# Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method

### Alicja Piasecka Belkhayat

,### Anna Korczak

Journal of Applied Mathematics and Computational Mechanics |
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@article{Belkhayat_2017, doi = {10.17512/jamcm.2017.4.06}, url = {https://doi.org/10.17512/jamcm.2017.4.06}, year = 2017, publisher = {The Publishing Office of Czestochowa University of Technology}, volume = {16}, number = {4}, pages = {57--65}, author = {Alicja Piasecka Belkhayat and Anna Korczak}, title = {Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method}, journal = {Journal of Applied Mathematics and Computational Mechanics} }

TY - JOUR DO - 10.17512/jamcm.2017.4.06 UR - https://doi.org/10.17512/jamcm.2017.4.06 TI - Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method T2 - Journal of Applied Mathematics and Computational Mechanics JA - J Appl Math Comput Mech AU - Belkhayat, Alicja Piasecka AU - Korczak, Anna PY - 2017 PB - The Publishing Office of Czestochowa University of Technology SP - 57 EP - 65 IS - 4 VL - 16 SN - 2299-9965 SN - 2353-0588 ER -

Belkhayat, A., & Korczak, A. (2017). Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method. Journal of Applied Mathematics and Computational Mechanics, 16(4), 57-65. doi:10.17512/jamcm.2017.4.06

Belkhayat, A. & Korczak, A., 2017. Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method. Journal of Applied Mathematics and Computational Mechanics, 16(4), pp.57-65. Available at: https://doi.org/10.17512/jamcm.2017.4.06

[1]A. Belkhayat and A. Korczak, "Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method," Journal of Applied Mathematics and Computational Mechanics, vol. 16, no. 4, pp. 57-65, 2017.

Belkhayat, Alicja Piasecka, and Anna Korczak. "Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method." Journal of Applied Mathematics and Computational Mechanics 16.4 (2017): 57-65. CrossRef. Web.

1. Belkhayat A, Korczak A. Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method. Journal of Applied Mathematics and Computational Mechanics. The Publishing Office of Czestochowa University of Technology; 2017;16(4):57-65. Available from: https://doi.org/10.17512/jamcm.2017.4.06

Belkhayat, Alicja Piasecka, and Anna Korczak. "Modelling of transient heat transport in two-layered crystalline solid films using the interval lattice Boltzmann method." Journal of Applied Mathematics and Computational Mechanics 16, no. 4 (2017): 57-65. doi:10.17512/jamcm.2017.4.06

MODELLING OF TRANSIENT HEAT TRANSPORT IN TWO-LAYERED CRYSTALLINE SOLID FILMS USING THE INTERVAL LATTICE BOLTZMANN METHOD

Alicja Piasecka Belkhayat, Anna Korczak

Institute of Computational Mechanics and Engineering, Silesian University of Technology

Gliwice, Poland

alicja.piasecka@polsl.pl, anna.korczak@polsl.pl

Received: 6 December 2017;
Accepted: 28 December 2017

**Abstract.** In the paper, the numerical modelling of heat transfer in
one-dimensional crystalline solid films is considered. A generalized two-layer
problem is described by the Boltz-
mann transport equations transformed in the
phonon energy density equations supplemented by the adequate
boundary-initial conditions. Such an approach in which the parameters
appearing in the problem analysed are treated as the constant values is widely
used, but
in this paper the interval values of relaxation time and the boundary condition
for silicon and diamond are taken into account. The problem formulated has been
solved by means
of the interval lattice Boltzmann method using the rules of directed interval
arithmetic.
In the final part of the paper the results of numerical computations are
presented.

*MSC 2010**: 65M99, 80A20*

*Keywords: **Boltzmann transport
equation, interval lattice Boltzmann method, directed
interval arithmetic*

1. Introduction

In dielectric materials and semiconductors, the heat transport is mainly realized by a quanta of lattice vibrations called phonons. The phonons represent the conduction of heat and electricity through solids. In non-metals, phonons as heat carriers always “move” from the part with the higher temperature to the part with the lower temperature and, during this move, phonons carry energy. This kind of phenomena can be described by the Boltzmann transport equation (BTE). It should be pointed out that taking into account the extremely short duration and the domain dimensions expressed in nanometers, the macroscopic heat conduction equation based on the Fourier law cannot be used [1, 2]. Such an approach in which the parameters appearing in the mathematical model are treated as the constant values is widely used [3, 4]. Here, the interval values of relaxation times and boundary conditions for successive sub-domains are taken into account. The relaxation time is estimated experimentally, and its actual value is still a subject of discussion [5]. In the paper the heat transport proceeding in a two-layered thin film is considered [6-9]. To solve the problem formulated, the interval version of the lattice Boltzmann method is applied using the rules of directed interval arithmetic [10, 11]. In the final part of the paper the examples of numerical computations are shown.

