Thermal networks for heat trasfer in buildings

The basic principles of thermal networks applied to modeling heat transfer in buildings are presented here. An example of the application of the thermal networks to modeling a wall is given in Simple wall and for a building is shown in Toy model house.

Thermal networks

Thermal networks (or thermal circuits) are weighted directed graphs (Figure 1) in which:

• the nodes (or vertices) represent homogenous temperatures, \(\theta_i\), of geometrical points (0D), lines (1D), surfaces (2D) or volumes (3D);

• the oriented branches (or edges) represent heat flow rates, \(q_j\), between the temperature nodes.

Figure 1. Basic thermal network.

The dependent variables of the thermal network are:

• \(\theta_i\) - temperatures, in °C or K, of a volume, surface, line or point (3D, 2D, 1D or 0D space, respectively) in which the temperature is considred homogeneous. Temperatures are grouped in vector \(\theta.\)

• \(q_j ≡ q_{k,i}\) - flow rates, in W, between two (0D, 1D, 2D or 3D) spaces characterized by homogeneous temperatures \(\theta_k\) and \(\theta_i,\) respectively. Flow rates are grouped in vector \(q.\)

The parameters of the thermal network are:

• \(G_j ≡ G_{k,i}\) - thermal conductance, in W/K, between the spaces characterized by temperatures \(\theta_k\) and \(\theta_i\). Conductances are grouped in diagonal matrix \(G.\)

• \(C_i\) - heat capacity, in J/K, of the volume characterized by the homogenous temperature \(\theta_i\). Note that the thermal capacities for surfaces, lines and points are zero because their volume is zero. Capacities are grouped in diagonal matrix \(C.\)

The sources (or the inputs, or the independent variables) of the thermal network are:

• \(T_j\) - temperature source, in °C or K, is a source of thermal energy for which the temperature changes arbitrarely and does not depend on the flow rate. They are grouped in vector \(b\). Typical examples of temperature sources are the temperatures of the outdoor air, soil, indoor air set point.

• \(\dot Q_i\) - flow rate source, in W, is a source of thermal energy for which the flow rate changes arbitrarely and does not depend on the temperature difference. They are grouped in vector \(f.\) Typical examples of flow rate sources are absorbed solar irradiance, electric appliances, basal metabolic rate of occupants.

In a node, there are:

• a heat capacity, \(C_i\), which can be positive or zero;

• a heat flow rate source, \(\dot Q_i\), which can be positive, negative or zero.

On a branch, there are:

• a conductance, \(G_j\), which needs to be strictly positive;

• a temperature source, \(T_j\), which can be positive, negative or zero.

The incidence matrix \(A\) shows the relation between oriented branches (i.e., flow rates) and nodes (i.e., temperatures). The rows in the incidence matrix \(A\) correspond to the branches containing the heat flow rates \(q_j\) across the conductances \(G_j\) and the columns correspond to the temperature nodes \(\theta_i\). In the row corresponding to the branch \(q_j\), in the position of the node \(\theta_i\) there is -1 if the flow \(q_j\) leaves the node, +1 if the flow \(q_j\) enters into the node \(\theta_i\) and 0 if the flow \(q_j\) is not connected to the temperature node \(\theta_i\) (Ghiaus, 2013):

\[\begin{split}A_{j,i} = \begin{cases}\phantom{-} 0 & \text{if branch } q_j \text{ is not connected to node } \theta_i \\ +1 & \text{if branch } q_j \text{ enters into node } \theta_i\\ -1 & \text{if branch } q_j \text{ gets out of node } \theta_i \end{cases}\end{split}\]

Every thermal network needs to have a reference temperature from which the temperatures are measured.

The problem of analysis of thermal circuits (or the simulation problem, or the direct problem) is:

given:

• incidence matrix \(A\) which indicates how the nodes are connected by oriented branches;

• conductance diagonal matrix \(G;\)

• capacity diagonal matrix \(C;\)

• temperature source vector \(b;\)

• heat flow source vector \(f;\)

find the temperature vector \(\theta\) and the flow rate vector \(q.\)

Thermal networks for heat equation

For heat conduction, the basic thermal network (Figure 1) can be deduced from the discretization of heat equation for a plan wall, from the integral form for a stram tube or from the application of finite element method (Naveros, Ghiaus, Ruiz, 2017).

Plan wall

Heat conduction in a plane wall without internal sources is modelled by 1D heat equation:

\[\rho c \frac{\partial \theta}{\partial t} = - \lambda \frac{\partial ^2 \theta}{\partial x^2}\]

By discretizing the second derivative in function of \(x\):

\[\frac{\partial ^2 \theta}{\partial x^2} = \frac{\partial \frac{\partial \theta}{\partial x}}{\partial x}\approx \frac{\frac{\theta_{i+1} - \theta_i}{\Delta x} - \frac{\theta_i - \theta_{i-1}}{\Delta x}}{\Delta x}\]

and expressing the heat equation for a volume \(V = \Delta x S\), we obtain:

\[\rho c \Delta x S = \frac{\lambda S}{\Delta x} (\theta_{i-1} - \theta_i) - \frac{\lambda S}{\Delta x} (\theta_i - \theta_{i+1})\]

where:

• \(C_i = \rho c \Delta x S\) is the heat capacity of mesh \(i\);

• \(G_j = G_{j+1} = \frac{\lambda S}{\Delta x}\) are the thermal conductances;

The same equation is obtained by energy balance on mesh \(i\), where:

• \(q_j = \frac{\lambda S}{\Delta x} (\theta_{i-1} - \theta_i)\)

• \(q_{j+1} = \frac{\lambda S}{\Delta x} (\theta_i - \theta_{i+1}).\)

The discrete form of the heat equation can be modeled by a thermal circuit (Figure 3).

