Orifice (IL)

Constant-area or variable-area orifice in an isothermal system

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Description

The Orifice (IL) block models the flow through a local restriction with a constant or variable opening area. For variable orifices, a control member connected to port S sets the opening position. The opening area is parametrized either linearly or by lookup table.

For orifices that open and close based on the displacement of a central spool, see Spool Orifice (IL).

You can enable faulty behavior by setting Enable faults to On.

Momentum is conserved through the orifice:

$${\dot{m}}_A+{\dot{m}}_B=\rho v_A{A}_A+\rho v_B{A}_B=0.$$

This momentum balance implies that there is an increase in velocity when there is a decrease in area, and a reduction in velocity when the flow discharges into a larger area. In accordance with the Bernoulli principle, this change in velocity results in a region of lower pressure in the orifice and a higher pressure in the expansion zone. The resulting increase in pressure, which is called pressure recovery, depends on the discharge coefficient of the orifice and the ratio of the orifice and port areas.

Constant Orifices

For constant orifices, the orifice area, A_orifice, does not change over the course of the simulation.

Using the Constant Area Parameterization

The volumetric flow rate is calculated by:

$$\dot{m}=\frac{C_d{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{A_{orifice}}{A_{port}}\right)}^2\right)}}\sqrt{p_A-p_B}\approx \frac{C_d{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{A_{orifice}}{A_{port}}\right)}^2\right)}}\frac{p_A-p_B}{{\left[{\left(p_A-p_B\right)}^2+\Delta p_{crit}\right]}^{1/4}},$$

where:

• C_d is the Discharge coefficient.

• A_orifice is the instantaneous orifice open area.

• A_port is the Cross-sectional area at ports A and B.

• $$\overline{\rho }$$ is the average fluid density.

PR_loss and Δp_crit are calculated in the same manner for constant and variable orifices and are defined in the Pressure Loss and Critical Pressure sections below.

This approximation for $$\dot{m}$$ and the Local Restriction (IL) block are the same.

Using the Tabulated data - Volumetric flow rate vs. pressure drop Parameterization

The volumetric flow rate is determined from the tabular values of the pressure differential, Δp, which you can provide. If only non-negative values are provided for both the volumetric flow rate and pressure drop vectors, the table will be extrapolated to contain negative values. The volumetric flow rate is interpolated from this extended table.

Variable Orifices

For variable orifices, setting Opening orientation to Positive control member displacement opens orifice opens the orifice when the signal at S is positive, while a Negative control member displacement opens orifice orientation opens the orifice when the signal at S is negative. In both cases, the signal is positive and the orifice opening is set by the magnitude of the signal.

Using the Linear - Area vs. control member position Parameterization

The orifice area A_orifice is based on the control member position and the ratio of orifice area and maximum control member position:

$$A_{orifice}=\frac{\left(A_{\max }-A_{leak}\right)}{\Delta S}\left(S-S_{\min }\right)\epsilon +A_{leak},$$

where:

• S_min is the control member position when the orifice is fully closed.

• ΔS is the Control member travel between closed and open orifice.

• A_max is the Maximum orifice area.

• A_leak is the Leakage area.

• ε is the Opening orientation.

The volumetric flow rate is determined by the pressure-flow rate equation:

$$\dot{m}=\frac{C_d{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{A_{orifice}}{A}\right)}^2\right)}}\frac{\Delta p}{{\left[\Delta p^2+\Delta p_{crit}^2\right]}^{1/4}},$$

where A is the Cross-sectional area at ports A and B.

Using the Tabulated data - Area vs. control member position Parameterization

When you use the Tabulated data - Area vs. control member position parameterization, the orifice area A_orifice is interpolated from the tabular values of opening area and the control member position, ΔS, which you can provide. As with the Linear - Area vs. control member position parameterization, the volumetric flow rate is determined by the pressure-flow rate equation:

$$\dot{m}=\frac{C_d{A}_{orifice}\sqrt{2\overline{\rho }}}{\sqrt{P{R}_{loss}\left(1-{\left(\frac{A_{orifice}}{A}\right)}^2\right)}}\frac{\Delta p}{{\left[\Delta p^2+\Delta p_{crit}^2\right]}^{1/4}},$$

where A_orifice is:

• A_max, the last element of the Orifice area vector, if the opening is larger than the maximum specified opening.

