ow> T m the melting temperature, and R is the reflectivity at the front surface.

One obtains Cattaneo equation in dimensionless form as:

φ τ + 2 φ = θ , τ = t t k (10)

the energy conservation equation in the dimensionless scale attains the form:

θ τ = φ (11)

and the hyperbolic heat conduction equation (HHCE) in the following dimensionless form is:

2 θ τ 2 + 2 θ τ = 2 θ X 2 (12)

Equation (12) is subjected to the following dimensionless initial and boundary conditions:

at τ = 0 , θ ( X , 0 ) = 0 (13)

at X = X d , θ X ( X d , τ ) = 0 (14)

at X = 0 , θ X ( 0 , τ ) = [ 2 φ ( 0 , τ ) + φ τ ( 0 , τ ) ] (15)

Equation (14) indicates that the rear surface of the target is insulated.

Laplace integral transform technique is applied to solve Equation (12). Taking Laplace transform on Equation (12) w.r.t. the time t one gets:

2 X 2 θ ¯ ( X , s ) ( s 2 + 2 s ) θ ¯ ( X , s ) = 0 (16)

The solution of equation (16) is as follows:

θ ¯ ( X , s ) = A e s ( s + 2 ) X + B e s ( s + 2 ) X (17)

at X = 0 one gets:

θ ¯ X | X = 0 = A s ( s + 2 ) B s ( s + 2 ) (18)

And at X = X d one gets:

θ ¯ X | X = X d = A s ( s + 2 ) e s ( s + 2 ) X d B s ( s + 2 ) e s ( s + 2 ) X d = 0

Thus: A e s ( s + 2 ) X d B e s ( s + 2 ) X d = 0 (19)

Taking Laplace transform to the boundary condition Equation (15) one gets:

θ ¯ X ( X , s ) = [ 2 φ ¯ ( X , s ) + { s φ ¯ ( X , s ) φ ( X , 0 ) } ] ,

At the front surface X = 0 this equation can be rewritten as:

θ ¯ X ( 0 , s ) | X = 0 = 2 φ ¯ ( 0 , s ) s φ ¯ ( 0 , s ) , Where φ ( X , 0 ) = 0

θ ¯ X ( 0 , s ) | X = 0 = ( s + 2 ) φ ¯ ( 0 , s ) (20)

Comparing the two Equations (18), (20) A and B can be obtained as follows:

A s ( s + 2 ) B s ( s + 2 ) = ( s + 2 ) φ ¯ ( 0 , s )

( A B ) = s ( s + 2 ) s φ ¯ ( 0 , s ) or: ( A B ) = s + 2 s φ ¯ ( 0 , s )

Multiplying both sides by e s ( s + 2 ) X d then by e s ( s + 2 ) X d respectively one gets:

A e s ( s + 2 ) X d B e s ( s + 2 ) X d = φ ¯ ( 0 , s ) s + 2 s e s ( s + 2 ) X d (21)

A e s ( s + 2 ) X d B e s ( s + 2 ) X d = φ ¯ ( 0 , s ) s + 2 s e s ( s + 2 ) X d (22)

Subtracting (19), (21) one gets:

B [ e + s ( s + 2 ) X d e s ( s + 2 ) X d ] = φ ¯ ( 0 , s ) s + 2 s e + s ( s + 2 ) X d

Thus:

B = φ ¯ ( 0 , s ) s + 2 s e + s ( s + 2 ) X d 2 sinh s ( s + 2 ) X d

Also from (19), (22) one gets:

A [ e + s ( s + 2 ) X d e s ( s + 2 ) X d ] = φ ¯ ( 0 , s ) s + 2 s e + s ( s + 2 ) X d

Thus:

A = φ ¯ ( 0 , s ) s + 2 s e s ( s + 2 ) X d 2 sinh s ( s + 2 ) X d

Substituting in Equation (17) one gets:

ϑ ¯ ( X , s ) = φ ¯ ( 0 , s ) s + 2 s [ e + s ( s + 2 ) ( X d X ) + e s ( s + 2 ) ( X d X ) ] e s ( s + 2 ) X d [ 1 e 2 s ( s + 2 ) X d ]