Figure 2. Discretization of heat equation: a) Space discretization; b) Thermal circuit.

Steady state conduction in a stream tube

Thermal conduction is the heat diffusion in solids in the direction of the temperature gradient (Figure 3a). Fourier law, the equation relating the thermal heat flow rate in a direction \(x\), \(q_x\), to the temperature gradient, \(\frac{d \theta}{d x}\), in the direction \(x\) is:

\[q_x = -\lambda S \frac{d \theta}{d x}\]

where:

• \(\lambda\) is the material conductivity, W/(m·K),

• \(S\) is the area of the surface perpendicular to the heat flow rate \(q_x\), m².

The minus sign shows that heat transfer is from high to low temperature.

Figure 3. Steady state thermal conduction: a) stream tube; b) thermal network model.

Conduction without internal sources

Let us consider stationary conduction in a stream tube in a homogenous and isotropic material without internal heat sources (Figure 3a). Since the heat flow rate \(q\) is conserved, Fourier law in section \(s\) of the streamline is

\[q = -\lambda S \frac{d \theta}{d x}\]

where the conductivity \(\lambda = \lambda(s)\) and the area surface \(S = S(s)\) depend on the curvilinear coordinate \(s\). By separating the variables, the above equation becomes:

\[q \frac{ds}{\lambda S} = -d \theta\]

By integrating from \(s_0\) to \(s_1\) and from \(\theta_0\) to \(\theta_1\),

\[q \int_{s_0}^{s_1} \frac{ds}{\lambda S} = -\int_{\theta_0}^{\theta_1} d \theta\]

the temperature variation with the distance can be written as

\[q R = \theta_0 - \theta\]

where

\[R \equiv \int_{s_0}^{s} \frac{1}{\lambda S} ds\]

is the thermal resistance of the stream tube between \(s_0\) and \(s.\)

Conduction with internal sources

If there are internal sources, the variation of the heat flow rate along the curvilinear coordinate \(ds\) is

\[dq = p \, dV\]

where \(p\) is the rate of heat generation per unit volume and \(dV\) is the infinitesimal volume. If \(ds \rightarrow 0\), then \(dV = S \, ds\). By integrating between \(s_0 \) and \(s\),

\[\int_{q_0}^{q} dq = \int_{s_0}^{s} p \, S \, ds\]

it becomes

\[q(s) = \int_{s_0}^{s} p \, S \, ds + q_0\]

where \(q_0\) is the flow rate entering through the surface \(S_0\).

The flow rate getting out through the surface \(S_1\) is

\[q_1 = \int_{s_0}^{s_1} p \, S \, ds + q_0\]

where \(q_0\) is the heat flow rate entering through the surface \(S_0\).

Substituting the above equation in the equation \(q = -\lambda S \frac{d \theta}{d x}\), we obtain, after integration,

\[\theta_1 = \int_{s_0}^{s_1} \frac{1}{\lambda S} \left ( \int_{s_0}^{s} -p \, S \, ds' \right ) ds - q_0\int_{s_0}^{s_1}\frac{1}{\lambda S} ds + \theta_0\]

By substituting the expression of thermal resistances given by \(R \equiv \int_{s_0}^{s} \frac{1}{\lambda S} ds\), we obtain

\[\theta_1 = -\int_{s_0}^{s_1} \frac{1}{\lambda S} \left ( \int_{s_0}^{s} p \, S \, ds' \right ) ds - R_0 \, q_0 + \theta_0\]

Equations

• \(q(s) = \int_{s_0}^{s} p \, S \, ds + q_0\) and

• \(\theta_1 = -\int_{s_0}^{s_1} \frac{1}{\lambda S} \left ( \int_{s_0}^{s} p \, S \, ds' \right ) ds - R_0 \, q_0 + \theta_0\)

can be represented by the thermal circuit presented in Figure 3b where the heat rate source is

\[f_1 = \int_{s_0}^{s} p \, S ds\]

and the temperature source is

\[b_0 = \int_{s_0}^{s_1} \frac{1}{\lambda S} \left ( \int_{s_0}^{s} p \, S \, ds' \right ) ds\]

The equation for \(b_0\) can be integrated by parts. By noting \(u \equiv \int_{s_0}^{s_1}p \, S \, ds\) and \(v' \equiv \frac{1}{\lambda S}\), integrating by parts \((\int uv'ds = uv - \int u'vds)\), we obtain:

\[b_0 = -\left ( \int_{s_0}^{s_1} p \, S \, ds \right )\left ( \int_{s_0}^{s_1} \frac{1}{\lambda S} ds \right ) + \int_{s_0}^{s_1} p \, S \left ( \int_{s_0}^{s_1} \frac{1}{\lambda S} ds' \right )ds\]

