Buildings.HeatTransfer.Conduction

Information

This package provides component models to compute heat conduction.

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

Package Content

Name	Description
SingleLayer	Model for single layer heat conductance
MultiLayer	Model for heat conductance through a solid with multiple material layers
BaseClasses	Package with base classes for Buildings.HeatTransfer.Conduction

Buildings.HeatTransfer.Conduction.SingleLayer

Information

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   (T_a-T_b),

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

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

Parameters

Type	Name	Default	Description
Area	A		Heat transfer area [m2]
ThermalResistance	R	if (material.R == 0) then ma...	Thermal resistance of construction [K/W]
Material	material	redeclare parameter Data.Bas...	Material from Data.Solids or Data.Resistances
Initialization
Boolean	steadyStateInitial	false	=true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start
Temperature	T_a_start	293.15	Initial temperature at port_a, used if steadyStateInitial = false [K]
Temperature	T_b_start	293.15	Initial temperature at port_b, used if steadyStateInitial = false [K]

Connectors

Type	Name	Description
HeatPort_a	port_a	Heat port at surface a
HeatPort_b	port_b	Heat 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.MultiLayer

Information

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

Type	Name	Default	Description
Area	A		Heat transfer area [m2]
ThermalResistance	R	sum(lay[:].R)	Thermal resistance of construction [K/W]
Generic	layers	redeclare parameter Building...	Construction definition from Data.OpaqueConstructions
Initialization
Boolean	steadyStateInitial	false	=true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start
Temperature	T_a_start	293.15	Initial temperature at port_a, used if steadyStateInitial = false [K]
Temperature	T_b_start	293.15	Initial temperature at port_b, used if steadyStateInitial = false [K]

Connectors

Type	Name	Description
HeatPort_a	port_a	Heat port at surface a
HeatPort_b	port_b	Heat 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;