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Buildings.HeatTransfer.Conduction

Package with models for heat conduction

Information

This package provides component models to compute heat conduction.

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

Package Content

NameDescription
Buildings.HeatTransfer.Conduction.SingleLayer SingleLayer Model for single layer heat conductance
Buildings.HeatTransfer.Conduction.SingleLayerCylinder SingleLayerCylinder Heat conduction in a cylinder
Buildings.HeatTransfer.Conduction.MultiLayer MultiLayer Model for heat conductance through a solid with multiple material layers
Buildings.HeatTransfer.Conduction.BaseClasses BaseClasses Package with base classes for Buildings.HeatTransfer.Conduction

Buildings.HeatTransfer.Conduction.SingleLayer Buildings.HeatTransfer.Conduction.SingleLayer

Model for single layer heat conductance

Buildings.HeatTransfer.Conduction.SingleLayer

Information

This is a model of a heat conductor for a single material that computes transient or steady-state heat conductions.

If the material is a record that extends Buildings.HeatTransfer.Data.Solids and its specific heat capacity (as defined by the record material.c) is non-zero, then this model computes transient heat conduction, i.e., it computes a numerical approximation to the solution of the heat equation

ρ c (∂ T(u,t) ⁄ ∂t) = k (∂² T(u,t) ⁄ ∂u²),

where ρ is the mass density, c is the specific heat capacity per unit mass, T is the temperature at location u and time t and k is the heat conductivity. At the locations u=0 and u=x, where x is the material thickness, the temperature and heat flow rate is equal to the temperature and heat flow rate of the heat heat ports.

To spatially discretize the heat equation, the construction is divided into compartments with material.nSta ≥ 1 state variables. The state variables are connected to each other through thermal conductors. There is also a thermal conductor between the surfaces and the outermost state variables. Thus, to obtain the surface temperature, use port_a.T (or port_b.T) and not the variable T[1]. Each compartment has the same material properties. To build multi-layer constructions, use Buildings.HeatTransfer.Conduction.MultiLayer instead of this model.

If material.c=0, or if the material extends Buildings.HeatTransfer.Data.Resistances, then steady-state heat conduction is computed. In this situation, the heat flow between its heat ports is

Q = A   k ⁄ x   (Ta-Tb),

where A is the cross sectional area, x is the layer thickness, Ta is the temperature at port a and Tb is the temperature at port b.

Extends from Buildings.HeatTransfer.Conduction.BaseClasses.PartialConductor (Partial model for heat conductor).

Parameters

TypeNameDefaultDescription
AreaA Heat transfer area [m2]
ThermalResistanceRif (material.R == 0) then ma...Thermal resistance of construction [K/W]
Materialmaterialredeclare parameter Data.Bas...Material from Data.Solids or Data.Resistances
Initialization
BooleansteadyStateInitialfalse=true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start
TemperatureT_a_start293.15Initial temperature at port_a, used if steadyStateInitial = false [K]
TemperatureT_b_start293.15Initial temperature at port_b, used if steadyStateInitial = false [K]

Connectors

TypeNameDescription
HeatPort_aport_aHeat port at surface a
HeatPort_bport_bHeat port at surface b

Modelica definition

model SingleLayer "Model for single layer heat conductance"
  extends Buildings.HeatTransfer.Conduction.BaseClasses.PartialConductor(
   final R=if (material.R == 0) then material.x/material.k/A else material.R/A);
   // if material.R == 0, then the material specifies material.k, and this model specifies x
   // For resistances, material.k need not be specified, and hence we use material.R
  // The value T[:].start is used by the solver when finding initial states
  // that satisfy dT/dt=0, which requires solving a system of nonlinear equations
  // if the convection coefficient is a function of temperature.
  Modelica.SIunits.Temperature T[nSta](start=
     {T_a_start+(T_b_start-T_a_start) * UA *
        sum(1/(if (k==1 or k==nSta+1) then UAnSta2 else UAnSta) for k in 1:i) for i in 1:nSta},
      each nominal = 300) "Temperature at the states";
  Modelica.SIunits.HeatFlowRate Q_flow[nSta+1] 
    "Heat flow rate from state i to i+1";

  replaceable parameter Data.BaseClasses.Material material 
    "Material from Data.Solids or Data.Resistances";