By substituting in the above equation \(R \equiv \int_{s_0}^{s} \frac{1}{\lambda S} ds\) and \(R_0 \equiv \int_{s_0}^{s_1} \frac{1}{\lambda S} ds\), we obtain

\[b_0 = -R_0 \int_{s_0}^{s_1} p \, S \, ds + \int_{s_0}^{s_1} R \, p \, S ds\]

Equation \(q_1 = \int_{s_0}^{s_1} p \, S \, ds + q_0\) becomes

\[q_1 = q_0 + f_1\]

and equation \(\theta_1 = -\int_{s_0}^{s_1} \frac{1}{\lambda S} \left ( \int_{s_0}^{s} p \, S \, ds' \right ) ds - R_0 \, q_0 + \theta_0\) becomes

\[\theta_0 - \theta_1 + b_0 = R_0 q_0\]

where

\[e_0 = \theta_0 - \theta_1 + b_0\]

is the temperature difference across the thermal resistance \(R_0.\)

Refering to the model from Figure 3b:

• \(b_0 = -R_0 \int_{s_0}^{s_1} p \, S \, ds + \int_{s_0}^{s_1} R \, p \, S ds\), where \(R = \int_{s_0}^{s} \frac{1}{\lambda S} ds\) is temperature source;

• \(f_1 = \int_{s_0}^{s} p \, S ds\) is flow rate source;

• \(R_0 = \int_{s_0}^{s_1} \frac{1}{\lambda S} ds\) is thermal resistance.

Example: Cylindrical wall without internal sources

Let’s consider cylindrical insulation of length \(L\) with internal radius \(r_0\) and external radius \(r_1\), without internal sources, \(p = 0\).

For a cylinder of width \(w = ds\), the thermal resistance is \(dR = \frac{1}{\lambda S}ds.\) The thermal resistance of the cylinder is

\[R_0 = \int_{s_0}^{s_1}\frac{1}{\lambda S}ds\]

where:

• \(s = r\), \(s_0 = r_0\), \(s_1 = r_1\), the radius;

• \(\lambda\) - conductivity is constant;

• \(S = 2 \pi r L\) - surface area of the cylinder of radius \(r\) and length \(L\).

It results

\[R_0 = \frac{1}{2 \pi L \lambda} \int_{r_0}^{r_1} \frac{1}{r} dr =\frac{1}{2 \pi L \lambda} (\ln r_1 - \ln r_0) = \frac{\ln(r_1 /r_0)}{2 \pi L \lambda}\]

The temperature source is

\[b_0 = -R_0 \int_{r_0}^{r_1} p \, S \, ds + \int_{r_0}^{r_1} R \, p \, S ds = 0\]

where \(R = \int_{r_0}^{r} \frac{1}{\lambda S} ds\)

The flow rate source is

\[f_1 = \int_{r_0}^{r_1} p \, S dr = 0\]

The heat flow rate is

\[q_0 = \frac{e_0}{R_0} = \frac{\theta_0 - \theta_1 + b_0}{R_0} = \frac{\theta_0 - \theta_1}{\frac{\ln (r_2 / r_1)}{2 \pi L \lambda}}\]

Thermal conductances

The three modes of heat transfer (conduction, convection and radiation) and the heat advection can be modelled by thermal resistances or conductances (Figure 4).

Figure 4. Conductances in heat transfer and energy advection: a) conduction; b) convection; c) radiation; d) advection.

Plan wall conduction

For a plan wall, conduction conductances, in W/K, are of the form (Figure 4a):

\[G_{cd} = \frac{\lambda}{w}S\]

where:

• \(\lambda\) - thermal conductvity, W/(m⋅K);

• \(w\) - width of the wall, m;

• \(S\) - surface area of the wall, m².

Convection

Convection conductances, in W/K, are of the form (Figure 4b):

\[G_{cv} = {h S}\]

where:

• \(h\) is the convection coefficient, W/(m²⋅K);

• \(S\) - surface area of the wall, m².

Table 1. Surface thermal conductances (adapted after Dal Zotto et al. 2014, p. 261)

Type of wall	Indoor surface	Outdoor surface
	\(h_i\)/(W·m⁻²·K⁻¹)	\(h_o\)/(W·m⁻²·K⁻¹)
Vertical (tilt > 60°): horizontal flow rate	7.7	25
Horizontal (tilt < 60°): vertical flow rate
- upward heat flow rate	10	25
- downward heat flow rate	5.9	25

Long wave radiation

View factors inside the building

The majority of methods used for modelling the radiative heat exchange use the view factors between surfaces. The view factor \(F_{i,j}\) is defined as the proportion of radiation leaving surface \(i\) that is intercepted by surface \(j\). The view factors can be estimated by differential areas or for different configurations of surfaces (Howell, 2010: C-2a, C-3, C-5, C-5a, C9, C-9b, C-10, C-11, C-13, etc.).