• A_leak, the first element of the Orifice area vector, if the orifice opening is less than the minimum opening.

• A_orifice if the calculated area is between the limits of the Orifice area vector.

A_open is a function of the control member position received at port S. The block queries between data points with linear interpolation and uses nearest extrapolation for points beyond the table boundaries.

Using the Tabulated data - Volumetric flow rate vs. control member position and pressure drop Parameterization

The Tabulated data - Volumetric flow rate vs. control member position and pressure drop parameterization interpolates the volumetric flow rate directly from a user-provided volumetric flow rate table, which is based on the control member position and pressure drop over the orifice. The block queries between data points with linear interpolation and uses linear extrapolation for points beyond the table boundaries.

This data can include negative pressure drops and negative opening values. If a negative pressure drop is included in the dataset, the volumetric flow rate will change direction. However, the flow rate will remain unchanged for negative opening values.

Numerically-Smoothed Valve Area

When a linearly-parameterized variable orifice is in the near-open or near-closed position, you can maintain numerical robustness in your simulation by adjusting the block Smoothing factor. A smoothing function is applied to all calculated areas, but primarily influences the simulation at the extremes of the valve area.

The normalized area is calculated as:

$$\widehat{A}=\frac{\left(A_{orifice}-A_{leak}\right)}{\left(A_{\max }-A_{leak}\right)}.$$

The Smoothing factor, s, is applied to the normalized area:

$${\widehat{A}}_{smoothed}=\frac{1}{2}+\frac{1}{2}\sqrt{{\widehat{A}}_{}^2+{\left(\frac{s}{4}\right)}^2}-\frac{1}{2}\sqrt{{\left(\widehat{A}-1\right)}^2+{\left(\frac{s}{4}\right)}^2}.$$

The smoothed valve area is:

$$A_{smoothed}={\widehat{A}}_{smoothed}\left(A_{\max }-A_{leak}\right)+A_{leak}.$$

Pressure Loss

Pressure loss describes the reduction of pressure in the valve due to a decrease in area. The pressure loss term, PR_loss is calculated as:

$$P{R}_{loss}=\frac{\sqrt{1-{\left(\frac{A_{orifice}}{A_{port}}\right)}^2\left(1-C_d^2\right)}-C_d\frac{A_{orifice}}{A_{port}}}{\sqrt{1-{\left(\frac{A_{orifice}}{A_{port}}\right)}^2\left(1-C_d^2\right)}+C_d\frac{A_{orifice}}{A_{port}}}.$$

Pressure recovery describes the positive pressure change in the valve due to an increase in area. If you do not wish to capture this increase in pressure, set Pressure recovery to Off. In this case, PR_loss is 1.

Critical Pressure

The critical pressure difference, Δp_crit, is the pressure differential associated with the Critical Reynolds number, Re_crit, which is the point of transition between laminar and turbulent flow in the fluid:

$$\Delta p_{crit}=\frac{\pi \overline{\rho }}{8A_{orifice}}{\left(\frac{\nu {\mathrm{Re}}_{crit}}{C_d}\right)}^2.$$

Faulty Behavior

When Orifice type is set to Variable and faults are enabled, the orifice open area becomes stuck at a specified value in response to one or both of these triggers:

• Simulation time — Faulting occurs at a specified time.

• Simulation behavior — Faulting occurs in response to an external trigger. This exposes port T.

Three fault options are available in the Opening area when faulted parameter:

• Closed — The orifice area seizes at the leakage area, which is the Leakage area in the linear parameterization, the first element of the Orifice area vector in the 1D parameterization, or the first row of the Volumetric flow rate table in the 2D parameterization.

• Open — The orifice area seizes at the maximum orifice area, which is the Maximum opening area in the linear parameterization, the last element of the Orifice area vector in the 1D parameterization, or the last row of the Volumetric flow rate table in the 2D parameterization.

• Maintain at last value — The orifice seizes at the open area when the trigger occurs.

Once triggered, the orifice remains at the faulted area for the rest of the simulation.

