Request: Mines 2020 exam, Physics II, MP (part 3)

Heya.

This is the last part of the exam.

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III - Experimental determination of the thermal conductivity of copper

To experimentally determine the conductivity of copper, it is useful to know its thermal massic capacity and its density ρ.

Question 15:

Suggest an experiment to determine the density ρ of copper, then another to determine its thermal massic capacity c.

To get the conductivity of copper, we use the "flash" method. In this emthod, we use a copper plate of thickness L = 3.12mm on the (Ox) axis and of dimensions far greater than L on the (Oy) and (Oz) axis - so that the temperature in the plate would only depend of x and t.

The plate is located between the abscissa x=0 and x=L and we will neglect lateral losses through convection or radiation. By linearity of the equation (question 16), we will suppose (without loss of generality) that the temperature (in °C) is null everywhere in the plate for t<0. At t=0, an infrared lamp, from the side x<0, sends a powerful luminous flash. It results, at t=0, in a temperature profile in the plate T(x, 0) which form will be mentioned later.

Question 16:

Establish the differential equation verified by T(x, t) in which we will display the thermal diffusivity D which we will express in function of parameters in this problem.

We are seeking solutions under the form: T(x, t) = f(x)g(t).

Question 17:

Determine two differential equations verified by f(x) and g(t). Infer the general form of T(x, t).

To model the effect of the flash lamp, we will use the following temperature profile:

Where Γ, δ, L are three constants. The evolution is fast enough to consider that the plate is isolated, in a first approximation, for t>0.

Question 18:

Justify that we are seeking solutions under the form:

Question 19:

Explicit the coefficients wn, then the coefficients kn and αn in function of n, L and D.

Question 20:

Establish the expression of the coefficients un and show that:

The thickness δ is supposed to be very small compared to L. An optical sensor can measure the temperature T(L, t) of the face behind the plate (at x = L) in function of the time t.

Question 21:

Infer from the expression from the previous question, that the approximated expression of T(L, t) is:

The following graph shows ζ(t) in function of α1t.

We will note t_1/2 the instant at which ζ(t_1/2) = 1/2.

Question 22:

Explicit a relationship between α1 and t_1/2.

The following figure shows the experimental graph T(L, t) obtained for the copper plate in question.

Question 23:

Estimate the valule of the thermal conductivity of copper.

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That's it folks! Thanks for reading! I'll be open for suggestions if anyone has questions or ideas about it.