Documentation

Local Restriction (2P)

Fixed flow resistance

Library

Two-Phase Fluid/Elements

Description

The Local Restriction (2P) block models the pressure drop due to a fixed flow resistance such as an orifice. Ports A and B represent the restriction inlet and outlet. The restriction area, specified in the block dialog box, remains constant during simulation.

The restriction consists of a contraction followed by a sudden expansion in flow area. The contraction causes the fluid to accelerate and its pressure to drop. The expansion recovers the lost pressure though only in part, as the flow separates from the wall, losing momentum in the process.

Local Restriction Schematic

Mass Balance

The mass balance equation is

m˙A+m˙B=0,

where:

  • m˙A and m˙B are the mass flow rates into the restriction through port A and port B.

Energy Balance

The energy balance equation is

ϕA+ϕB=0,

where:

  • ϕA and ϕB are the energy flow rates into the restriction through port A and port B.

The local restriction is assumed to be adiabatic and the change in specific total enthalpy is therefore zero. At port A,

uA+pAνA+wA22=uR+pRνR+wR22,

while at port B,

uB+pBνB+wB22=uR+pRνR+wR22,

where:

  • uA, uB, and uR are the specific internal energies at port A, at port B, and the restriction aperture.

  • pA, pB, and pR are the pressures at port A, port B, and the restriction aperture.

  • νA, νB, and νR are the specific volumes at port A, port B, and the restriction aperture.

  • wA, wB, and wR are the ideal flow velocities at port A, port B, and the restriction aperture.

The ideal flow velocity is computed as

wA=m˙idealνAS

at port A, as

wB=m˙idealνBS

at port B, and as

wR=m˙idealνRSR,

inside the restriction, where:

  • m˙ideal is the ideal mass flow rate through the restriction.

  • S is the flow area at port A and port B.

  • SR is the flow area of the restriction aperture.

The ideal mass flow rate through the restriction is computed as:

m˙ideal=m˙ACD,

where:

  • CD is the flow discharge coefficient for the local restriction.

Local Restriction Variables

Momentum Balance

The pressure difference between the ports is derived from the momentum balances in the contraction zone (the region between the inlet and the restriction aperture) and expansion zone (the region between the restriction aperture and the outlet). In the turbulent flow regime, with the flow directed from port A to port B:

ΔpAB=wR|wR|νR[(1+r2)(1rνAνR)r(1rνBνR)],

where the parameter r is defined as the flow area ratio

SRS.

With the flow directed from port B to port A:

ΔpBA=wR|wR|νR[(1+r2)(1rνBνR)r(1rνAνR)],

The equations indicate that the pressure difference between the ports varies with the square of the flow rate through the restriction. This relationship is characteristic of turbulent flows only. In the laminar regime, where the relationship becomes linear, the pressure difference is approximated as:

Δplam=wRΔptransition2νR(1r),

where Δptransition is the pressure difference threshold between the laminar and turbulent flow regimes:

Δptransition=pavg(1Blam),

with:

  • pavg as the average of the pressures at port A and port B:

    pavg=pA+pB2

  • Blam as the Laminar flow pressure ratio parameter.

The laminar pressure difference equation is the same for both flow directions—from port A to port B or from port B to port A.

The pressure at the restriction aperture is computed from the momentum balance in the flow contraction zone. In the turbulent flow regime, with the flow directed from port A to port B:

pR,AB=pAwR|wR|νR(1+r2)(1rνAνR)

With the flow directed from port B to port A:

pR,BA=pB+wR|wR|νR(1+r2)(1rνBνR)

In the laminar regime, the restriction pressure becomes linear with respect to the flow rate and is approximated as:

pR,lam=pavgwR2νR(1r22)

A cubic polynomial function is used to blend the pressure difference between the ports as well as the pressure at the restriction aperture between the laminar and turbulent flow regimes:

  • When ΔptransitionpApB, then pApB=ΔpAB and pR=pR,AB.

  • When 0pApBΔptransition, then pApB is smoothly blended between ΔpAB and Δplam and pR is smoothly blended between pR,AB and pR,lam.

  • When ΔptransitionpApB0, then pApB is smoothly blended between ΔpBA and Δplam and pR is smoothly blended between pR,BA and pR,lam.

  • When 0pApBΔptransition, then pApB=ΔpBA and pR=pR,BA.

Variables

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.

Assumptions and Limitations

  • The restriction is adiabatic. It does not exchange heat with its surroundings.

Parameters

Restriction area

Area normal to the flow path at the restriction aperture—the narrow orifice located between the ports. The default value, 0.01 m^2, is the same as the port areas.

Cross-sectional area at ports A and B

Area normal to the flow path at the restriction ports. The ports are assumed to be identical in cross-section. The default value, 0.01 m^2, is the same as the restriction aperture area.

Flow discharge coefficient

Ratio of the actual to the theoretical mass flow rate through the restriction. The discharge coefficient is an empirical parameter used to account for non-ideal effects such as those due to restriction geometry. The default value is 0.64.

Laminar flow pressure ratio

Ratio of the outlet to the inlet port pressure at which the flow regime is assumed to switch from laminar to turbulent. The prevailing flow regime determines the equations used in simulation. The pressure drop across the restriction is linear with respect to the mass flow rate if the flow is laminar and quadratic (with respect to the mass flow rate) if the flow is turbulent. The default value is 0.999.

Ports

A pair of two-phase fluid conserving ports labeled A and B represent the restriction inlet and outlet.

Introduced in R2015b

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