  parameter Boolean steadyStateInitial=false 
    "=true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start";
  parameter Modelica.SIunits.Temperature T_a_start=293.15 
    "Initial temperature at port_a, used if steadyStateInitial = false";
  parameter Modelica.SIunits.Temperature T_b_start=293.15 
    "Initial temperature at port_b, used if steadyStateInitial = false";
protected 
  parameter Modelica.SIunits.HeatCapacity C = A*material.x*material.d*material.c/material.nSta 
    "Heat capacity associated with the temperature state";
  // nodes at surface have only 1/2 the layer thickness
//  final parameter Modelica.SIunits.ThermalConductance G[nSta+1](each fixed=false)
 //   "Thermal conductance of layer between the states";
  Modelica.SIunits.TemperatureSlope der_T[nSta] 
    "Time derivative of temperature (= der(T))";
  final parameter Integer nSta(min=1) = material.nSta 
    "Number of state variables";
  final parameter Modelica.SIunits.ThermalConductance UAnSta = UA*nSta 
    "Thermal conductance between nodes";
  final parameter Modelica.SIunits.ThermalConductance UAnSta2 = 2*UAnSta 
    "Thermal conductance between nodes and surface boundary";
initial equation 
 // G={UA*nSta * (if (i==1 or i==nSta+1) then 2 else 1) for i in 1:nSta+1};
  // The initialization is only done for materials that store energy.
  if not material.steadyState then
    if steadyStateInitial then
      der_T = zeros(nSta);
    else
      for i in 1:nSta loop
        T[i] = T_a_start+(T_b_start-T_a_start) * UA *
          sum(1/(if (k==1 or k==nSta+1) then UAnSta2 else UAnSta) for k in 1:i);
      end for;
      end if;
   end if;

equation 
    port_a.Q_flow = +Q_flow[1];
    port_b.Q_flow = -Q_flow[nSta+1];

    port_a.T-T[1] = Q_flow[1]/UAnSta2;
    T[nSta] -port_b.T = Q_flow[nSta+1]/UAnSta2;
    for i in 2:nSta loop
       // Q_flow[i] is heat flowing from (i-1) to (i)
       T[i-1]-T[i] = Q_flow[i]/UAnSta;
    end for;
    if material.steadyState then
      der_T = zeros(nSta);
      for i in 2:nSta+1 loop
        Q_flow[i] = Q_flow[1];
      end for;
      else
        for i in 1:nSta loop
          der(T[i]) = (Q_flow[i]-Q_flow[i+1])/C;
          der_T[i] = der(T[i]);
        end for;
    end if;
end SingleLayer;

Buildings.HeatTransfer.Conduction.SingleLayerCylinder Buildings.HeatTransfer.Conduction.SingleLayerCylinder

Heat conduction in a cylinder

Buildings.HeatTransfer.Conduction.SingleLayerCylinder

Information

Model for radial heat transfer in a hollow cylinder.

If the heat capacity of the material is non-zero, then this model computes transient heat conduction, i.e., it computes a numerical approximation to the solution of the heat equation

ρ c ( ∂ T(r,t) ⁄ ∂t ) = k ( ∂² T(r,t) ⁄ ∂r² + 1 ⁄ r   ∂ T(r,t) ⁄ ∂r ),

where ρ is the mass density, c is the specific heat capacity per unit mass, T is the temperature at location r and time t and k is the heat conductivity. At the locations r=ra and r=rb, the temperature and heat flow rate are equal to the temperature and heat flow rate of the heat ports.

If the heat capacity of the material is set to zero, then steady-state heat flow is computed using

Q = 2 π k (Ta-Tb)⁄ ln(ra ⁄ rb),

where ra is the internal radius, rb is the external radius, Ta is the temperature at port a and Tb is the temperature at port b.

Implementation

To spatially discretize the heat equation, the construction is divided into compartments with material.nSta ≥ 1 state variables. The state variables are connected to each other through thermal conductors. There is also a thermal conductor between the surfaces and the outermost state variables. Thus, to obtain the surface temperature, use port_a.T (or port_b.T) and not the variable T[1].

Parameters

TypeNameDefaultDescription
Genericmaterialredeclare parameter Building...Material thermal properties
Heighth Height of the cylinder [m]
Radiusr_a Internal radius [m]
Radiusr_b External radius [m]
IntegernSta Number of state variables
RealgriFac2Grid factor for spacing
Initialization
TemperatureTInt_start293.15Initial temperature at port_a, used if steadyStateInitial = false [K]
TemperatureTExt_start293.15Initial temperature at port_b, used if steadyStateInitial = false [K]
BooleansteadyStateInitialfalsetrue initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start

Connectors

TypeNameDescription
HeatPort_aport_aHeat port at surface a
HeatPort_bport_bHeat port at surface b

