Description: "Package with models for fixed flow resistances (pipes, diffusers etc.)"

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## Buildings.Fluid.FixedResistances

Package with models for fixed flow resistances (pipes, diffusers etc.)

### Information

This package contains components models for fixed flow resistances. By fixed flow resistance, we mean resistances that do not change the flow coefficient

k = m ⁄ √ΔP.

For models of valves and air dampers, see Buildings.Fluid.Actuators. For models of flow resistances as part of the building constructions, see Buildings.Airflow.Multizone.

The model Buildings.Fluid.FixedResistances.FixedResistanceDpM is a fixed flow resistance that takes as parameter a nominal flow rate and a nominal pressure drop. The actual resistance is scaled using the above equation.

The model Buildings.Fluid.FixedResistances.LosslessPipe is an ideal pipe segment with no pressure drop. It is primarily used in models in which the above pressure drop model need to be replaced by a model with no pressure drop.

The model Buildings.Fluid.FixedResistances.SplitterFixedResistanceDpM can be used to model flow splitters or flow merges.

Extends from Modelica.Icons.VariantsPackage (Icon for package containing variants).

### Package Content

Name | Description
--- | ---
FixedResistanceDpM | Fixed flow resistance with dp and m_flow as parameter
LosslessPipe | Pipe with no flow friction and no heat transfer
Pipe | Pipe with finite volume discretization along flow path
SplitterFixedResistanceDpM | Flow splitter with fixed resistance at each port
Examples | Collection of models that illustrate model use and test models
BaseClasses |

## Buildings.Fluid.FixedResistances.FixedResistanceDpM

Fixed flow resistance with dp and m_flow as parameter

### Information

This is a model of a resistance with a fixed flow coefficient. The mass flow rate is computed as

ṁ = k √ΔP,

where k is a constant and ΔP is the pressure drop. The constant k is equal to `k=m_flow_nominal/sqrt(dp_nominal)`, where `m_flow_nominal` and `dp_nominal` are parameters. In the region `abs(m_flow) < m_flow_turbulent`, the square root is replaced by a differentiable function with finite slope. The value of `m_flow_turbulent` is computed as follows:

- If the parameter `use_dh` is `false` (the default setting), the equation `m_flow_turbulent = deltaM * abs(m_flow_nominal)`, where `deltaM=0.3` and `m_flow_nominal` are parameters that can be set by the user.
- Otherwise, the equation `m_flow_turbulent = eta_nominal*dh/4*π*ReC` is used, where `eta_nominal` is the dynamic viscosity, obtained from the medium model. The parameter `dh` is the hydraulic diameter and `ReC=4000` is the critical Reynolds number, which both can be set by the user.

The figure below shows the pressure drop for the parameters `m_flow_nominal=5` kg/s, `dp_nominal=10` Pa and `deltaM=0.3`.

If the parameter `show_T` is set to `true`, then the model will compute the temperature at its ports. Note that this can lead to state events when the mass flow rate approaches zero, which can increase computing time.

The parameter `from_dp` is used to determine whether the mass flow rate is computed as a function of the pressure drop (if `from_dp=true`), or vice versa. This setting can affect the size of the nonlinear system of equations.

If the parameter `linearized` is set to `true`, then the pressure drop is computed as a linear function of the mass flow rate.

Setting `allowFlowReversal=false` can lead to simpler equations. However, this should only be set to `false` if one can guarantee that the flow never reverses its direction. This can be difficult to guarantee, as pressure imbalance after the initialization, or due to medium expansion and contraction, can lead to reverse flow.

#### Notes

For more detailed models that compute the actual flow friction, models from the package Modelica.Fluid can be used and combined with models from the `Buildings` library.

#### Implementation

The pressure drop is computed by calling a function in the package Buildings.Fluid.BaseClasses.FlowModels, This package contains regularized implementations of the equation

m = sign(Δp) k √ Δp

and its inverse function.

To decouple the energy equation from the mass equations, the pressure drop is a function of the mass flow rate, and not the volume flow rate. This leads to simpler equations.

Extends from Buildings.Fluid.BaseClasses.PartialResistance (Partial model for a hydraulic resistance).