You can set the block to issue a fault report as a warning or error message in the Simulink Diagnostic Viewer with the Reporting when fault occurs parameter.

Ports

Conserving

A — Liquid port

isothermal liquid

Liquid entry or exit point to the orifice.

B — Liquid port

isothermal liquid

Liquid entry or exit point to the orifice.

Input

S — Control member displacement

physical signal

Control member displacement that sets orifice opening.

T — External fault trigger

physical signal

Physical signal port for an external fault trigger. Triggering occurs when the value is greater than 0.5. There is no unit associated with the trigger value.

Dependencies

This port is visible when Enable faults is set to On and Enable external fault trigger is set to On.

Parameters

Parameters

Orifice type — Type of orifice

Variable (default) | Constant

Type of orifice, defined by the orifice area. When set to Variable, the orifice area varies according to the input signal received at port S.

Orifice parameterization — Method of calculating orifice area

Orifice Area (default) | Tabulated data - volumetric flow rate vs. pressure drop | Linear - area vs. control member position | Tabulated data - Area vs. control member position | Tabulated data - Volumetric flow rate vs. control member position and pressure drop

Method of calculating orifice area over simulation. When Orifice type is set to Constant, there are two parameterization options.

• Orifice area (default). The assigned area does not change over the simulation.

• Tabulated data - Volumetric flow rate vs. pressure drop. The area remains constant, but the volumetric flow rate through the orifice can vary. This value is interpolated directly from the Volumetric flow rate vector and the Pressure drop vector dataset.

When the Orifice type is set to Variable, there are three parameterization options.

• Linear - Area vs. control member position. Area is determined by a linear relationship to the control member position with respect to a fully open or fully closed orifice. The position is set by a varying physical signal at port S.

• Tabulated data - Area vs. control member position. The opening area is interpolated from the Control member position vector and the Orifice area vector based on the control member position received at port S.

• Tabulated data - Volumetric flow rate vs. control member position and pressure drop. The volumetric flow rate is directly interpolated from the user- provided Control member position vector, s, Pressure drop vector, dp, and Volumetric flow rate table, q(s,dp) parameters based on the control member position received at port S and the pressure drop across ports A and B.

Orifice area — Area of orifice opening

1e-3 m^2 (default) | positive scalar

Cross-sectional area of the orifice opening.

Dependencies

To enable this parameter, set Orifice type to Constant and Orifice parameterization to Orifice area.

Pressure drop vector — Vector of differential pressure values for tabular parameterization

[-4, -3, -2, -1, -.5, 0, .5, 1, 2, 3, 4] MPa (default) | 1-by-n vector

Vector of pressure differential values for the tabular parameterization of volumetric flow rate. The values in this vector correspond one-to-one to values in the Volumetric flow rate vector parameter. The pressure drop vector values are listed in ascending order. The volumetric flow rate is interpolated directly from the Volumetric flow rate vector, which depends on the Pressure drop vector parameter.

Dependencies

To enable this parameter, set Orifice type to Constant and Orifice parameterization to Tabulated data - Volumetric flow rate vs. pressure drop.

Volumetric flow rate vector — Vector of volumetric flow rate values

[-2.44, -2.12, -1.68, -1.22, -.84, 0, .85, 1.21, 1.7, 2.09, 2.41] .* 1e-3 m^3/s (default) | 1-by-n vector

Vector of volumetric flow rate values for the tabular parameterization of volumetric flow rate. The values in this vector correspond one-to-one to values in the Pressure drop vector. The volumetric flow rate is interpolated directly from the provided Volumetric flow rate vector, which depends on the Pressure drop vector parameter.

Dependencies

To enable this parameter, set Orifice type to Constant and Orifice parameterization to Tabulated data - Volumetric flow rate vs. pressure drop.

Control member position at closed orifice — Control member offset

0 m (default) | scalar

Initial control member offset when the variable orifice is fully closed.

Dependencies

To enable this parameter, set Orifice type to Variable.

Control member travel between closed and open orifice — Control member maximum stroke

5e-3 m (default) | positive scalar

Maximum distance the control member travels between closed and open orifice. The orifice is fully open at the sum of the Control member position at closed orifice and Control member travel between closed and open orifice,

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Linear - Area vs. control member position.