Considering that: 1 1 a = n = 0 a n , | a | < 1 [39]

Thus:

1 [ 1 e 2 s ( s + 2 ) X d ] = n = 0 e 2 n s ( s + 2 ) X d

This gives:

ϑ ¯ ( X , s ) = φ ¯ ( 0 , s ) s + 2 s [ n = 0 e ( X + 2 n X d ) s ( s + 2 ) + n = 0 e ( 2 ( 1 + n ) X d X ) s ( s + 2 ) ]

Neglecting the last term with respect to the first one, one gets:

ϑ ¯ ( X , s ) = n = 0 φ ¯ ( 0 , s ) s + 2 s ( s + 2 ) e ( X + 2 n X d ) s ( s + 2 )

This equation can be rewritten in the form:

ϑ ¯ ( X , s ) = n = 0 [ φ ¯ ( 0 , s ) s s ( s + 2 ) e ( X + 2 n X d ) s ( s + 2 ) + φ ¯ ( 0 , s ) 2 s ( s + 2 ) e ( X + 2 n X d ) s ( s + 2 ) ]

Let, f ( s ) = 1 s ( s + 2 ) e ( X + 2 n X d ) s ( s + 2 ) we get:

ϑ ¯ ( X , s ) = n = 0 [ φ ¯ ( 0 , s ) { s f ( s ) } + 2 φ ¯ ( 0 , s ) f ( s ) ] Put : 2 φ ¯ ( 0 , s ) f ( s ) = M 1 , φ ¯ ( 0 , s ) { s f ( s ) } = M 2 (23)

From Equation (9):

L 1 { φ ¯ ( s ) } = q ( t ) ( 1 R ) W ρ c p ( T m T 0 ) (24)

L 1 { f ( s ) } = L 1 { 1 s ( s + 2 ) e ( X + 2 n X d ) s ( s + 2 ) } (25)

We have: [40]

L 1 e c [ ( s + a ) 1 2 ( s + b ) 1 2 ] ( s + a ) 1 2 ( s + b ) 1 2 = e ( a + b 2 ) τ I 0 [ ( a b 2 ) ( τ 2 c 2 ) 1 2 ] , τ > c , c > 0

Put: c = X + 2 n X d , b = 0 , a = 2 , τ = t 2 t k one gets:

L 1 e ( X + 2 n X d ) s ( s + 2 ) s ( s + 2 ) = e τ I 0 τ 2 c 2 (26)

The modified Bessel function I η ( x ) is written in the following form: [41]

I η ( x ) = m = 1 ( x 2 ) ( η + 2 m ) m ! Γ ( η + m + 1 ) (27)

Comparing (25), (26) (at η = 0 ) one gets:

L 1 { f s } = e τ m = 1 [ τ 2 ( X + 2 n X d ) 2 ] m 2 2 m m ! Γ ( m + 1 ) (28)

Thus the inverse in time domain is as follows:

L 1 { f s } = e t 2 t k m = 1 [ ( t 2 t k ) 2 ( W 2 α ) 2 ( x + 2 n x d ) 2 ] m 2 2 m m ! Γ ( m + 1 ) (29)

where: Γ ( m + 1 ) = m !

L 1 { φ ¯ ( s ) } = q ( t ) W ρ c p ( T m T 0 ) (30)

The convolution theorem:

L 1 [ f 1 ( s ) f 2 ( s ) ] = 0 t F 1 ( t u ) F s ( u ) d u (31)

Thus

L 1 [ M 1 ] = L 1 [ 2 φ ( s ) f ( s ) ] = 2 0 t q ( t u ) W ρ c p ( T m T 0 ) e u 2 t k m = 1 [ ( u 2 t k ) 2 ( W 2 α ) 2 ( x + 2 n x d ) 2 ] m 2 2 m (32)

Also we have:

L [ d d t F ( t ) ] = s f ( s ) f ( 0 ) ,

For such a case one has f ( 0 ) = 0 , thus:

