A Thermal and Thermal Stress Analysis in Thermoelectric Solid ...

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013A Thermal and Thermal Stress Analysis inThermoelectric Solid under the Influence ofThomson EffectMagdy A. Ezzat and H. M YoussefAbstract—Our purpose in this work is to examine the influence ofthe Thomson heating and the Fourier’s heat conduction, in thepresence of magnetic field, on half-space thermoelectricity solidwhose surface is subjected to a thermal shock. A constantThomson coefficient, instead of traditionally a constant Seebeckcoefficient, is assumed. The charge density of the induced electriccurrent is taken as a function of time. The Maxwell’s equationsare formulated and the thermoelectricity solid coupled governingequations are established. Laplace and Fourier transforms areused to get the solution in direct approach for a two-dimensionalproblem. The problem is in the context of the Lord andShulman’s generalized thermoelasticity with one relaxation time.The distributions of the considered variables are representedgraphically. The comparisons are made with the results obtainedin the case of absence Peltier coefficient.Index Terms—Magneto-thermoelectric materials; Thomsoneffect; Thermal analysis; Thermal stressI. INTRODUCTIONThermoelectric currents in the presence of magnetic fieldscan cause pumping or stirring of liquid-metal coolants innuclear reactors or stirring of molten metal in industrialmetallurgy. The thermal and mechanical properties of materialsvary with temperature, so that the temperature dependence ofmaterial properties must be taken into consideration in thethermal stress analysis of these elements [1]. The interactionbetween the thermal and magneto-hydrodynamic fields is amutual one owing to alterations in the thermal convection and toManuscript received April 1, 2013.Magdy A. Ezzat, Department of Mathematics, Faculty of Education,Alexandria University, Alexandria, Egypt(E-mail: maezzat2000@yahoo.com).H. M. Youssef, current address is Faculty of Engineering, Umm Al-QuraUniversity, P.O. 5555, Makkah, Saudi Arabia(E-mail: yousefanne2005@gmail.com). Home address: Department ofMathematics, Faculty of Education, Alexandria University, Alexandria,Egypt.4the Peltier and Thomson effects (although these are usuallysmall).During the 1990s there was a heightened interest in the fieldof thermoelectrics driven by the need for more efficientmaterials for electronic refrigeration and power generation[2-4] .Proposed industrial and military applications ofthermoelectric materials are generating increased activity in thisfield by demanding higher performance, near roomtemperaturethermoelectric materials than those presently in use.A direct conversion between electricity and heat by usingthermoelectric materials has attracted much attention because oftheir potential applications in Peltier coolers and thermoelectricpower generators [5].A discussion of thermoelectric effects and devices shouldstart with the most fundamental phenomena, the Seebeck effector thermo-power [6]. In the early 1800s Seebeck observed thatif two dissimilar materials were joined together and thejunctions were held at different temperatures (T and ‏+T T) avoltage difference ( V) was developed that was proportionalto the temperature difference ( T). The ratio of the voltagedeveloped to the temperature gradient ( V/ T) is related toan intrinsic property of the materials called the Seebeckcoefficient S or the thermo-power. The Seebeck coefficient isvery low for metals (only a few mVK -1 ) and much larger forsemiconductors (typically a few 100 mV K -1 ). A related effect(the Peltier effect) was discovered a few years later by Peltier,who observed that if an electrical current is passed through thejunction