Pressure Relief Valve (G)

Valve for venting fluid when at excessive pressure

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Description

The Pressure Relief Valve (G) block models an orifice that opens with inlet pressure so as to keep its level from reaching extreme levels. At normal pressures, the valve is closed and only leakage flow develops. Above a specified pressure setting, the valve begins to open, allowing the gas to vent from its region of high pressure. The opening area increases with inlet pressure up to a maximum value, at which point the valve is fully open and pressure is again free to rise unabated.

The valve responds to either of two inlet pressure measurements—one relative to the outlet or to the environment. The chosen pressure measurement is referred to as the control pressure of the valve and it impacts the calculations of the block. The Control pressure specification block parameter determines which of the measurements the block uses during simulation.

The relationship between the opening area and the pressure drop depends on the parameterization of the valve. That relationship can take the form of a linear analytical expression or of a (generally nonlinear) tabulated function.

The flow can be laminar or turbulent, and it can reach (up to) sonic speeds. This happens at the vena contracta, a point just past the throat of the valve where the flow is both its narrowest and fastest. The flow then chokes and its velocity saturates, with a drop in downstream pressure no longer sufficing to increase its velocity. Choking occurs when the back-pressure ratio hits a critical value characteristic of the valve. Supersonic flow is not captured by the block.

Control and Other Pressures

The pressure at the inlet serves as the control signal of the valve. The greater its rise over the valve's pressure setting, the greater the opening area becomes. The inlet pressure measurement, however, can be relative to the outlet or the environment. The choice of reference depends on the setting of the Pressure control specification block parameter—either Pressure differential or Pressure at port A. The term control pressure is used here to refer to both measurements.

Pressure differential

When the Pressure control specification parameter is set to Pressure differential (the default setting), the control pressure is computed as:

pCtl=pApB,

where p is instantaneous pressure. The subscript Ctl denotes the control value and the subscripts A and B the inlet and outlet, respectively. The port pressures are instantaneous values determined during simulation. The pressure setting is likewise defined as:

PSet=[PAPB]Set,

where P is a constant pressure parameter. The subscript Set denotes the valve setting (here a differential). The term in parentheses is obtained as a constant from the Set pressure differential block parameter. Similarly for the maximum pressure of the valve (at which the valve is fully open):

PMax=PSet+ΔP,

where ΔP refers specifically to the pressure regulation range of the valve, obtained as a constant from the block parameter of the same name. The subscript Max denotes the maximum value.

Pressure at port A

When the Pressure control specification parameter is set to Pressure at port A, the control pressure is computed as:

pCtl=pA+pAtm,

where the subscript Atm denotes the atmospheric value (specified for the model as a constant in the Gas Properties (G) block of the Simscape Foundation library). The port pressure is an instantaneous value determined during simulation. For the pressure setting:

PSet=PA,Set+PAtm,

where the subscript A,Set denotes the valve setting, specified as a gauge pressure at port A. This value is obtained as a constant from the Set pressure (gauge) block parameter. The maximum pressure of the valve is:

PMax=PSet+ΔP,

(where, as before, ΔP refers specifically to the pressure regulation range of the valve).

Control Pressure Overshoot

The degree to which the control pressure exceeds the pressure setting determines how much the valve will open. The pressure overshoot is expressed here as a fraction of the (width of the) pressure regulation range:

p^=pCtlpSetΔP.

The control pressure (pCtl), pressure setting (pSet), and pressure regulation range (ΔP) are those defined for the control pressure specification chosen (Pressure differential or Pressure at port A).

The fraction—technically, the overshoot normalized—is valued at 0 in the fully closed valve and 1 in the fully open valve. If the calculation should return a value outside of these bounds, the nearest of the two is used instead. (In other words, the fraction saturates at 0 and 1.)

Numerical Smoothing

The normalized (and saturated) control pressure overshoot spans three regions. Below the pressure setting of the valve, its value is a constant zero. Above the maximum pressure—the sum of the pressure setting and the pressure regulation range—it is 1. In between, it varies, as a linear function of the control pressure measurement, pCtl.

The transitions between the regions are sharp and their slopes discontinuous. These pose a challenge to variable-step solvers (the sort commonly used with Simscape models). To precisely capture discontinuities, referred to in some contexts as zero crossing events, the solver must reduce its time step, pausing briefly at the time of the crossing in order to recompute its Jacobian matrix (a representation of the dependencies between the state variables of the model and their time derivatives).

This solver strategy is efficient and robust when discontinuities are present. It makes the solver less prone to convergence errors—but it can considerably extend the time needed to finish the simulation run, perhaps excessively so for practical use in real-time simulation. An alternative approach, used here, is to remove the discontinuities altogether.

