Variable Orifice ISO 6358 (G)

Flow restriction of variable area modeled per ISO 6358

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Description

The Variable Orifice ISO 6358 (G) block models the pressure loss incurred in a gas network due to a purely resistive element of variable size—such as a controlled flow restriction, orifice, or valve—using the methods outlined in the ISO 6358 standard. These methods are used in industry in the measurement and reporting of gas flow characteristics. The availability of data on the coefficients of the ISO formulas makes the ISO parameterizations useful when component geometries are unavailable or cumbersome to specify.

Orifice Parameterizations

The default orifice parameterization is based on the most recommended of the ISO 6358 methods: one based on the sonic conductance of the resistive element at steady state. The sonic conductance measures the ease with which a gas can flow when choked, a condition in which the flow velocity is at its theoretical maximum (the local speed of sound). Choking occurs when the ratio between downstream and upstream pressures reaches a critical value known as the critical pressure ratio.

The remaining parameterizations are formulated in terms of alternative measures of flow capacity: the flow coefficient (in either of its forms, Cv or Kv) or the size of the flow restriction. The flow coefficient measures the ease with which a gas can flow when driven by a certain pressure differential. The definition of Cv differs from that of Kv in the standard pressure and temperature established in its measurement and in the physical units used in its expression:

• Cv is measured at a generally accepted temperature of `60°F` and pressure drop of `1 PSI`; it is expressed in imperial units of `US gpm`. This is the flow coefficient used in the model when the Orifice parameterization block parameter is set to `Cv coefficient (USCS)`.

• Kv is measured at a generally accepted temperature of `15°C` and pressure drop of `1 bar`; it is expressed in metric units of `m^3/h`. This is the flow coefficient used in the model when the Orifice parameterization block parameter is set to ```Kv coefficient (SI)```.

Two values are required for the chosen measure of flow capacity (that for which the orifice parameterization is named): a maximum and a minimum. The maximum corresponds to a valve open to full capacity; this is the value for which the coefficient data are frequently reported in valve data sheets. The minimum corresponds to a valve closed tight, when only leakage flow remains, if any at all. This lower bound serves primarily to ensure the numerical robustness of the model. Its exact value is less important than its being a (generally very small) number greater than zero.

Opening Parameterizations

The sonic conductance and (in certain settings) the critical pressure ratio are determined during simulation from the input at port L. This input is the control signal and it is, in some valves, associated with stroke or lift percent. The control signal can range in value from `0` to `1`. If a lesser or greater value is specified, it is adjusted to the nearest of the two limits. In other words, the signal is saturated at `0` and `1`.

If the orifice parameterization is changed from its default of ```Sonic conductance```, the sonic conductance and critical pressure ratio are determined as linear functions of the chosen measure of flow capacity. This alternative measure is in turn obtained from the control signal. Calculations of mass flow rate are carried out as before, using the equations described in ``Sonic Conductance Parameterization''.

The conversion from a control signal to the chosen measure of flow capacity depends on the opening parameterization selected in the block. Flow is always maximally restricted when the control signal is `0` and minimally so when the control signal is `1`. However, in between, the flow rate achieved within the resistive element depends on whether the opening parameterization is linear or based on tabulated data:

• `Linear` — The measure of flow capacity (sonic conductance, Cv coefficient, other) is proportional to the control signal at port L. The two vary in tandem until the control signal either drops below `0` (flow is maximally restricted) or rises above `1` (flow is minimally restricted). As the control signal rises from `0` to `1`, the measure of flow capacity scales from its specified minimum to its specified maximum.

In the conversion to the parameters of the sonic conductance parameterization, both the critical pressure ratio and the subsonic index are treated as constants, each independent of the control signal.

• `Tabulated data` — The measure of flow capacity is a tabulated function of the control signal at port L. This function is based on a one-way lookup table with the control signal corresponding to the abscissa and the measure of flow capacity to the ordinate. The tabulated data must be specified such that the measure of flow capacity increases monotonically with the control signal.

In the conversion to the parameters of the sonic conductance parameterization, the critical pressure ratio is treated as a function of the control signal while the subsonic index is treated as a constant.

Mass Balance

The volume of fluid inside the resistive element, 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 valve through one port must therefore equal that out of the valve through the other port:

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

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

Momentum Balance

The causes of the pressure losses incurred in the passages of the resistive element are ignored in the block. Whatever their natures—sudden area changes, flow passage contortions—only their cumulative effect is considered during simulation. It is this cumulative effect that the sonic conductance in the default orifice parameterization captures in a model. If a different parameterization is selected, the coefficients on which it is based are converted into the parameters of the default parameterization; the mass flow rate calculation is then carried out as described in Sonic Conductance Parameterization.