The view factors need to satisfy the summation rule

\[\sum_{j=0}^{n-1} F_{i,j} = 1\]

and the reciprocity theorem:

\[F_{i,j} S_i = F_{j,i} S_j\]

where \(S_{i}\) and \(S_{j}\) are the surface areas.

For a convex surface \(i\), the self-viewing factor is zero,

\[F_{i,i} = 0\]

Two simplified relations are used to calculate the view factors for buildings.

In the first one, the view factors are defined by:

\[\begin{split}\begin{cases} F_{i,j} = \frac{S_i}{S_T}\\ F_{i,i} = 0 \end{cases}\end{split}\]

where \(S_{T} = \sum_{j=0}^{n-1} S_j\), i.e. the surface \(S_j\) is included in the total surface \(S_T\). In this method, the reciprocity theorem is satisfied,

\[F_{i,j} S_i = F_{j,i} S_j = \frac{S_i S_j}{S_T}\]

but summation rule isn’t,

\[\sum_{j=0}^{n-1} F_{i,j} = \sum_{j=0, j \neq i}^{n-1} \frac{S_j}{S_T} = \frac {S_T - S_i}{S_T} \neq 1\]

In this case, the heat balance for each surface would be wrong.

In the second one, the view factors are defined by:

\[\begin{split}\begin{cases} F_{i,j} = \frac{S_j}{S_{T,i}} = \frac{S_j}{S_{T} -S_i}\\ F_{i,i} = 0 \end{cases}\end{split}\]

where \(S_{T,i} = \sum_{j=0, j \neq i}^{n-1} S_j\), i.e. the surface \(S_i\) is not included in the total surface \(S_{T,i} = S_T - S_i\).

In this case, the reciprocity theorem is generally not respected:

\[F_{i, j} S_i = \frac{S_j}{S_{T} - S_i} S_i \neq F_{j, i} S_j = \frac{S_i}{S_{T} - S_j} S_j\]

but the summation rule is respected:

\[\begin{split} \sum_{i=0}^{n-1} F_{i, j} = \frac{1}{S_{T} - S_i} \sum_{\substack{j=0\\i\neq j}}^{n-1} S_j = 1 \end{split}\]

Note: The view factor between two surfaces, \(j, k\), that are in the same plane (e.g. a window and a wall) is zero,

\[F_{j,k} = F_{k,j}=0\]

Therefore the total surface \(S_{T,i}\) should be:

\[S_{T,i} = \sum_{j=0}^{n-1} S_j - \sum_k S_k\]

i.e., the surfaces \(S_k\) in the same plane with the surface \(S_i\) are not included in \(S_{T,i}\).

View factor between tilted outdoor walls and sky

The view factor between the top surface of finite wall \(w\) tilted relative to an infinite plane of the ground \(g\) is (Widén & Munkhammar 2019, eq. 4.18):

\[ F_{w,g} = \frac {1 - \cos \beta}{2}\]

Therefore, the view factor between the tilted wall \(w\) and the sky dome \(s\) is (Widén & Munkhammar 2019, eq. 4.17):

\[ F_{w,s} = 1 - F_{w,g} = \frac {1 + \cos \beta}{2}\]

Thermal network for long wave radiation

The long-wave heat exchange between surfaces may be modelled by using the concept of radiosity and then linearizing the radiative heat exchange.

Figure 5. Radiative long-wave heat exchange between two surfaces: a) modeled by emmitance (sources) and radiosity (nodes); b) modeled by linearization of emmitance (temperature sources) and radiosity (temperature nodes).

For two surfaces, shown by temperature nodes 1 and 2 in Figure 5, the conductances, in m², for radiative heat exchange expressed by using the emmitance (or the radiant excitance) of the black body, the radiosity, and the reciprocity of view factors are:

\[G_{1}^{r} = \frac{\varepsilon_1}{1 - \varepsilon_1} S_1\]

\[G_{1,2}^{r} = F_{1,2} S_1 = F_{2,1} S_2\]

\[G_{2}^{r} = \frac{\varepsilon_2}{1 - \varepsilon_2} S_2\]

where:

• \(\varepsilon_1\) and \(\varepsilon_2\) are the emmisivities of the surfaces 1 and 2;

• \(S_1\) and \(S_2\) - areas of the surfaces 1 and 2, m²;

• \(F_{1,2}\) - view factor between surfaces 1 and 2.

The net flows leaving the surfaces 1 and 2 are:

\[q_{net,1} = \frac{\varepsilon_1}{1 - \varepsilon_1} S_1 (M^o_1 - J_1)= G^r_1 (M_1^o - J_1)\]

\[q_{net,2} = \frac{\varepsilon_2}{1 - \varepsilon_2} S_2 (M^o_2 - J_2)= G^r_2 (M_2^o - J_2)\]

respectively, where:

• \(M^o_1\) and \(M^o_2\) are the emmitances of the surfaces 1 and 2 when emmiting as black bodies, \(M^o = \sigma T^4\), W/m²;

• \(J_1\) and \(J_2\) - radiosities of surfaces 1 and 2, W/m²;

• \(G^r_1\) and \(G^r_2\) - conductances for long wave radiative heat exchange, m².

