# Thermal Conductivity

To make an hypothesis of how the thermal conductivity in our insulated box will evolve, we're basing in Fourier's Law and the heat convection-diffusion equation.

$\vec{q}=-\kappa \nabla T$ $$$$ $\frac{\partial Q}{\partial t}=-\kappa \oint \limits_S\nabla T \vec{dA}$

$_{General Fourier's law}$

Where for the first form, $\vec{q}$ is the local heat flux density, $\kappa$ is the material's conductivity, $\nabla T$ is the gradient of the temperature. And for the second form $\frac{\partial Q}{\partial t}$ is the amount of heat transferred per unit time, and $\vec{dA}$ is an oriented surface area element.

$\frac{\Delta Q}{\Delta t} = -\kappa A \frac{\Delta T}{\Delta x}$ $$$$ $_{Fourier's law applied to an homogeneous material of 1D geometry between two endpoints at constant temperature}$

Where $\Delta T$ is the difference of temperature between the edges, and $\Delta x$ is the distance between them.

$\frac{\partial T}{\partial t}=\nabla.(\frac{\kappa}{\rho c_{\rho}} \nabla T) - \nabla. (\frac{\partial T}{\partial x}) + R$

$$$$ $_{heat convection-difussion equation}$

Where $\frac{\partial T}{\partial t}$ is the temperature transfer per unit time, $\rho$ is the density, $c_{\rho}$ is specific heat capacity, $\frac{\partial T}{\partial x}$ is the speed of the temperature transfer in the outside direction, and $R$ is the source of heat. $\nabla$ stands for the gradient and $\nabla.$ the divergence operators.

# Experiment

Objective: We want to know the behavior of the rock wool and polyurethane foam used as insulation materials during a long period of time.

Materials: Sensirion SHT21 Cardboard box (20x20x14 cm) Rock wool (density 50kg/m3, Thermal conductivity (R) =0.039, specific heat 1500 J/Kg°C) Polyurethane Foam (density 150kg/m3, Thermal conductivity (R) =0.04, specific heat 1500 J/Kg°C)

Procedure: We cover the interior of the cardboard boxes with a 40mm homogeneous layer of the insulation material, the sensor inside will recovered the data (temperature, humidity, and time)

Results: