Local Restriction (G)

Restriction in flow area in gas network

• 

Libraries:
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 Type	Block 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:

$${\dot{m}}_A+{\dot{m}}_B=0$$

where $$\dot{m}$$_A and $$\dot{m}$$_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 Φ_A and Φ_B 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:

$$\begin{array}{l}{w}_A=\frac{{\dot{m}}_{ideal}}{{\rho }_{A}S}\\ w_B=\frac{{\dot{m}}_{ideal}}{{\rho }_{B}S}\\ w_R=\frac{{\dot{m}}_{ideal}}{{\rho }_R{S}_R}\end{array}$$

where:

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

• S_R 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:

$${\dot{m}}_{ideal}=\frac{{\dot{m}}_A}{C_d}$$

where C_d 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:

$$\Delta p_{AB}={\rho }_R\cdot w_R\cdot \left|w_R\right|\cdot \left(\frac{1+r}{2}\left(1-r\frac{{\rho }_R}{{\rho }_A}\right)-r\left(1-r\frac{{\rho }_R}{{\rho }_B}\right)\right)$$

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

For flow from port B to port A:

$$\Delta p_{BA}={\rho }_R\cdot w_R\cdot \left|w_R\right|\cdot \left(\frac{1+r}{2}\left(1-r\frac{{\rho }_R}{{\rho }_B}\right)-r\left(1-r\frac{{\rho }_R}{{\rho }_A}\right)\right)$$

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 Δp_transition = p_avg(1 — B_lam), where B_lam is the pressure ratio at the transition between laminar and turbulent regimes (Laminar flow pressure ratio parameter value) and p_avg = (p_A + p_B)/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:

$$p_{R_{AB}}=p_A-{\rho }_R\cdot w_R\cdot \left|w_R\right|\cdot \frac{1+r}{2}\left(1-r\frac{{\rho }_R}{{\rho }_A}\right)$$

For flow from port B to port A:

$$p_{R_{BA}}=p_B+{\rho }_R\cdot w_R\cdot \left|w_R\right|\cdot \frac{1+r}{2}\left(1-r\frac{{\rho }_R}{{\rho }_B}\right)$$

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 (p_A – p_B) to smoothly blend the pressure difference and the restriction pressure between the turbulent regime and the laminar regime:

• When Δp_transition ≤ p_A – p_B,

  then p_A – p_B = Δp_AB

  and p_R = p_RAB.

• When 0 ≤ p_A – p_B < Δp_transition,

  then p_A – p_B is smoothly blended between Δp_AB and Δp_lam

  and p_R is smoothly blended between p_RAB and p_Rlam.

• When –Δp_transition < p_A – p_B ≤ 0,

  then p_A – p_B is smoothly blended between Δp_BA and Δp_lam

  and p_R is smoothly blended between p_RBA and p_Rlam.

• When p_A – p_B ≤ –Δp_transition,

  then p_A – p_B = Δp_BA

  and p_R = p_RBA.

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 p_Bchoked 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 p_Achoked 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, p_A and p_B, respectively, depend on whether choking has occurred.

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

$$p_B=\{\begin{array}{ll}B.p,\hfill & \text{if }B.p\ge p_{B_{choked}}\hfill \\ p_{B_{choked}},\hfill & \text{if }B.p<p_{B_{choked}}\hfill \end{array}$$

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

$$p_A=\{\begin{array}{ll}A.p,\hfill & \text{if }A.p\ge p_{A_{choked}}\hfill \\ p_{A_{choked}},\hfill & \text{if }A.p<p_{A_{choked}}\hfill \end{array}$$

Assumptions and Limitations

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

• This block does not model supersonic flow.

Examples

Choked Flow in Gas Orifice

The choking behavior of a gas orifice modeled by the Local Restriction (G) block. The Controlled Reservoir (G) blocks are used to set up controlled pressure and temperature boundary conditions from the Reservoir Inputs subsystem to test the gas orifice.

• Open Model

Pneumatic Actuation Circuit

How the Foundation Library gas components can be used to model a controlled pneumatic actuator. The Directional Valve is a masked subsystem created from Variable Local Restriction (G) blocks to model the opening and closing of the flow paths. The Double-Acting Actuator is a masked subsystem created from Translational Mechanical Converter (G) blocks to model the interface between the gas network and the mechanical translational network.

• Open Model

Pneumatic Motor Circuit

How a pneumatic vane motor can be modeled using the Simscape™ language. The Pneumatic Motor component is built using the Simscape Foundation gas domain. It inherits from the foundation.gas.two_port_steady base class, which contains common equations that implement the upwind energy flow rate and the gas properties at the ports. The Pneumatic Motor subclass implements equations that describe behaviors specific to the component, such as the motor torque and flow rate characteristics and the mass and energy balance. The Pneumatic Motor block is inserted into the model using the Simscape Component block without the need to generate a separate library.

• Open Model

Ports

Input

AR — Restriction area control signal, m^2

physical signal

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

A — Inlet or outlet

gas

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

B — Inlet or outlet

gas

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

Parameters

Restriction type — Specify whether restriction area can change during simulation

Variable (default) | Fixed

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.

Minimum restriction area — Lower bound for the restriction cross-sectional area

1e-10 m^2 (default)

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.

Maximum restriction area — Upper bound for the restriction cross-sectional area

0.005 m^2 (default)

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.

Restriction area — Area normal to flow path at the restriction

1e-3 m^2 (default)

Area normal to flow path at the restriction.

Dependencies

Enabled when the Restriction type parameter is set to Fixed.

Cross-sectional area at ports A and B — Area normal to flow path at the ports

0.01 m^2 (default)

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

Discharge coefficient — Ratio of actual mass flow rate to theoretical mass flow rate through restriction

0.64 (default)

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.

Laminar flow pressure ratio — Pressure ratio at which gas flow transitions between laminar and turbulent regimes

0.999 (default)

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.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using Simulink® Coder™.

Version History

Introduced in R2016b