidal with 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 transmitted unchanged to port R_B. The force in N is input to port R_B and the output force is calculated and transmitted 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 orifice area.
The flow rate is calculated taking into account the spool movement.
Equations
The overlap value is defined as
where
- Overlap corresponding to zero displacement Underlap corresponding to zero displacement;
- is the spool displacement, 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 Maximum flow coefficient parameters;
- critical flow coefficient, the value of the Critical flow number parameters.
For , the value of does not change much. For low , the value of varies linearly with the change in .
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 at port A is:
where
- is the area of the hole;
- is the density of the liquid at atmospheric pressure.
The volumetric flow rate at the port B, taking into account the contribution to the flow rate due to the spool movement, is calculated as:
where is the speed of 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 for the spool with sharp edges is considered constant and is set in the Jet angle parameters, and for the spool with rounded edges is determined from experimental data.
The force in the port R_A is calculated taking into account the force in the port R_B, pressure force and hydrodynamic force as follows
where is the value of hydrodynamic force depending on the overlap .
Cylindrical spool with sharp edges and round bore in the sleeve
If Sharp is selected for the Edges geometry parameter and Cylindrical is selected for the Effective flow area model parameter, the equations in this section are used to calculate the effective area and hydraulic diameter.
When the orifice is fully open, the flow path is the intersection of two cylinders with perpendicular axes, as shown in the figure.
For each overlap value , an iterative method is used to calculate the corresponding area by numerical integration.
The area is calculated as:
where
- is the overlap value;
- the length of the elementary area of width (highlighted in red in the figure above).
The area of the hole is calculated as:
By substituting the variables, we end up with:
where
- radius of the spool;
- radius of the bore.
Similarly, the perimeter is calculated using the iterative method:
where
- is the projection on the plane perpendicular to the hole axis of the angle between the plane in which the spool and hole axes lie and the radius-vector drawn from the hole centre to the point of intersection of the spool edge with the hole line;
- length of the elementary section of the perimeter of the angular width ;
- length of the red line (see figure above) at .
The hydraulic diameter is:
The number of steps for the iterative method used to calculate the area and hydraulic diameter is given as an integer in the Number of iterations parameters.
The overlap value is limited between and .
The flow area is also limited by the cross-sectional area of the orifice:
where is the diameter of the orifice.
The dependence of the hydrodynamic force on the overlap is determined as follows:
In this case there is no reactive force in the slab.
Cylindrical spool valve with rounded edges and round bore in the sleeve.
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.
Positive overlap
When the overlap value is positive , if the bore diameter is , and the spool offset from the closed position is , the area of the open bore is:
where
and the hydraulic diameter is:
Note that the area obtained by this formula cannot exceed the orifice area given by the formula:
which corresponds to .
The overlap value is bounded from above by the smaller of and . 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 between the spool and its housing, it can be expressed based on the Poiseuille flow approximation.
The layer on which the Poiseuille flow acts depends on and must be considered as a function of the orifice shape.
The simplified Poiseuille equation can be written as follows:
where
- is the absolute average viscosity of the liquid;
;
.
Poiseuille’s equation is transformed to:
which gives
where
;
- is a correction term to ensure continuity of the flow:
where is the base flow rate:
The area and flow rate are then multiplied by the number of orifices.
The cosine of the jet 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 is determined as follows:
Cylindrical spool with sharp edges and holes of different geometry in the sleeve
If the Edges geometry parameter is set to Sharp and the Effective flow area model parameter is set to Planar, it is possible to set the geometry of holes in the Geometry of section orifice parameter. The open orifice area and hydraulic diameter values are calculated depending on the selected orifice geometry.
If the Geometry of section orifice parameters is set to Circular, the open orifice area and hydraulic diameter are calculated according to the formulas:
where
The hydraulic diameter is:
In these formulas is the bore diameter, the value of the parameter Hole diameter.
Note that the area obtained by this formula cannot exceed the hole area given by the formula:
which corresponds to .
If the Geometry of section orifice parameters is set to Triangular, the open orifice area and hydraulic diameter are calculated using the formulas:
The hydraulic diameter is:
In these formulas is the angle of the triangular opening, the value of the parameter Aperture angle of the triangular opening.
Note that the area obtained by this formula cannot exceed the aperture area given by the formula:
which corresponds to .
If the Geometry of section orifice parameters is set to Rectangular, the open orifice area and hydraulic diameter are calculated using the formulas:
The hydraulic diameter is:
In these formulas is the width of the rectangular opening, the value of the parameter Width of the rectangular opening.
Note that the area obtained by this formula cannot exceed the area of the hole given by the formula:
which corresponds to .
If the parameter Geometry of section orifice is set to Trapezoidal, the open orifice area and hydraulic diameter are calculated using the formulas:
The hydraulic diameter is:
In these formulas, is the width of the trapezoidal opening, the value of the parameter Initial width of the trapezoidal opening, and is the angle of the trapezoidal opening, the value of the parameter Angle of the trapezoidal opening .
Note that the area obtained by this formula cannot exceed the area of the hole given by the formula:
which corresponds to .
If the Geometry of section orifice parameters is set to Trapezoidal with rounded edges, the open orifice area and hydraulic diameter are calculated using the formulas:
where
The hydraulic diameter is:
In these formulas, is the width of the trapezoidal opening, the value of the parameter Initial width of the trapezoidal opening, and is the diameter of the rounding of the sides of the trapezoidal opening, the value of the parameter Diameter of the round edges .
Note that the area obtained by this formula cannot exceed the area of the hole corresponding to .