Normalized pressure overshoot with sharp transitions

The block removes the discontinuities by smoothing them over a specified time scale. The smoothing, which adds a slight distortion to the control pressure overshoot, ensures that the valve eases into its limiting positions rather than snap (abruptly) into them. The smoothing is optional: you can disable it by setting its time scale to zero. The shape and scale of the smoothing, when applied, derives in part from the cubic polynomials:

λL=3p¯L22p¯L3

and

λR=3p¯R22p¯R3,

where

p¯L=p^Δp*

and

p¯R=p^(1Δp*)Δp*.

In the equations:

  • ƛL is the smoothing expression for the transition from the maximally closed position.

  • ƛR is the smoothing expression for the transition from the fully open position.

  • Δp* is the (unitless) characteristic width of the pressure smoothing region:

    Δp*=f*12,

    where f* is a smoothing factor valued between 0 and 1 and obtained from the block parameter of the same name.

    When the smoothing factor is 0, the control pressure overshoot stays in its original form—no smoothing applied—and its transitions remain abrupt. When it is 1, the smoothing spans the whole of the pressure regulation range (with the control pressure overshoot taking the shape of an S-curve).

    At intermediate values, the smoothing is limited to a fraction of that range. A value of 0.5, for example, will smooth the transitions over a quarter of the pressure regulation range on each side (for a total smooth region of half the regulation range).

The smoothing adds two new regions to the control pressure overshoot—one for the smooth transition on the left, another for that on the right, giving a total of five regions. These are expressed in the piecewise function:

p^*={0,p^0p^λL,p^<ΔP*p^,p^1ΔP*p^(1λR)+λR,p^<11p^1,

where the asterisk denotes a smoothed variable. The figure shows the effect of smoothing on the sharpness of the transitions.

Sonic Conductance

As the normalized control pressure varies during simulation, so does the mass flow rate through the valve. The relationship between the two variables, however, is indirect. The mass flow rate is defined in terms of the valve's sonic conductance and it is this quantity that the normalized inlet pressure determines.

Sonic conductance, if you are unfamiliar with it, describes the ease with which a gas will flow when it is choked—when its velocity is at its theoretical maximum (the local speed of sound). Its measurement and calculation are covered in detail in the ISO 6358 standard (on which this block is based).

Only one value is commonly reported in valve data sheets: one taken at steady state in the fully open position. This is the same specified in the Sonic conductance at maximum flow parameter when the Valve parameterization setting is Sonic conductance. For values across the opening range of the valve, this maximum is scaled by the normalized pressure overshoot:

C=(CMaxCMin)p^+CMin,

where C is sonic conductance and the subscripts Max and Min denote its values in the fully open and fully closed valve.

Other Parameterizations

Because sonic conductance may not be available (or the most convenient choice for your model), the block provides several equivalent parameterizations. Use the Valve parameterization drop-down list to select the best for the data at hand. The parameterizations are:

  • Restriction area

  • Sonic conductance

  • Cv coefficient (USCS)

  • Kv coefficient (SI)

The parameterizations differ only in the data that they require of you. Their mass flow rate calculations are still based on sonic conductance. If you select a parameterization other than Sonic conductance, then the block converts the alternate data—the (computed) opening area or a (specified) flow coefficient—into an equivalent sonic conductance.

Flow Coefficients

The flow coefficients measure what is, at bottom, the same quantity—the flow rate through the valve at some agreed-upon temperature and pressure differential. They differ only in the standard conditions used in their definition and in the physical units used in their expression:

  • Cv is measured at a generally accepted temperature of 60 ℉ 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 Valve parameterization block parameter is set to Cv coefficient (USCS).

  • Kv is measured at a generally accepted temperature of 15 ℃ and pressure drop of 1 bar; it is expressed in metric units of m3/h. This is the flow coefficient used in the model when the Valve parameterization block parameter is set to Kv coefficient (SI).

Sonic Conductance Conversions

If the valve parameterization is set to Cv Coefficient (USCS), the sonic conductance is computed at the maximally closed and fully open valve positions from the Cv coefficient (SI) at maximum flow and Cv coefficient (SI) at leakage flow block parameters:

C=(4×108Cv)m3/(s Pa),

where Cv is the flow coefficient value at maximum or leakage flow. The subsonic index, m, is set to 0.5 and the critical pressure ratio, bcr, is set to 0.3. (These are used in the mass flow rate calculations given in the Momentum Balance section.)

