# Local Restriction (G)

Restriction in flow area in gas network

• Library:
• Simscape / Foundation Library / Gas / Elements

## Description

The Local Restriction (G) block models the pressure drop due to a localized reduction in flow area, such as a valve or an orifice, in a gas network. Choking occurs when the restriction reaches the sonic condition.

Ports A and B represent the restriction inlet and outlet. The input physical signal at port AR specifies the restriction area. Alternatively, you can specify a fixed restriction area as a block parameter.

The block icon changes depending on the value of the Restriction type parameter.

Restriction TypeBlock Icon

`Variable`

`Fixed`

The restriction is adiabatic. It does not exchange heat with the environment.

The restriction consists of a contraction followed by a sudden expansion in flow area. The gas accelerates during the contraction, causing the pressure to drop. The gas separates from the wall during the sudden expansion, causing the pressure to recover only partially due to the loss of momentum.

Local Restriction Schematic

Caution

Gas flow through this block can choke. If a Mass Flow Rate Source (G) block or a Controlled Mass Flow Rate Source (G) block connected to the Local Restriction (G) block specifies a greater mass flow rate than the possible choked mass flow rate, the simulation generates an error. For more information, see Choked Flow.

### Mass Balance

The mass balance equation is:

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

where $\stackrel{˙}{m}$A and $\stackrel{˙}{m}$B are the mass flow rates at ports A and B, respectively. Flow rate associated with a port is positive when it flows into the block.

### Energy Balance

The energy balance equation is:

`${\Phi }_{A}+{\Phi }_{B}=0$`

where ΦA and ΦB are energy flow rates at ports A and B, respectively.

The block is assumed adiabatic. Therefore, there is no change in specific total enthalpy between port A, port B, and the restriction:

`$\begin{array}{l}{h}_{A}+\frac{{w}_{A}^{2}}{2}={h}_{R}+\frac{{w}_{R}^{2}}{2}\\ {h}_{B}+\frac{{w}_{B}^{2}}{2}={h}_{R}+\frac{{w}_{R}^{2}}{2}\end{array}$`

where h is the specific enthalpy at port A, port B, or restriction R, as indicated by the subscript.

The ideal flow velocities at port A, port B, and the restriction are:

`$\begin{array}{l}{w}_{A}=\frac{{\stackrel{˙}{m}}_{ideal}}{{\rho }_{A}S}\\ {w}_{B}=\frac{{\stackrel{˙}{m}}_{ideal}}{{\rho }_{B}S}\\ {w}_{R}=\frac{{\stackrel{˙}{m}}_{ideal}}{{\rho }_{R}{S}_{R}}\end{array}$`

where:

• S is the cross-sectional area at ports A and B.

• SR is the cross-sectional area at the restriction.

• ρ is the density of gas volume at port A, port B, or restriction R, as indicated by the subscript.

The theoretical mass flow rate without nonideal effects is:

`${\stackrel{˙}{m}}_{ideal}=\frac{{\stackrel{˙}{m}}_{A}}{{C}_{d}}$`

where Cd is the discharge coefficient.

### Momentum Balance

The pressure difference between ports A and B is based on a momentum balance for flow area contraction between the inlet and the restriction, plus a momentum balance for sudden flow area expansion between the restriction and the outlet.

For flow from port A to port B:

`$\Delta {p}_{AB}={\rho }_{R}\cdot {w}_{R}\cdot |{w}_{R}|\cdot \left(\frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{A}}\right)-r\left(1-r\frac{{\rho }_{R}}{{\rho }_{B}}\right)\right)$`

where r is the area ratio, r = SR/S.

For flow from port B to port A:

`$\Delta {p}_{BA}={\rho }_{R}\cdot {w}_{R}\cdot |{w}_{R}|\cdot \left(\frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{B}}\right)-r\left(1-r\frac{{\rho }_{R}}{{\rho }_{A}}\right)\right)$`

The pressure differences in the two preceding equations are proportional to the square of the flow rate. This is the typical behavior for turbulent flow. However, for laminar flow, the pressure difference becomes linear with respect to flow rate. The laminar approximation for the pressure difference is:

`$\Delta {p}_{lam}=\sqrt{\frac{{\rho }_{R}\Delta {p}_{transition}}{2}}\left(1-r\right)$`

The threshold for transition from turbulent flow to laminar flow is defined as Δptransition = pavg(1 — Blam), where Blam is the pressure ratio at the transition between laminar and turbulent regimes (Laminar flow pressure ratio parameter value) and pavg = (pA + pB)/2.