Modelica definition

model SingleLayerCylinder "Heat conduction in a cylinder"
  replaceable parameter Buildings.HeatTransfer.Data.Soil.Generic material 
    "Material thermal properties";

  parameter Modelica.SIunits.Height h "Height of the cylinder";
  parameter Modelica.SIunits.Radius r_a "Internal radius";
  parameter Modelica.SIunits.Radius r_b "External radius";
  parameter Integer nSta(min=1) "Number of state variables";
  parameter Modelica.SIunits.Temperature TInt_start=293.15 
    "Initial temperature at port_a, used if steadyStateInitial = false";
  parameter Modelica.SIunits.Temperature TExt_start=293.15 
    "Initial temperature at port_b, used if steadyStateInitial = false";
  parameter Boolean steadyStateInitial=false 
    "true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start";

  parameter Real griFac(min=1) = 2 "Grid factor for spacing";

  Modelica.SIunits.TemperatureDifference dT "port_a.T - port_b.T";

  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a port_a 
    "Heat port at surface a";
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_b port_b 
    "Heat port at surface b";

  Modelica.SIunits.Temperature T[nSta](start=
     {TInt_start+ (TExt_start-TInt_start)/Modelica.Math.log(r_b/r_a)
       *Modelica.Math.log((r_a+(r_b-r_a)/(nSta)*(i-0.5))/r_a) for i in 1:nSta}) 
    "Temperature of the states";
  Modelica.SIunits.HeatFlowRate Q_flow[nSta+1] 
    "Heat flow rate from state i to i+1";

protected 
  Modelica.SIunits.TemperatureSlope der_T[nSta] 
    "Time derivative of temperature (= der(T))";
  parameter Modelica.SIunits.Radius r[nSta+1](each fixed=false) 
    "Radius to the boundary of the i-th domain";
  parameter Modelica.SIunits.Radius rC[nSta](each fixed=false) 
    "Radius to the center of the i-th domain";

  final parameter Modelica.SIunits.SpecificHeatCapacity c= material.c 
    "Specific heat capacity";
  final parameter Modelica.SIunits.ThermalConductivity k= material.k 
    "Thermal conductivity of the material";
  final parameter Modelica.SIunits.Density d = material.d 
    "Density of the material";

  parameter Modelica.SIunits.ThermalConductance G[nSta+1](each fixed=false) 
    "Heat conductance between the temperature nodes";
  parameter Modelica.SIunits.HeatCapacity C[nSta](each fixed=false) 
    "Heat capacity of each state";

initial equation 
  assert(r_a < r_b, "Error: Model requires r_a < r_b.");
  assert(0 < r_a,   "Error: Model requires 0 < r_a.");
  r[1] = r_a;
  for i in 2:nSta+1 loop
     r[i]= r[i-1] + ( r_b - r_a)  * (1-griFac)/(1-griFac^(nSta)) * griFac^(i-2);
  end for;
  assert(abs(r[nSta+1]-r_b) < 1E-10, "Error: Wrong computation of radius. r[nSta+1]=" + String(r[nSta+1]));

  // Radii to the center of the domain
  for i in 1:nSta loop
    rC[i] = (r[i]+r[i+1])/2;
  end for;
  // Conductance between nodes (which are in the center of the domain)
  G[1]      = 2*Modelica.Constants.pi*k*h/Modelica.Math.log(rC[1]/r_a);
  G[nSta+1] = 2*Modelica.Constants.pi*k*h/Modelica.Math.log(r_b/rC[nSta]);
  for i in 2:nSta loop
    G[i]      = 2*Modelica.Constants.pi*k*h/Modelica.Math.log(rC[i]/rC[i-1]);
  end for;

  // Heat capacity of each segment
  for i in 1:nSta loop
   C[i] = (d*Modelica.Constants.pi*c*h*((r[i+1])^2-(r[i])^2));
  end for;
  // The initialization is only done for materials that store energy.
  if not material.steadyState then
    if steadyStateInitial then
      der_T = zeros(nSta);
    else
      for i in 1:nSta loop
        T[i]=TInt_start+ (TExt_start-TInt_start)/Modelica.Math.log(r_b/r_a) * Modelica.Math.log(rC[i]/r_a);
      end for;
    end if;
  end if;
equation 
  dT = port_a.T - port_b.T;
  port_a.Q_flow = +Q_flow[1];
  port_b.Q_flow = -Q_flow[nSta+1];
  Q_flow[1]      = G[1]      * (port_a.T-T[1]);
  Q_flow[nSta+1] = G[nSta+1] * (T[nSta]-port_b.T);
  for i in 2:nSta loop
    Q_flow[i]=G[i]*(T[i-1]-T[i]);
    // Q_flow[i] represents the heat flowing betwen two nodes
  end for;
  if material.steadyState then
    der_T = zeros(nSta);
    for i in 2:nSta+1 loop
      Q_flow[i]=Q_flow[1];
    end for;
  else
    for i in 1:nSta loop
        der(T[i])= (Q_flow[i]-Q_flow[i+1])/C[i];
        der_T[i] = der(T[i]);
    end for;
  end if;
end SingleLayerCylinder;

