# Variable Area Orifice (TL)

Local flow restriction with a variable cross-sectional area

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• Simscape / Fluids / Thermal Liquid / Valves & Orifices

## Description

The Variable Area Orifice (TL) block models the flow through a local restriction with variable opening area. The orifice contains a control member—such as a ball, spool, or diaphragm—which determines by its displacement the instantaneous opening area. Elements such as this are characteristic of valves and are, in the Thermal Liquid library, the foundation upon which all directional valve blocks are based. See, for example, the 2-Way Directional Valve (TL) block. Use this block to create a custom component with variable orifices if such is not provided in the Thermal Liquid library.

The orifice is assumed to consist of a contraction followed by a sudden expansion in flow area. The contraction causes the flow rate to rise and the pressure to drop. The expansion allows the pressure to recover, though only in part: past the vena contracta, where the flow is at its narrowest, the flow generally separates from the wall, causing it to lose some energy. The extent of the pressure recovery depends on the discharge coefficient of the orifice and on the ratio of the orifice and port areas. Set the Pressure recovery to `Off` to ignore this effect if necessary.

The effect that the motion of the control member has on the opening area of the orifice depends on the setting of the Opening orientation block parameter. In the default setting of `Positive`, the orifice (if within its opening range) opens when the control member moves in the positive direction. In the alternate setting of `Negative`, the orifice opens with motion in the negative direction.

### Orifice Positions

The orifice is continuously variable. It shifts smoothly between positions, of which it has two. One—the normal position—is that to which the orifice reverts when its control signal falls to zero. Unless a control member offset has been specified, the AB orifice is always fully closed in this position. Another—the working position—is that to which the orifice moves when its control signal rises to a maximum. The orifice is generally fully open in this position. Note that whether the orifice is in fact open and how open it is both depend on the value of the control member offset.

### Orifice Opening

Which position the orifice is in depends on the control member coordinate—a length that, in the valve blocks based on this orifice model, is often referred to as the orifice opening. This variable is calculated during simulation from the control member offset, specified via the block parameter of the same name, and from the control member displacement, a variable obtained from the physical signal specified at port S:

`$h={h}_{\text{0}}+\delta x,$`

where:

• h is the AB orifice opening.

• h0 is the AB opening offset.

• δ is the orifice orientation, `+1` if `Positive`, `-1` if `Negative`.

• x is the control member displacement.

A control member displacement of zero corresponds to a valve that is in its normal position. The orifice begins to open when the orifice opening (h) rises above zero and it continues to open until the orifice opening is at a maximum value. This maximum is obtained from the Maximum control displacement block parameter, in the linear orifice parameterization, or from the specified data vectors, in the tabulated orifice parameterizations.

### Opening Offsets

The orifice is by default configured so that it is fully closed when the control member displacement is zero. Such an orifice, when it represents a valve, is often described as being zero-lapped. It is possible, by applying an offset to the control member, to model an orifice that is underlapped—that is, partially open when in the normal position. The orifice can also be overlapped—fully closed over a range of control member displacements extending past the normal closed position.

The figure graphs the orifice opening—h(x)—in the cases of zero-lapped (I), underlapped (II), and overlapped (III) orifices. The opening offset—hs0—is zero in the first case, greater than zero in the second, and smaller than zero in the third. The control member must move right of its normal position (in the positive direction along the x-axis) for the overlapped orifice to crack open; it must move left of its normal position for the underlapped orifice to shut tight.

### Opening Characteristics

The orifice opening serves during simulation to calculate the mass flow rate through the orifice. The calculation can be a direct mapping from opening to flow rate or an indirect conversion, first from opening to orifice area and then from orifice area to mass flow rate. The calculation, and the data required for it, depend on the setting of the Valve parameterization block parameter:

• `Linear area-opening relationship` — Calculate the valve opening area from the control member position and from it obtain the mass flow rate through the valve. The opening area is assumed to vary linearly with the control member position. The slope of the linear expression is determined from the Maximum valve opening and Maximum opening area block parameters:

`${S}_{\text{Lin}}=\frac{{S}_{\text{Max}}}{{h}_{\text{Max}}}h,$`

where SLin is the linear form of the opening area, SMax is the value of the Maximum orifice area block parameter, hMax is the value of the Maximum control displacement block parameter. This expression is reformulated as a piecewise conditional expression so as to saturate the opening area at a small leakage value and ensure that transitions to the normal and working positions are smooth.

• `Tabulated data - Area vs. opening` — Calculate the valve opening area from the control member position and from it obtain the mass flow rate through the valve. The opening area can vary nonlinearly with the control member position. The relationship between the two is given by the tabulated data in the Valve opening vector and Opening area vector block parameters:

`${S}_{\text{Tab}}=S\left(h\right),$`

where STab is the tabulated form of the opening area, a function of the orifice opening, h.

• ```Tabulated data - Mass flow rate vs. opening and pressure drop``` — Calculate the mass flow rate directly from the control member position and the pressure drop across the valve. The relationship between the three variables can be nonlinear and it is given by the tabulated data in the Valve opening vector, Pressure drop vector, and Mass flow rate table block parameters:

`${\stackrel{˙}{m}}_{\text{Tab}}=\frac{{\rho }_{\text{Ref}}}{{\rho }_{\text{Avg}}}\stackrel{˙}{m}\left(h,\Delta p\right),$`

where $\stackrel{˙}{m}$ is the tabulated form of the mass flow rate, a function of the orifice opening, h, and of the pressure drop across the orifice, Δp. The mass flow rate is adjusted for temperature and pressure by the ratio ρRef/ρAvg, where ρ is the fluid density at some reference temperature and pressure (subscript `Ref`) or at the averages of those variables within the orifice.

### Numerical Smoothing

To ensure adequate simulation performance, the orifice opening area is smoothed over two small regions of the orifice opening, one near the fully closed state, the other near the fully open state. The smoothing is accomplished by means of polynomial expressions (to be incorporated into the final form of the opening area expression):

where ƛ is the smoothing factor applied at the minimum (subscript `Min`) and maximum (subscript `Max`) portions of the opening area expression. The smoothing factors are calculated as:

where hMin is the minimum orifice opening and ΔhSmooth is the range of orifice openings over which to smooth the linear form of the opening area. The value of SMin is calculated as:

`${h}_{\text{Min}}={h}_{\text{Max}}\frac{{S}_{\text{Leak}}}{{S}_{\text{Max}}},$`

where SLeak is the value of the Leakage area block parameter. The value of SSmooth is calculated as:

`$\Delta {h}_{\text{Smooth}}={f}_{\text{Smooth}}\frac{{h}_{\text{Max}}-{h}_{\text{Min}}}{2},$`

where fSmooth is the value of the Smoothing factor block parameter—a fraction between `0` and `1`, with `0` indicating zero smoothing and `1` maximum smoothing. The final, smoothed, orifice opening area is given by the piecewise expression:

Orifice Area Smoothing

### Leakage Flow

The primary purpose of the leakage flow rate of a closed orifice is to ensure that at no time a portion of the thermal liquid network becomes isolated from the remainder of the model. Such isolated portions reduce the numerical robustness of the model and can slow down simulation or cause it to fail. Leakage flow is generally present in real orifices but in a model its exact value is less important than its being a small number greater than zero. The leakage flow rate is determined from the Leakage area block parameter.

### Mass Balance

The volume of fluid inside the orifice, and therefore the mass of the same, is assumed to be very small and it is, for modeling purposes, ignored. As a result, no amount of fluid can accumulate there. By the principle of conservation of mass, the mass flow rate into the orifice through one port must therefore equal that out of the orifice through the other port:

`${\stackrel{˙}{m}}_{A}+{\stackrel{˙}{m}}_{B}=0,$`

where $\stackrel{˙}{m}$ is defined as the mass flow rate into the orifice through the port indicated by the subscript (A or B).