### Parameters

Type | Name | Default | Description
--- | --- | --- | ---
replaceable package Medium | PartialMedium | Medium in the component
MassFlowRate | m_flow_turbulent | if (computeFlowResistance an... | Turbulent flow if |m_flow| >= m_flow_turbulent [kg/s]
Boolean | use_dh | false | Set to true to specify hydraulic diameter
Length | dh | 1 | Hydraulic diameter [m]
Real | ReC | 4000 | Reynolds number where transition to turbulent starts
Real | deltaM | 0.3 | Fraction of nominal mass flow rate where transition to turbulent occurs
Nominal condition
MassFlowRate | m_flow_nominal |  | Nominal mass flow rate [kg/s]
Pressure | dp_nominal |  | Pressure drop at nominal mass flow rate [Pa]
Initialization
MassFlowRate | m_flow.start | 0 | Mass flow rate from port_a to port_b (m_flow > 0 is design flow direction) [kg/s]
Pressure | dp.start | 0 | Pressure difference between port_a and port_b [Pa]
Assumptions
Boolean | allowFlowReversal | true | = true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Advanced
Diagnostics
Boolean | show_T | false | = true, if actual temperature at port is computed
Boolean | from_dp | false | = true, use m_flow = f(dp) else dp = f(m_flow)
Boolean | homotopyInitialization | true | = true, use homotopy method
Boolean | linearized | false | = true, use linear relation between m_flow and dp for any flow rate

### Connectors

Type | Name | Description
--- | --- | ---
FluidPort_a | port_a | Fluid connector a (positive design flow direction is from port_a to port_b)
FluidPort_b | port_b | Fluid connector b (positive design flow direction is from port_a to port_b)

### Modelica definition

model FixedResistanceDpM "Fixed flow resistance with dp and m_flow as parameter" extends Buildings.Fluid.BaseClasses.PartialResistance( final m_flow_turbulent = if (computeFlowResistance and use_dh) then eta_default*dh/4*Modelica.Constants.pi*ReC elseif (computeFlowResistance) then deltaM * m_flow_nominal_pos else 0); parameter Boolean use_dh = false "Set to true to specify hydraulic diameter"; parameter Modelica.SIunits.Length dh=1 "Hydraulic diameter"; parameter Real ReC(min=0)=4000 "Reynolds number where transition to turbulent starts"; parameter Real deltaM(min=0.01) = 0.3 "Fraction of nominal mass flow rate where transition to turbulent occurs"; final parameter Real k(unit="") = if computeFlowResistance then m_flow_nominal_pos / sqrt(dp_nominal_pos) else 0 "Flow coefficient, k=m_flow/sqrt(dp), with unit=(kg.m)^(1/2)"; protected final parameter Boolean computeFlowResistance=(dp_nominal_pos > Modelica.Constants.eps) "Flag to enable/disable computation of flow resistance"; initial equation if computeFlowResistance then assert(m_flow_turbulent > 0, "m_flow_turbulent must be bigger than zero."); end if; assert(m_flow_nominal_pos > 0, "m_flow_nominal_pos must be non-zero. Check parameters."); assert(m_flow_nominal_pos > m_flow_turbulent, "In FixedResistanceDpM, m_flow_nominal is smaller than m_flow_turbulent. m_flow_nominal = " + String(m_flow_nominal) + " dh = " + String(dh) + " To correct it, set dh < " + String( 4*m_flow_nominal/eta_default/Modelica.Constants.pi/ReC) + " Suggested value: dh = " + String(1/10*4*m_flow_nominal/eta_default/Modelica.Constants.pi/ReC), AssertionLevel.warning); equation // Pressure drop calculation if computeFlowResistance then if linearized then m_flow*m_flow_nominal_pos = k^2*dp; else if homotopyInitialization then if from_dp then m_flow=homotopy(actual=Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp(dp=dp, k=k, m_flow_turbulent=m_flow_turbulent), simplified=m_flow_nominal_pos*dp/dp_nominal_pos); else dp=homotopy(actual=Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow(m_flow=m_flow, k=k, m_flow_turbulent=m_flow_turbulent), simplified=dp_nominal_pos*m_flow/m_flow_nominal_pos); end if; // from_dp else // do not use homotopy if from_dp then m_flow=Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp(dp=dp, k=k, m_flow_turbulent=m_flow_turbulent); else dp=Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow(m_flow=m_flow, k=k, m_flow_turbulent=m_flow_turbulent); end if; // from_dp end if; // homotopyInitialization end if; // linearized else // do not compute flow resistance dp = 0; end if; // computeFlowResistance end FixedResistanceDpM;