of two dissimilar materials, heat is either absorbed orrejected at the junction depending on the direction of the current.This effect is due to the difference in Fermi energies of the twomaterials. The absolute temperature T, the Seebeck coefficient Sand the Peltier coefficient are related by the first Thomsonrelation ST .Several generalizations to the coupled theory of elasticityhave been introduced. One can refer to Hetnarski and Ignaczak[7] for a review, presentation and some important resultsobtained in this field. They pointed out five generalizations tothe coupled theory. Lord and Shulman [8] introduced thegeneralized theory of thermoelasticity with one relaxation, inwhich the acceleration of heat flux was taken into account andthe Fourier's law of heat conduction was replaced by ageneralized version including a constant represents the time-lagneeded to establish steady state of heat conduction in an element

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013of volume when a temperature gradient is suddenly imposed onthat element, this constant was called relaxation time.Investigation of the interaction between magnetic field andstress and strain in a thermoelastic solid is very important due toits many applications in the field of geophysics, plasma physicsand related topics. Especially in nuclear fields, the extremelyhigh temperatures and temperature gradients, as well as themagnetic fields originating inside nuclear reactors, influencetheir design and operations.In recently Sherief and Ezzat [9] solved a thermal shockhalf space problem using asymptotic expansion. Ezzat andYoussef [10-12] solved two-dimensional problems ofgeneralized magneto-thermoelastic half-space of finiteconductivity by Fourier and Laplace transforms. Ignaczak [13]studied a strong discontinuity wave and obtained adecomposition theorem for this theory [14]. Ezzat andEl-Karamany [15] studied the propagation of discontinuities inmagneto-thermoelastic half space.In the present work we shall formulateelectromagneto-thermoelastic coupled two-dimensionalproblem of a thermoelectricity half-space solid subjected to athermal shock on its surface. The Thomson coefficient assumedto be a constant but the charge density of the induced electriccurrent considered as a function of time. The governingequations are established and Laplace and Fourier transformsused to obtain the solution with direct method. The distributionsof the considered variables are represented graphically. Thecomparisons are made with the results obtained in the case ofabsence Peltier coefficient.II. PROBLEM FORMULATIONWe consider the problem of a half-space thermoelectricity solid( x 0 ). A magnetic field with constant intensityH (0,0,H ο)acts parallel to the bounding plane (take as thedirection of the z-axis). The surface of the half-space issubjected at time t 0 to a thermal shock that is a function of yand t. Thus, all quantities considered will be functions of thetime variable t and of the coordinates x and y.Due to the application of initial magnetic field H, there resultsan induced magnetic field h and an induced electric field E. Thesimplified linear equations of electrodynamics of slowlymoving medium for a homogeneous, thermally and electricallyconducting elastic solid arecurl h = J+D , (1)curl E = B , (2)div B =0, div D =ρ e , (3)B = μ ( H h),D = ε E . (4)οοThe above equations are supplemented by the Ohm's law formedia with finite conductivity.J = σ ( E μ u H )(5)οοIn the absence of the body force and inner heat source, theconstitutive law for the theory of generalized thermoelasticityis:σ i j2 μ e i jλ e δ γ δ T , (6)ij ijwhereT T T o .The generalized Fourier's law with Thomson effect is given byShercliff [16]:qi oqi kT,i Ji, (7|)which givesq