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. See this model for an explanation of the equations that are applied to each material layer.

Important parameters

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.

The parameters stateAtSurface_a and stateAtSurface_b determine whether there is a state variable at these surfaces, as described above. Note that if stateAtSurface_a = true, then there is temperature state on the surface a with prescribed value, as determined by the differential equation of the heat conduction. Hence, in this situation, it is not possible to connect a temperature boundary condition such as Buildings.HeatTransfer.Sources.FixedTemperature as this would yield to specifying the same temperature twice. To avoid this, either set stateAtSurface_a = false, or place a thermal resistance between the boundary condition and the surface of this model. The same applies for surface b. See the examples in Buildings.HeatTransfer.Examples.

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

Parameters

Connectors

Modelica definition

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.

Main equations

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   (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.

Spatial discretization

To spatially discretize the heat equation, the construction is divided into compartments (control volumes) with material.nSta ≥ 1 state variables. Each control volume has the same material properties. The state variables are connected to each other through thermal resistances. If stateAtSurface_a = true, a state is placed at the surface a, and similarly, if stateAtSurface_b = true, a state is placed at the surface b. Otherwise, these states are placed inside the material, away from the surface. Thus, to obtain the surface temperature, use port_a.T (or port_b.T) and not the variable T[1].

• If stateAtSurface_a = false and stateAtSurface_b = false, then each of the four state variables is placed in the middle of a control volume with length l=x/material.nSta.

  

• If stateAtSurface_a = true or stateAtSurface_b = true, then one state is placed on the surface of the material. Each of the remaining three states is placed in the middle of a control volume with length l=x/(material.nSta-1).

  

• If stateAtSurface_a = true and stateAtSurface_b = true, then two states are placed on the surfaces of the material. Each of the remaining two states is placed in the middle of a control volume with length l=x/(material.nSta-2).

  

To build multi-layer constructions, use Buildings.HeatTransfer.Conduction.MultiLayer instead of this model.

Important parameters

The parameters stateAtSurface_a and stateAtSurface_b determine whether there is a state variable at these surfaces, as described above. Note that if stateAtSurface_a = true, then there is temperature state on the surface a with prescribed value, as determined by the differential equation of the heat conduction. Hence, in this situation, it is not possible to connect a temperature boundary condition such as Buildings.HeatTransfer.Sources.FixedTemperature as this would yield to specifying the same temperature twice. To avoid this, either set stateAtSurface_a = false, or place a thermal resistance between the boundary condition and the surface of this model. The same applies for surface b. See the examples in Buildings.HeatTransfer.Examples.

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

Parameters

Connectors

Modelica definition

Buildings.HeatTransfer.Conduction.SingleLayerCylinder

Heat conduction in a cylinder

Information

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=r_a and r=r_b, 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 (T_a-T_b)⁄ ln(r_a ⁄ r_b),

where r_a is the internal radius, r_b is the external radius, T_a is the temperature at port a and T_b 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

Connectors

Modelica definition