# Heat transfer and diffusion of particles

# Experimental measurement of a thermal conductivity

Take 15 minutes to prepare this exercise.

Then, if you lack ideas to begin, look at the given clue and start searching for the solution.

A detailed solution is then proposed to you.

If you have more questions, feel free to ask them on the forum.

We put the following montage into place in order to measure the thermal conductivity of copper, .

The left edge of the cylindrical copper bar, of , is placed in an oven at the temperature .

We note the density of the copper and its mass thermal capacity.

The other edge is in contact with a coil in which water circulates (with its mass thermal capacity) with a mass flow .

The heat exchanger so made allows water to be overheated from to .

The bar is laterally insulated, with insulating covers.

We measure the temperatures and of the bar, at two different points of it, with between them, and also the temperatures and of water just before and after the heat exchange zone.

The problem is one-dimensional, the material is homogeneous and any convection or radiation phenomenon on the bar will be neglected.

## Question

Why is the bar laterally insulated ? Give an example of insulating material that could be used for this experiment.

### Hint

### Solution

Insulating material : glass wool or polystyrene foam.

## Question

We wait for the stationary mode to be put into place. After applying the heat conduction equation, express the temperature along the bar using , the constants , , and .

### Hint

### Solution

The heat conduction equation is :

In stationary mode :

## Question

Determine the thermal power through the section in the bar. Does it depend on ?

### Hint

### Solution

The thermal power is a constante :

## Question

Determine the thermal power received by the water in the coil using and (you can first make a balance on the water mass circulating in the coil during )

### Hint

### Solution

## Question

Deduce the expression of the thermal conductivity of copper .

Experimentally, we obtain . Give its units.

### Hint

### Solution

The exchanger is ideal :

Thus :