Local Restriction (G)
Local narrowing of the flow in the gas network.
blockType: AcausalFoundation.Gas.Elements.LocalRestriction
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Local Restriction (G) Path in the library: |
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Variable Local Restriction (G) Path in the library: |
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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 area of narrowing as a block parameter.
The block icon changes depending on the value of the Restriction type parameter.
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, causing a pressure drop. It then separates from the wall during a sudden expansion, as a result of which the pressure is only partially restored due to loss of momentum.
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 the value is selected for the Restriction type parameter
Fixed, then this is the value of the Restriction area parameter. If the value is selected for the Restriction type parameterVariable, 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 — the flow coefficient, the value of the Discharge coefficient parameter.
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 (value of the parameter 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 — 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
Ports
Entrance
AR is the value of the narrowing cross—sectional area, m2
scalar
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 set by the block parameters.
Dependencies
This port is used only if the Restriction type parameter is set to Variable.
Non-directional
A — gas inlet or outlet
gas
The gas port corresponds to the entrance or exit of the local constriction. This block has no internal orientation.
B — gas inlet or outlet
gas
The gas port corresponds to the entrance or exit of the local constriction. This block has no internal orientation.
Parameters
Restriction type — the ability to change the cross-sectional area of the narrowing
Variable (by default) | Fixed
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 can change during the simulation. The Minimum restriction area and Maximum restriction area parameters set the lower and upper boundaries of the constriction area. -
Fixed— the area of constriction, set by the value of the block parameter Restriction area, remains constant during the simulation. The AR port is hidden.
Minimum restriction area — the lower boundary of the cross-sectional area of the narrowing
1e-10 m2 (default)
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 the Restriction type parameter to Variable.
Maximum restriction area — the upper limit of the cross-sectional area of the narrowing
0.005 m2 (default)
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 the Restriction type parameter to Variable.
Restriction area — the cross-sectional area of the constriction normal to the flow direction of
1e−3 m2 (default)
Narrowing cross-sectional area normal to the direction of the flow.
Dependencies
To use this parameter, set the Restriction type parameter to Fixed.
Cross-sectional area at ports A and B — the cross-sectional area is normal to the flow path at ports
0.005 m2 (default)
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.
Discharge coefficient — the ratio of the actual mass flow to the theoretical mass flow through the narrowing of the
0.64 (default)
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 discharge coefficient is an empirical parameter that takes into account the effects associated with the imperfection of the actual flow.
Laminar flow pressure ratio — the pressure coefficient at which the gas flow passes between laminar and turbulent modes
0.999 (default)
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.