Local Restriction (G)
Localised flow constriction in the gas network.
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Description
The Local Restriction (G) block simulates the pressure drop due to a localised constriction, such as a valve or orifice, 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 Local Restriction (G) block. The input signal on the AR port defines the area of the constriction. It is also possible to specify a fixed constriction area as a block parameter.
The block icon changes depending on the value of the Restriction type parameters.
The flow constriction is considered an adiabatic system, i.e. it does not exchange heat with the environment.
The block model consists of a sudden constriction followed by a sudden expansion of the flow-through section. The gas accelerates as it passes through the constriction, causing a drop in pressure. It then separates from the wall during the sudden expansion, causing the pressure to recover only partially due to loss of momentum.
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Preservation of mass
Mass conservation equation:
,
where and are the mass flow rates at ports A and B, respectively. The flow rate through the port is positive when the flow is flowing into the block.
Energy conservation
Energy conservation equation:
,
where and are the energy flow at ports A and B respectively.
The unit is assumed to be adiabatic. Consequently, the specific total enthalpy between port A, port B and the constriction does not change:
,
,
where is the specific enthalpy at port A, port B or at the constriction (lower index ).
Ideal flow velocities at port A, port B and constriction:
,
,
,
where
-
- is the cross-sectional area of ports A and B, the value of the parameter Cross-sectional area at ports A and B;
-
- is the cross-sectional area at the point of constriction. If
Fixedis selected for the Restriction type parameter, this is the value of the Restriction area parameter. IfVariableis selected for the Restriction type parameters, the cross-sectional area is defined as:where
-
- is the minimum cross-sectional area at the point of constriction, the value of the Minimum restriction area parameters;
-
- the maximum cross-sectional area at the constriction point, the value of the Maximum restriction area parameters;
-
- signal value on the AR port.
-
-
- gas volume density at port A, port B or at the constriction.
Theoretical mass flow rate excluding non-ideality effects:
,
where is the flow coefficient, the value of the Discharge coefficient parameters.
Storing the pulse
The pressure difference between ports A and B is based on momentum conservation to reduce the flow cross-section between inlet and constriction, and on momentum conservation to dramatically expand the flow cross-section between constriction and outlet.
For flow from port A to port B:
,
where is the ratio of areas .
For the flow from port B to port A:
.
The pressure difference in the previous two equations is proportional to the square of the flow velocity. This is typical behaviour for turbulent flow. However, for laminar flow, the pressure difference becomes linear with respect to the flow rate. The pressure difference for the laminar case can be approximated as:
.
The threshold of transition from turbulent to laminar flow is defined as ,
where
-
- is the pressure ratio at the transition between laminar and turbulent regimes (value of the Laminar flow pressure ratio parameters);
-
.
The pressure at the constriction is based on the law of conservation of momentum to reduce the cross-section between the inlet and the constriction.
For flow from port A to port B:
.
For a flow from port B to port A:
.
For laminar flow, the pressure at the constriction is approximated to be:
.
The block uses a cubic polynomial from for smooth transition between turbulent and laminar regimes:
-
If
then
и
-
If
then smoothly transitions between and
and seamlessly transitions between and
-
If
then seamlessly transitions between and
and seamlessly transitions between and
-
If
then
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Critical flow
When the flow through a constriction becomes critical, further changes in flow depend on upstream conditions and are independent of downstream conditions.
If is the pressure at port A and is the hypothetical pressure at port B, assuming critical flow from port A to port B, then
,
where is the speed of sound.
Where is the pressure at port B and is the hypothetical pressure at port A, assuming critical flow from port B to port A:
The actual pressures at ports A and B, and respectively, depend on whether the transition to critical flow has occurred.
For flow from port A to port B and
For flow from port B to port A and
Ports
Input
AR - value of the cross-sectional area of the constriction, m²
scalar
Input port that controls the local constriction area. The port is saturated when its value is outside the minimum and maximum limits of the constriction area set by the block parameters.
Dependencies
This port is only used when the Restriction type parameter is set to Variable.
Parameters
Restriction type - possibility to change the cross sectional area of the constriction
Variable (by default) | Fixed
Select whether the cross-sectional area of the constriction can be changed during the simulation:
-
Variable- the input signal on the AR port defines the constriction area that can be varied during the simulation. The parameters Minimum restriction area and Maximum restriction area set the lower and upper limits of the constriction area. -
Fixed- the constriction area defined by the value of the Restriction area block parameters remains constant during the simulation. The AR port is hidden.
Minimum restriction area - lower limit of the constriction cross-sectional area
1e-10 m² (by default).
Lower limit of the constriction cross-sectional area . You can use this parameter to represent the area of the constriction. The AR input signal saturates at this value to prevent further reduction of the constriction area.
Dependencies
To use this parameter, set the Restriction type parameter to Variable.
Maximum restriction area - upper limit of the cross sectional area of the constriction
0.005 m² (by default).
Upper limit of the constriction cross-sectional area . The AR input signal is saturated at this value to prevent further increase of the constriction cross-sectional area.
Dependencies
To use this parameter, set the Restriction type parameter to Variable.
Restriction area - cross-sectional area of the constriction normal to the flow direction
1e-3 m² (by default).
Cross-sectional area of the constriction normal to the flow direction.
Dependencies
To use this parameter, set the Restriction type parameters to Fixed.
Cross-sectional area at ports A and B - cross-sectional area normal to the flow path at the ports
0.005 m² (by default).
Cross-sectional area normal to the flow path at ports A and B. This area is assumed to be the same for the two ports.
Discharge coefficient - ratio of actual mass flow rate to theoretical mass flow rate through the constriction
`0.64 (by default).
Discharge coefficient is a semi-empirical parameter defined as the ratio of actual mass flow rate to theoretical mass flow rate through the constriction.
The unloading coefficient is an empirical parameter that accounts for the effects associated with the non-ideality of the actual flow.
Laminar flow pressure ratio is the pressure coefficient at which the gas flow transitions between laminar and turbulent regimes
`0.999 (by default).
Pressure ratio , at which the gas flow transitions from laminar to turbulent mode. The pressure loss is linear with respect to the mass flow rate in the laminar regime and quadratic with respect to the mass flow rate in the turbulent regime.