## Buildings.Fluid.FixedResistances.LosslessPipe

Pipe with no flow friction and no heat transfer

### Information

Model of a pipe with no flow resistance, no heat loss and no transport delay. This model can be used to replace a `replaceable` pipe model in flow legs in which no friction should be modeled. This is for example done in the outlet port of the base class for three way valves, Buildings.Fluid.Actuators.BaseClasses.PartialThreeWayValve.

Extends from Buildings.Fluid.Interfaces.PartialTwoPortInterface (Partial model transporting fluid between two ports without storing mass or energy).

### Parameters

Type | Name | Default | Description
--- | --- | --- | ---
replaceable package Medium | PartialMedium | Medium in the component
Nominal condition
MassFlowRate | m_flow_nominal |  | Nominal mass flow rate [kg/s]
Initialization
MassFlowRate | m_flow.start | 0 | Mass flow rate from port_a to port_b (m_flow > 0 is design flow direction) [kg/s]
Pressure | dp.start | 0 | Pressure difference between port_a and port_b [Pa]
Assumptions
Boolean | allowFlowReversal | true | = true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Advanced
MassFlowRate | m_flow_small | 1E-4*abs(m_flow_nominal) | Small mass flow rate for regularization of zero flow [kg/s]
Diagnostics
Boolean | show_T | false | = true, if actual temperature at port is computed

### Connectors

Type | Name | Description
--- | --- | ---
FluidPort_a | port_a | Fluid connector a (positive design flow direction is from port_a to port_b)
FluidPort_b | port_b | Fluid connector b (positive design flow direction is from port_a to port_b)

### Modelica definition

model LosslessPipe "Pipe with no flow friction and no heat transfer" extends Buildings.Fluid.Interfaces.PartialTwoPortInterface; final parameter Boolean from_dp=true "Used to satisfy replaceable models"; equation dp=0; // Isenthalpic state transformation (no storage and no loss of energy) port_a.h_outflow = inStream(port_b.h_outflow); port_b.h_outflow = inStream(port_a.h_outflow); // Mass balance (no storage) port_a.m_flow + port_b.m_flow = 0; // Transport of substances port_a.Xi_outflow = inStream(port_b.Xi_outflow); port_b.Xi_outflow = inStream(port_a.Xi_outflow); port_a.C_outflow = inStream(port_b.C_outflow); port_b.C_outflow = inStream(port_a.C_outflow); end LosslessPipe;

## Buildings.Fluid.FixedResistances.Pipe

Pipe with finite volume discretization along flow path

### Information

Model of a pipe with flow resistance and optional heat exchange with environment.

If `useMultipleHeatPorts=false` (default option), the pipe uses a single heat port for the heat exchange with the environment. If `useMultipleHeatPorts=true`, then one heat port for each segment of the pipe is used for the heat exchange with the environment. If the heat port is unconnected, then the pipe has no heat loss.

The default value for the parameter `diameter` is computed such that the flow velocity is equal to `v_nominal=0.15` for a mass flow rate of `m_flow_nominal`. Both parameters, `diameter` and `v_nominal`, can be overwritten by the user. The default value for `dp_nominal` is two times the pressure drop that the pipe would have if it were straight with no fittings. The factor of two that takes into account the pressure loss of fittings can be overwritten. These fittings could also be explicitly modeled outside of this component using models from the package Modelica.Fluid.Fittings. For mass flow rates other than `m_flow_nominal`, the model Buildings.Fluid.FixedResistances.FixedResistanceDpM is used to compute the pressure drop.

For a steady-state model of a flow resistance, use Buildings.Fluid.FixedResistances.FixedResistanceDpM instead of this model.

Extends from Buildings.Fluid.FixedResistances.BaseClasses.Pipe (Model of a pipe with finite volume discretization along the flow path).

