Heat transfer spray model: An improved theoretical thermal time-response to uniform layers deposit using Bessel and Boubaker polynomials

doi:10.1016/j.cap.2008.05.016

Current Applied Physics

Volume 9, Issue 3, May 2009, Pages 622-624

https://doi.org/10.1016/j.cap.2008.05.016 Get rights and content

Abstract

This study presents temperature profiling theoretical investigations in a pyrolysis spray model. Calculations are based on heat transfer equation resolution in cylindrical coordinates. Boundary conditions are taken into account in the resolution algorithm at different stages.

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 T_b. 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: $\nabla^{2} T_{g} (x, y, z, t) = \frac{1}{D_{g}} \frac{\partial T_{g} (x, y, z, t)}{\partial t} - \frac{1}{k_{g}} (P_{b} - P_{s}),$ $\nabla^{2} T_{s} (x, y, z, t) = \frac{1}{D_{s}} \frac{\partial T_{g} (x, y, z, t)}{\partial t} - \frac{1}{k_{s}} P_{s} .$ As we consider that the heat is transmitted integrally from bulk to deposited material, we have P_s ≈ P_b. We have hence to solve the Eq. (3), expressed in cylindrical coordinates: $\nabla^{2} T_{g} = \frac{\partial^{2} T_{g}}{\partial r^{2}} + \frac{1}{r} \frac{\partial T_{g}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} T_{g}}{\partial ϑ^{2}} = \frac{1}{D_{g}} \frac{\partial T_{g}}{\partial t},$ where T_g = T_g (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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Current Applied Physics

Heat transfer spray model: An improved theoretical thermal time-response to uniform layers deposit using Bessel and Boubaker polynomials

Abstract

Introduction

Section snippets

The uniform layer model

Theoretical calculations

Conclusion

Sens. & Actuat. B: Chem.

App. Math. & Comp.

J. Approx. Theory

App. Math. & Comp.

J. Nanotech.

J. Mater Res.

Int. J. Energy Res.

Heat Transf.-Asian Res.

Particle & Part. Syst. Character.

Lattice-related understanding regarding V-doping induced ZnO n-to-p type shift within LCT scope

Non-stationary random vibration analysis of multi degree systems using auto-covariance orthogonal decomposition

Perspective and potential of CO<inf>2</inf>: A focus on potentials for renewable energy conversion in the Mediterranean basin

An attempt to give exact solitary and periodic wave polynomial solutions to the nonlinear Klein-Gordon-Schrödinger equations

Lattice Compatibility Theory LCT investigations on physical and optical constants of antimony processed antimonite nano-films

A model for musculoskeletal disorder-related fatigue in upper limb manipulation during industrial vegetables sorting