If the Kv coefficient (SI) parameterization is used instead, the sonic conductance is computed at the same valve positions (maximally closed and fully open) from the Kv coefficient (USCS) at maximum flow and Kv coefficient (USCS) at leakage flow block parameters:

C=(4.758×108Kv)m3/(s Pa),

where Kv is the flow coefficient value at maximum or leakage flow. The subsonic index, m, is set to 0.5 and the critical pressure ratio, bcr, is set to 0.3.

For the Restriction area parameterization, the sonic conductance is computed (at the same valve positions) from the Maximum opening area, and Leakage area block parameters:

C=(0.128×4S/π)L/(s bar),

where S is the opening area at maximum or leakage flow. The subsonic index, m, is set to 0.5 while the critical pressure ratio, bcr is computed from the expression:

0.41+0.272[p^(SMaxSLeak)+SLeakS]0.25.

Momentum Balance

The causes of those pressure losses incurred in the passages of the valve 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 assumed to reflect entirely in the sonic conductance of the valve (or in the data of the alternate valve parameterizations).

Mass Flow Rate

When the flow is choked, the mass flow rate is a function of the sonic conductance of the valve and of the thermodynamic conditions (pressure and temperature) established at the inlet. The function is linear with respect to pressure:

m˙ch=Cρ0pinT0Tin,

where:

  • C is the sonic conductance inside the valve. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

  • ρ is the gas density, here at standard conditions (subscript 0), obtained from the Reference density block parameter.

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

  • T is the gas temperature at the inlet (in) or at standard conditions (0), the latter obtained from the Reference temperature block parameter.

When the flow is subsonic, and therefore no longer choked, the mass flow rate becomes a nonlinear function of pressure—both that at the inlet as well as the reduced value at the outlet. In the turbulent flow regime (with the outlet pressure contained in the back-pressure ratio of the valve), the mass flow rate expression is:

m˙tur=Cρ0pinT0Tin[1(prbcr1bcr)2]m,

where:

  • pr is the back-pressure ratio, or that between the outlet pressure (pout) and the inlet pressure (pin):

    Pr=poutpin

  • bcr is the critical pressure ratio at which the flow becomes choked. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

  • m is the subsonic index, an empirical coefficient used to more accurately characterize the behavior of subsonic flows. Its value is obtained from the block parameter of the same name or by conversion of other block parameters (the exact source depending on the Valve parameterization setting).

When the flow is laminar (and still subsonic), the mass flow rate expression changes to:

m˙lam=Cρ0pin[1pr1blam]T0Tin[1(blambcr1bcr)2]m

where blam is the critical pressure ratio at which the flow transitions between laminar and turbulent regimes (obtained from the Laminar flow pressure ratio block parameter). Combining the mass flow rate expressions into a single (piecewise) function, gives:

m˙={m˙lam,blampr<1m˙tur,bcrpr<plamm˙ch,pr<bCr,

with the top row corresponding to subsonic and laminar flow, the middle row to subsonic and turbulent flow, and the bottom row to choked (and therefore sonic) flow.

Mass Balance

The volume of fluid inside the valve, 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 gas 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:

m˙A+m˙B=0,

where m˙ is defined as the mass flow rate into the valve through port A or B. Note that in this block the flow can reach but not exceed sonic speeds.

Energy Balance

The valve is modeled as an adiabatic component. No heat exchange can occur between the gas and the wall that surrounds it. No work is done on or by the gas 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:

ϕA+ϕB=0,

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

Ports

Conserving

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Opening through which the working fluid must enter the valve.

Opening through which the working fluid must exit the valve.

Parameters

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Choice of pressure measurement to use as valve control signal. The block uses this setting to determine when the valve should begin to open. In the default setting (Pressure differential), the opening pressure of the valve is expressed as a pressure drop from inlet to outlet. In the alternative setting (Pressure at port A), it is expressed as a gauge inlet pressure. This parameter determines which of the two measures appears as a parameter in the block dialog box.

Pressure drop, from inlet to outlet, required to open the valve. This value marks the beginning of the pressure regulation range, over which the valve progressively opens in an effort to contain a pressure rise. The opening area of the valve depends on the difference between the actual pressure drop to the value specified here.

Dependencies

This parameter is active and exposed in the block dialog box when the Pressure control specification parameter is set to Pressure differential.

Inlet gauge pressure required to open the valve. This parameter marks the beginning of the pressure regulation range, over which the valve progressively opens in an effort to contain a pressure rise The opening area of the valve depends on the difference between the actual inlet pressure to the value specified here.

Dependencies

This parameter is active and exposed in the block dialog box when the Pressure control specification parameter is set to Pressure at port A.

