The Laplace transformation is commonly used to get analytical solutions for transient conduction
problems where the heat equation and its associated conditions are Linear with Time Invariant coefficients
(LTI) and where the geometry is simple [1]. In that case, the temperature solution at a given point in the
material system is a convolution product between the intensity of the heat or temperature source and a
corresponding impulse response, which is the inverse Laplace transform of a transfer function whose
expression is analytical in the Laplace domain. Inversion of such an analytical transfer function can now be
achieved using numerical inversion algorithms. Once the impulse function retrieved, the temperature solution
of the heat transfer problem can be found by implementation of the convolution product for any time shape of
the source. It short-circuits the need for finding more than once the solution of the Partial Differential
Equation (PDE) problem, that is the “detailed model”: the convolution product is its corresponding “reduced
model” and its structure is not biased.
The Laplace transformation can also be applied to LTI systems where heat transfer occurs in a domain
characterized by any non simple 3D geometry. It is the case for a heterogeneous physical system (including
solids and flowing fluids, even with linearized radiation in a cavity configuration) if the coefficients of the heat
equation system do not vary with time, without being necessarily uniform. In particular, the velocity field can
be 3D but in a steady state regime. This also requires each single transient source (the input) to be
separable, which means it can be written as a product of a time part, its intensity, by a space part, its
geometrical support [2]. The temperature response at any point (output) is still a convolution product in time
between its cause (source or input), and the impulse response.
In practice, the impulse response has to be found through solving an inverse problem, here a deconvolution.
One can either use a numerical temperature solution of a Finite Element simulation code for a given source
(model reduction), or the experimental noisy temperature signal delivered by a local sensor for a measured
source in a calibration experiment (model identification, which requires some kind of regularization in the
inversion). Examples of experimentally identified impulse responses, for characterizing heat exchangers [3,
4] are given in this presentation: impedances, between a temperature and a thermal power, or
transmittances, linking temperatures at two different points.