Buildings.Fluid.HeatExchangers.Radiators

Information

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

Package Content

Name	Description
RadiatorEN442_2	Dynamic radiator for space heating
Examples	Collection of models that illustrate model use and test models

Buildings.Fluid.HeatExchangers.Radiators.RadiatorEN442_2

Information

However, to allow for varying mass flow rates, the transferred heat is computed using a discretization along the water flow path, and heat is exchanged between each compartment and a uniform room air and radiation temperature. This discretization is different from the computation in EN 442-2, which may yield water outlet temperatures that are below the room temperature at low mass flow rates. Furthermore, rather than using only one room temperature, this model uses a room air and room radiation temperature.

The transferred heat is modeled as follows: Let N denote the number of elements used to discretize the radiator model. For each element i ∈ {1, … , N}, the convective and radiative heat transfer Qⁱ_c and Qⁱ_r from the radiator to the room is

Qⁱ_c = sign(Tⁱ-T_a) (1-f_r) UA ⁄ N |Tⁱ-T_a|ⁿ

Qⁱ_r = sign(Tⁱ-T_r) f_r UA ⁄ N |Tⁱ-T_r|ⁿ

The parameter energyDynamics (in the Assumptions tab), determines whether the model computes the dynamic or the steady-state response. For the transient response, heat storage is computed using a finite volume approach for the water and the metal mass, which are both assumed to be at the same temperature.

The default parameters for the heat capacities are valid for a flat plate radiator without fins, with one plate of water carying fluid, and a height of 0.42 meters.

Extends from Fluid.Interfaces.PartialTwoPortInterface (Partial model transporting fluid between two ports without storing mass or energy), Buildings.Fluid.Interfaces.LumpedVolumeDeclarations (Declarations for lumped volumes).

Parameters

Type	Name	Default	Description
replaceable package Medium		PartialMedium	Medium in the component
Integer	nEle	5	Number of elements used in the discretization
Real	fraRad	0.35	Fraction radiant heat transfer
Real	n	1.24	Exponent for heat transfer
Nominal condition
MassFlowRate	m_flow_nominal	abs(Q_flow_nominal/cp_nomina...	Nominal mass flow rate [kg/s]
Power	Q_flow_nominal		Nominal heating power (positive for heating) [W]
Temperature	T_a_nominal		Water inlet temperature at nominal condition [K]
Temperature	T_b_nominal		Water outlet temperature at nominal condition [K]
Temperature	TAir_nominal	293.15	Air temperature at nominal condition [K]
Temperature	TRad_nominal	TAir_nominal	Radiative temperature at nominal condition [K]
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	system.allowFlowReversal	= 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
Diagnostics
Boolean	show_V_flow	false	= true, if volume flow rate at inflowing port is computed
Boolean	show_T	true	= true, if actual temperature at port is computed (may lead to events)
Dynamics
Equations
Dynamics	energyDynamics	Modelica.Fluid.Types.Dynamic...	Formulation of energy balance
Dynamics	massDynamics	energyDynamics	Formulation of mass balance
Volume	VWat	5.8E-6*abs(Q_flow_nominal)	Water volume of radiator [m3]
Mass	mDry	0.0263*abs(Q_flow_nominal)	Dry mass of radiator that will be lumped to water heat capacity [kg]
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.)

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	heatPortCon	Heat port for convective heat transfer with room air temperature
HeatPort_a	heatPortRad	Heat port for radiative heat transfer with room radiation temperature

Modelica definition

model RadiatorEN442_2 "Dynamic radiator for space heating"
   extends Fluid.Interfaces.PartialTwoPortInterface(
   showDesignFlowDirection = false,
   show_T=true,
   m_flow_nominal=abs(Q_flow_nominal/cp_nominal/(T_a_nominal-T_b_nominal)));
   extends Buildings.Fluid.Interfaces.LumpedVolumeDeclarations(
     final X_start = Medium.X_default,
     final C_start = fill(0, Medium.nC),
     final C_nominal = fill(1E-2, Medium.nC));

  parameter Integer nEle(min=1) = 5 
    "Number of elements used in the discretization";
  parameter Real fraRad(min=0, max=1) = 0.35 "Fraction radiant heat transfer";
  // Assumptions