2. Boltzmann transport equation

The unsteady BTE in a phonon energy density formulation using the simplifying assumptions of the Debye model for one-dimensional two-layered analysis [3, 12] can be written as

where *s* = 1, 2 corresponds to the successive layers of the thin
film (silicon, diamond), is the phonon energy density, is the equilibrium phonon energy density, is the frequency-dependent phonon propagation
speed, is the frequency-dependent phonon relaxation time,
*t* denotes the time and is the external heat generation rate related to a unit of volume.

Using the Debye model, the dependence between phonon energy and lattice temperature can be calculated from the following formula

(2) |

where is the Debye temperature of the
solid, is the Boltzmann constant, *T*_{s} is the lattice temperature while is the number density of
oscillators [3].

The equations (1) should be supplemented by the boundary and initial conditions.

3. Interval lattice Boltzmann equation

The lattice Boltzmann method (LBM) is a numerical technique for the simulation of heat transfer. The LBM solves a discretized set of the BTE known as the lattice Boltzmann equations. The phonon energy density is defined as the sum

(3) |

where is the phonon energy density in the positive *x*
direction for *s*^{th} layer while is the phonon energy density in
the negative *x* direction and *d* signifies the
lattice direction.

The interval Boltzmann transport equations for the one-dimensional problem take the form [13]

(4) |

where is the component of velocity along the x-axis, is the lattice distance from site to site, is the time step needed for a phonon to travel from one lattice site to the neighboring lattice site, is the interval relaxation time and .

The set of equations (4) must be supplemented by the boundary-initial conditions [6, 9]

(5) |

where and are the interval boundary temperatures and is the initial temperature. Between the successive sub-domains the continuity condition can be taken into account [9]

(6) |

The interval LBM algorithm has been used to solve the problem analysed [6, 10]. The approximate form of the equations (4) is of the following form

(7) |

Taking into account the assumption that

(8) |

and

(9) |

the product (*s* = 1, *i* = 3,
*f* = 4) is calculated
using the rules of
directed interval arithmetic according to the following
formula

(10) |

As a result, the interval obtained is improper.

After subsequent computations the interval lattice temperature is determined using the formula (see eq. (2))

(11) |

4. Results of computations

As a numerical
example, the heat transport in a silicon-diamond film of the
dimension *L* = 200 nm has been analysed. The following input data have been
introduced for a silicon-diamond film
respectively: , , , , , , , , and .

Figure 1 illustrates the interval temperature distribution in the domain considered for the chosen times. Figure 2 presents the courses of the temperature function at the internal nodes nm (1) and nm (2) for the silicon and diamond layer respectively.

Fig. 1. The interval temperature distribution

In the second analysed example it is assumed that the external heat generation rate related to an unit of volume is the interval number

Fig. 2. The interval heating curves at internal nodes

Similar to the previous example, Figure 3 illustrates the interval temperature distribution in the domain considered for the chosen times, and Figure 4 presents the courses of the temperature function at the same internal nodes.

Fig. 3. The interval temperature distribution

Fig. 4. The interval heating curves at internal nodes

Fig. 5. The interval temperature distribution

Fig. 6. The interval heating curves at internal nodes

In the last numerical example, accurate boundary temperatures and have been introduced. In Figure 5, the interval temperature distribu- tion in the domain considered for the chosen times are shown and Figure 6 presents the courses of the temperature function at the same internal nodes.

It should be pointed out, that for each node of the domain considered there are two curves representing the beginning and end of temperatures intervals. The interval solution is illustrated by the area between these two curves.

Additionally one can see, that for longer calculation time, the temperature inter- vals are wider (see Figs. 2, 4 and 6). It is visible that a greater number of operations in the set of interval numbers impacts on the increase of the width of the obtained intervals.

5. Conclusions

In the paper a interval version of the lattice Boltzmann method for solving 1D problems in two-layered crystalline solid films has been presented. A model with interval values of relaxation times, boundary conditions and the external heat generation rate related to a unit of volume for a silicon-diamond film has been proposed.

The generalization of LBM allows one to find the numerical solution in the interval form, and such information may be important, especially for the parameters that are estimated experimentally, for example the relaxation time. The problem analysed can be extended to multi-layered thin films.

References

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[2] Smith A.N., Norris P.M., in Heat Transfer Handbook, A. Bejan, D. Kraus, Eds. 2003, 1309.

[3] Escobar R.A., Ghai S.S., Jhon M.S., Amon C.H., Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling, Journal of Heat and Mass Transfer 2006, 49, 97-107.

[4] Joshi A.A., Majumdar A., Transient ballistic and diffusive phonon heat transport in thin films, Journal of Applied Physics 1993, 74(1), 31-39.

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[10] Markov S.M., On directed interval arithmetic and its applications, Journal of Universal Computer Science 1995, 1, 514-526.

[11] Piasecka-Belkhayat A., Interval boundary element method for transient diffusion problem in two layered domain, Journal of Theoretical and Applied Mechanics 2011, 49, 1, 265-276.

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[13] Piasecka-Belkhayat A., Korczak A., Modelling of transient heat transport in one-dimensional crystalline solids using the interval lattice Boltzmann method, Recent Advances in Computational Mechanics, Taylor & Francis Group, A Balkema Book, London 2014, 363-368.