How to estimate dynamic effects in a more or less simplified way

## Video

Explanation of thermal inertia and the differences between internal and external insulation. PDF slides

## Formulas

Heat stored in a building component is defined by the difference between its temperature $T_i$ and a reference temperature $T_0$. In a wall of thickness $e_i$, mass density $\rho_i$ and heat capacity $c_{p,i}$, it can be estimated in the unit [J/m$^2$] as:

The total inertia of an $n$-layered wall is the sum of these accumulated energies over all layers: In a transient state, we can calculate the evolution of temperatures in time as a function of R and C values of a building component. These values depend on the discretization size and material properties. In the example shown here, the following equation is to be solved: $$C \dfrac{\partial T}{\partial t} = \frac{1}{R} (T_1-T) + \frac{1}{R} (T_2-T)$$

## Exercise

Consider a wall made of three layers:

Layer 1 (concrete)  Layer 2 (insulatin) Layer 3 (finishing)
$e_c=15$ cm $e_i=4$ cm $e_f=1,5$ cm
$\lambda_c=1,5$ W/m.K $\lambda_i=0,04$ W/m.K $\lambda_f=1,5$ W/m.K
$c_c=920$ J/kg.K $c_i=920$ J/kg.K $c_f=920$ J/kg.K
$\rho_c=2700$ kg/m$^3$ $\rho_i=75$ kg/m$^3$ $\rho_f=2700$ kg/m$^3$

The outdoor temperature is $T_e=2°C$ and the indoor temperature is $T_i=20°C$. The outdoor heat transfer coefficient is $h_e=16,7$ W/m$^2$.K and the indoor one is $h_i=9,1$ W/m$^2$.K

Exercise 1: inertia

Suppose an internal insulation.

• Calculate the thermal resistance of each layer and the heat flux $\varphi$ through the wall.
• Calculate the temperature distribution across the wall and the average temperature of each material.
• Calculate the total stored heat and estimate the heat inertia.

Answer the same questions as above in the case of an external insulation

Exercise 2: transient simulations

Simulate the evolution of temperature inside the wall by following these steps:

• Discretise each material layer with a sufficient number of resistances and capacities (this is similar to a finite difference scheme)
• Write the equations for the temporal evolution of temperature on each point. You can select either an implicit or an explicit scheme for time discretization.
• Solve and plot the evolution of temperature at the interface between concrete and insulation, by supposing a uniform initial temperature $T=10°C$

The solution will be shown here soon.