Sonic Conductance Parameterization

In a choked flow, the mass flow rate through the resistive element is calculated as:

`${\stackrel{˙}{m}}_{\text{ch}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}},$`

where:

• C is the sonic conductance inside the resistive element.

• ρ is the gas density, here at standard conditions (subscript `0`, ```1.185 kg/m^3```).

• p is the absolute gas pressure, here corresponding to the inlet (subscript `in`).

• T is the gas temperature at the inlet (subscript `in`) or at standard conditions (`0`, `293.15 K`).

In a subsonic and turbulent flow, the mass flow rate calculation becomes:

`${\stackrel{˙}{m}}_{\text{tur}}=C{\rho }_{\text{0}}{p}_{\text{in}}\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}}{\left[1-{\left(\frac{{p}_{\text{r}}-{b}_{\text{cr}}}{1-{b}_{\text{cr}}}\right)}^{2}\right]}^{m},$`

where:

• pr is the ratio between downstream pressure (pout) and upstream pressure (pin) (each measured against absolute zero):

`${p}_{\text{r}}=\frac{{p}_{\text{out}}}{{p}_{\text{in}}}$`
• bcr is the critical pressure ratio at which the gas flow first begins to choke.

• m is the subsonic index, an empirical coefficient used to more accurately characterize the behavior of subsonic flows.

In a subsonic and laminar flow, the mass flow rate calculation changes to:

`${\stackrel{˙}{m}}_{\text{lam}}=C{\rho }_{\text{0}}\left(\frac{{p}_{\text{out}}-{p}_{\text{in}}}{1-{b}_{\text{lam}}}\right)\sqrt{\frac{{T}_{\text{0}}}{{T}_{\text{in}}}}{\left[1-{\left(\frac{{b}_{\text{lam}}-{b}_{\text{cr}}}{1-{b}_{\text{cr}}}\right)}^{2}\right]}^{m},$`

where blam is the critical pressure ratio at which the flow transitions between laminar and turbulent regimes. Combining the calculations for the three flow regimes into a piecewise function gives across all pressure ratios:

Conversions to Sonic Conductance

If the orifice parameterization is set to ```Cv coefficient (USCS)```, the parameters of the mass flow rate calculation are set as follows:

• Sonic conductance: C = 4E-8 * Cv m^3/(s*Pa)

• Critical pressure ratio: bcr = 0.3

• Subsonic index: m = 0.5

If the `Kv coefficient (SI)` parameterization is used:

• Sonic conductance: C = 4.78E-8 * Kv m^3/(s*Pa)

• Critical pressure ratio: bcr = 0.3

• Subsonic index: m = 0.5

For the `Restriction area` parameterization:

• Sonic conductance: C = 0.128 * 4 SR/π L/(s*bar), where S is the flow area in the resistive element (subscript `R`).

• Critical pressure ratio: bcr = 0.41 + 0.272 (SR/SP)^0.25

• Subsonic index: m = `0.5`

Energy Balance

The resistive element 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 then always equal zero:

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

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

Ports

Input

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Control signal by means of which to specify the opening (in some valves associated with stroke or lift percent) of the orifice. The orifice is fully closed at a value of `0` and fully open at a value of `1`.

Conserving

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Opening through which the flow can enter or exit the flow resistance. Which of the ports serves as inlet and which as outlet depends on the direction of flow.

Opening through which the flow can enter or exit the flow resistance. Which of the ports serves as inlet and which as outlet depends on the direction of flow.

Parameters

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Choice of ISO method to use in the calculation of mass flow rate. All calculations are based on the `Sonic conductance` parameterization. If a different parameterization is selected, the coefficients that characterize it are converted into sonic conductance, critical pressure ratio, and subsonic index.

Method by which to convert the control signal specified at port L to the chosen measure of flow capacity (sonic conductance, either of the flow coefficients, or restriction area). See the block description for more detail on the opening parameterizations.

Value of the sonic conductance when the control signal specified at port L is `1`. The cross-sectional area available for flow is then at a maximum. During simulation the sonic conductance at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Sonic conductance at leakage flow parameter.

Value of the sonic conductance when the control signal specified at port L is `0`. The cross-sectional area available for flow is then at a minimum, with only a negligible leakage flow remaining between the ports. The primary purpose of this parameter is to ensure the numerical robustness of the model during simulation. Its exact value is less important than its being a very small number greater than zero.

During simulation the sonic conductance at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Sonic conductance at maximum flow parameter.

Pressure ratio at which flow first begins to choke and the flow velocity reaches its maximum, given by the local speed of sound. The pressure ratio is the fraction of the absolute pressure downstream of the resistive element over the absolute pressure upstream of the same component.

