# Flow Restriction

Isentropic ideal gas flow through an orifice

• Library:
• Powertrain Blockset / Propulsion / Combustion Engine Components / Fundamental Flow

• ## Description

The Flow Restriction block models isentropic ideal gas flow through an orifice. The block uses the conservation of mass and energy to determine the mass flow rate. The flow velocity is limited by choked flow.

You can specify these orifice area models:

• Constant

• External input

• Throttle body geometry

### Equations

The Flow Restriction block implements these equations.

CalculationEquations

Standard orifice

${\stackrel{˙}{m}}_{orf}=\Gamma \cdot \Psi \left({P}_{ratio}\right)$

${P}_{ratio}=\frac{{P}_{downstr}}{{P}_{upstr}}$

$\Gamma =\frac{{A}_{eff}\cdot {P}_{upstr}}{\sqrt{R\cdot {T}_{upstr}}}$

${P}_{cr}={\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma }{\gamma -1}}$

$\Psi =\left\{\begin{array}{cc}\sqrt{\gamma {\left(\frac{2}{\gamma +1}\right)}^{\frac{\gamma +1}{\gamma -1}}}& {P}_{ratio}<{P}_{cr}\\ \sqrt{\frac{2\gamma }{\gamma -1}\left({P}_{ratio}{}^{\frac{2}{\gamma }}-{P}_{ratio}{}^{\frac{\gamma +1}{\gamma }}\right)}& {P}_{cr}\le {P}_{ratio}\le {P}_{lim}\\ \frac{{P}_{ratio}-1}{{P}_{lim}-1}\sqrt{\frac{2\gamma }{\gamma -1}\left({P}_{lim}{}^{\frac{2}{\gamma }}-{P}_{lim}{}^{\frac{\gamma +1}{\gamma }}\right)}& {P}_{lim}<{P}_{ratio}\end{array}$

Constituent mass flow rates

${\stackrel{˙}{m}}_{i}={\stackrel{˙}{m}}_{orf}{y}_{upstr,i}$

Constant orifice area

${A}_{eff}={A}_{orf_cnst}\cdot C{d}_{cnst}$

External input orifice area

${A}_{eff}={A}_{orf_ext}\cdot C{d}_{ext}$

Throttle body geometry

${\theta }_{thr}=Pc{t}_{thr}\cdot \frac{90}{100}$

${A}_{eff_thr}=\frac{\pi }{4}{D}_{thr}{}^{2}{C}_{d_thr}\left({\theta }_{thr}\right)$

Heat flow rate

${q}_{orf}={\stackrel{˙}{m}}_{orf}{h}_{upstr}$

The equations use these variables.

 ${A}_{eff}$, ${A}_{eff_thr}$ Effective orifice cross-sectional area ${A}_{orf_cnst}$, ${A}_{orf_ext}$ Orifice area $C{d}_{cnst}$, $C{d}_{ext}$ Discharge coefficient $R$ Ideal gas constant ${P}_{cr}$ Critical pressure at which choked flow occurs γ Ratio of specific heats $\Gamma$ Flow function based on pressure ratio ${P}_{ratio}$ Pressure ratio ${P}_{upstr}$ Upstream orifice pressure ${P}_{downstr}$ Downstream orifice pressure ${P}_{lim}$ Pressure ratio limit to avoid singularities as the pressure ratio approaches 1 yupstr,i Upstream species mass fraction for i = O2, N2, unburned fuel, CO2, H2O, CO, NO, NO2, PM, air, and burned gas ${\stackrel{˙}{m}}_{i}$ Mass flow rate for i = O2, N2, unburned fuel, CO2, H2O, CO, NO, NO2, PM, air, and burned gas ${\theta }_{thr}$ Throttle angle $Pc{t}_{thr}$ Percentage of throttle body that is open Cd_thr Throttle discharge coefficient ${D}_{thr}$ Throttle body diameter at opening ${\stackrel{˙}{m}}_{orf}$ Orifice mass flow hupstr Upstream specific enthalpy qorf Heat flow rate

The block uses the internal signal `FlwDir` to track the direction of the flow.