q kT J . (8)i,i o i,i ,ii i,iwhereqi,i CET ToeFrom equations (1) and (3), we haveJ D e . (9)Substituting from equations (9) into equation (8), we getq q kT . (10)i,i o i,i ,ii eThus, the heat conduction equation will take the form k T = ρC T T + γ T e τe- . (11),ii ο o eE οThe continuity equation of the charges in the medium ise e i tdiv v 0 , (12)where viare the components of the velocity of the charges.Now, we will consider the charge density is function only intime i.e. . e e tThus, the continuity equation will take the form div v . (13)e e iIf we consider that, the velocity of the chargesv i isproportional to the velocity of the particles U , we can set thatvi Ui, (14)which gives i idiv v div U e(15)where is small non-dimensional constant.Hence, we have (16)e eewhich givesde de . (17)eHence, we obtain 0 exp e 0 1 e . (18) e e e0where e is the charge density when the strain vanishes.Then, we obtaini5

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 20130e e 1 e e . (19)Hence, heat equation will take the formT, t T,tt e, τe,k T = ρC + γ T,ii ο t o ttE ο0e+ 1 e ewhere we can approximate the above equation to the followingform0T tTtt e, τe,k T = ρC , , + γ T + e (20),ii ο t o tt eE οThe constitutive equationsσ F ρ u , (21)i j ,j i iand strain-displacement relations1e i j= ( u i ,j+ u j,i ) . (22)2The body force is given byF μi ο iJH . (23)In the above equations, a comma followed by a suffix denotesmaterial derivative and a superposed dot denotes the derivativewith respect to time, i,j x,y .The displacement components have the following formu x = u( x,y,t ), u y = v( x,y,t ), u z = 0 , (24)hence, from equations (22) and (24), we obtain the straincomponents u v 1 u ve x x = , e yy = , e x y = ( ), x y2 y x (25)e xz= eyz ezz 0.From equations (6) and (25), the stress components are given byγTσ =( λ 2μ ) uxx,x + λ v ,y (26)γTσ yy =(λ 2μ) v ,y + λ u ,x (27)σ zz =λ(u,x v ,y) γT(28)xy ,y ,xσ =μ u v . (29)The components of the magnetic intensity vector in the mediumareH x = 0, H y = 0, H z = Hο h( x,y,t ) . (30)The electric intensity vector is normal to both the magneticintensity and the displacement vectors. Thus, it has thecomponentsE x = E 1 , E y = E 2 , E z = 0 (31),The current density vector J be parallel to E, thusJ x = J 1 , J y = J 2 , J z = 0 . (32)The Ohm’s law (5) after linearization givesvuJ= 1 σ ο(E1 μο H ο ) , J 2 = σ ο(E2 μο H ο ).ttEquations (1), (4) and (33) give the two equationshv E1= σ ο(E1 μο H ο ) ε ο , y t thu E= σ ο(E2 μο H ο ) εο x t t2(33)(34). (35)From equations (2), (5) we can get one-vanishing component,namely E E h1 2 = μο y x t. (36)From equations (23) and (33), we obtain uF= x μο σοH ο(E2 μο H ο ) , (37) t vF= y μο σοH ο(E1 μο H ο ) t, (38)F z = 0 . (39)From equations (6), (26)-(29) and (37)-(39), we get2 T γ x u e2ρ =( λ μ ) μ u2 t x, (40) u+ μοσοH ο ( E2 μοH ο ) t2 T v e2ρ =( λ μ ) μ v γ2 t y y, (41) v μοσοH ο ( E1 μοH ο ) twhere,uve xy(42)For convenience, the following non-dimensional variables areused:x c1ηx, y c1ηy, u c1ηu, v c1ηv,t c 21 ηt, γ (TT ο)σ2ijθ , τ=c1ητ, σ ,λ2μij μηhηEh E i 1,2(43) i, i ,2σομοH ο σομοHοc16

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013In terms of the non-dimensional quantities defined in equation(43), the above governing equations reduce to (dropping thedashed for convenience)2 2 2 2 2,x ,x 1 1 2 ,t ,tt(β 1) e u β θ β αβ (β E u ) β u , (44) 2 s s 2 θ ε 21+ o s ε1s e where all the initial state of the material is at rest(59)2 2 2 2 2,y ,y 1 1 1 ,t ,tt(β 1) e v β θ β αβ (β E v ) β v , (45)2 θ (θ τθ ) ε (e τ e ) e , (46),t ,tt 1 ,t ,tt oh E1v= β1E1 ε2 , (47) y t th E2u= β1E2 ε2 x t t, (48) E1 E2h = y x t. (49)The constitutive equations reduce to2 2x x ,xσ (β 2)e 2 u β θ , (50)2 2y y ,yσ (β 2)e 2 v β θ , (51)zz2 2σ (β 2)e β θ , (52)σx y u,y v,x. (53)Differentiating equation (44) with respect to x, and equation (45)with respect to y, and then sum, we obtain22 2 2 h β1α e θ β21 α 0t t t(54)Differentiating equation (47) with respect to y, and equation (48)with respect to x, and then sum, we obtain22 e β1 ε2 h 2t t , (55) tThe heat equation will take the form2 22 1 o 2 12θ ε + ε e . (56) tt ttIII. FORMULATION IN THE LAPLACE TRANSFORM DOMAINWe will apply Laplace transform defined asst f s f t e d t ,0hence, the above equations will take the forms2 2 2 21 1 β αs s e θ β αsh 0(57)2 21 2 β s ε s h se(58)IV. FORMULATION IN THE FOURIER TRANSFORM DOMAINWe will apply Fourier transform defined as1 iqyf (q) e f (y) d y2 ,where1 iqyf (y) e f (q) dq2 .we get2d edx22d θdx2 2 2 q β1α+ε 1+ o s 1 ε1s e2 2 s s θ + β1αsh 2 2 21 o 1, (60) ε + s ε s e q s s θ , (61)2dh2 2se q +β21s ε2s hdx . (62)We can write the above system as follows:2d 2 1 e 2θ + 3hdx , (63) 2ddx2 5 θ 4e, (64)2d 2 7 h 6edx where2 2 q β α+ε + s 1 ε s , 1 1 1 o 123 14 1 o 16 s , 7 q+β1s ε2s .2 β αs , ε + s ε s ,(65)22 s s ,2 25 q s s ,The above system gives the following ordinary differentialequations6 4 2d d d L M N θ 0 6 4 2d x d x d x , (66)6 4 2d d d L M N e 0 6 4 2d x d x d x , (67)and6 4 2d d d L M N h 0 6 4 2d x d x d x , (68)whereL= ,1 5 7M= ,1 5 1 7 5 7 2 4 3 67

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013h0,q,s h0 0. (90)Using the conditions (86), (88) and (90), we get the solution ofthe problem in as follow:32 sin qa e x,q,s = 1 ki 5 e qs42ki 1 ki 7 3 6 2 kxi 3 , (91)i=12 2ki kjj1, ij32 sin qa θ x,q,s =1 qs32ki 1 ki 7 3 6 kxie , (92)i=12 2ki kjj1, ij2ki 1k i 7 2k3i 52 6 sin qa 36kxh ix,q,s = 1 e qs34 i=12 2 2ki 7 ki k j j1, ij. (93)VI. THE INVERSION OF LAPLACE AND FOURIER TRANSFORMSTo obtain the solution of the problem in the physicaldomain, we have to invert the double transforms in equations(91)-(93), these expressions can be formally expressed asfunctions of x and the parameters of the Fourier and Laplacetransforms q and s, of the form f x,q,s .First, we invert the Fourier transform using the inversionformula given previously. This gives the Laplace transformexpression f x, y,sof the function f x,q,s as1 iq yf x, y,s e f x,q,sdq 2where f eand2o0 cos q y f isin q y f dq,eo(94)f denote the even and the odd parts of thefunction f x,q,s respectively.We shall now outline the numerical inversion method used tofind the solution in the physical domain. For fixed values of x, y,and q the function inside braces in the last integral can beconsidered as a Laplace transform g s of some function g(t).The inversion formula for the Laplace transform can be writtenascist1gt e g sds2 i , (95)ciwhere c is an arbitrary real number greater than all the real partsof the singularities of g s . Taking s = c +i y, the above integraltakes the form cte it ygt e g c iydy2 , (96)Expanding the function h t exp ctg t in a Fourierseries in the interval [0,2L], we obtain the approximate formula[16]:g t g t E , (97) Dwherek1cte i k t / Lgt Re e g c ikt / LL for 0 t 2L , (98)E D, the discretization error, can be made arbitrary small bychoosing c large enough [16].Since the infinite series in equation (98) can be summed up to afinite number N of terms, the approximate value of g(t) becomesNctegNt Re e g c ik t / LL i k t / L for 0 t 2L .