Thermal inertia | EBE clim

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:

$Q_i = \rho_i . e_i . c_{p,i} (T_i-T_0)$

The total inertia of an $n$ -layered wall is the sum of these accumulated energies over all layers:

$Q = \sum_{i=1}^n Q_i = \sum_{i=1}^n \rho_i . e_i . c_{p,i} (T_i-T_0)$

Warning: Always check for units! It’s very easy to get them wrong but it’s also easy to make sure they’re right.

	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: $$ C \dfrac{\partial T}{\partial t} = \frac{1}{R} (T_1-T) + \frac{1}{R} (T_2-T) $$

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.