### Parameters

Type | Name | Default | Description
--- | --- | --- | ---
replaceable package Medium | PartialMedium | Medium in the component
Integer | nSeg | 10 | Number of volume segments
Length | thicknessIns |  | Thickness of insulation [m]
ThermalConductivity | lambdaIns |  | Heat conductivity of insulation [W/(m.K)]
Length | diameter | sqrt(4*m_flow_nominal/rho_de... | Pipe diameter (without insulation) [m]
Length | length |  | Length of the pipe [m]
Velocity | v_nominal | 0.15 | Velocity at m_flow_nominal (used to compute default diameter) [m/s]
Length | roughness | 2.5e-5 | Absolute roughness of pipe, with a default for a smooth steel pipe (dummy if use_roughness = false) [m]
Boolean | useMultipleHeatPorts | false | = true to use one heat port for each segment of the pipe, false to use a single heat port for the entire pipe
Nominal condition
MassFlowRate | m_flow_nominal |  | Nominal mass flow rate [kg/s]
Pressure | dp_nominal | 2*dpStraightPipe_nominal | Pressure difference [Pa]
Initialization
MassFlowRate | m_flow.start | 0 | Mass flow rate from port_a to port_b (m_flow > 0 is design flow direction) [kg/s]
Pressure | dp.start | 0 | Pressure difference between port_a and port_b [Pa]
Dynamics
Equations
Dynamics | energyDynamics | Modelica.Fluid.Types.Dynamic... | Formulation of energy balance
Dynamics | massDynamics | energyDynamics | Formulation of mass balance
Real | mSenFac | 1 | Factor for scaling the sensible thermal mass of the volume
Initialization
AbsolutePressure | p_start | Medium.p_default | Start value of pressure [Pa]
Temperature | T_start | Medium.T_default | Start value of temperature [K]
MassFraction | X_start[Medium.nX] | Medium.X_default | Start value of mass fractions m_i/m [kg/kg]
ExtraProperty | C_start[Medium.nC] | fill(0, Medium.nC) | Start value of trace substances
ExtraProperty | C_nominal[Medium.nC] | fill(1E-2, Medium.nC) | Nominal value of trace substances. (Set to typical order of magnitude.)
Assumptions
Boolean | allowFlowReversal | true | = true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Advanced
MassFlowRate | m_flow_small | 1E-4*abs(m_flow_nominal) | Small mass flow rate for regularization of zero flow [kg/s]
Boolean | homotopyInitialization | true | = true, use homotopy method
Flow resistance
Boolean | from_dp | false | = true, use m_flow = f(dp) else dp = f(m_flow)
Boolean | linearizeFlowResistance | false | = true, use linear relation between m_flow and dp for any flow rate
Real | deltaM | 0.1 | Fraction of nominal flow rate where flow transitions to laminar
Real | ReC | 4000 | Reynolds number where transition to turbulent starts

### Connectors

Type | Name | Description
--- | --- | ---
FluidPort_a | port_a | Fluid connector a (positive design flow direction is from port_a to port_b)
FluidPort_b | port_b | Fluid connector b (positive design flow direction is from port_a to port_b)
HeatPort_a | heatPort | Single heat port that connects to outside of pipe wall (default, enabled when useMultipleHeatPorts=false)
HeatPorts_a | heatPorts[nSeg] | Multiple heat ports that connect to outside of pipe wall (enabled if useMultipleHeatPorts=true)

