Engee documentation

Local Restriction (G)

Local narrowing of the flow in the gas network.

blockType: AcausalFoundation.Gas.Elements.LocalRestriction

local restriction (g)

Local Restriction (G)

Path in the library:

/Physical Modeling/Fundamental/Gas/Elements/Local Restriction (G)

variable local restriction (g)

Variable Local Restriction (G)

Path in the library:

/Physical Modeling/Fundamental/Gas/Elements/Variable Local Restriction (G)

Description

Block Local Restriction (G) simulates a pressure drop due to a local constriction, such as a valve or an opening, in a gas network. The flow through the constriction becomes critical when the gas reaches the speed of sound.

Ports A and B represent the input and output of the unit Local Restriction (G). The input signal on the AR port determines the area of the constriction. In addition, you can specify a fixed narrowing area as a block parameter.

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

The narrowing of the flow is considered an adiabatic system, that is, it does not exchange heat with the environment.

The block model consists of a sudden narrowing, followed by a sudden expansion of the passage section. The gas accelerates as it passes through the constriction, resulting in a pressure drop. Then it separates from the wall during a sudden expansion, as a result of which the pressure is restored only partially due to loss of momentum.

The scheme of local narrowing

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Conservation of mass

The equation of conservation of mass:

where and — mass flow rate at ports A and B respectively. The flow rate through the port is positive when the flow flows into the unit.

Energy conservation

The energy conservation equation:

where and — energy flow at ports A and B respectively.

The block is considered adiabatic. Therefore, the specific total enthalpy between port A, port B and the constriction does not change.:



where — specific enthalpy at port A, port B or at the constriction (subscript ).

Ideal flow rates at port A, port B and constriction:







where

  • — the cross-sectional area of ports A and B, the value of the parameter Cross-sectional area at ports A and B;

  • — the cross-sectional area at the point of constriction. If for the parameter Restriction type value selected Fixed, then this is the value of the parameter Restriction area. If for the parameter Restriction type value selected Variable, then the cross-sectional area is defined as:

    where

    • — the minimum cross-sectional area at the point of narrowing, the value of the parameter Minimum restriction area;

    • — the maximum cross-sectional area at the point of narrowing, the value of the parameter Maximum restriction area;

    • — the value of the signal on the AR port.

  • — the volume density of the gas at port A, port B or at the constriction.

Theoretical mass consumption without taking into account the effects of imperfection:

where — flow coefficient, the value of the parameter Discharge coefficient.

Conservation of momentum

The pressure difference between ports A and B is based on conservation of momentum to reduce the passage section between the inlet and the constriction, as well as conservation of momentum to dramatically expand the passage section between the constriction and the outlet.

For the flow from port A to port B:

where — area ratio .

For the flow from port B to port A:

The pressure difference in the two previous equations is proportional to the square of the flow velocity. This is typical behavior for a turbulent flow. However, for a laminar flow, the pressure drop becomes linear with respect to the flow rate. The pressure difference for the laminar case can be approximately calculated as:

The threshold of transition from turbulent to laminar flow is defined as

where

  • — pressure ratio during the transition between laminar and turbulent modes (parameter value Laminar flow pressure ratio);

  • .

The pressure at the constriction is based on the law of conservation of momentum to reduce the passage section between the inlet and the constriction.

For the flow from port A to port B:

For the flow from port B to port A:

For a laminar flow, the pressure at the constriction is approximately equal to:

The block uses a cubic polynomial from for a smooth transition between turbulent and laminar modes:

  • If

    then

    and

  • If

    then smoothly transitions between and

    and smoothly transitions between and

  • If

    then smoothly transitions between and

    and smoothly transitions between and

  • If

    then

    and

Critical flow

When the flow through the constriction becomes critical, further flow changes depend on upstream conditions and are independent of downstream conditions.

If — this is the pressure on the port A, and — this is the hypothetical pressure at port B, assuming a critical flow from port A to port B, then

where а — the speed of sound.

If — this is the pressure on port B, and — this is a hypothetical pressure at port A, assuming a critical flow from port B to port A:

Actual pressures at ports A and B, and accordingly, they depend on whether there has been a transition to a critical flow.