The net flow between surfaces 1 and 2 is:

\[q_{1,2} = F_{1,2} S_1 (J_1 - J_2) = F_{2,1} S_2 (J_1 - J_2)= G_{1,2}^r (J_1 - J_2)\]

In order to express the long-wave radiative exchange as a function of temperature differences, a linearization of the difference of temperatures \(T_1^4 - T_2^4\) may be used:

\[T_1^4 - T_2^4 = (T_1^2 + T_2^2)(T_1^2 - T_2^2) = (T_1^2 + T_2^2)(T_1 + T_2)(T_1 - T_2) = 4 \bar{T}^3 (T_1 - T_2)\]

where the mean temperature \(\bar{T}\), measured in kelvin, is:

\[\bar{T} =\sqrt[3]{ \frac{(T_1^2 + T_2^2)(T_1 + T_2)}{4}}\]

The evaluation of mean temperaure, \(\bar{T}\), requires the values of the surface tempetratures, \(T_1\) and \(T_2\). An initial guess can be used (and then an iterative process, for a more precise evaluation). For temperatures that are usual in buildings, i.e., \(10 \, \mathrm{ °C} \leq \bar T - 273.15 \leq 30 \,\mathrm{ °C}\), the values of \(4 \sigma \bar T^3\) vary between 5 and 6 W/(m²·K). Practically, \(\bar T - 273.15 = 20 \,\mathrm{ °C}\) may be assumed, for which \(4 \sigma \bar T^3 = 5.7 \,\mathrm{W/(m^2K)}\).

After linearization, the conductances, in W/K, for radiative heat exchange are (Figure 4c):

\[G_{1} = 4 \sigma \bar{T}^3 \frac{\varepsilon_1}{1 - \varepsilon_1} S_1\]

\[G_{1,2} = 4 \sigma \bar{T}^3 F_{1,2} S_1 = 4 \sigma \bar{T}^3 F_{2,1} S_2\]

\[G_{2} = 4 \sigma \bar{T}^3 \frac{\varepsilon_2}{1 - \varepsilon_2} S_2\]

Convection and radiation

Since the calculation of long-wave radiative exchange is very laborious, a simplfied approach is used to estimate the surfce heat coefficient for convection and long-wave radiation.

The net radiatiative flow rate between a surface and its closed surrundings is (Figure 6):

\[q_{rad} = (\varepsilon M_s^o - \alpha E_s) S_s = \varepsilon \sigma (T_s^4 - T_{sur}^4) S_s\]

where:

• \(\varepsilon M_s^o\) is the heat flux emitted by the surface, where \(M_s^o = \sigma T_s^4\) is the radiant emmitance.

• \(\alpha E_s\) is the heat flux abosrbed by the surface, where \(E_s = \sigma T_{sur}^4\) is the radiant emmitance of the surroundings.

Figure 6. Radiation and convection between a surface and its surroundings.

It is convenient to express the radiation heat flow rate as a function of temperature differences (Bergman, T. L. et al. 2011):

\[q_{rad} = h_r S_s (\theta_s - \theta_{sur})\]

where the radiative heat transfer coefficient is

\[h_r = \varepsilon \sigma (T_s + T_{sur})(T_s^2 + T_{sur}^2) = 4 \varepsilon \sigma \bar{T}^3\]

with:

\[\bar{T} =\sqrt[3]{ \frac{(T_s + T_{sur})(T_s^2 + T_{sur}^2)}{4}}\]

The total heat rate flow by convection and radiation is

\[q = q_{cv} + q_{rad} = (h_{cv} + h_r) S_s (\theta_s - \theta_{sur})\]

where

\[q_{cv} = h_{cv}S_s (\theta_s - \theta_{sur})\]

is the heat flow rate by convection (Newton’s law).

The surface conductance for convetion and radiation is (CSTB 2017 § 3.1.1.3)

\[G_s = (h_{cv} + h_r) S_s\]

where:

• \(h_r = 4 \varepsilon \sigma \bar{T}^3\) is the radiative heat transfer coefficent where \(\bar{T}\) is the mean temperature of the surfaces and the air;

• \(h_{cv}\) is the convection heat transfer coefficient.

Advection

The volumetric flow rate of the air, in m³/s, is:

\[\dot{V}_a = \frac{\mathrm{ACH}}{3600} V_a\]

where:

• \(\mathrm{ACH}\) (air changes per hour) is the air infiltration rate, 1/h;

• \(3600\) - number of seconds in one hour, s/h;

• \(V_a\) - volume of the air in the thermal zone, m³.

The net flow rate that the building receives by advection, i.e., introducing outdoor air at temperature \(T_o\) and extracting indoor air at temperature \(\theta_i\) by ventilation and/or air infiltration, is:

\[q_v = \dot{m}_a c_a (T_o - \theta_i) = \rho_a c_a \dot{V}_a (T_o - \theta_i)\]

where:

• \(\dot{m}_a\) is the mass flow rate of air, kg/s;

• \(\dot{V}_a\) - volumetric flow rate, m³/s;

• \(c_a\) - specific heat capacity of the air, J/kg·K;

• \(\rho_a\) - density of air, kg/m³;

• \(T_o\) - outdoor air temperature, °C (noted in majuscule because it is a temperature source or input variable);

• \(\theta_i\) - indoor air temperature, °C (noted in minuscule because it is a dependent temperature or output variable).