#Number of iterations —
number of iterations to calculate the hydraulic diameter and effective flow area
Details
The flow area and hydraulic diameter are calculated from the spool position and the number of iterations. This parameter determines the number of flow area and hydraulic diameter values depending on the spool position. The number of iterations is used to calculate the sampling step between two offset values.
Dependencies
To use this parameter, set the Effective flow area model parameters to `Cylindrical'.
Default value
1000
Program usage name
discretization_interval_count
Evaluatable
Yes
#Geometry of section orifice —
hole section geometry
Circular | Triangular | Rectangular | Trapezoidal | Trapezoidal with rounded edges
Details
Hole geometry, options to choose from:
Circular;
Triangular;
Rectangular;
`Trapezoidal';
`Trapezoidal with rounded edges'.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp' and the Effective flow area model parameter to `Planar'.
#Spool diameter —
spool diameter
m | cm | ft | in | km | mi | mm | um | yd
Details
The spool diameter must be larger than the stem diameter.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
15.0 mm
Program usage name
spool_diameter
Evaluatable
Yes
#Rod diameter —
stem diameter
m | cm | ft | in | km | mi | mm | um | yd
Details
Stem diameter.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
5 mm
Program usage name
rod_diameter
Evaluatable
Yes
#Hole diameter —
bore diameter
m | cm | ft | in | km | mi | mm | um | yd
Details
Hole diameter.
Dependencies
To use this parameters:
set the Edges geometry parameter to Sharp and the Effective flow area model parameter to Cylindrical;
set the Edges geometry parameters to `Rounded';
set the Edges geometry parameter to Sharp, the Effective flow area model parameter to Planar, and the Geometry of section orifice parameter to Circular.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
1.0 mm
Program usage name
circular_hole_diameter
Evaluatable
Yes
#Aperture angle of the triangular opening —
triangular hole angle
deg | rad | rev | mrad
Details
Inner angle of a triangular hole.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp', the Effective flow area model parameter to `Planar', and the Geometry of section orifice parameter to `Triangular'.
Values
deg | rad | rev | mrad
Default value
30.0 deg
Program usage name
triangular_hole_angle
Evaluatable
Yes
#Width of the rectangular opening —
rectangular hole width
m | cm | ft | in | km | mi | mm | um | yd
Details
Width of the rectangular hole.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp', the Effective flow area model parameter to `Planar', and the Geometry of section orifice parameter to `Rectangular'.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
1.0 mm
Program usage name
rectangular_hole_width
Evaluatable
Yes
#Initial width of the trapezoidal opening —
trapezoidal width
m | cm | ft | in | km | mi | mm | um | yd
Details
Width of the trapezoidal bore.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp', the Effective flow area model parameter to `Planar', and the Geometry of section orifice parameter to `Trapezoidal' or `Trapezoidal with rounded edges'.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
1.0 mm
Program usage name
trapezoidal_hole_width
Evaluatable
Yes
#Angle of the trapezoidal opening —
trapezoidal hole angle
deg | rad | rev | mrad
Details
Inner angle of the trapezoidal hole.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp', the Effective flow area model parameter to `Planar', and the Geometry of section orifice parameter to `Trapezoidal'.
Values
deg | rad | rev | mrad
Default value
45.0 deg
Program usage name
trapezoidal_hole_angle
Evaluatable
Yes
#Diameter of the round edges —
rounding diameter of the sides of the trapezoidal hole
m | cm | ft | in | km | mi | mm | um | yd
Details
Rounding diameter of the corners of the trapezoidal hole.
Dependencies
To use this parameter, set the Edges geometry parameter to `Sharp', the Effective flow area model parameter to `Planar', and the Geometry of section orifice parameter to `Trapezoidal with rounded edges'.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
1.0 mm
Program usage name
trapezoidal_hole_side_diameter
Evaluatable
Yes
#Chamber length at zero displacement —
chamber length at zero offset
m | cm | ft | in | km | mi | mm | um | yd
Details
Chamber length at zero offset (within the calculated chamber volume).
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
0.0 mm
Program usage name
chamber_length_offset
Evaluatable
Yes
Underlap Definition
#Underlap corresponding to zero displacement —
overlap corresponding to zero offset
m | cm | ft | in | km | mi | mm | um | yd
Details
Overlap corresponding to zero offset.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
0.0 mm
Program usage name
offset
Evaluatable
Yes
#Underlap corresponding to minimum area —
overlap corresponding to the minimum area
m | cm | ft | in | km | mi | mm | um | yd
Details
Overlap corresponding to the minimum area of the hole.
Dependencies
To use this parameter, set the Edges geometry parameters to Sharp.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
0.0 mm
Program usage name
orifice_opening_at_min_area
Evaluatable
Yes
#Underlap corresponding to maximum area —
overlap corresponding to the maximum area
m | cm | ft | in | km | mi | mm | um | yd
Details
Overlap corresponding to the maximum area of the hole.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
Inf mm
Program usage name
orifice_opening_at_max_area
Evaluatable
Yes
Leakages on the Spool
#Rounded corner radius —
corner radius
m | cm | ft | in | km | mi | mm | um | yd
Details
Rounding radius of the spool edge corner.
Dependencies
To use this parameter, set the Edges geometry parameters to Rounded.
Values
m | cm | ft | in | km | mi | mm | um | yd
Default value
0.005 mm
Program usage name
edge_radius
Evaluatable
Yes
#Clearance on diameter —
diametral clearance
m | cm | ft | in | km | mi | mm | um | yd
Details
The diameter gap between the spool and the sleeve, due to the fact that the edges of the spool are rounded. Note that for diametral gaps greater than 60 µm, the leakage must be overestimated.
Dependencies
To use this parameter, set the Edges geometry parameters to `Rounded'.
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 parameters 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.