### Modelica definition

model Pipe "Pipe with finite volume discretization along flow path" extends Buildings.Fluid.FixedResistances.BaseClasses.Pipe( diameter=sqrt(4*m_flow_nominal/rho_default/v_nominal/Modelica.Constants.pi), dp_nominal=2*dpStraightPipe_nominal, preDro(dp(nominal=length*10))); // Because dp_nominal is a non-literal value, we set // dp.nominal=100 instead of the default dp.nominal=dp_nominal, // because the latter is ignored by Dymola 2012 FD 01. parameter Modelica.SIunits.Velocity v_nominal = 0.15 "Velocity at m_flow_nominal (used to compute default diameter)"; parameter Modelica.SIunits.Length roughness(min=0) = 2.5e-5 "Absolute roughness of pipe, with a default for a smooth steel pipe (dummy if use_roughness = false)"; final parameter Modelica.SIunits.Pressure dpStraightPipe_nominal= Modelica.Fluid.Pipes.BaseClasses.WallFriction.Detailed.pressureLoss_m_flow( m_flow=m_flow_nominal, rho_a=rho_default, rho_b=rho_default, mu_a=mu_default, mu_b=mu_default, length=length, diameter=diameter, roughness=roughness, m_flow_small=m_flow_small) "Pressure loss of a straight pipe at m_flow_nominal"; parameter Boolean useMultipleHeatPorts=false "= true to use one heat port for each segment of the pipe, false to use a single heat port for the entire pipe"; Modelica.Thermal.HeatTransfer.Components.ThermalConductor conPipWal[nSeg]( each G=2*Modelica.Constants.pi*lambdaIns*length/nSeg/Modelica.Math.log(( diameter/2.0 + thicknessIns)/(diameter/2.0))) "Thermal conductance through pipe wall"; Modelica.Thermal.HeatTransfer.Components.ThermalCollector colAllToOne(m=nSeg) if not useMultipleHeatPorts "Connector to assign multiple heat ports to one heat port"; Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a heatPort if not useMultipleHeatPorts "Single heat port that connects to outside of pipe wall (default, enabled when useMultipleHeatPorts=false)"; Modelica.Fluid.Interfaces.HeatPorts_a heatPorts[nSeg] if useMultipleHeatPorts "Multiple heat ports that connect to outside of pipe wall (enabled if useMultipleHeatPorts=true)"; equation connect(conPipWal.port_b, vol.heatPort); if useMultipleHeatPorts then connect(heatPorts, conPipWal.port_a); else connect(colAllToOne.port_a, conPipWal.port_a); connect(colAllToOne.port_b, heatPort); end if; end Pipe;

## Buildings.Fluid.FixedResistances.SplitterFixedResistanceDpM

Flow splitter with fixed resistance at each port

### Information

Model of a flow splitter or mixer with a fixed resistance in each flow leg. In each flow lag, a pressure drop can be modeled, and at the fluid junction, a mixing volume can be modeled.

The pressure drop is implemented using the model Buildings.Fluid.FixedResistances.FixedResistanceDpM. If its nominal pressure drop is set to zero, then the pressure drop model will be removed. For example, the pressure drop declaration

      m_flow_nominal={ 0.1, 0.1,  -0.2},
      dp_nominal =   {500,    0, -6000}

would model a mixer that has the nominal flow rates and associated pressure drops as shown in the figure below. Note that `port_3` is set to negative values. The negative values indicate that at the nominal conditions, fluid is leaving the component.

Optionally, at the fluid junction, a control volume can be modeled. This is implemented using the model Buildings.Fluid.Delays.DelayFirstOrder. The fluid volume is modeled if `dynamicBalance=true`, and it is removed if `dynamicBalance=false`. The control volume has the size

      V = sum(abs(m_flow_nominal[:])/3)*tau/rho_nominal

where `tau` is a parameter and `rho_nominal` is the density of the medium in the volume at nominal condition. Setting `dynamicBalance=true` can help reducing the size of the nonlinear system of equations.

Extends from Buildings.BaseClasses.BaseIcon (Base icon), Buildings.Fluid.BaseClasses.PartialThreeWayResistance (Flow splitter with partial resistance model at each port).

