Buildings.Fluid.HeatExchangers.Radiators

This is a model of a radiator that can be used as a dynamic or steady-state model. The required parameters are data that are typically available from manufacturers that follow the European Norm EN 442-2.

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.

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

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

where Tⁱ is the water temperature of the element, T_a is the temperature of the room air, T_r is the radiative temperature, 0 < f_r < 1 is the fraction of radiant to total heat transfer, UA is the UA-value of the radiator, and n is an exponent for the heat transfer. The model computes the UA-value by numerically solving the above equations for given nominal heating power, nominal temperatures, fraction radiant to total heat transfer and exponent for heat transfer.

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.

Parameters

Connectors

Modelica definition

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]
PressureDifference	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]
Boolean	homotopyInitialization	true	= true, use homotopy method
Diagnostics
Boolean	show_T	true	= true, if actual temperature at port is computed
Dynamics
Equations
Dynamics	energyDynamics	Modelica.Fluid.Types.Dynamic...	Type of energy balance: dynamic (3 initialization options) or steady state
Dynamics	massDynamics	energyDynamics	Type of mass balance: dynamic (3 initialization options) or steady state
Real	mSenFac	1 + 500mDry/(VWatcp_nomina...	Factor for scaling the sensible thermal mass of the volume
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.)

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

model RadiatorEN442_2 "Dynamic radiator for space heating" extends Buildings.Fluid.Interfaces.PartialTwoPortInterface( 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), final mSenFac = 1 + 500*mDry/(VWat*cp_nominal*Medium.density( Medium.setState_pTX(Medium.p_default, Medium.T_default, Medium.X_default)))); 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) "Dry mass of radiator that will be lumped to water heat capacity"; parameter Boolean homotopyInitialization = true "= true, use homotopy method"; // Heat flow rates Modelica.SIunits.HeatFlowRate QCon_flow = heatPortCon.Q_flow "Heat input into the water due to convective heat transfer with room air"; Modelica.SIunits.HeatFlowRate QRad_flow = heatPortRad.Q_flow "Heat input into the water due to radiative heat transfer with room"; Modelica.SIunits.HeatFlowRate Q_flow = QCon_flow + QRad_flow "Heat input into the water"; // Heat ports 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"; 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=massDynamics, each final p_start=p_start, each final T_start=T_start, each final X_start=X_start, each final C_start=C_start, each final mSenFac=mSenFac) "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"; Buildings.HeatTransfer.Sources.PrescribedHeatFlow[nEle] preCon "Heat input into radiator from convective heat transfer"; Buildings.HeatTransfer.Sources.PrescribedHeatFlow[nEle] preRad "Heat input into radiator from radiative heat transfer"; Modelica.SIunits.TemperatureDifference dTCon[nEle] = {heatPortCon.T - vol[i].T for i in 1:nEle} "Temperature difference for convective heat transfer"; Modelica.SIunits.TemperatureDifference dTRad[nEle] = {heatPortRad.T - vol[i].T for i in 1:nEle} "Temperature difference for radiative heat transfer"; Modelica.Blocks.Sources.RealExpression QCon[nEle](y={if homotopyInitialization then homotopy(actual=(1 - fraRad) * UAEle * (heatPortCon.T - vol[i].T) * Buildings.Utilities.Math.Functions.regNonZeroPower( x=(heatPortCon.T - vol[i].T), n=n - 1, delta=0.05), simplified=(1 - fraRad) * UAEle .* abs(dTCon_nominal[i]) ^ ( n - 1) * (heatPortCon.T - vol[i].T)) else (1 - fraRad) * UAEle * (heatPortCon.T - vol[i].T) * Buildings.Utilities.Math.Functions.regNonZeroPower( x=(heatPortCon.T - vol[i].T), n=n - 1, delta=0.05) for i in 1:nEle}) "Convective heat flow rate"; Modelica.Blocks.Sources.RealExpression QRad[nEle](y={if homotopyInitialization then homotopy(actual=fraRad * UAEle * (heatPortRad.T - vol[i].T) * Buildings.Utilities.Math.Functions.regNonZeroPower( x=(heatPortRad.T - vol[i].T), n=n - 1, delta=0.05), simplified=fraRad * UAEle * abs(dTRad_nominal[i]) ^ (n - 1) * (heatPortRad.T - vol[i].T)) else fraRad * UAEle * (heatPortRad.T - vol[i].T) * Buildings.Utilities.Math.Functions.regNonZeroPower( x=(heatPortRad.T - vol[i].T), n=n - 1, delta=0.05) for i in 1:nEle}) "Radiative heat flow rate"; Buildings.HeatTransfer.Sources.PrescribedHeatFlow preSumCon "Heat input into radiator from convective heat transfer"; Modelica.Blocks.Math.Sum sumCon(nin=nEle, k=-ones(nEle)) "Sum of convective heat flow rate"; Modelica.Blocks.Math.Sum sumRad(nin=nEle, k=-ones(nEle)) "Sum of radiative heat flow rate"; Buildings.HeatTransfer.Sources.PrescribedHeatFlow preSumRad "Heat input into radiator from 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 // Use difference, TWat_nominal[i] - TRad/Air_nominal, to avoid larger system of equations QEle_flow_nominal[i] = k * UAEle * (fraRad * Buildings.Utilities.Math.Functions.powerLinearized( x=k*TWat_nominal[i] - TRad_nominal, n=n, x0=0.1*k*(T_b_nominal-TRad_nominal)) + (1-fraRad) * Buildings.Utilities.Math.Functions.powerLinearized( x=k*TWat_nominal[i] - TAir_nominal, n=n, x0=0.1*k*(T_b_nominal-TAir_nominal))); end for; equation connect(preCon.port, vol.heatPort); connect(preRad.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; connect(QCon.y, preCon.Q_flow); connect(sumCon.u, QCon.y); connect(sumCon.y, preSumCon.Q_flow); connect(preSumCon.port, heatPortCon); connect(QRad.y, preRad.Q_flow); connect(QRad.y, sumRad.u); connect(sumRad.y, preSumRad.Q_flow); connect(preSumRad.port, heatPortRad); end RadiatorEN442_2;