### Power Accounting

For the power accounting, the block implements these equations.

Bus Signal DescriptionEquations

`PwrInfo`

`PwrTrnsfrd` — Power transferred between blocks

• Positive signals indicate flow into block

• Negative signals indicate flow out of block

`PwrHeatFlwIn`

Heat flow rate at port A

qorf

`PwrHeatFlwOut`

Heat flow rate at port B

-qorf

`PwrNotTrnsfrd` — Power crossing the block boundary, but not transferred

• Positive signals indicate an input

• Negative signals indicate a loss

Not used

`PwrStored` — Stored energy rate of change

• Positive signals indicate an increase

• Negative signals indicate a decrease

Not used

## Ports

### Input

expand all

Bus containing orifice:

• `Prs` — Pressure, in Pa

• `Temp` — Temperature, in K

• `Enth` — Specific enthalpy, in J/kg

• `MassFrac` — Inlet mass fractions, dimensionless.

Specifically, a bus with these mass fractions:

• `O2MassFrac` — Oxygen

• `N2MassFrac` — Nitrogen

• `UnbrndFuelMassFrac` — Unburned fuel

• `CO2MassFrac` — Carbon dioxide

• `H2OMassFrac` — Water

• `COMassFrac` — Carbon monoxide

• `NOMassFrac` — Nitric oxide

• `NO2MassFrac` — Nitrogen dioxide

• `NOxMassFrac` — Nitric oxide and nitrogen dioxide

• `PmMassFrac` — Particulate matter

• `AirMassFrac` — Air

• `BrndGasMassFrac` — Burned gas

Bus containing orifice:

• `Prs` — Pressure, in Pa

• `Temp` — Temperature, in K

• `Enth` — Specific enthalpy, in J/kg

• `MassFrac` — Outlet mass fractions, dimensionless.

Specifically, a bus with these mass fractions:

• `O2MassFrac` — Oxygen

• `N2MassFrac` — Nitrogen

• `UnbrndFuelMassFrac` — Unburned fuel

• `CO2MassFrac` — Carbon dioxide

• `H2OMassFrac` — Water

• `COMassFrac` — Carbon monoxide

• `NOMassFrac` — Nitric oxide

• `NO2MassFrac` — Nitrogen dioxide

• `NOxMassFrac` — Nitric oxide and nitrogen dioxide

• `PmMassFrac` — Particulate matter

• `AirMassFrac` — Air

• `BrndGasMassFrac` — Burned gas

External area input for orifice area, ${A}_{orf_ext}$, in m^2.

#### Dependencies

To create this port, select `External input` for the Orifice area model parameter.

Percentage of throttle body that is open, $Pc{t}_{thr}$.

#### Dependencies

To create this port, select `Throttle body geometry` for the Orifice area model parameter.

### Output

expand all

Bus containing:

• `MassFlw` — Mass flow rate through inlet, in kg/s

• `HeatFlw` — Inlet heat flow rate, in J/s

• `Temp` — Inlet temperature, in K

• `MassFrac` — Inlet mass fractions, dimensionless.

Specifically, a bus with these mass fractions:

• `O2MassFrac` — Oxygen

• `N2MassFrac` — Nitrogen

• `UnbrndFuelMassFrac` — Unburned fuel

• `CO2MassFrac` — Carbon dioxide

• `H2OMassFrac` — Water

• `COMassFrac` — Carbon monoxide

• `NOMassFrac` — Nitric oxide

• `NO2MassFrac` — Nitrogen dioxide

• `NOxMassFrac` — Nitric oxide and nitrogen dioxide

• `PmMassFrac` — Particulate matter

• `AirMassFrac` — Air

• `BrndGasMassFrac` — Burned gas

Bus containing:

• `MassFlw` — Outlet mass flow rate, in kg/s

• `HeatFlw` — Outlet heat flow rate, in J/s

• `Temp` — Outlet temperature, in K

• `MassFrac` — Outlet mass fractions, dimensionless.

