Heat transfer spray model: An improved theoretical thermal time-response to uniform layers deposit using Bessel and Boubaker polynomials
Introduction
Spray pyrolysis is a powerful and very low cost technique to synthesize varieties of thin films from several materials. In the last decades, many models [1], [8] have been established in order to predict deposited material composition and behaviour.
Eslamian et al. [1] proposed, i.e., a mathematical model based on Boltzmann equation numerical solution for the transition regime with the continuum based governing equations. This model was successfully tested on submicron zirconium particles, prepared by spray pyrolysis. The model presented by Grader et al. [2] was rather based on particle and gas temperatures alterations during the spraying process, and tried to explain the incomplete reactions obtained under particular conditions. Other models were mainly based on numerical and empirical simulation [3], [4], [5], [6], [7], [8].
In this paper, we propose an analytical solution to the heat equation. The studied model is a vertical gas–solution spray depositor under the presumption of a uniform deposited layer. The boundary conditions are introduced at the final solving stage by the mean of polynomial expansions, essentially Bessel polynomials [9], [10], [11] and Boubaker polynomials [12], [13], [14], [15], [16].
Section snippets
The uniform layer model
In the presented model, the targeted glass layer is a 2.0 cm × 2.0 cm × 0.2 cm parallelepiped sample fixed on a wide heating bulk. The bulk is maintained at constant temperature Tb. The efficient targeted zone for the study is limited to a 2.0 cm – diameter, 0.2 cm – thick cylinder (of glass sample). This means that the deposited layer would be spread out on this cylinder (Fig. 1).
If we then place the point A on the z-axis (Fig. 2), the problem can be considered as a cylindrical one, in which the
Theoretical calculations
The temperature at any point, expressed in the cartesian coordinates system is: T(x, y, z, t).
Considering temperature response at the median plane, we have the following equations:As we consider that the heat is transmitted integrally from bulk to deposited material, we have Ps ≈ Pb. We have hence to solve the Eq. (3), expressed in cylindrical coordinates:where Tg = Tg (r, ϑ, z,
Conclusion
An analytic solution to the heat equation has been presented in the particular case of pyrolysis spray vertical setup. This paper yields a complete theoretical expression of the temperature distribution in the related cylindrical coordinates system with respect to time-dependent variations. The result presents a meaningful supply to the investigations [17], [18] of predicting the thermal behaviour and the related chemical alterations inside deposited layer components.
Actually, this model is
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