Flow restriction of variable area modeled per ISO 6358

**Library:**Simscape / Fluids / Gas / Valves & Orifices

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.

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,
*C*_{v} or
*K*_{v}) 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
*C*_{v} differs from
that of *K*_{v} in the
standard pressure and temperature established in its measurement and in the physical
units used in its expression:

*C*_{v}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)`

.*K*_{v}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.

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.

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:

$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$

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

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.

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

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

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:

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

where:

*p*_{r}is the ratio between downstream pressure (*p*_{out}) and upstream pressure (*p*_{in}) (each measured against absolute zero):$${p}_{\text{r}}=\frac{{p}_{\text{out}}}{{p}_{\text{in}}}$$

*b*_{cr}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:

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

where *b*_{lam} 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:

$$\dot{m}=\{\begin{array}{ll}{\dot{m}}_{\text{lam}},\hfill & \text{if}{b}_{\text{lam}}\le {p}_{\text{r}}1\hfill \\ {\dot{m}}_{\text{tur}},\hfill & \text{if}{b}_{\text{cr}}\le {p}_{\text{r}}{b}_{\text{lam}}\hfill \\ {\dot{m}}_{\text{ch}},\hfill & \text{if}{p}_{\text{r}}{b}_{\text{cr}}\hfill \end{array},$$

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 **C*_{v}m^3/(s*Pa)**Critical pressure ratio**:*b*_{cr}= 0.3**Subsonic index**:*m*= 0.5

If the `Kv coefficient (SI)`

parameterization is
used:

**Sonic conductance**:*C*= 4.78E-8 **K*_{v}m^3/(s*Pa)**Critical pressure ratio**:*b*_{cr}= 0.3**Subsonic index**:*m*= 0.5

For the `Restriction area`

parameterization:

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

).**Critical pressure ratio**:*b*_{cr}= 0.41 + 0.272 (*S*_{R}/*S*_{P})^0.25**Subsonic index**:*m*=`0.5`

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**).

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