Therefore, the conductance of advection by ventilation and/or infiltration, in W/K, is (Figure 4d):

\[G_v = \rho_a c_a \dot{V}_a\]

Table 2. Typical values for the ventilation rates (in air changes per hour, ACH) as a function of the position of windows (Recknagel et al. 2013, Table 1.12.1-4)

Position of windows	Ventilation rate, ACH / (h⁻¹)
Windows closed, doors closed	0 to 0.5
Tilted window, venetian blind closed	0.3 to 1.5
Tilted window, whitout venetian blind	0.8 to 4.0
Window half open	5 to 10
Window fully open	9 to 15
Cross ventilation, window and French window fully open	about 40

Proportional controller

In the simplest representation, the HVAC system can be considered as a proportional controller that adjusts the heat flow rate \(q_{HVAC}\) in order to control the indoor temperature \(\theta_i\) at its setpoint value \(T_{i,sp}\) (Figure 7a).

Figure 7. HVAC systems as proportional controller of indoor air. a) Block diagramme. b) Thermal circuit representation.

The heat flow-rate, in W, injected by the HVAC system into the controlled space is (Figure 7):

\[ q_{HVAC} = K_p (T_{i, sp} - \theta_i)\]

where:

• \(K_p\) is the proportional gain, W/K;

• \(T_{i, sp}\) - indoor temperature setpoint, °C (noted in majuscule because it is an input, i.e. independent variable);

• \(\theta_i\) - indoor temperature, °C (noted in minuscule because it is a output, i.e. dependent variable).

This equation shows that the proportional controller can be modelled by a source of temperature, \(T_{i, sp}\), and a conductance, \(K_p\). If the controller gain tends towards:

• infinity, \(K_p \rightarrow \infty\), then the controller is perfect, \(\theta_i \rightarrow T_{i, sp}\).

• zero, \(K_p \rightarrow 0\), then the controller is not acting and the building is in free-running, i.e. \(q_{HVAC} = 0\) (Ghiaus 2003).

Note: Respecting the sign convention, the flow rate \(q_{HVAC}\) is oriented from the lower to the higher potential of the temperature source \(T_{i,sp}\).

Thermal capacities

Walls

The thermal capacity of a wall, in J/kg, is:

\[C_w= m_w c_w= \rho_w c_w w_w S_w\]

where:

• \(m_w = \rho_w w_w S_w\) is the mass of the wall, kg;

• \(c_w\) - specific heat capacity, J/(kg⋅K);

• \(\rho_w\) - density, kg/m³;

• \(w_w\) - width of the wall, m;

• \(S_w\) - surface area of the wall, m².

Air

Similarly, the thermal capacity of the air, in J/kg, is:

\[C_a = m_a c_a = \rho_a c_a V_a\]

where:

• \(m_a = \rho_a V_a\) is the mass of the air, kg;

• \(\rho_a\) - density of the air, kg/m³;

• \(c_a\) - specific heat capacity of the air, J/(kg⋅K);

• \(V_a\) - volume of the air in the thermal zone, m³.

Temperature sources

The temperature sources model temperatures which vary independently of what happens in the themal circuit; they are inputs of the physical model. Generally, the temperature sources are:

• outdoor air and soil (ground) temperature;

• temperature of adjacent spaces which have controlled temperature;

• setpoint temperature.

Adjacent spaces with controlled temperature

If the adjacent spaces are controlled by a HVAC system, it means that their temperature can be considered independent of the studied thermal zone(s); therefore, they can be modelled by a temperature source.

Setpoint temperature

Setpoint temperature does not depend on the heat transfer processes of the analyzed thermal zone. If the HVAC system can deliver the heat flow rate:

\[ q_{HVAC} = K_p (T_{i, sp} - \theta_i)\]

where:

• \(K_p\) is the proportional gain, W/K;

• \(T_{i, sp}\) - indoor temperature setpoint, °C;

• \(\theta_i\) - indoor temperature, °C,

then the setpoint for indoor temperature, \(T_{i, sp}\), may be modelled by a source of temperature.

Heat flow rate sources

The heat flow rate sources model flow rates which vary idependently of what happens in the themal circuit. They are inputs of the physical model. Generally, the heat flow rate sources are:

• solar radiation absorbed by the walls;

• internal auxiliary sources.

Solar radiation absorbed by the walls

The direct, diffuse and reflected components of the solar radiation on a tilted surface can be estimated from weather data by using the function sol_rad_tilt_surf from the module dm4bem (see the tutorial on weather data and solar radiation).

External wall

The radiation absorbed by the outdoor surface of the wall,, in W, is:

\[\Phi_o = \alpha_{w,SW} S_w E_{tot}\]

where:

• \(\alpha_{w,SW}\) is the absorptance of the outdoor surface of the wall in short wave, \(0 \leqslant \alpha_{w,SW} \leqslant 1\);

• \(S_w\) - surface area of the wall, m²;

• \(E_{tot}\) - total solar irradiance on the wall, W/m².