### Momentum Balance

The causes of the pressure losses incurred in the orifice are ignored in the block. Whatever their natures—sudden area changes, flow passage contortions—only their cumulative effect is considered during simulation. This effect is captured in the block by the discharge coefficient, a measure of the mass flow rate through the orifice relative to the theoretical value that it would have in an ideal orifice. Expressing the momentum balance in the orifice in terms of the pressure drop induced in the flow:

`${p}_{\text{A}}-{p}_{\text{B}}=\frac{{\stackrel{˙}{m}}_{\text{Avg}}\sqrt{{\stackrel{˙}{m}}_{\text{Avg}}^{2}+{\stackrel{˙}{m}}_{\text{Crit}}^{2}}}{2{\rho }_{\text{Avg}}{C}_{\text{D}}{S}_{\text{Smooth}}^{2}}\left[1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\right]{\xi }_{\text{p}},$`

where CD is the discharge coefficient, and ξp is the pressure drop ratio—a measure of the effect impressed by the pressure recovery that in real orifices occurs between the vena contracta (the point at which the flow is at its narrowest) and the outlet, assumed to be a small distance away. The subscript `Avg` denotes an average of the values at the thermal liquid ports. The critical mass flow rate ${\stackrel{˙}{m}}_{\text{Crit}}$ is calculated from the critical Reynolds number—that at which the flow in the orifice is assumed to transition from laminar to turbulent:

`${\stackrel{˙}{m}}_{\text{Crit}}={\text{Re}}_{\text{Crit}}{\mu }_{\text{Avg}}\sqrt{\frac{\pi }{4}{S}_{\text{Lin}}},$`

where μ denotes dynamic viscosity. The value of the pressure ratio depends on the setting of the Pressure recovery block parameter. In the default setting of `Off`:

`${\xi }_{\text{p}}=1.$`

If `On` is selected instead:

`${\xi }_{\text{p}}=\frac{\sqrt{1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\left(1-{C}_{\text{D}}^{2}\right)}-{C}_{\text{D}}\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}}{\sqrt{1-{\left(\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}\right)}^{2}\left(1-{C}_{\text{D}}^{2}\right)}+{C}_{\text{D}}\frac{{S}_{\text{Smooth}}}{{S}_{\text{Lin}}}}.$`

### Energy Balance

The orifice is modeled as an adiabatic component. No heat exchange can occur between the fluid and the wall that surrounds it. No work is done on or by the fluid as it traverses from inlet to outlet. With these assumptions, energy can flow by advection only, through ports A and B. By the principle of conservation of energy, the sum of the port energy flows must always equal zero:

`${\varphi }_{\text{A}}+{\varphi }_{\text{B}}=0,$`

where ϕ is defined as the energy flow rate into the orifice through one of the ports (A or B).

## Ports

### Input

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Instantaneous displacement of the valve control member.

### Conserving

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Opening through which the flow can enter or exit the valve.

Opening through which the flow can enter or exit the valve.

## Parameters

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Method by which to model the opening characteristics of the orifice. The default setting prescribes a linear relationship between the orifice opening area and the orifice opening. The alternative settings allow for a general, nonlinear relationship to be specified in tabulated form, in one case between the opening area and the orifice opening, in the other case between the mass flow rate and both the orifice opening and the pressure drop between the ports.

Orifice opening at which the orifice is fully open and its opening area is therefore at a maximum. This parameter is used to calculate the slope of the linear expression relating the opening area to the orifice opening.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Linear area-opening relationship`.

Opening area of the orifice in the fully open position, when the orifice opening is that specified in the maximum valve opening block parameter. This parameter is used to calculate the slope of the linear expression relating the opening area to the orifice opening.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Linear area-opening relationship`.

Opening area of the orifice in the fully closed position, when only internal leakage between its ports remains. This parameter serves primarily to ensure that closure of the orifice does not cause portions of the thermal liquid network to become isolated. The exact value specified here is less important than its being a small number greater than zero.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Linear area-opening relationship`.