L 1 [ M 2 ] = L 1 [ φ ¯ s f ( s ) ] = 0 t φ ( t u ) d d u f ( u ) d u = 0 t A q ( t u ) W ρ c p ( T m T 0 ) d d u { e u 2 t k m = 1 { ( u 2 t k ) 2 W 2 4 α 2 ( x + 2 n x d ) 2 } m 2 2 m } d u (33)

L 1 [ ϑ ¯ ( X , s ) ] = L 1 [ M 1 ] + L 1 [ M 2 ]

ϑ ( x , t ) = n = 1 [ ( 2 1 2 t k ) 0 t A q ( t u ) W ρ c p ( T m T 0 ) e u 2 t k m = 1 { ( u 2 t k ) 2 W 2 4 α 2 ( x + 2 n x d ) 2 } m 2 2 m d u + 0 t A q ( t u ) W ρ c p ( T m T 0 ) 1 2 t k 2 e u 2 t k m = 1 m { ( u 2 t k ) 2 W 2 4 α 2 ( x + 2 n x d ) 2 } m 1 2 2 m u d u ] (34)

where: q ( t u ) = q max e ( ( t u ) t 0 γ ) 2 , q max , W/m2 is the laser maximum power density

At x = 0 we get:

ϑ ( 0 , t ) = n = 1 [ ( 2 1 2 t k ) 0 t A q ( t u ) W ρ c p ( T m T 0 ) e u 2 t k m = 1 { ( u 2 t k ) 2 W 2 4 α 2 ( 2 n x d ) 2 } m 2 2 m d u + 0 t A q ( t u ) W ρ c p ( T m T 0 ) 1 2 t k 2 e u 2 t k m = 1 m { ( u 2 t k ) 2 W 2 4 α 2 ( 2 n x d ) 2 } m 1 2 2 m u d u ] (35)

3. Computations

The obtained thermal profile is computed for different laser pulses with different maximum power densities as follows:

q max = 2E7 , 1 . 5E7 , 1E7 , 0. 8E7 , 0. 7E7 , 0. 6E7 , 0. 5E7 , 0. 4E7 , 0. 3E7 , 0. 2E7 , W / m 2 .

The laser pulse shape is taken of Gaussian form as: q ( t ) = q max e ( t t 0 γ ) 2 , γ is

the full width at half maximum of the suggested pulse, t0 in seconds is the time required for q ( t ) to reach the maximum value q max .

The other laser pulse parameters are as follows: t0 = 6 μsec, γ = 6 μ sec. pulse duration = 12 μsec, the slab thickness = 300 μm, the absorption coefficient A = 0.67, tk = 1 μsec.

The physical and thermal properties of the silver selenide slab material [42] [43] are given in Table 1.

The temperature of the irradiated front surface ϑ ( 0 , t ) for the different pulses is computed and are illustrated graphically in Figure 1 and Figure 2 and are tabulated in Table 2 for m = 1, n = 1, and in Table 3 for m = 1, n = 1.2.

The critical time tph required to start phase transition, and the critical time tm required to initiate melting at the front surface are obtained for the considered pulses and are tabulated in Table 4 for m = 1, n = 1, and in Table 5 for m = 1, n = 1.2 and are illustrated graphically in Figure 3 and Figure 4.

4. Conclusions

As a result of the study the following conclusions can be made:

1) The temperature of the irradiated surface does depend linearly on the maximum power density qmax of the laser pulse and also on the absorption coefficient (1 − R).

2) The dependence on the slab thickness and the pulse parameters γ & t0 are no longer linear.

3) The dependence of the tph and tm on qmax is not linear.

Table 1. The physical and thermal properties of the silver selenide slab material [42] [43] .

Table 2. The temperature of the front surface as a function of the qmax, W/m2 and exposure time, (m = 1, n = 1).

Table 3. The temperature of the front surface as a function of the qmax, W/m2 and exposure time, (m = 1, n = 1, 2)

Table 4. The critical time tph required to start phase transition, and the critical time tm required to initiate melting at the front surface are obtained for the considered pulses for m = 1, n = 1.