Specifically, a bus with these mass fractions:

• `O2MassFrac` — Oxygen

• `N2MassFrac` — Nitrogen

• `UnbrndFuelMassFrac` — Unburned fuel

• `CO2MassFrac` — Carbon dioxide

• `H2OMassFrac` — Water

• `COMassFrac` — Carbon monoxide

• `NOMassFrac` — Nitric oxide

• `NO2MassFrac` — Nitrogen dioxide

• `NOxMassFrac` — Nitric oxide and nitrogen dioxide

• `PmMassFrac` — Particulate matter

• `AirMassFrac` — Air

• `BrndGasMassFrac` — Burned gas

Bus signal containing these block calculations.

SignalDescriptionUnits

`Flw`

`PrsAdj`

`DwnstrmPrs`

Downstream pressure

Pa

`UpstrmPrs`

Upstream pressure

Pa

`PrsRatio`

Pressure ratio

NA

`DwnstrmTemp`

Downstream temperature

K

`UpstrmTemp`

Upstream temperature

K

`OrfMassFlw`

Mass flow rate through orifice

kg/s

`Species`

`O2MassFlw`

Oxygen mass flow rate

kg/s

`N2MassFlw`

Nitrogen mass flow rate

kg/s

`UnbrndFuelMassFlw`

Unburned gas mass flow rate

kg/s

`CO2MassFlw`

Carbon dioxide mass flow rate

kg/s

`H2OMassFlw`

Water mass flow rate

kg/s

`COMassFlw`

Carbon monoxide mass flow rate

kg/s

`NOMassFlw`

Nitric oxide mass flow rate

kg/s

`NO2MassFlw`

Nitrogen dioxide mass flow rate

kg/s

`NOxMassFlw`

Nitric oxide and nitrogen dioxide mass flow rate

kg/s

`PmMassFlw`

Particulate matter mass flow rate

kg/s

`AirMassFlw`

Air mass flow rate

kg/s

`BrnedGasMassFlw`

Burned gas mass flow rate

kg/s

`PwrInfo`

`PwrTrnsfrd`

`PwrHeatFlwIn`

Heat flow rate at port A

W

`PwrHeatFlwOut`

Heat flow rate at port B

W

`PwrNotTrnsfrd`

Not used

`PwrStored`

Not used

`Area`

`FlwArea`

Cross-sectional flow area

m^2

`EffctArea`

Effective orifice cross-sectional area

m^2

`ThrAng`

Throttle area, if applicable

deg

## Parameters

expand all

Block Options

Orifice area model.

#### Dependencies

The orifice area model enables the parameters on the Area Parameters tab.

Block icon color:

• `Cold` for blue.

• `Hot` for red.

General

Ratio of specific heats, γ.

Ideal gas constant, $R$, in J/(kg·K).

Pressure ratio limit to avoid singularities as the pressure ratio approaches 1, ${P}_{lim}$.

Area

Constant area value, ${A}_{orf_cnst}$, in m^2.

#### Dependencies

To enable this parameter, select `Constant` for the Orifice area model parameter.

Discharge coefficient for constant area, $C{d}_{cnst}$.

#### Dependencies

To enable this parameter, select `Constant` for the Orifice area model parameter.

Discharge coefficient for external area input, $C{d}_{ext}$.

#### Dependencies

To enable this parameter, select ```External input``` for the Orifice area model parameter.

Throttle body diameter at opening, ${D}_{thr}$, in mm.

#### Dependencies

To enable this parameter, select ```Throttle body geometry``` for the Orifice area model parameter.

Discharge coefficient table, Cd_thr.

#### Dependencies

To enable this parameter, select ```Throttle body geometry``` for the Orifice area model parameter.

Angle breakpoints, $Th{r}_{ang_bpts}$, in deg.

#### Dependencies

To enable this parameter, select ```Throttle body geometry``` for the Orifice area model parameter.

 Heywood, John B. Internal Combustion Engine Fundamentals. New York: McGraw-Hill, 1988.