Measure of the amount of smoothing to apply to the opening area function. This parameter determines the widths of the regions to be smoothed, one being at the fully open position, the other at the fully closed position. The smoothing superposes on the linear opening area function two nonlinear segments, one for each region of smoothing. The greater the value specified, the greater the smoothing and the broader the nonlinear segments.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Linear area-opening relationship`.

Vector of orifice openings at which to specify—dependent on the valve parameterization—the opening area of the orifice or its mass flow rate. The vector elements must increase monotonically from left to right. This vector must be equal in size to that specified in the Opening area vector block parameter or to the number of rows in the Mass flow rate table block parameter.

This data serves to construct a one-way lookup table by which to determine, from the orifice opening, the opening area of the orifice or a two-way lookup table by which to determine, from the orifice opening and pressure drop, the mass flow rate of the orifice. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the data range).

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Tabulated data - Area vs. opening`.

Vector of opening areas corresponding to the breakpoints defined in the Opening vector block parameter. The vector elements must increase monotonically from left to right (with increasing values of the orifice opening). This vector must be equal in size to the number of orifice opening breakpoints.

This data serves to construct a one-way lookup table by which to determine from the orifice opening the opening area of the orifice. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the data range).

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to `Tabulated data - Area vs. opening`.

Vector of pressure differentials from port A to port B at which to specify the mass flow rate of the orifice. The vector elements must increase monotonically from left to right. This vector must be equal in size to the number of columns in the Mass flow rate table block parameter.

This data serves to construct a two-way lookup table by which to determine, from the orifice opening and pressure drop, the opening area of the orifice. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the data range).

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to ```Tabulated data - Mass flow rate vs. opening and pressure drop```.

Matrix of mass flow rates corresponding to the breakpoints defined in the Opening vector and Pressure drop vector block parameters. The orifice opening increases from row to row from top to bottom. The pressure drop increases from column to column from left to right. The mass flow rate must increase monotonically in the same directions (with increasing control member position and increasing pressure drop).

This data serves to construct a two-way lookup table by which to determine, from the orifice opening and pressure drop, the opening area of the orifice. Data is handled with linear interpolation (within the tabulated data range) and nearest-neighbor extrapolation (outside of the data range). Ensure that the number of rows is equal to the size of the Opening area vector block parameter and that the number of columns is equal to the size of the Pressure drop vector block parameter.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to ```Tabulated data - Mass flow rate vs. opening and pressure drop```.

Nominal inlet temperature, with reference to absolute zero, at which to specify the tabulated data. This parameter is used to adjust the mass flow rate according to the temperature measured during simulation.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to ```Tabulated data - Mass flow rate vs. opening and pressure drop```.

Nominal inlet pressure, with reference to absolute zero, at which to specify the tabulated data. This parameter is used to adjust the mass flow rate according to the pressure measured during simulation.

#### Dependencies

This parameter is active when the Orifice parameterization block parameter is set to ```Tabulated data - Mass flow rate vs. opening and pressure drop```.

Offset between the control member and the location at which, in the normal orifice position, it would completely cover the orifice. Specify a positive offset to model an underlapped orifice or a negative offset to model an overlapped orifice. For detail on how the opening offsets impact the block calculations, see the block description.

Area normal to the flow path at each port. The ports are assumed to be equal in size. The flow area specified here should match those of the inlets of those components to which the orifice connects.

Average distance traversed by the fluid as it moves from inlet to outlet. This distance is used in the calculation of the internal thermal conduction that occurs between the two ports (as part of the smoothed upwind energy scheme employed in the thermal liquid domain).

Ratio of the actual flow rate through the orifice to the theoretical value that it would have in an ideal valve. This semi-empirical parameter measures the flow allowed through the orifice: the greater its value, the greater the flow rate. Refer to the valve data sheet, if available, for this parameter.

Reynolds number at which the flow is assumed to transition between laminar and turbulent regimes.

## References

[1] Measurement of fluid flow by means of pressure differential devices inserted in circular cross-section conduits running full — Part 2: Orifice plates (ISO 5167–2:2003). 2003.