The pressure at the restriction is based on a momentum balance for flow area contraction between the inlet and the restriction.

For flow from port A to port B:

`${p}_{{R}_{AB}}={p}_{A}-{\rho }_{R}\cdot {w}_{R}\cdot |{w}_{R}|\cdot \frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{A}}\right)$`

For flow from port B to port A:

`${p}_{{R}_{BA}}={p}_{B}+{\rho }_{R}\cdot {w}_{R}\cdot |{w}_{R}|\cdot \frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{B}}\right)$`

For laminar flow, the pressure at the restriction is approximately

`${p}_{{R}_{lam}}={p}_{avg}-{\rho }_{R}\cdot {w}_{R}^{2}\frac{1-{r}^{2}}{2}$`

The block uses a cubic polynomial in terms of (pApB) to smoothly blend the pressure difference and the restriction pressure between the turbulent regime and the laminar regime:

• When ΔptransitionpApB,

then pApB = ΔpAB

and pR = pRAB.

• When 0 ≤ pApB < Δptransition,

then pApB is smoothly blended between ΔpAB and Δplam

and pR is smoothly blended between pRAB and pRlam.

• When Δptransition < pApB ≤ 0,

then pApB is smoothly blended between ΔpBA and Δplam

and pR is smoothly blended between pRBA and pRlam.

• When pApB ≤ –Δptransition,

then pApB = ΔpBA

and pR = pRBA.

### Choked Flow

When the flow through the restriction becomes choked, further changes to the flow are dependent on the upstream conditions and are independent of the downstream conditions.

If A.p is the Across variable at port A and pBchoked is the hypothetical pressure at port B, assuming choked flow from port A to port B, then

`$A.p-{p}_{{B}_{choked}}={\rho }_{R}\cdot {a}_{R}^{2}\left(\frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{A}}\right)-r\left(1-r\frac{{\rho }_{R}}{{\rho }_{B}}\right)\right)$`

where a is speed of sound.

If B.p is the Across variable at port B and pAchoked is the hypothetical pressure at port A, assuming choked flow from port B to port A, then

`$B.p-{p}_{{A}_{choked}}={\rho }_{R}\cdot {a}_{R}^{2}\left(\frac{1+r}{2}\left(1-r\frac{{\rho }_{R}}{{\rho }_{B}}\right)-r\left(1-r\frac{{\rho }_{R}}{{\rho }_{A}}\right)\right)$`

The actual pressures at ports A and B, pA and pB, respectively, depend on whether choking has occurred.

For flow from port A to port B, pA = A.p and

For flow from port B to port A, pB = B.p and

### Assumptions and Limitations

• The restriction is adiabatic. It does not exchange heat with the environment.

• This block does not model supersonic flow.

## Ports

### Input

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Input physical signal that controls the gas flow restriction area. The signal saturates when its value is outside the minimum and maximum restriction area limits, specified by the block parameters.

#### Dependencies

This port is visible only if you set the Restriction type parameter to `Variable`.

### Conserving

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Gas conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

Gas conserving port associated with the inlet or outlet of the local restriction. This block has no intrinsic directionality.

## Parameters

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Select whether the restriction area can change during simulation:

• `Variable` — The input physical signal at port AR specifies the restriction area, which can vary during simulation. The Minimum restriction area and Maximum restriction area parameters specify the lower and upper bounds for the restriction area.

• `Fixed` — The restriction area, specified by the Restriction area block parameter value, remains constant during simulation. Port AR is hidden.

The lower bound for the restriction cross-sectional area. You can use this parameter to represent the leakage area. The input signal AR saturates at this value to prevent the restriction area from decreasing any further.

#### Dependencies

Enabled when the Restriction type parameter is set to `Variable`.

The upper bound for the restriction cross-sectional area. The input signal AR saturates at this value to prevent the restriction area from increasing any further.

#### Dependencies

Enabled when the Restriction type parameter is set to `Variable`.

Area normal to flow path at the restriction.

#### Dependencies

Enabled when the Restriction type parameter is set to `Fixed`.

Area normal to flow path at ports A and B. This area is assumed the same for the two ports.

Ratio of actual mass flow rate to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter that accounts for nonideal effects.

Pressure ratio at which the gas flow transitions between laminar and turbulent regimes. The pressure loss is linear with respect to mass flow rate in the laminar regime and quadratic with respect to mass flow rate in the turbulent regime.

## Version History

Introduced in R2016b