.Buildings.HeatTransfer.Conduction.SingleLayer

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.temperature_u 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 (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].

As an example, we assume a material with a length of x and a discretization with four state variables.

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.

Revisions

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