Vector of control signal values at which to specify the chosen measure of flow capacity (sonic conductance, either of the flow coefficients, or restriction area). The control signal is bounded at `0` and `1`, with each value in this range corresponding to an opening fraction of the resistive element. The greater the value, the greater the opening and (generally) the easier the flow.

The opening fractions must increase monotonically across the vector from left to right. The size of this vector must be the same as that of the chosen measure of flow capacity (Sonic conductance vector, Cv coefficient (USCS) vector, other).

Vector of sonic conductances inside the resistive element, with each conductance corresponding to a value in the Opening fraction vector parameter. The sonic conductances must increase monotonically from left to right, with greater opening fractions generally translating into greater sonic conductances. The size of the vector must be the same as that of the Opening fraction vector.

Vector of critical pressure ratios at which the flow first chokes, with each critical pressure ratio corresponding to a value in the Opening fraction vector parameter. The critical pressure ratio is the fraction of downstream to upstream pressures at which the flow velocity reaches the local speed of sound. The size of the vector must be the same as that of the Opening fraction vector.

Exponent used to more accurately calculate the mass flow rate in the subsonic flow regime as described in ISO 6358. This parameter is treated as a constant independent of the opening fraction specified by the control signal at port L.

Value of the Cv flow coefficient when the control signal specified at port L is `1`. The cross-sectional area available for flow is then at a maximum. During simulation the flow coefficient at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Cv coefficient (USCS) at leakage flow parameter.

Value of the Cv flow coefficient when the control signal specified at port L is `0`. The cross-sectional area available for flow is then at a minimum, with only a negligible leakage flow remaining between the ports. The primary purpose of this parameter is to ensure the numerical robustness of the model during simulation. Its exact value is less important than its being a very small number greater than zero.

During simulation the flow coefficient at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Cv coefficient (USCS) at maximum flow parameter.

Vector of Cv flow coefficients expressed in US customary units of ft^3}/min, with each coefficient corresponding to a value in the Opening fraction vector parameter. The flow coefficients must increase monotonically from left to right, with greater opening fractions generally translating into greater flow coefficients. The size of the vector must be the same as that of the Opening fraction vector.

Value of the Kv flow coefficient when the control signal specified at port L is `1`. The cross-sectional area available for flow is then at a maximum. During simulation the flow coefficient at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Kv coefficient (SI) at leakage flow parameter.

Value of the Kv flow coefficient when the control signal specified at port L is `0`. The cross-sectional area available for flow is then at a minimum, with only a negligible leakage flow remaining between the ports. The primary purpose of this parameter is to ensure the numerical robustness of the model during simulation. Its exact value is less important than its being a very small number greater than zero.

During simulation the flow coefficient at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Kv coefficient (SI) at maximum flow parameter.

Vector of Kv flow coefficients expressed in SI units of m^3/hr, with each coefficient corresponding to a value in the Opening fraction vector parameter. The flow coefficients must increase monotonically from left to right, with greater opening fractions generally translating into greater flow coefficients. The size of the vector must be the same as that of the Opening fraction vector.

Flow coefficient expressed in SI units of m^3/hr. This parameter measures the ease with which the gas traverses the resistive element when driven by a pressure differential. See the block description for the correspondence between these parameters.

Value of the flow area at the point of shortest aperture when the control signal specified at port L is `1`. The cross-sectional area available for flow is then at a maximum. During simulation the flow area at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Restriction area at leakage flow parameter.

Value of the flow area at the point of shortest aperture when the control signal specified at port L is `0`. The cross-sectional area available for flow is then at a minimum, with only a negligible leakage flow remaining between the ports. The primary purpose of this parameter is to ensure the numerical robustness of the model during simulation. Its exact value is less important than its being a very small number greater than zero.

During simulation the flow area at intermediate control signals (those valued between `0` and `1`) is set by linear interpolation between this value and that of the Restriction area at maximum flow parameter.

Vector of flow areas as measured at the point of shortest aperture, with each flow area corresponding to a value in the Opening fraction vector parameter. The flow areas must increase monotonically from left to right, with greater opening fractions generally translating into greater flow areas. The size of the vector must be the same as that of the Opening fraction vector.

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 resistive element connects.

Pressure ratio at which flow transitions between laminar and turbulent flow regimes. The pressure ratio is the fraction of the absolute pressure downstream of the resistive element over the absolute pressure upstream of the same component. Typical values range from `0.995` to `0.999`.

Temperature at standard reference atmosphere, defined as 293.15 K in ISO 8778.

Density at standard reference atmosphere, defined as 1.185 kg/m3 in ISO 8778.

References

[1] P. Beater, Pneumatic Drives, Springer-Verlag Berlin Heidelberg, 2007.