For the flow from port A to port B and

For the flow from port B to port A and

Assumptions and limitations

  • The narrowing of the flow is considered an adiabatic system, that is, it does not exchange heat with the environment.

  • This block does not simulate supersonic flow.

Ports

Conserving

# A — Gas Input or output
gas

Details

The gas port corresponds to the Input or output of the local constriction. This unit has no internal orientation.

Program usage name

port_a

# B — Gas Input or output
gas

Details

The gas port corresponds to the Input or output of the local constriction. This unit has no internal orientation.

Program usage name

port_b

Input

# AR — the value of the narrowing cross-sectional area, m2
scalar

Details

The input port that controls the area of the local constriction. The port is saturated when its value is outside the minimum and maximum limits of the narrowing area specified by the block parameters.

Dependencies

To use this port, set the parameter Restriction type value Variable.

Data types

Float64
Complex numbers support

I don’t

Parameters

Parameters

# Restriction type — a method for modeling the narrowing cross-sectional area
Fixed | Variable

Details

Select whether the cross-sectional area of the constriction can change during the simulation.:

  • Variable — the input signal on the AR port determines the area of the constriction, which may change during the simulation. Parameters Minimum restriction area and Maximum restriction area set the lower and upper boundaries of the narrowing area.

  • Fixed — the area of the narrowing, set by the value of the block parameter Restriction area, remains constant during simulation. The AR port is hidden.

Values

Fixed | Variable
Default value

—
Program usage name

type
Evaluatable

No

# Restriction area — The cross-sectional area of the constriction is normal to the flow direction
m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac

Details

Narrowing cross-sectional area normal to the direction of the flow.

Dependencies

To use this parameter, set for the parameter Restriction type meaning Fixed.

Units

m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac
Default value

0.001 m^2
Program usage name

fixed_restriction_area
Evaluatable

Yes

# Minimum restriction area — the lower boundary of the narrowing cross-sectional area
m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac

Details

The lower boundary of the narrowing cross-sectional area . You can use this parameter to represent the area of the constriction. The input signal AR is saturated at this value to prevent further reduction of the narrowing area.

Dependencies

To use this parameter, set for the parameter Restriction type meaning Variable.

Units

m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac
Default value

1e-10 m^2
Program usage name

min_restriction_area
Evaluatable

Yes

# Maximum restriction area — upper limit of the narrowing cross-sectional area
m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac

Details

Upper limit of the narrowing cross-sectional area . The input signal AR is saturated at this value to prevent a further increase in the narrowing cross-section.

Dependencies

To use this parameter, set for the parameter Restriction type meaning Variable.

Units

m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac
Default value

0.005 m^2
Program usage name

max_restriction_area
Evaluatable

Yes

# Cross-sectional area at ports A and B — the cross-sectional area is normal to the flow path at the ports
m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac

Details

Cross-sectional area normal to the flow path on ports A and B. It is assumed that this area is the same for the two ports.

Units

m^2 | um^2 | mm^2 | cm^2 | km^2 | in^2 | ft^2 | yd^2 | mi^2 | ha | ac
Default value

0.01 m^2
Program usage name

port_area
Evaluatable

Yes

# Discharge coefficient — the ratio of the actual mass flow to the theoretical mass flow through the taper

Details

Expense ratio is a semi—empirical parameter defined as the ratio of the actual mass flow to the theoretical mass flow through the constriction.

The flow coefficient is an empirical parameter that takes into account the effects associated with the imperfection of the actual flow.

Default value

0.64
Program usage name

C_d
Evaluatable

Yes

# Laminar flow pressure ratio — the pressure ratio at which the gas flow transitions between laminar and turbulent modes

Details

Pressure ratio , at which the gas flow passes from a laminar mode to a turbulent one. The pressure loss is linear with respect to the mass flow in the laminar mode and quadratic with respect to the mass flow in the turbulent mode.

Default value

0.999
Program usage name

B_laminar
Evaluatable

Yes

Examples