### Parameters

Type | Name | Default | Description
--- | --- | --- | ---
replaceable package Medium | PartialMedium | Medium in the component
Boolean | use_dh | false | Set to true to specify hydraulic diameter
Real | deltaM | 0.3 | Fraction of nominal mass flow rate where transition to turbulent occurs
Length | dh[3] | {1,1,1} | Hydraulic diameter [m]
Real | ReC[3] | {4000,4000,4000} | Reynolds number where transition to turbulent starts
Nominal condition
MassFlowRate | m_flow_nominal[3] |  | Mass flow rate. Set negative at outflowing ports. [kg/s]
Pressure | dp_nominal[3] |  | Pressure. Set negative at outflowing ports. [Pa]
Dynamics
Equations
Dynamics | energyDynamics | Modelica.Fluid.Types.Dynamic... | Formulation of energy balance
Dynamics | massDynamics | energyDynamics | Formulation of mass balance
Boolean | dynamicBalance | true | Set to true to use a dynamic balance, which often leads to smaller systems of equations
MassFlowRate | mDyn_flow_nominal | sum(abs(m_flow_nominal[:])/3) | Nominal mass flow rate for dynamic momentum and energy balance [kg/s]
Nominal condition
Time | tau | 10 | Time constant at nominal flow for dynamic energy and momentum balance [s]
Initialization
AbsolutePressure | p_start | Medium.p_default | Start value of pressure [Pa]
Temperature | T_start | Medium.T_default | Start value of temperature [K]
MassFraction | X_start[Medium.nX] | Medium.X_default | Start value of mass fractions m_i/m [kg/kg]
ExtraProperty | C_start[Medium.nC] | fill(0, Medium.nC) | Start value of trace substances
ExtraProperty | C_nominal[Medium.nC] | fill(1E-2, Medium.nC) | Nominal value of trace substances. (Set to typical order of magnitude.)
Advanced
Boolean | from_dp | true | = true, use m_flow = f(dp) else dp = f(m_flow)
PortFlowDirection | portFlowDirection_1 | Modelica.Fluid.Types.PortFlo... | Flow direction for port_1
PortFlowDirection | portFlowDirection_2 | Modelica.Fluid.Types.PortFlo... | Flow direction for port_2
PortFlowDirection | portFlowDirection_3 | Modelica.Fluid.Types.PortFlo... | Flow direction for port_3
Boolean | linearized | false | = true, use linear relation between m_flow and dp for any flow rate
Boolean | homotopyInitialization | true | = true, use homotopy method

### Connectors

Type | Name | Description
--- | --- | ---
FluidPort_a | port_1 | First port, typically inlet
FluidPort_b | port_2 | Second port, typically outlet
FluidPort_a | port_3 | Third port, can be either inlet or outlet

### Modelica definition

model SplitterFixedResistanceDpM "Flow splitter with fixed resistance at each port" extends Buildings.BaseClasses.BaseIcon; extends Buildings.Fluid.BaseClasses.PartialThreeWayResistance( mDyn_flow_nominal = sum(abs(m_flow_nominal[:])/3), redeclare Buildings.Fluid.FixedResistances.FixedResistanceDpM res1( redeclare package Medium=Medium, final allowFlowReversal=true, from_dp=from_dp, final m_flow_nominal=m_flow_nominal[1], final dp_nominal=dp_nominal[1], final ReC=ReC[1], final dh=dh[1], linearized=linearized, homotopyInitialization=homotopyInitialization, deltaM=deltaM), redeclare Buildings.Fluid.FixedResistances.FixedResistanceDpM res2( redeclare package Medium=Medium, final allowFlowReversal=true, from_dp=from_dp, final m_flow_nominal=m_flow_nominal[2], final dp_nominal=dp_nominal[2], final ReC=ReC[2], final dh=dh[2], linearized=linearized, homotopyInitialization=homotopyInitialization, deltaM=deltaM), redeclare Buildings.Fluid.FixedResistances.FixedResistanceDpM res3( redeclare package Medium=Medium, final allowFlowReversal=true, from_dp=from_dp, final m_flow_nominal=m_flow_nominal[3], final dp_nominal=dp_nominal[3], final ReC=ReC[3], final dh=dh[3], linearized=linearized, homotopyInitialization=homotopyInitialization, deltaM=deltaM)); parameter Boolean use_dh = false "Set to true to specify hydraulic diameter"; parameter Modelica.SIunits.MassFlowRate[3] m_flow_nominal "Mass flow rate. Set negative at outflowing ports."; parameter Modelica.SIunits.Pressure[3] dp_nominal(each displayUnit = "Pa") "Pressure. Set negative at outflowing ports."; parameter Real deltaM(min=0) = 0.3 "Fraction of nominal mass flow rate where transition to turbulent occurs"; parameter Modelica.SIunits.Length[3] dh={1, 1, 1} "Hydraulic diameter"; parameter Real[3] ReC(each min=0)={4000, 4000, 4000} "Reynolds number where transition to turbulent starts"; parameter Boolean linearized = false "= true, use linear relation between m_flow and dp for any flow rate"; parameter Boolean homotopyInitialization = true "= true, use homotopy method"; end SplitterFixedResistanceDpM;