Description: "Package with models for heat conduction"

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## Buildings.HeatTransfer.Conduction

Package with models for heat conduction

### Information

This package provides component models to compute heat conduction.

#### Implementation

The package declares the constant `nSupPCM`, which is equal to the number of support points that are used to approximate the specific internal energy versus temperature relation. This approximation is used by `Buildings.HeatTransfer.Conduction.SingleLayer` to replace the piece-wise linear function by a cubic hermite spline, with linear extrapolation, in order to avoid state events during the simulation.

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

### Package Content

Name | Description
--- | ---
MultiLayer | Model for heat conductance through a solid with multiple material layers
SingleLayer | Model for single layer heat conductance
SingleLayerCylinder | Heat conduction in a cylinder
nSupPCM=6 | Number of support points to approximate u(T) releation, used only for phase change material
BaseClasses | Package with base classes for Buildings.HeatTransfer.Conduction

### Types and constants

      constant Integer nSupPCM = 6
      "Number of support points to approximate u(T) releation, used only for phase change material";

## Buildings.HeatTransfer.Conduction.MultiLayer

Model for heat conductance through a solid with multiple material layers

### 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 assign the material properties to this model, do the following:

1. Create an instance of a record of Buildings.HeatTransfer.Data.OpaqueConstructions, for example by dragging the record into the schematic model editor.
2. Make sure the instance has the attribute `parameter`, which may not be assigned automatically when you drop the model in a graphical editor. For example, an instanciation may look like

        parameter Data.OpaqueConstructions.Insulation100Concrete200 layers
          "Material layers of construction"
          annotation (Placement(transformation(extent={{-80,60},{-60,80}})));

3. Assign the instance of the material to the instance of the heat transfer model as shown in Buildings.HeatTransfer.Examples.ConductorMultiLayer.

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(layers.material[i].R for... | 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(layers.material[i].R for i in 1:size(layers.material, 1))); 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[i] for i in 1:size(layers.material, 1)}, T_a_start = { T_b_start+(T_a_start-T_b_start) * 1/R * sum(layers.material[k].R for k in i:size(layers.material, 1)) for i in 1:size(layers.material, 1)}, T_b_start = { T_a_start+(T_b_start-T_a_start) * 1/R * sum(layers.material[k].R for k in 1:i) for i in 1:size(layers.material, 1)}, each steadyStateInitial = steadyStateInitial) "Material layer"; 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;

## Buildings.HeatTransfer.Conduction.SingleLayer

Model for single layer heat conductance

### Information

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

#### Transient heat conduction in materials without phase change

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(s,t) ⁄ ∂t) = k (∂² T(s,t) ⁄ ∂s²),

where ρ is the mass density, c is the specific heat capacity per unit mass, T is the temperature at location s and time t and k is the heat conductivity. At the locations s=0 and s=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 ports.

#### Transient heat conduction in phase change materials

If the material is declared using a record of type Buildings.HeatTransfer.Data.SolidsPCM, the heat transfer in a phase change material is computed. The record Buildings.HeatTransfer.Data.SolidsPCM declares the solidus temperature `TSol`, the liquidus temperature `TLiq` and the latent heat of phase transformation `LHea`. For heat transfer with phase change, the specific internal energy u is the dependent variable, rather than the temperature. Therefore, the governing equation is

ρ (∂ u(s,t) ⁄ ∂t) = k (∂² T(s,t) ⁄ ∂s²).

The constitutive relation between specific internal energy u and temperature T is defined in Buildings.HeatTransfer.Conduction.BaseClasses.enthalyTemperature by using cubic hermite spline interpolation with linear extrapolation.

#### Steady-state heat conduction

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.

#### Spatial discretization

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.

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, Data.SolidsPCM 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"; Modelica.SIunits.SpecificInternalEnergy u[nSta](start= material.c*{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 = 270000) "Definition of specific internal energy (enthalpy in solids)!"; replaceable parameter Data.BaseClasses.Material material "Material from Data.Solids, Data.SolidsPCM 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 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"; parameter Modelica.SIunits.Mass m = A*material.x*material.d/material.nSta "Mass associated with the temperature state"; parameter Modelica.SIunits.HeatCapacity C = m*material.c "Heat capacity associated with the temperature state"; parameter Modelica.SIunits.SpecificInternalEnergy ud[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points for derivatives (used for PCM)"; parameter Modelica.SIunits.Temperature Td[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false) "Support points for derivatives (used for PCM)"; parameter Real dT_du[Buildings.HeatTransfer.Conduction.nSupPCM](each fixed=false, each unit="kg.K2/J") "Derivatives dT/du at the support points (used for PCM)"; initial equation // The initialization is only done for materials that store energy. if not material.steadyState then if steadyStateInitial then if material.phasechange then der(u) = zeros(nSta); else der(T) = zeros(nSta); end if; 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; if material.phasechange then (ud, Td, dT_du) = Buildings.HeatTransfer.Conduction.BaseClasses.der_temperature_u( c= material.c, TSol=material.TSol, TLiq=material.TLiq, LHea=material.LHea, ensureMonotonicity=material.ensureMonotonicity); else ud = zeros(Buildings.HeatTransfer.Conduction.nSupPCM); Td = zeros(Buildings.HeatTransfer.Conduction.nSupPCM); dT_du = zeros(Buildings.HeatTransfer.Conduction.nSupPCM); 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; // Steady-state heat balance if material.steadyState then for i in 2:nSta+1 loop Q_flow[i] = Q_flow[1]; if material.phasechange then // Phase change material T[i-1]=Buildings.HeatTransfer.Conduction.BaseClasses.temperature_u( ud=ud, Td=Td, dT_du=dT_du, u=u[i-1]); else // Regular material u[i-1]=material.c*T[i-1]; end if; end for; else // Transient heat conduction if material.phasechange then // Phase change material for i in 1:nSta loop der(u[i]) = (Q_flow[i]-Q_flow[i+1])/m; // Recalculation of temperature based on specific internal energy T[i]=Buildings.HeatTransfer.Conduction.BaseClasses.temperature_u( ud=ud, Td=Td, dT_du=dT_du, u=u[i]); end for; else // Regular material for i in 1:nSta loop der(T[i]) = (Q_flow[i]-Q_flow[i+1])/C; u[i]=material.c*T[i]; end for; end if; end if; end SingleLayer;

## Buildings.HeatTransfer.Conduction.SingleLayerCylinder

Heat conduction in a cylinder

### 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

Type | Name | Default | Description
--- | --- | --- | ---
Generic | material | redeclare parameter Building... | Material thermal properties
Height | h |  | Height of the cylinder [m]
Radius | r_a |  | Internal radius [m]
Radius | r_b |  | External radius [m]
Integer | nSta |  | Number of state variables
Real | griFac | 2 | Grid factor for spacing
Initialization
Temperature | TInt_start | 293.15 | Initial temperature at port_a, used if steadyStateInitial = false [K]
Temperature | TExt_start | 293.15 | Initial temperature at port_b, used if steadyStateInitial = false [K]
Boolean | steadyStateInitial | false | true initializes dT(0)/dt=0, false initializes T(0) at fixed temperature using T_a_start and T_b_start

### 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 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"; // Modelica.SIunits.TemperatureSlope der_T[nSta] // "Time derivative of temperature (= der(T))"; protected 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 between two nodes end for; if material.steadyState then 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]; end for; end if; end SingleLayerCylinder;