Table 5. The critical time tph required to start phase transition, and the critical time tm required to initiate melting at the front surface are obtained for the considered pulses for m = 1, n = 1, 2.

Figure 1. The temperature of the irradiated front surface as a function of the laser exposure time “t”. qmax = 0.2 × 107 W/m2, m = 1 and n = 1.

Figure 2. The temperature of the irradiated front surface as a function of the laser exposure time “t”. qmax = 0.2 × 107 W/m2, m = 1 and n = 1.

Figure 3. The critical time tph required to start phase transition as a function of the maximum power density of the laser pulse for m = 1 and n = 1. The other parameters are kept constant.

Figure 4. The critical time tph required to start phase transition and the critical time tm required to initiate melting as a function of the maximum power density of the laser pulse for m = 1 and n = 1, 2. The other parameters are kept constant.

4) Successive pulses may be required to initiate phase transition or melting.

5) For the summation over “m” and “n” one gets acceptable values for θ(0, t) only for m = 1.

But for m > 1 one gets negative values.

Increasing values of θ(0, t) are obtained with increasing n values.

6) The extension of the present technique to other materials makes it possible to specify the optimum operation conditions to attain and maintain a certain phase for specific medical and technological applications.

Cite this paper
EL-Adawi, M. and Shalaby, S. (2018) Pulsed Laser Heating of a Finite Silver Selenide Slab Using (HHCE) Model. Applied Mathematics, 9, 355-368. doi: 10.4236/am.2018.94027.
References

[1]   Ready, J.F. (1965) Effects Due to Absorption of Laser Radiation. Journal of Applied Physics, 36, 462-468.
https://doi.org/10.1063/1.1714012

[2]   Ready, J.F. (1978) Industrial Applications of Laser. Academic Press, New York.

[3]   He, X. and Debroy, T. (2003) Probing Temperature during Laser Spot Welding from Vapor Composition and Modeling. Journal of Applied Physics, 94, 6949-6958.
https://doi.org/10.1063/1.1622118

[4]   Zhang, Y.M., Shen, Z.H. and Ni, X.W. (2014) Modeling and Simulation on Long Pulse Laser Drilling Processing. International Journal of Heat and Mass Transfer, 73, 429-437.
https://doi.org/10.1016/j.ijheatmasstransfer.2014.02.037

[5]   Zhou, B.H. and Mahdavian, S.M. (2004) Experimental and Theoretical Analyses of Cutting Nonmetallic Materials by Low Power CO2-Laser. Journal of Materials Processing, 146, 188-192.
https://doi.org/10.1016/j.jmatprotec.2003.10.017

[6]   Liu, B.A., Hu, G.H., Zhao, Y.A., Zhang, L.S., Sun, X., Wang, Z.P. and Xu, X.G. (2013) Laser Induced Damage of DKDP Crystals with Different Deuterated Degrees. Optics & Laser Technology, 45, 469-472.
https://doi.org/10.1016/j.optlastec.2012.06.008

[7]   Abd El-Gany, S.E.-S. (2009) A Theoretical Study on the Melting of a Finite Slab with a Pulsed Laser. Optik, 120, 890-897.
https://doi.org/10.1016/j.ijleo.2008.04.002

[8]   Detlev, R., Marco, J. and Kai S., (2009) Laser Damage Thresholds of Optical Coatings. Thin Solid Films, 518, 1607-1613.
https://doi.org/10.1016/j.tsf.2009.07.150

[9]   Bo F., Dayon Z.,Fei L.,Chenxing L.,Guodong L., (2012) Pulsed Laser Induced Damage in SOI Material. Journal of Physics and Chemistry of Solids, 73, 969-972.
https://doi.org/10.1016/j.jpcs.2012.03.002

[10]   Hassan, A.F., El-Niclawy, M.M., El-Adawi, M.K. and Hemida, A.A. (1993) A General Problem of Pulse Heating Of a Slab. Optics &Laser Technology, 25, 155-162.
https://doi.org/10.1016/0030-3992(93)90059-O