Internal walls

The total shortwave incident irradiance on the wall \(i\), \(E_i\), may be estimated as a function of the direct solar irradiance incident on the surface of the walls, \(E_{i}^{o}\):

\[S_i E_i = S_i E_{i}^{o} + \sum_{j=1}^{n} F_{j,i} S_j \rho_j E_j\]

where:

• \(S_i\) is the area of the surface of the wall \(i\), m²;

• \(E_i\) - total irradiance received directly and by multiple reflections on surface \(i\), W/m²;

• \(E_{i}^{o}\) - irradiance received directly from the sun on surface \(i\), W/m²;

• \(F_{j, i}\) - view factor between surface \(j\) and surface \(i\), \(0 ⩽ F_{j,i} ⩽ 1\);

• \(\rho_j\) - reflectance of surface \(j\), \(0 ⩽ \rho_j ⩽ 1\).

By taking into account the reciprocity of the view factors: \(S_i F_{i,j} = S_j F_{j,i}\), the set of previous equation becomes:

\[\begin{split} \begin{bmatrix} 1 - \rho_1 F_{1,1} & - \rho_2 F_{1,2} & ... & - \rho_n F_{1,n}\\ - \rho_1 F_{2,1} & 1 - \rho_2 F_{2,2} & ... & - \rho_n F_{2,n} \\ ... & ... & ... & ... \\ - \rho_1 F_{n,1} & - \rho_2 F_{n,1} & ... & 1 - \rho_n F_{n,n} \end{bmatrix} \begin{bmatrix} E_1\\ E_2\\ ...\\ E_n \end{bmatrix} = \begin{bmatrix} E_{1}^{o}\\ E_{2}^{o}\\ ...\\ E_{n}^{o} \end{bmatrix} \end{split}\]

or

\[(I - \rho \circ F) E = E^o\]

The unknown total irradiances on walls, in W/m², are then

\[ E = (I - \rho \circ F)^{-1} E^o\]

where:

the symbol \(\circ\) represents the Hadamard (or element-wise) product;

\(I =\begin{bmatrix} 1 & 0 & ... & 0 \\ 0 & 1 & ... & 0 \\ ... & ... & ... & ...\\ 0 & 0 & ... & 1 \end{bmatrix}, \) is the identity matrix;

\(\rho = \begin{bmatrix} \rho_1\\ \rho_2\\ ...\\ \rho_n \end{bmatrix}\) - vector of reflectances, \(0 \le \rho_{i,j} \le 1\);

\(F = \begin{bmatrix} F_{1,1} & F_{1,2} & ... & F_{1,n}\\ F_{2,1} & F_{2,2} & ... & F_{2,n} \\ ... & ... & ... & ...\\ F_{n,1} & F_{n,2} & ... & F_{n,n} \end{bmatrix}\) - matrix of view factors, \(0 \le F_{i,j} \le 1\);

\(E^o = \begin{bmatrix} E_{1}^{o}\\ E_{2}^{o}\\ ...\\ E_{n}^{o} \end{bmatrix}\) - vector of direct solar irradiances, W/m²;

\(E = \begin{bmatrix} E_1\\ E_2\\ ...\\ E_n \end{bmatrix}\) - vector of unknown total irradiances, W/m².

The radiative short wave (i.e. solar) heat flow rate on each surface is:

\[ \Phi = S E \]

where:

\(\Phi = \begin{bmatrix} \Phi_1\\ \Phi_2\\ ...\\ \Phi_n \end{bmatrix}\) - vector of total heat flow rates due to solar radiation, W;

\(S =\begin{bmatrix} S_1 & 0 & ... & 0 \\ 0 & S_2 & ... & 0 \\ ... & ... & ... & ...\\ 0 & 0 & ... & S_n \end{bmatrix}\) - matrix of surface areas of walls, m².

Internal sources

Internal flow rates are generated by occupants and by the electrical equipment (with values given for offices, commercial spaces, etc.).

System of algebraic-differential equations (DAE)

The analysis of a thermal circuit, or the direct problem (Ghiaus 2022), means to find the temperatures in the nodes, \(\theta\), and the heat flows on the branches, \(q\), i.e. to solve for \(\theta\) and \(q\) the system of Differential-Algebraic Equations (DAE) (see Jupyter Notebook From thermal networks to differential-algebraic equations):

\[\begin{split}\left\{\begin{array}{ll} C \dot{\theta} = -(A^T G A) \theta + A^T G b + f\\ q = G (-A \theta + b) \end{array}\right.\end{split}\]

where:

• \(\theta\) is the temperature vector of size \(n_\theta\) equal to the number of nodes;

• \(q\) - heat flow vector of size \(n_q\) equal to the number of branches;

• \(A\) - incidence matrix of size \(n_q\) rows and \(n_{\theta}\) columns, where \(n_q\) is the number of flow branches and \(n_{\theta}\) is the number of temperature nodes. It shows how the temperature nodes are connected by oriented branches of heat flows:

  • if flow m enters into node n, then the element (m, n) of the matrix \(A\) is 1, i.e., \(A_{m,n} = 1\);

  • if flow m exits from node n, then the element (m, n) of the matrix \(A\) is -1, i.e., \(A_{m,n} = -1\), ;

  • if flow m is not connected to node n, then the element (m, n) of the matrix \(A\) is 0, i.e., \(A_{m,n} = 0\).

