This paper presents a set of fully analytical solutions, together with explicit expressions, in the time and frequency domain for the heat conduction response of homogeneous unbounded and of bounded rectangular spaces (three-, two-, and one-dimensional spaces) subjected to point, line, and plane heat diffusion sources. Particular attention is given to the case of spatially sinusoidal, harmonic line sources. In the literature this problem is often referred to as the two-and-a-half-dimensional
Problems in thermodynamics can often be solved with the aid of formulas or expressions known as Green’s functions. These functions, or
The fundamental solutions most often used are point sources in a three-dimensional (3D), infinite homogeneous space; line sources acting within two-dimensional (2D) spaces; and plane sources heating one-dimensional (1D) spaces. The reason for these choices is that these three fundamental solutions are known in closed-form in time domain and have a relatively simple structure [
They are frequently combined to simulate heat conduction in the time domain or in a transform space defined by the Laplace transform, in half-spaces, infinite plates, rectangular 2D spaces, wedges, and rectangular 3D spaces [
This paper compiles alternative fundamental solutions in explicit form, specifically Green’s functions for harmonic 2D and 3D and harmonic (steady state) line sources whose amplitude varies sinusoidally in the third dimension. This last solution, which is often referred to in the literature as the 2.5D problem, can be of significant value when formulating 3D thermodynamics problems via boundary elements together with integral transforms. In addition, the proposed Green’s functions are combined using an image source technique to model a half-space, a corner, a layer system, a laterally confined layer system, a solid rectangular column, a solid rectangular column with an end cross section, and a 3D parallelepiped inclusion. To the best of our knowledge, this is the first such derivation that promises to be efficient for formulating 3D thermodynamics problems using boundary elements and integral transforms.
Time domain solutions are obtained by applying inverse Fourier transforms, using complex frequencies to avoid aliasing phenomena. These solutions are validated by comparing computed responses with those obtained directly in the time domain.
The transient heat transfer by conduction in an infinite, homogeneous space can be described by the diffusion equation in Cartesian coordinates:
The solution of (
Consider first an infinite, homogeneous space subjected at
Consider next an infinite, homogeneous space subjected to a spatially varying line heat source of the form
Applying a Fourier transformation in the
The full 3D solution can then be achieved by applying an inverse Fourier transform in the
Equation (
Assuming the existence of an infinite number of virtual sources, we can discretize these continuous integrals. The integral in the above equation can be transformed into a summation if an infinite number of such sources are distributed along the
Next, the above Green’s functions are combined so as to define Green’s functions for a half-space, a corner, a single layer system, a U system, a solid rectangular pipe, a solid open box, and a 3D parallelepiped box. Expressions in frequency and time solutions are provided. The time solutions obtained after the application of inverse spatial and frequency Fourier transforms are compared with those given by Green's functions defined directly in the time domain.
Green’s functions for the different spaces are determined using the image source method. By this method a distribution of virtual sources and sinks are combined so as to give null temperatures (Dirichlet boundary conditions) or heat fluxes on the required boundaries (Neumann boundary conditions). Other boundary conditions, such as Robin, are not studied in this paper. In the case of solid bodies bounded by two parallel surfaces the number of sources, placed perpendicular to the surfaces, is theoretically infinite. The use of complex frequencies allows the contribution of the sources placed at greater distances to vanish and so to limit the number of the virtual sources. The use of complex frequencies with a small imaginary part, taking the form
Green’s functions are validated assuming that the medium is subject to a Dirac delta source. This type of source would require the solution to be computed in the frequency domain
The number of virtual sources used depends directly on the predefined convergence criterion. As we move from one dimension to two dimensions and then to three dimensions, the number of sources grows significantly. Thus, although the method converges rapidly, the cost of computation grows significantly as we move from a one-dimensional to a three-dimensional problem.
Green’s functions in the time and frequency domain will be grouped for the following three cases: unbounded space, which includes Green’s functions for 1D, 2D, and 3D sources; two-dimensional space, which contains Green’s functions for a half-space, a space bounded by two perpendicular planes, a single layer system, a U system, and a solid rectangular pipe, when subjected to 2D and 3D sources; three-dimensional space, which compiles Green’s functions for point sources placed in a solid open box and in a 3D parallelepiped box.
Special attention is given to the 2.5D solution in all cases since it enables the computation of the 3D heat field as a summation of 2D sources with varying spatial wavenumbers. Different boundary conditions are assumed and combined, namely, null temperatures or null heat fluxes. For each case, a scheme of the geometry is first illustrated (Figures
Responses for an unbounded space: (a) 1D heat source; (b) 2D heat source; (c) 3D heat source.
Response for the half-space (Case
Response for the bounded space defined by
Response for the horizontal layer (Case
Response for the U system (Case
Response for the solid rectangular pipe (Case
Response for the solid open box: (a) Case
Responses for the 3D parallelepiped box: (a) Case
Unbounded space.
Boundary conditions prescribed for the half-space (Cases
Boundary conditions prescribed for the bounded space defined by
Boundary conditions prescribed for the horizontal layer (Cases
Boundary conditions prescribed for the U system (Cases
Boundary conditions prescribed for the solid rectangular pipe (Cases
Boundary conditions prescribed for the solid open box (Cases
Boundary conditions for the 3D parallelepiped box (Cases
All results showed a good agreement between the different formulations for all cases.
In the examples provided the Dirac delta source is positioned at the coordinate
The notation in Table
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The superscripts
See Figure
2D source is as follows:
3D source is as follows:
Equations for the half-space (Cases 1 and 2), subjected to a 2D heat source, are as follows.
Consider
Consider
Equations for the half-space (Cases 1 and 2), subjected to a 3D heat source, are as follows.
Consider
Consider
Equations for the bounded space defined by
Consider
Consider
Consider
Consider
Equations defined for the bounded space defined by
Consider
Consider
Consider
Consider
Equations for the horizontal layer (Cases 1–4), subjected to a 2D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the horizontal layer (Cases 1–4), subjected to a 3D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the U system (Cases 1–4), subjected to a 2D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the U system (Cases 1–4), subjected to a 3D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the solid rectangular pipe (Cases 1–4), subjected to a 2D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the solid rectangular pipe (Cases 1–4), subjected to a 3D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the solid open box (Cases 1–4), subjected to a 3D heat source, are as follows.
Consider
Consider
Consider
Consider
Equations for the 3D parallelepiped box (Cases 1–4), subjected to a 3D heat source, are as follows.
Consider
Consider
Consider
Consider
Fully analytical solutions for heat conduction for unbounded and rectangular spaces subjected to point, line, and plane sources have been presented. Two boundary conditions were assumed, namely, the Dirichlet and Neumann boundary conditions. Particular attention was given to the two-and-a-half-dimensional fundamental solution or 2.5D Green’s functions defined for spatially sinusoidal, harmonic line sources. The final expressions were validated by applying the equations to the problem of a Dirac delta source, for which the solutions in the time domain are known in analytical form. Excellent agreement was found between the numerical solutions given by Fourier synthesis and the exact solutions.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work has been framed under the Initiative Energy for Sustainability of the University of Coimbra and supported by the Energy and Mobility for Sustainable Regions (EMSURE) Project (CENTRO-07-0224-FEDER-002004) and was supported in part by the POCI-01-0247-FEDER-003179 (Revi Clean Facade) Project funded by Portugal 2020 through the Operational Programme for Competitiveness Factors (COMPETE 2020).