Abstract
Usually, measurements are adversely affected by several influence factors, and adequate efforts have to be spent to control environmental conditions. We explain how it is possible to keep thermal fluctuations and thermal gradients around the X-ray interferometer less than 1 mK/h and 10 μK/cm, respectively. The Fourier equation of heat propagation is derived rigorously and an analytical solution of a simplified model in one spatial dimension is obtained by means of the method of the separation of variables. Some simulations are carried out to supply useful information about the dimensions of the real thermal shield.
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Notes
- 1.
In a storage system, passive components are those that do not require electrical power to operate.
- 2.
It is a passive system and is described in [BWZ83].
- 3.
- 4.
Also known as mixed conditions when a linear combination of T(r,t) and its directional derivative is present.
- 5.
More appropriate than amount of heat; In fact the internal energy is related to the position, while the heat is related to the flux and doesn’t include sources/well but only heat transfer relative to any kind of gradient that induces diffusion.
- 6.
For example, a laser source, a Peltier cell…
- 7.
We also are assuming that the medium is isotropic, that is, the thermal energy flow spreads out equally in all directions.
- 8.
The dimensions of k,ρ,c v are (\(\mathrm{W/(m K)} \)), (\(\mathrm{kg/m^{3}} \)) and (\(\mathrm{J/(kg K)} \)), respectively.
- 9.
Whereas in the case of ordinary differential equations the solution depends, in general, on n arbitrary constants, here the general solutions are found to involve arbitrary function of specific functions.
- 10.
A more realistic model should consider also the velocity of the fluid, the shape of the surfaces, …
- 11.
Among other methods, we have: integral transform, distribution method, substitution by series. The book by [CC95] illustrates a rather wide spectrum of methods. However it must be underlined that while for linear ODE there is a general method for determining the solution, for PDE the solution depends on the nature of the equation (first order, second order).
- 12.
If we differentiate Eq. (4.2.6) with respect to t, we obtain
$$ \frac{ d}{ dt} \biggl(\frac{ 1}{ T} \frac{ dT}{ dt}\biggr) =0 $$(4.2.7)which implies that \(\frac{ 1}{ T} \frac{ dT}{ dt}\) is equal to a numerical constant.
- 13.
For the time being, it is enough to say that this function is well behaved, for example, C 0 or piecewise continuous; at the end of the Sect. 4.2.1.2 we shall give a link to a notebook which considers a case when the BC and the IC are non consistent.
- 14.
The solution must decays exponentially as t increases.
- 15.
In [GG72] you can read that the committee consisting of Lagrange, Laplace, among others, concluded that …the manner in which the Author arrives at his equations…leaves something to be desired in the realm of both generality and even rigor.
- 16.
All the equations treated in the following with the method of separation of variable admit a real solution.
- 17.
It is possible, because we can apply the separation variables method seen in Sect. 4.2.1.2.
- 18.
We are facing a first order linear ODE, non-homogeneous with constant coefficients.
- 19.
The materials forming the walls must be sufficiently rigid leaving enough space to contain the necessary instruments; at the same time, they have to protect the interferometers from external fluctuations of the environment which can induce unacceptable gradients; finally, the cost should be reasonable.
- 20.
For simplicity we have neglected the term Q(r,t) which appears in Eq. (4.0.1).
- 21.
A simple derivation is given in [Hil76].
- 22.
Notwithstanding the adjective Advanced the Tutorial Advanced Numerical Differential Equation Solving in Mathematica is a good introduction to this method extensively used by MATHEMATICA ®.
- 23.
The stiffness is defined by Lambert [Lam92] as follows: If a numerical method with a finite region of absolute stability, applied to a system with any initial conditions, is forced to use in a certain interval of integration a step-length which is excessively small in relation to the smoothness of the exact solution in that interval, then the system is said to be stiff in that interval.
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Barbero, N., Delfino, M., Palmisano, C., Zosi, G. (2014). Propagation of Thermal Energy. In: Pathways Through Applied and Computational Physics. Undergraduate Lecture Notes in Physics. Springer, Milano. https://doi.org/10.1007/978-88-470-5220-8_4
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