Width of the pressure range over which the opening area of the valve is variable. The pressure range begins at the set pressure measurement specified in the block (Set pressure differential or Pressure at port A), depending on the setting of the Pressure control specification parameter). It ends at the sum of the set pressure with the value specified here. The smaller the pressure regulation range, the faster the valve reaches its maximum opening area.

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 option is selected, the data specified in converted into equivalent sonic conductance, critical pressure ratio, and subsonic index. See the block description for more information on the conversion.

This parameter determines which measures of valve opening you must specify—and therefore which of those measures appear as parameters in the block dialog box.

Method by which to calculate the opening area of the valve. The default setting treats the opening area as a linear function of the orifice opening fraction. The alternative setting allows for a general, nonlinear relationship to be specified (in tabulated form).

Equivalent measure of the maximum flow rate allowed through the valve at some reference inlet conditions, generally those outlined in ISO 8778. The flow is at a maximum when the valve is fully open and the flow velocity is choked (it being saturated at the local speed of sound). This is the value generally reported by manufacturers in technical data sheets.

Sonic conductance is defined as the ratio of the mass flow rate through the valve to the product of the pressure and density upstream of the valve inlet. This parameter is often referred to in the literature as the C-value.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Equivalent measure of the minimum flow rate allowed through the valve at some reference inlet conditions, generally those outlined in ISO 8778. The flow is at a minimum when the valve is maximally closed and only a small leakage area—due to sealing imperfections, say, or natural valve tolerances—remains between its ports.

Sonic conductance is defined as the ratio of the mass flow rate through the valve to the product of the pressure and density upstream of the valve inlet. This parameter is often referred to in the literature as the C-value.

This parameter serves primarily to ensure that closure of the valve does not cause portions of the gas network to become isolated (a condition known to cause problems in simulation). The exact value specified here is less important that its being a (very small) number greater than zero.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Ratio of downstream to upstream absolute pressures at which the flow becomes choked (and its velocity becomes saturated at the local speed of sound). This parameter is often referred to in the literature as the b-value. Enter a number greater than or equal to zero and smaller than the Laminar flow pressure ratio block parameter.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Empirical exponent used to more accurately calculate the mass flow rate through the valve when the flow is subsonic. This parameter is sometimes referred to as the m-index. Its value is approximately 0.5 for valves (and other components) whose flow paths are fixed.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance.

Flow coefficient of the fully open valve, expressed in the US customary units of ft3/min (as described in NFPA T3.21.3). This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential. This is the value generally reported by manufacturers in technical data sheets.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Cv coefficient (USCS).

Flow coefficient of the maximally closed valve, expressed in the US customary units of ft3/min (as described in NFPA T3.21.3). This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential.

The purpose of this parameter is primarily to ensure that closure of the valve does not cause portions of the gas network to become isolated (a condition known to cause problems in simulation). The exact value specified here is less important that its being a (very small) number greater than zero.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Cv coefficient (USCS).

Flow coefficient of the fully open valve, expressed in the SI units of L/min. This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential. This is the value generally reported by manufacturers in technical data sheets.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Kv coefficient (SI).

Flow coefficient of the maximally closed valve, expressed in the SI units of L/min. This parameter measures the relative ease with which the gas will traverse the valve when driven by a given pressure differential.

The purpose of this parameter is primarily to ensure that closure of the valve does not cause portions of the gas network to become isolated (a condition known to cause problems in simulation). The exact value specified here is less important that its being a (very small) number greater than zero.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Kv coefficient (SI).

Opening area of the valve in the fully open position, when the valve is at the upper limit of the pressure regulation range. The block uses this parameter to scale the chosen measure of valve opening—sonic conductance, say, or CV flow coefficient—throughout the pressure regulation range.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Restriction area.

Opening area of the valve in the maximally closed position, when only internal leakage between the ports remains. This parameter serves primarily to ensure that closure of the valve does not cause portions of the gas network to become isolated (a condition known to cause problems in simulation). The exact value specified here is less important that its being a (very small) number greater than zero.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Restriction area.

Vector of control pressures at which to specify the chosen measure of valve opening—sonic conductance, flow coefficient (in SI or USCS forms), or opening area. This vector must be equal in size to that (or those) containing the valve opening data. The vector elements must be positive and increase monotonically in value from left to right.

The first vector element gives the pressure setting of the valve (at which the valve begins to open). This element is equivalent to the Set pressure parameter of the Linear opening parameterizations. The last element gives the maximum pressure, at which the valve is fully open. The difference between the two gives the pressure regulation range of the valve.

The definition of control pressure depends on the setting of the Pressure control specification parameter. It is either the pressure drop from inlet to outlet (Pressure differential setting) or the gauge pressure at the inlet (Pressure at port A setting).

