Heating of a Finite Slab with CW Laser in Relation to Cooling Conditions- at the Rear Surface

M.K. El-Adawi^1* and I.A.Al-Nuaim²

¹Department of Physics, Faculty of Education, Ain Shams University, Helliopolis, Cairo Egypt

²Physics Department, Faculty of Science for Girls, King Faisal University P.O. Box 838 Dammam 31113 , Saudi Arabia.

ABSTRACT:

Heating a slab induced by laser irradiance is studied. The heat diffusion equation is solved using the Laplace integral transform method. The critical time tm required to initiate melting is obtained for the elements: Aluminum (Al), Gold (Au), Germanium (Ge) and Silicon (Si). Good cooling conditions at the rear surface of the slab are assumed. The obtained results show that for the considered elements and for such sources of high power density, cooling conditions at the rear surface are not of pronouncing effect. The effect of laser power density on the critical time required to initiate melting is predominant.

KEYWORDS: Laser heating; Laser damage; Heat diffusion equation; Laplace transform

Copy the following to cite this article: El-Adawi M. K, Al-Nuaim I. A. Heating of a Finite Slab with CW Laser in Relation to Cooling Conditions- at the Rear Surface. Mat.Sci.Res.India;10(1)

Copy the following to cite this URL: El-Adawi M. K, Al-Nuaim I. A. Heating of a Finite Slab with CW Laser in Relation to Cooling Conditions- at the Rear Surface. Mat.Sci.Res.India;10(1). Available from: http://www.materialsciencejournal.org/?p=281

Introduction

Laser heating of materials has aroused  considerable interest^1-23 due to its demonstrated importance in many processes for industrial applications such as spot welding, scribing and hole drilling, optical recording, semiconductor materials processing.

Serious trials are made to solve the diffusion equation describing the heating problem subjected to different boundary and initial conditions. The bulk of such trials have been numerically based. Some attempts are treated analytically.^12-23

The present treatment represents one of such analytical trials oriented to study the problem of heating a slab considering the cooling conditions at its rear surface using Laplace integral transform technique.

Theory

In setting up the problem it is assumed that the slab is subjected to a constant flux of laser irradiance q_o (W/m²) incident on the front surface and perpendicular to it. The diameter of the laser beam is assumed to be large compared to the thermal penetration depth within the slab, and thus the problem is solved as a one-dimensional problem.¹ The thermal and optical properties are assumed to be temperature independent.

The heat-flow equation for the temperature rise T(x,t) has the form :

where T(x,t) is the excess temperature with respect to the ambient temperature T_o, α = λ/ρC_p is the thermal diffusion coefficient of the slab in terms of the thermal conductivity λ and the heat capacity per unit volume (ρC_p). Eq. (1) is subjected to the initial condition :

And the boundary conditions :

 Derivation of the required solution

The Laplace transform of equation (1) with respect to the time variable “t” gives :

where

 

(x,s) is the Laplace transform of T(x,t).

One must also write the Laplace transform of the initial and boundary conditions this gives :

Substituting equation (2) into equation (5) gives :

The solution of equation (9) can be written in the form :

Applying the conditions (6-8) to equation (10) one gets :

One can determine the values of c₁ and c_2. The solution can finally be written in the form.

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Where

It is worth noting that for positive arguments the hyperbolic functions coth(x) and tanh(x) are positive. Moreover, as x increases from zero to + ∞, tanh(x) increases from 0 to +1 while coth(x) decreases from ∞ to 1. This behavior leads to the following conditions :

The equality represents the case when h = 0, i.e., the case of a thermally insulated rear surface.¹⁹

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The temperature of the front surface T(0,t) at x = 0 and that of the rear surface at x=d can be easily obtained by substituting x=0 and x=d respectively in equation).¹⁸

Using the standard tables of the inverse Laplace transform²⁴ and taking into consideration the convolution property. One can finally get the required solution in the form :

Computations

As illustrative examples computations are made to obtain the critical time required to initiate melting t_m for the considered elements.

Each target is assumed to be of thickness d = 5 mm and is subjected to constant laser flux of irradiance q_o = 10¹², W/m² and q₀=1E 14 W/m². To ensure good cooling conditions high value h = 10⁵

W/m²K of the heat transfer coefficient is also considered at the rear surface.

The physical and optical parameters of the chosen elements are obtained from refs,^{25, 26} and are given in Table 1

The computed values t_m for the considered elements are given in table 2.

Conclusions

The obtained values for t_m makes it possible to conclude that: 

1. Cooling conditions at the rear surface of a target exposed to light of high power density are not effective.

2. The difference in the critical time required to initiate damage for h=0 and h=10⁵W/m²K is almost undetectable for such sources of high power densities.

3. The values of the critical time required to initiate melting is strongly dependent on the incident power density and the value of the absorption coefficient of the irradiated surface.

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