  parameter Modelica.SIunits.Power Q_flow_nominal 
    "Nominal heating power (positive for heating)";
  parameter Modelica.SIunits.Temperature T_a_nominal 
    "Water inlet temperature at nominal condition";
  parameter Modelica.SIunits.Temperature T_b_nominal 
    "Water outlet temperature at nominal condition";
  parameter Modelica.SIunits.Temperature TAir_nominal = 293.15 
    "Air temperature at nominal condition";
  parameter Modelica.SIunits.Temperature TRad_nominal = TAir_nominal 
    "Radiative temperature at nominal condition";

  parameter Real n = 1.24 "Exponent for heat transfer";
  parameter Modelica.SIunits.Volume VWat = 5.8E-6*abs(Q_flow_nominal) 
    "Water volume of radiator";
  parameter Modelica.SIunits.Mass mDry = 0.0263*abs(Q_flow_nominal) if 
        not (energyDynamics == Modelica.Fluid.Types.Dynamics.SteadyState) 
    "Dry mass of radiator that will be lumped to water heat capacity";
  parameter Boolean homotopyInitialization = true "= true, use homotopy method";
  Modelica.SIunits.HeatFlowRate QCon_flow 
    "Heat input into the water due to convective heat transfer with room air";
  Modelica.SIunits.HeatFlowRate QRad_flow 
    "Heat input into the water due to radiative heat transfer with room";
  Modelica.SIunits.HeatFlowRate Q_flow "Heat input into the water";
public 
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a heatPortCon 
    "Heat port for convective heat transfer with room air temperature";
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a heatPortRad 
    "Heat port for radiative heat transfer with room radiation temperature";

  Modelica.Thermal.HeatTransfer.Components.HeatCapacitor[nEle] heaCap(
    each C=500*mDry/nEle,
    each T(start=T_start)) if 
     not (energyDynamics == Modelica.Fluid.Types.Dynamics.SteadyState) 
    "heat capacity of radiator metal";

  Buildings.HeatTransfer.Sources.PrescribedHeatFlow[nEle] preHeaFloCon 
    "Heat input into radiator from convective heat transfer";
  Buildings.HeatTransfer.Sources.PrescribedHeatFlow[nEle] preHeaFloRad 
    "Heat input into radiator from radiative heat transfer";

  Fluid.MixingVolumes.MixingVolume[nEle] vol(
    redeclare each package Medium = Medium,
    each nPorts = 2,
    each V=VWat/nEle,
    each final m_flow_nominal = m_flow_nominal,
    each final energyDynamics=energyDynamics,
    each final massDynamics=energyDynamics,
    each final p_start=p_start,
    each final T_start=T_start,
    each final X_start=X_start,
    each final C_start=C_start) "Volume for fluid stream";
protected 
   parameter Modelica.SIunits.SpecificHeatCapacity cp_nominal=
      Medium.specificHeatCapacityCp(
        Medium.setState_pTX(Medium.p_default, T_a_nominal, Medium.X_default)) 
    "Specific heat capacity at nominal conditions";
   parameter Modelica.SIunits.HeatFlowRate QEle_flow_nominal[nEle](
      each fixed=false, each start=Q_flow_nominal/nEle) 
    "Nominal heating power of each element";
   parameter Modelica.SIunits.Temperature TWat_nominal[nEle](
      each fixed=false,
      start={T_a_nominal - i/nEle * (T_a_nominal-T_b_nominal) for i in 1:nEle}) 
    "Water temperature in each element at nominal conditions";
   parameter Modelica.SIunits.TemperatureDifference[nEle] dTRad_nominal(
    each fixed=false, start={T_a_nominal - i/nEle * (T_a_nominal-T_b_nominal) - TRad_nominal
    for i in 1:nEle}) 
    "Temperature difference for radiative heat transfer at nominal conditions";
   parameter Modelica.SIunits.TemperatureDifference[nEle] dTCon_nominal(
    each fixed=false, start={T_a_nominal - i/nEle * (T_a_nominal-T_b_nominal) - TAir_nominal
    for i in 1:nEle}) 
    "Temperature difference for convective heat transfer at nominal conditions";

   parameter Modelica.SIunits.ThermalConductance UAEle(fixed=false, min=0,
     start=Q_flow_nominal/((T_a_nominal+T_b_nominal)/2-((1-fraRad)*TAir_nominal+fraRad*TRad_nominal))/nEle) 
    "UA value at nominal condition for each element";

