Fixed restriction in flow area

**Library:**Simscape / Foundation Library / Gas / Elements

The Local Restriction (G) block models the pressure drop due to a temporary 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 restriction area, specified as a block parameter, remains constant during simulation. 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**

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) specifies a greater mass flow rate than the possible choked mass flow rate, you get a simulation error. For more information, see Choked Flow.

The mass balance equation is:

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

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

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.

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}≤ 0then

*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}

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

$${p}_{B}=\{\begin{array}{ll}B.p,\hfill & \text{if}B.p\ge {p}_{{B}_{choked}}\hfill \\ {p}_{{B}_{choked}},\hfill & \text{if}B.p{p}_{{B}_{choked}}\text{}\hfill \end{array}$$

For flow from port B to port A, *p*_{B} = *B.p* and

$${p}_{A}=\{\begin{array}{ll}A.p,\hfill & \text{if}A.p\ge {p}_{{A}_{choked}}\hfill \\ {p}_{{A}_{choked}},\hfill & \text{if}A.p{p}_{{A}_{choked}}\text{}\hfill \end{array}$$

Use the **Variables** tab in the block dialog
box (or the **Variables** section in the block Property
Inspector) to set the priority and initial target values for the block
variables prior to simulation. For more information, see Set Priority and Initial Target for Block Variables and Initial Conditions for Blocks with Finite Gas Volume.

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

This block does not model supersonic flow.

Was this topic helpful?