Maximum orifice area — Maximum orifice opening area

1e-4 m^2 (default) | positive scalar

Maximum orifice area experienced during simulation. When using Tabulated data - Area vs. control member position, the maximum orifice area is the last element of the Orifice area vector.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Linear - Area vs. control member position.

Leakage area — Orifice gap area when in fully closed position

1e-10 m^2 (default) | positive scalar

Sum of all gaps when the valve is in fully closed position. Any area smaller than this value is maintained at the specified leakage area. This parameter contributes to numerical stability by maintaining continuity in the flow.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Linear - Area vs. control member position.

.

Control member position vector — Vector of control member positions

[0, .002, .004, .007, .017] m (default) | 1-by-n vector

Vector of control member positions for the tabular parameterization of orifice area. The vector elements correspond one-to-one to the values in the Orifice area vector. The vector elements are listed in ascending order and the first element must be 0. The orifice opening area is interpolated from the Orifice area vector, which depends on the Control member position vector parameter.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Tabulated data - Area vs. control member position.

Orifice area vector — Vector of orifice opening area values

[1e-09, 2.0352e-07, 4.0736e-05, .00011438, .00034356] m^2 (default) | 1-by-n vector

Vector of orifice area values for the tabular parameterization of orifice area. The values in this vector correspond one-to-one with the elements in the Control member position vector. The first element of this vector is the orifice leakage area and the last element is the maximum orifice area. The orifice opening area is interpolated from the Orifice area vector, which depends on the Control member position vector.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Tabulated data - Area vs. control member position.

Cross-sectional area at ports A and B — Area at orifice entry or exit

inf m^2 (default) | positive scalar

Cross-sectional area at the entry and exit ports A and B. This area is used in the pressure-flow rate equation that determines the mass flow rate through the orifice.

Dependencies

To enable this parameter, set either:

• Orifice type to Variable and Orifice parameterization to either Linear - Area vs. control member position or Tabulated data - Area vs. control member position.

• Orifice type to Constant and Orifice parameterization to Orifice area.

Opening orientation — Direction of member movement that opens the orifice

Positive control member displacement opens orifice (default) | Negative control member displacement opens orifice

Direction of member displacement that opens a variable orifice. A positive orientation means that a positive signal at S opens the orifice. A negative orientation means that a negative signal at S opens the orifice.

Dependencies

To enable this parameter, set Orifice type to Variable.

Discharge coefficient — Discharge coefficient

0.64 (default) | positive scalar in the range of [0,1]

Correction factor that accounts for discharge losses in theoretical flows.

Dependencies

To enable this parameter, set either:

• Orifice type to Variable and Orifice parameterization to either Linear - Area vs. control member position or Tabulated data - Area vs. control member position.

• Orifice type to Constant and Orifice parameterization to Orifice area.

Critical Reynolds number — Upper Reynolds number limit for laminar flow

150 (default) | positive scalar

Upper Reynolds number limit for laminar flow through the orifice.

Dependencies

To enable this parameter, set either:

• Orifice type to Variable and Orifice parameterization to either Linear - Area vs. control member position or Tabulated data - Area vs. control member position.

• Orifice type to Constant and Orifice parameterization to Orifice area.

Smoothing factor — Numerical smoothing factor

0.01 (default) | positive scalar in the range [0,1]

Continuous smoothing factor that introduces a layer of gradual change to the flow response when the orifice is in near-open or near-closed positions. Set this value to a nonzero value less than one to increase the stability of your simulation in these regimes.

Pressure recovery — Whether to account for pressure increase in area expansions

On (default) | Off

Whether to account for pressure increase when fluid flows from a region of smaller cross-sectional area to a region of larger cross-sectional area. This increase in pressure is not captured when Pressure recovery is set on Off.

Dependencies

To enable this parameter, set either:

• Orifice type to Variable and Orifice parameterization to either Linear - Area vs. control member position or Tabulated data - Area vs. control member position.

• Orifice type to Constant and Orifice parameterization to Orifice area.