[11]   El-Adawi, M.K., Abdel-Naby, M.A. and Shalaby, S.A. (1994) Laser Heating of a Slab Having Temperature Dependent Surface Absorbtance. Vacuum, 45, 65-70.
https://doi.org/10.1016/0042-207X(94)90344-1

[12]   El-Adawi, M.K., Abdel-Naby, M.A. and Shalaby, S.A. (1995) Laser Heating of a Two-Layer System with Constant Surface Absorptance: An Exact Solution. International Journal Heat Mass Transfer, 38, 947-952.
https://doi.org/10.1016/0017-9310(94)00162-O

[13]   El-Adawi, M.K., Shalaby, S.A. and Abdel-Naby, M.A. (1995) Laser Heating of a Two-Layer System with Temperature Dependent Front Surface Absorptance. Vacuum, 64, 37-42.
https://doi.org/10.1016/0042-207X(93)E0005-6

[14]   Hassan A.F., El-Niclawy, M.M., El-Adawi, M.K., Nasr, A.E., Hemida M.A. and Abd El-Ghaffar, O.A. (1996) Heating Effects Induced by a Pulsed Laser in a Semi-Infinite Target in View of The theory of Linear Systems. Journal of Optics & Laser Technology, 28, 337-343.
https://doi.org/10.1016/0030-3992(95)00083-6

[15]   El-Adawi, M.K., Shalaby, S.A. and Abd El-Naby, M.A. (1996) Laser Heating of a Slab with Temperature Dependent Absorptance In Relation to the Cooling Conditions. Vacuum, 47, 1167-1173.
https://doi.org/10.1016/0042-207X(96)00152-2

[16]   Hassan, A.F., El-Niclawy, M.M., El-Adawi, M.K., Nasr, A.E., Hemida, M.A. and Abd El-Ghaffar, O.A. (1997) Heating a Slab of Finite Thickness with a Pulsed Laser in View of the Theory of Linear Systems. Pure and Applied Optics, 6, 67-84.
https://doi.org/10.1088/0963-9659/6/1/009

[17]   El-Adawi, M.K., Shalaby, S.A., Mostafa, S.S. and Kotkata, M.K. (2007) Laser Thermal Response of a Finite Slab as a Function of the Laser Pulse Parameters. Optics and Laser Technology, 39, 424-429.
https://doi.org/10.1016/j.optlastec.2005.06.038

[18]   El-Adawi, M.K. and Al-Theeb, H.M. (2008) Heating One-Dimensional Homogeneous Slab with Pulsed Laser in Relation to the Laser Pulse Parameters. AJSE, 33, 147-162.

[19]   El-Adawi, M.K. and Al-Fanakh, H.S. (2013) Laser Heating of a Finite Silver Selenide Slab. Journal of Optics & Laser Technology, 49, 231-236.
https://doi.org/10.1016/j.optlastec.2012.12.031

[20]   El-Adawi, M.K. and Al-Nuaim, I.A. (2013) Heating of a Finite Slab with CW Laser in Relation to Cooling Conditions at the Rear Surface. Material Science Research India, 10, 17-22.

[21]   El-Adawi, M.K., Shalaby, S.A. and Abd El-Ghany, S.E.S. (2015) Interaction of Laser Radiation with Solids-Review Articles. International Journal of Natural Sciences Research, 3, 83-100.
https://doi.org/10.18488/journal.63/2015.3.6/63.6.83.100

[22]   El-Adawi, M.K. and Al-Theeb, H.M. (2015) Heating Two-Layer System with Pulsed Laser. International Journal of Science and Research, 4, 247-254.