(99)k1Using the above formula to evaluate g(t), we introduce atruncation error E which, must be added to the discretizationTerror to produce the total approximation error.Two methods are used to reduce the total error. First, the“Korrecktur” method Honig and Hirdes [17] is used to reducethe discretization error. Next, the -algorithm is used to reducethe truncation error and hence to accelerate convergence.The Korrecktur method uses the following formula to evaluatethe function g(t)2cLgt gt e g2L t ED ,where the discretization error ED ED. Thus, theapproximate value of g(t) becomesg2cL(t) g (t) e g 2L t , (100)NK N N where N is an integer such that N N.We shall now describe the ε-algorithm that is used to acceleratethe convergence of the series in equation (99). Let N=2q+1where q is a natural number, and letms c ,mk1kbe the sequence of partial sums of a series (99). We define theε-sequence byo, m 0 , 1, m s m and p1,m11 p1,m1 , p 1,2,3,... .It can be shown that the sequence, , ,......... , converges to D1,1 3,1 5,1 N,1g tp ,m1 p ,m E faster thanthe sequence of partial sums s m , m 1,2,3,... .The actual procedure used to invert the Laplace transformconsists of using equation (99) together with the ε-algorithm.The values of c and L are chosen according to the criteriaoutlined in [17].9

JOURNAL OF THERMOELASTICITY ISSN 2328-241X VOL.1 NO. 2 June 2013 oF i ijElectric conductivityComponents of Lorentz forceKronecker delta functionγ =( 3λ 2 μ)ε 1 =α =β 1 =ε 2 =ηE2γ Tορ C ( λ 2μ )2μHο ολ2μσμccο212ηρCE=k2β = ( λ 2 μ ) o=kομ0e 2 T, Peltier coefficient at T o[10] M. A. Ezzat, H. M. Youssef, “Generation of generalizedmagneto-thermoelastic waves by thermal shock in ahalf-space of finite conductivity,” Ital. J. Pure Appl. Math.vol.19, pp. 9–26, 2006.[11] M. A. Ezzat, H. M. Youssef, “GeneralizedMagneto-Thermoelasticity in Perfectly ConductingMedium,” Int. J. Solids and Struc. vol.42, pp. 6319-6334,2005.[12] M. A. Ezzat, H. M. Youssef, “State space approach forconducting magneto-thermoelastic medium with variableelectrical and thermal conductivity subjected to ramp-typeheating,” J. Thermal Stresses. vol. 32, pp. 414-427, 2009.[13] J. Ignaczak, “A strong discontinuity wave inthermoelasticity with relaxation times,” J. Thermal Stresses,vol. 8, pp. 25- 40, 1985.[14] J. Ignaczak, “Decomposition theorem for thermoelasticitywith finite wave speed,” J. Thermal Stresses. vol. 1, pp.41-52, 1978.[15] M. A. Ezzat, A. S. El Karamany, Propagation ofdiscontinuities in magneto-thermoelastic half-space, J.Thermal Stresses. vol. 28. pp. 331-358, 2008.[16] J.A. Shercliff, “Thermoelectric magnetohydrodynamics,” J.Fluid Mech. vol.191, pp. 231-251, 1979.[17] G. Honig, U. Hirdes, “A method for the numericalinversion of the Laplace transform,“ J. Comp. Appl. Math.vol.10, pp.113 –132, 1984.[18] M. A. Ezzat, E. S. Awad, “Micropolar generalizedmagneto-thermoelasticity with modified Ohm’s andFourier’s laws,” J. Math. Anal. Appl. vol. 99, pp. 353-364,2009.REFERENCES[1] R. B. Hetnarski, Thermal Stresses I, 2nd series, vol. 1,North-Holland, 1986.[2] G. S. Nolas, D. Johnson, D. G. Mandrus, Thermoelectricmaterials and devices. In: Proc. (2002), MaterialsResearch Society. Warrendale, PA, 691, 2002.[3] T. M. Tritt, Semiconductors and semimetals, recent trendsin thermoelectric materials, Research, Academic Press,San Diego, 69–71, 2000.[4] T. M. Tritt, M. G. Kanatzidis, H. B. Lyon, G. D. Mahan,New material for small-scale thermoelectric refrigerationand power generation applications, Materials ResearchSociety, Warrendale, PA, 545, 1998.[5] D. T. Morelli, Thermoelectric devices. In: Trigg G L,Immergut E H (eds.) Encyclopedia of Applied Physics,Wiley-VCH, New York, 21 (1997) 339.[6] D. M. Rowe, CRC Handbook of Thermoelectrics, CRCPress, 1995.[7] R. B. Hetnarski and J. Ignaczak, “Generalizedthermoelasticity,” J. Thermal Stresses, vol. 22, pp.451–476, 1999.[8] 8. H. Lord, Y. Shulman, “Generalized dynamic theory ofthermoelasticity,” J. Mech. Phys. Solids, vol.15, pp.299-309, 1967.[9] H. H. Sherief, M. A. Ezzat, “A thermal shock problem inmagneto-thermo-elasticity with thermal relaxation,” Int. J.Solids and Struct. vol. 33, pp. 4449-4459, 1996.11

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