   final parameter Real k = if T_b_nominal > TAir_nominal then 1 else -1 
    "Parameter that is used to compute QEle_flow_nominal for heating or cooling mode";
   Modelica.SIunits.TemperatureDifference dTCon[nEle] 
    "Temperature difference for convective heat transfer";
   Modelica.SIunits.TemperatureDifference dTRad[nEle] 
    "Temperature difference for radiative heat transfer";

initial equation 
  if T_b_nominal > TAir_nominal then
     assert(T_a_nominal > T_b_nominal, "In RadiatorEN442_2, T_a_nominal must be higher than T_b_nominal");
     assert(Q_flow_nominal > 0, "In RadiatorEN442_2, nominal power must be bigger than zero if T_b_nominal > TAir_nominal");
  else
     assert(T_a_nominal < T_b_nominal, "In RadiatorEN442_2, T_a_nominal must be lower than T_b_nominal");
     assert(Q_flow_nominal < 0, "In RadiatorEN442_2, nominal power must be smaller than zero if T_b_nominal < TAir_nominal");
  end if;
  TWat_nominal[1] = T_a_nominal - QEle_flow_nominal[1]/m_flow_nominal/
  Medium.specificHeatCapacityCp(
        Medium.setState_pTX(Medium.p_default, T_a_nominal, Medium.X_default));
  for i in 2:nEle loop
    TWat_nominal[i] = TWat_nominal[i-1] - QEle_flow_nominal[i]/m_flow_nominal/
    Medium.specificHeatCapacityCp(
        Medium.setState_pTX(Medium.p_default, TWat_nominal[i-1], Medium.X_default));
  end for;
  dTRad_nominal = TWat_nominal .- TRad_nominal;
  dTCon_nominal = TWat_nominal .- TAir_nominal;
  Q_flow_nominal = sum(QEle_flow_nominal);

  for i in 1:nEle loop
    QEle_flow_nominal[i] = k * UAEle * ((1-fraRad) *
                   Buildings.Utilities.Math.Functions.powerLinearized(x=k*dTRad_nominal[i],
                   n=n,
                   x0=0.1*k*(T_b_nominal-TRad_nominal))
                   + fraRad *
                   Buildings.Utilities.Math.Functions.powerLinearized(x=k*dTCon_nominal[i],
                   n=n,
                   x0=0.1*k*(T_b_nominal-TAir_nominal)));
   end for;

equation 
  dTCon = heatPortCon.T .- vol.T;
  dTRad = heatPortRad.T .- vol.T;
  if homotopyInitialization then
    preHeaFloCon.Q_flow = homotopy(actual=(1-fraRad) .* UAEle .* dTCon .*
                                          Buildings.Utilities.Math.Functions.regNonZeroPower(x=dTCon, n=n-1, delta=0.05),
                                   simplified= (1-fraRad) .* UAEle .* abs(dTCon_nominal).^(n-1) .* dTCon);
    preHeaFloRad.Q_flow = homotopy(actual=fraRad     .* UAEle .* dTRad .*
                                          Buildings.Utilities.Math.Functions.regNonZeroPower(x=dTRad, n=n-1, delta=0.05),
                                   simplified=fraRad .* UAEle .* abs(dTRad_nominal).^(n-1) .* dTRad);
  else
    preHeaFloCon.Q_flow = (1-fraRad) .* UAEle .* dTCon .* Buildings.Utilities.Math.Functions.regNonZeroPower(x=dTCon, n=n-1, delta=0.05);
    preHeaFloRad.Q_flow = fraRad     .* UAEle .* dTRad .* Buildings.Utilities.Math.Functions.regNonZeroPower(x=dTRad, n=n-1, delta=0.05);
  end if;

  QCon_flow = sum(preHeaFloCon.Q_flow);
  QRad_flow = sum(preHeaFloRad.Q_flow);
  Q_flow = QCon_flow + QRad_flow;
  heatPortCon.Q_flow = QCon_flow;
  heatPortRad.Q_flow = QRad_flow;

  connect(preHeaFloCon.port, vol.heatPort);
  connect(preHeaFloRad.port, vol.heatPort);
  connect(heaCap.port, vol.heatPort);
  connect(port_a, vol[1].ports[1]);
  connect(vol[nEle].ports[2], port_b);
  for i in 1:nEle-1 loop
    connect(vol[i].ports[2], vol[i+1].ports[1]);
  end for;
end RadiatorEN442_2;