[23]   Glass, D.E., Ozisik, M.N. and Vick, B. (1985) Hyperbolic Heat Conduction with Surface Radiation. International Journal of Heat and Mass Transfer, 28, 1823-1830.
https://doi.org/10.1016/0017-9310(85)90204-2

[24]   Frankel, J.I., Vic, B. and Ozisk, M.N. (1987) General Formation and Analysis of Hyperbolic Heat Conduction in Composite Media. International Journal of Heat and Mass Transfer, 30, 1293-1305.
https://doi.org/10.1016/0017-9310(87)90162-1

[25]   Gembarovic, J. and Majernik, V. (1988) Non-Fourier Propagation of Heat Pulses in Finite Medium. International Journal of Heat and Mass Transfer, 31, 1073-1080.
https://doi.org/10.1016/0017-9310(88)90095-6

[26]   Kaminski, W. (1990) Hyperbolic Heat Conduction Equation for Materials with a Nonhomogeneous Inner Structure. Journal of Heat Transfer, 112, 555-560.
https://doi.org/10.1115/1.2910422

[27]   Lewandowska, M. (2001) Hyperbolic Heat Conduction in the Semi-Infinite Body with a Time-Dependent Laser Source. Heat and Mass Transfer, 37, 333-342.
https://doi.org/10.1007/s002310000176

[28]   Peshkov, V. (1944) Second Sound in Helium II. Journal of Physics, 8, 381-389.

[29]   Bertman, B. and Sandiford, D.J. (1970) Second Sound in Solid Helium. Scientific American, 222, 92-101.
https://doi.org/10.1038/scientificamerican0570-92

[30]   Wiggert, D.C. (1970) Analysis of Early-Time Transient Heat Conduction by Method of Characteristics. Journal of Heat Transfer, 99, 35-40.
https://doi.org/10.1115/1.3450651

[31]   Carey, G.F. and Tsai, F.M. (1982) Hyperbolic Heat Transfer with Reflection. Numerical Heat Transfer, 5, 309-327.
https://doi.org/10.1080/10407788208913451

[32]   Ozisik, M.N. and Vick, B. (1984) Propagation and Reflection of Thermal Waves in a Finite Medium. International Journal of Heat and Mass Transfer, 27, 1845-1854.
https://doi.org/10.1016/0017-9310(84)90166-2

[33]   Miki, H. (1991) Superlattice Formation and Phase Transition in Silver Selenide. Journal of Applied Physics, 30, 1765-1869.
https://doi.org/10.1143/JJAP.30.1765

[34]   Kumar, M. and Pradeep, B. (2002) Structural Electrical and Optical Properties of Silver Selenide Thin Films. Semiconductor Science and Technology, 7, 261-265.
https://doi.org/10.1088/0268-1242/17/3/314

[35]   Schoen, D., Yie, C. and Cui, Y. (2007) Electrical Switching and Phase Transformation in Silver Selenide Nanowiers. Journal of the American Chemical Society, 129, 4116-4117.

[36]   Banus, M.D. (1965) Pressure Dependence of Alpha-Beta Transition Temperature in Silver Selenide. Science, 147, 732-733.
https://doi.org/10.1126/science.147.3659.732

[37]   El-Adawi, M.K. and El-Shehawy, E.F. (1986) Heating a Slab Induced by a Time-Dependent Laser Irradiance—An Exact Solution. Journal of Applied Physics, 60, 2250-2255.
https://doi.org/10.1063/1.337184

[38]   Aivazov, A.A., Okhotin, A.S. and Gutsev, A.F. (1971) Thermal Conductivity of Silver Selenide in the Solid and Liquid States. Fizika, 4, 128-130.
https://doi.org/10.1007/BF00817988

[39]   Rainville, E.D. and Bedient, P.E. (1974) Elementary Differential Equations. 5th Edition, Macmillan Publishing Co., New York.

[40]   Roberts, G.E. and Kauvman, H. (1966) Tables of Laplace Transforms. Saunders Company, London, 254.

[41]   Gradshtegn, I.S. and Ryzhik, I.M. (1980) Tables of Integrals, Series and Products. 4th Edition, Academic Press, New York.

[42]   West, A.R. (1999) Basic Solid State Chemistry. John Wiley & Sons, Hoboken.

[43]   Dean, J.A. (1978) Lange’s Handbook of Chemistry. McGraw Hill Book Company, New York, 126-127.

 
 
Top