Dependencies

This parameter is active and exposed in the block dialog box when the Opening parameterization setting is Tabulated data.

Vector of sonic conductances at the given valve control pressures. The sonic conductances each correspond to an element in the Control pressure (gauge) vector parameter, with the latter serving (in a Cartesian plot) as the abscissa and the former as the ordinate. The two vectors—of sonic conductances and control pressures—must be the same in size.

Sonic conductance measures the flow rate through the fully open valve at the onset of choking (when the flow reaches its maximum velocity, the local speed of sound). Measurements are carried out at some reference inlet conditions (generally those outlined in ISO 8778). Here, those conditions include the pressures given (as interpolation breakpoints) in the control pressure vector.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance and the Opening parameterization setting is Tabulated data.

Vector of critical pressure ratios at the given valve control pressures. The ratios each correspond to an element in the Control pressure (gauge) vector parameter, with the latter serving (in a Cartesian plot) as the abscissa and the former as the ordinate. The two vectors—of critical pressure ratios and control pressures—must be the same in size.

The critical pressure ratio is the fraction of downstream over upstream absolute pressures at which the flow becomes choked (and its velocity becomes saturated at the local speed of sound). Measurements are carried out at some reference inlet conditions (generally those outlined in ISO 8778). Here, those conditions include the pressures given (as interpolation breakpoints) in the control pressure vector.

The values specified here must each be greater than or equal to zero and smaller than the Laminar flow pressure ratio block parameter.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Sonic conductance and the Opening parameterization setting is Tabulated data.

Vector of flow coefficients, expressed in US customary units of ft3/min (as described in NFPA T3.21.3), at the given valve control pressures. The flow coefficients each correspond to an element in the Control pressure (gauge) vector parameter, with the latter serving (in a Cartesian plot) as the abscissa and the former as the ordinate. The two vectors—of flow coefficients and control pressures—must be the same in size.

The flow coefficient measures the relative ease with which the flow, when subjected to some reference pressure condition, can traverse the valve. Its value is commonly reported in technical data sheets for the fully open valve (alongside the pressure condition to which the measurement corresponds). The vector elements specified here each correspond to a different measurement.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Cv coefficient (USCS) and the Opening parameterization setting is Tabulated data.

Vector of flow coefficients, expressed in SI units of L/min (as described in NFPA T3.21.3), at the given valve control pressures. The flow coefficients each correspond to an element in the Control pressure (gauge) vector parameter, with the latter serving (in a Cartesian plot) as the abscissa and the former as the ordinate. The two vectors—of flow coefficients and control pressures—must be the same in size.

The flow coefficient measures the relative ease with which the flow, when subjected to some reference pressure condition, can traverse the valve. Its value is commonly reported in technical data sheets for the fully open valve (alongside the pressure condition to which the measurement corresponds). The vector elements specified here each correspond to a different measurement.

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Kv coefficient (SI) and the Opening parameterization setting is Tabulated data.

Vector of valve opening areas at the given control pressure breakpoints. The opening areas each correspond to an element in the Control pressure (gauge) vector parameter, with the latter serving (in a Cartesian plot) as the abscissa and the former as the ordinate. The two vectors—of opening areas and control pressures—must be the same in size.

The opening area is that normal to the direction of flow at the point of narrowest aperture (the location of the valve control member).

Dependencies

This parameter is active and exposed in the block dialog box when the Valve parameterization setting is Restriction area and the Opening parameterization setting is Tabulated data.

Area normal to the flow path at the valve ports. The ports are assumed to be of the same size. The flow area specified here should (ideally) match those of the inlets of adjoining components.

Amount of smoothing to apply to the opening function of the valve. This parameter determines the widths of the regions to be smoothed—one located at the fully open position, the other at the fully closed position.

The smoothing superposes on each region of the opening function a nonlinear segment (a third-order polynomial function, from which the smoothing arises). The greater the value specified here, the greater the smoothing is, and the broader the nonlinear segments become.

At the default value of 0, no smoothing is applied. The transitions to the maximally closed and fully open positions then introduce discontinuities (associated with zero-crossings). These can slow down the rate of simulation.

Pressure ratio at which the flow transitions between laminar and turbulent flow regimes. The pressure ratio is the fraction of the absolute pressure downstream of the valve over that just upstream of it. The flow is laminar when the actual pressure ratio is above the threshold specified here and turbulent when it is below. Typical values range from 0.995 to 0.999.

Absolute temperature used at the inlet in the measurement of sonic conductance (as defined in ISO 8778).

Gas density established at the inlet in the measurement of sonic conductance (as defined in ISO 8778).

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