Buildings.HeatTransfer.Conduction.MultiLayer Buildings.HeatTransfer.Conduction.MultiLayer

Model for heat conductance through a solid with multiple material layers

Buildings.HeatTransfer.Conduction.MultiLayer

Information

This is a model of a heat conductor with multiple material layers and energy storage. The construction has at least one material layer, and each layer has at least one temperature node. The layers are modeled using an instance of Buildings.HeatTransfer.Conduction.SingleLayer.

The construction material is defined by a record of the package Buildings.HeatTransfer.Data.OpaqueConstructions. This record allows specifying materials that store energy, and material that are a thermal conductor only with no heat storage.

To obtain the surface temperature of the construction, use port_a.T (or port_b.T) and not the variable T[1] because there is a thermal resistance between the surface and the temperature state.

Extends from Buildings.HeatTransfer.Conduction.BaseClasses.PartialConductor (Partial model for heat conductor), Buildings.HeatTransfer.Conduction.BaseClasses.PartialConstruction (Partial model for multi-layer constructions).

Parameters

TypeNameDefaultDescription
AreaA Heat transfer area [m2]
ThermalResistanceRsum(lay[:].R)Thermal resistance of construction [K/W]
Genericlayersredeclare parameter Building...Construction definition from Data.OpaqueConstructions
Initialization
BooleansteadyStateInitialfalse=true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start
TemperatureT_a_start293.15Initial temperature at port_a, used if steadyStateInitial = false [K]
TemperatureT_b_start293.15Initial temperature at port_b, used if steadyStateInitial = false [K]

Connectors

TypeNameDescription
HeatPort_aport_aHeat port at surface a
HeatPort_bport_bHeat port at surface b

Modelica definition

model MultiLayer 
  "Model for heat conductance through a solid with multiple material layers"
  extends Buildings.HeatTransfer.Conduction.BaseClasses.PartialConductor(
   final R=sum(lay[:].R));
  Modelica.SIunits.Temperature T[sum(nSta)](each nominal = 300) 
    "Temperature at the states";
  Modelica.SIunits.HeatFlowRate Q_flow[sum(nSta)+nLay] 
    "Heat flow rate from state i to i+1";
  extends Buildings.HeatTransfer.Conduction.BaseClasses.PartialConstruction;

protected 
  Buildings.HeatTransfer.Conduction.SingleLayer[nLay] lay(
   each final A=A,
   material = layers.material,
   T_a_start = _T_a_start,
   T_b_start = _T_b_start,
   each steadyStateInitial = steadyStateInitial) "Material layer";

protected 
  parameter Modelica.SIunits.Temperature _T_a_start[nLay](fixed=false) 
    "Initial temperature at port_a of respective layer, used if steadyStateInitial = false";
  parameter Modelica.SIunits.Temperature _T_b_start[nLay](fixed=false) 
    "Initial temperature at port_b of respective layer, used if steadyStateInitial = false";

initial equation 
  for i in 1:nLay loop
    _T_a_start[i] = T_b_start+(T_a_start-T_b_start) * 1/R * sum(lay[k].R for k in i:nLay);
    _T_b_start[i] =  T_a_start+(T_b_start-T_a_start) * 1/R * sum(lay[k].R for k in 1:i);
  end for;

equation 
  // This section assigns the temperatures and heat flow rates of the layer models to
  // an array that makes plotting the results easier.
  for i in 1:nLay loop
    for j in 1:nSta[i] loop
      T[sum(nSta[k] for k in 1:(i-1)) +j] = lay[i].T[j];
    end for;
    for j in 1:nSta[i]+1 loop
      Q_flow[sum(nSta[k] for k in 1:i-1)+(i-1)+j] = lay[i].Q_flow[j];
    end for;
  end for;
  connect(port_a, lay[1].port_a);
  for i in 1:nLay-1 loop
  connect(lay[i].port_b, lay[i+1].port_a);
  end for;
  connect(lay[nLay].port_b, port_b);

end MultiLayer;
Automatically generated Wed Feb 29 16:53:47 2012.