Control member position vector, s — Vector of opening values

[0, .002, .004, .007, .017] m (default) | 1-by-n vector

Vector of control member positions for tabular parametrization of volumetric flow rate. The control member position vector forms an independent axis with the Pressure drop vector, dp parameter for the 2-D dependent Volumetric flow rate table, q(s,dp) parameter. A positive displacement corresponds to valve opening. The values are listed in ascending order and the first element must be 0. Linear interpolation is employed between table data points.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Tabulated data - Volumetric flow rate vs. control member position and pressure drop.

Pressure drop vector, dp — Vector of pressure drop values

[.3, .5, .7] MPa (default) | 1-by-n vector

Vector of pressure drop values for tabular parametrization of volumetric flow rate. The pressure drop vector forms an independent axis with the Control member position vector, s parameter for the 2-D dependent Volumetric flow rate table, q(s,dp) parameter. The values are listed in ascending order and must be greater than 0. Linear interpolation is employed between table data points.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Tabulated data - Volumetric flow rate vs. control member position and pressure drop.

Volumetric flow rate table, q(s,dp) — Array of volumetric flow rate values

[1.7e-05, 2e-05, 2.6e-05; .0035, .0045, .0053; .7, .9, 1.06; 1.96, 2.5, 3; 6, 7.7, 9.13] .* 1e-3 m^3/s (default) | M-by-N matrix

M-by-N matrix of volumetric flow rates based on independent values of pressure drop and control member position. M and N are the sizes of the corresponding vectors:

• M is the number of elements in the Pressure drop vector, dp parameter.

• N is the number of elements in the Control member position vector, s parameter.

Dependencies

To enable this parameter, set Orifice type to Variable and Orifice parameterization to Tabulated data - Volumetric flow rate vs. control member position and pressure drop.

Faults

Enable faults — Fault option

Off (default) | On

Enable externally or temporally triggered faults. When faulting occurs, the valve area normally set by the opening parameterization will be set to the value specified in the Opening area when faulted parameter.

Dependencies

To enable this parameter, set Orifice type to Variable.

Opening area when faulted — Set faulted area

Closed (default) | Open | Maintain at last value

Sets the faulted orifice area. You can choose for the orifice to seize at the fully closed or fully open position, or at the valve area when faulting is triggered.

Dependencies

To enable this parameter, set Orifice type to VariableEnable faults to On.

Enable external fault trigger — External trigger option

Off (default) | On

Enables port T. A physical signal at port T that is greater than 0.5 triggers faulting.

Dependencies

To enable this parameter, set Orifice type to VariableEnable faults to On.

Enable temporal fault trigger — Temporal trigger option

Off (default) | On

Enables fault triggering at a specified time. When the Simulation time for fault event is reached, the orifice area will be set to the value specified in the Opening area when faulted parameter.

Dependencies

To enable this parameter, set Orifice type to VariableEnable faults to On.

Simulation time for fault event — Time at which faulting is triggered

5 s (default) | positive scalar

When the Simulation time for fault event is reached, the orifice area is set to the value specified in the Opening area when faulted parameter.

Dependencies

To enable this parameter, set set Orifice type to Variable, Enable faults to On, and Enable temporal fault trigger to On.

Reporting when fault occurs — Fault condition report

None (default) | Warning | Error

Reporting preference for the fault condition. When reporting is set to Warning or Error, a message is displayed in the Simulink Diagnostic Viewer. When Error is selected, the simulation will stop if faulting occurs.

Dependencies

To enable this parameter, set Orifice type to VariableEnable faults to On.

Model Examples

Hydraulic Flow Rectifier Circuit

A flow rectifier circuit with four check valves and a flow control valve. It is used to allow a single flow control valve to control fluid flow in both directions. Similar to a Graetz circuit implemented with diodes, the check valves are arranged in such a way that flow always passes through the flow control valve in the same direction. In the Orifices subsystem, there are two more check valves that are used to select the orifice that the flow passes through depending on the flow direction.

Flow Divider

A simple way of modeling of the flow divider and using it with the load harness. The flow divider helps in dividing the input flow in desired percentage when the loads connected to the two lines are unequal. Flow divider allows flow only in forward direction. Therefore when it is used with the loads where unloading is done through reverse flows, then it should be used in conjunction with the direction control valves.