1.2. Heat Transfer in Composite Walls

The image below shows an example of a composite wall used for heat transfer simulation:

Example of heat transfer simulation in a composite wall:

from HeatTransfer import CompositeWall

# Create a composite wall with external and internal convection coefficients
wall = CompositeWall.Object(he=23, hi=8, Ti=20, Te=-10, A=10)

# Add layers to the wall using material names
wall.add_layer(thickness=0.20, material='Parpaings creux')  # Hollow concrete blocks
wall.add_layer(thickness=0.05, material='Polystyrène')  # Polystyrene
wall.add_layer(thickness=0.02, material='Plâtre')  # Plaster

# Calculate heat transfer and temperatures at each layer interface
wall.calculate()
wall.df
print(f"df = {wall.df}")
print(f"R_total = {wall.R_total} m².°C/W")

Expected result:

Thickness (m)	Material	Conductivity (W/m.°C)	Resistance (m².°C/W)	Entry Temperature (°C)	Exit Temperature (°C)	Q (W)	A (m²)
NaN	Outdoor air	NaN	0.043478	-10.000000	-9.353644	148.661889	10
0.20	Hollow concrete blocks	1.40	0.142857	-9.353644	-7.229903	148.661889	10
0.05	Polystyrene	0.03	1.666667	-7.229903	17.547079	148.661889	10
0.02	Plaster	0.50	0.040000	17.547079	18.141726	148.661889	10
NaN	Indoor air	NaN	0.125000	18.141726	20.000000	148.661889	10

List of Available Materials

Anglais	Français	Conductivité thermique (W/m.°C)
Glass wool	Laine de verre	0.034
Expanded cork agglomerated with pitch	Liège expansé aggloméré au brai	0.048
Pure expanded cork	Liège expansé pur	0.043
Hollow concrete blocks	Parpaings creux	1.4
Hard limestone (marble)	Pierre calcaire dure (marbre)	2.9
Soft limestone	Pierre calcaire tendre	0.95
Granite	Pierre granit	3.5
Expanded polystyrene	Polystyrène expansé	0.047
Polystyrene	Polystyrène	0.03
Extruded polystyrene	Polystyrène extrudé	0.035
Polyurethane foam	Mousse de polyuréthane	0.03
Plaster	Plâtre	0.5
Glass	Verre	1.0
Air	Air	None

Explanation of the Equations Used

The composite wall heat transfer model uses the following equations to calculate total thermal resistance, heat flux, and temperatures at layer interfaces:

1. Convective thermal resistance: - External resistance:

   \[R_e = \frac{1}{h_e}\]

   • Internal resistance: .. math:

     R_i = \frac{1}{h_i}
     

2. Thermal resistance of layers: - For each layer, thermal resistance is calculated as follows:

   \[R_{\text{layer}} = \frac{\text{thickness}}{\text{conductivity}}\]

3. Total thermal resistance: - The total thermal resistance of the composite wall is the sum of convective resistances and layer resistances:

   \[R_{\text{total}} = R_e + R_i + \sum R_{\text{layers}}\]

4. Heat transfer coefficient: - The heat transfer coefficient is the inverse of total thermal resistance:

   \[U = \frac{1}{R_{\text{total}}}\]

5. Heat flux: - Heat flux through the composite wall is calculated using Fourier’s law:

   \[Q = U \cdot A \cdot (T_i - T_e)\]

   where ( A ) is the wall surface area, ( T_i ) is the indoor temperature, and ( T_e ) is the outdoor temperature.

6. Temperatures at layer interfaces: - The external wall temperature after convective resistance is calculated as follows:

   \[T_{\text{external wall}} = T_e + \frac{Q \cdot R_e}{A}\]

   • Les températures aux interfaces des couches sont ensuite calculées en utilisant le flux thermique et les résistances thermiques : .. math:

     T_{\text{interface}} = T_{\text{précédente}} + \frac{Q \cdot R_{\text{couche}}}{A}
     

Ces équations permettent de déterminer la distribution de température à travers le mur composite et le flux thermique total traversant le mur.