• \(G\) - conductance matrix of size \(n_q \times n_q\), where \(n_q\) is the number of flow branches: diagonal matrix containing the conductances. Each branch \(k\) needs to contain a conductance \(0 < G_{k,k} < \infty \).

• \(C\) - capacity matrix of size \(n_θ \times n_θ\), where \(n_θ\) is the number of temperature nodes: diagonal matrix containing the capacities. If there is no capacity in the node n, then \(C_{n, n} = 0\).

• \(b\) - temperature source vector of size \(n_q\): if there is no temperature source on branch m, then \(b_m = 0\).

• \(f\) - heat flow source vector of size \(n_θ\): if there is no heat flow source in node n, then \(f_n = 0\).

Note: The incidence matrix \(A\) is related to difference operator (applied to temperatures) \(\Delta = -A\) and to summation operator (applied to heat flow rates) \(\Sigma = A^T\). Note that \(\Sigma = -\Delta^T\).

The resolution is first done for temperatures, \(\theta\), by solving the equation

\[C \dot{\theta} = -(A^T G A) \theta + A^T G b + f\]

which, generally, is a system of differential-algebraic equations (DAE). Then, the heat flow rates are found from the equation

\[q = G (-A \theta + b)\]

The vector of outputs is \(y\), of size \(n_{\theta}\), the number of nodes. The non-zero values of \(y\) indicate the teperature nodes which are the outputs of the model.

State-space representation

The differential-algebraic system of equations (DAE)

\[C \dot{\theta} = -(A^T G A) \theta + A^T G b + f\]

is transformed in state-space representation (Ghiaus 2013) (see Jupyter Notebook From differential-algebraic equations to state-space representation):

\[\begin{split}\left\{\begin{array}{rr} \dot{\theta}_s=A_s \theta_s + B_s u\\ y = C_s \theta_s + D_s u \end{array}\right.\end{split}\]

where:

• \(\theta_s\) is the vector of state variables which are the temperatures of nodes containing capacities; the elements are in the same order as in the vector of temperatures, \(\theta\); its dimension, \(\dim \theta_s\), is equal to the number of capacities from the thermal network.

• \(u = \begin{bmatrix} b_T \\ f_Q\end{bmatrix}\) - vector of inputs of dimension \(\dim u\) equal to the number of sources (of temperaure, \(b_T\), and heat flows, \(f_Q\)) of the thermal network, where:

  • \(b_T\): vector of nonzero elements of vector \(b\) of temperature sources;

  • \(f_Q\): vector of nonzero elements of vector \(f\) of flow sources.

• \(y\) - vector of outputs, a subset of vector \(\theta\) representing temperature nodes which are of interest.

• \(A_s\) - state matrix, of dimension \(\dim A_s = \dim {\theta_s} \times \dim {\theta_s}\).

• \(B_s\) - input matrix, of dimension \(\dim B_s = \dim {\theta_s} \times \dim u\).

• \(C_s\) - output matrix, of dimension \(\dim C_s = \dim y \times \dim {\theta_s}\).

• \(D_s\) - feedthrough (or feedforward) matrix, of dimension \(\dim D_s = \dim y \times \dim u\).

Note: The subscript \(s\) of the matrices \(A_s, B_s, C_s, D_s\) is used to differentiante the matrices \(A_s, C_s\) of the state-space represenation of the matrices \(A, C\) of the system of DAE.

References

Ghiaus, C. (2021). Dynamic Models for Energy Control of Smart Homes, in S. Ploix M. Amayri, N. Bouguila (eds.) Towards Energy Smart Homes, Online ISBN: 978-3-030-76477-7, Print ISBN: 978-3-030-76476-0, Springer, pp. 163-198, HAL 03578578

Dal Zotto et al.,(2014) Mémotech. Génie énergétique, 5 edition, Casteilla, ISBN-13: 978-2-206-10018-0

Howell, J. R. (2010). A catalog of radiation heat transfer configuration factors 3rd edition. University of Texas at Austin

Naveros, I., Ghiaus, C., & Ruíz, D. P. (2017). Frequency response limitation of heat flux meters. Building and Environment, 114, 233-245.

Widén, J., & Munkhammar, J. (2019). Solar radiation theory. Uppsala University.

RE2020 (2021). Annexe IV : Règles « Th-Bat 2020 » - données d’entrée au calcul de la performance énergétique

Bergman, T. L. et al. (2011). Fundamentals of heat and mass transfer. Chapter 1. John Wiley & Sons. ISBN 13 978-0470-50197-9

CSTB (2012). RT 2012 Th-U Parois opaques, § 3.1.1.3

Clarke, J (2001). Energy simulation in building design. 2nd ed. Butterworth-Heinemann, ISBN 0 7506 5082 6

Beausoleil-Morrison, I. (2021). Fundamentals of building performance simulation. Routledge, ISBN 978-0-367-51805-9