On Bessel Composite Function of the Third Order and First Kind Solution to Damped Wave Conduction and Relaxation Equation

Transcription

1 Jornal of Basi and Applied Physis Feb., ol. ss., PP. 6- On Bessel Composite Fntion of the Third Order and First Kind Soltion to Damped ave Condtion and elaxation Eqation Kal enganathan Sharma Lone Star College, Hoston, T, USA Abstrat- Damped wave ondtion and relaxation eqation is solved in three dimensions in order to gage the ampaity risks in PCB interonnetions. /r sbstittion is sed and the temperatre profile is obtained in three dimensions and in one dimension. n the reeping transfer limit, the spatio-temporal profile is given as a modified Bessel omposite fntion in spae and time of the third order. Three regimes of soltion are identified; (i) lag regime; (ii) rising regime given by Bessel omposite fntion in spae and time and; (iii) rising regime given by modified Bessel omposite fntion in spae and time. n the general ase, the order of the soltion is fond to be 7/ and the order of the soltion is two in the ase of one dimension. Keywords- Damped ave Condtion and elaxation; elativisti Transformation; Bessel Composite Fntion of Third Order; Ampaity isks; Seond Law of Thermodynamis. NTODUCTON nformation handling systems (HS) are sed to proess, ompile, store and ommniate data for bsiness personnel or other similar prposes. Appliation speifi reqirements are sed to onfigre HS. t may se one or more ompter systems, data storage systems and networking systems. Depending on the information handled, the HS may differ from one appliation to another. They may differ as to what information is handled, how the information is proessed, stored, ommniated and how soon and effiiently the information may be proessed, stored or ommniated. HS may be onfigred for general se or for speifi se sh as passenger reservation in the airlines, railways, redit ard transations, global ommniations, and enterprise data storage. A variety of hardware and software omponents may be onfigred to perform different tasks sh as storage, ommniation, proessing. Dring the design and manfatre of HS, one salient onsideration is the detetion of areas of the system or irit that are prone to ertain risks. Corretive steps have to be taken in order to minimize these risks. One example is the ampaity risks on irit boards. The irit or interonnetion may go bak to the drawing board stage for reroting. Ampaity is defined [] as the rrent in amperes that a ondtor an arry ontinosly nder the ondition of se withot exeeding the temperatre rating or fse point. The simlation, analysis, validation to laboratory data, diret rrent (DC) and short dration, alternating rrent (AC) transient plse effets on the ampaity risks on PCB interonnetivity are needed. This wold reslt in assessment of ampaity risks on PCB interonnetions and improvement of the reliability of the operation of HS. One ritial aspet is the omptation of heat ondtion at short times where non-forier effets an be expeted. The damped wave ondtion eqation may be appliable here. The damped wave ondtion and relaxation eqation was soght over Forier s law of heat ondtion for eight reasons by Sharma []. The damped wave ondtion and relaxation eqation was originally sggested by Maxwell [], and postlated independently by Cattaneo [, 5] and ernotte [6]. The damped wave ondtion and relaxation eqation in one dimension aross onstant area may be written as follows; q x T q x ka r A () x t here q x is the heat transfer rate in x diretion in (watts, w), A is the ross-setional area aross whih the heat ondtion ors in (m ), k is the thermal ondtivity of the material in (w.m -.K - ), r is the relaxation time (s). eviews of the se of this eqation have been presented by Joseph and Preziosi [7, 8] and Ozisik and Tzo [9]. Extensive theoretial treatments of the eqation have been reported by Tzo [9] and Sharma []. Experimental measrement of relaxation times has been reported by Mitra et al. [] reently for biologial materials. Taitel [] fond an overshoot in his transient temperatre soltion for a finite slab sbjet to onstant wall temperatre bondary ondition. Bai and Lavine [] was onerned - 6 -

2 Jornal of Basi and Applied Physis Feb., ol. ss., PP. 6- abot Eq. () violating the seond law of thermodynamis. Zanhini [], Barletta and Zanhini [5] allated an entropy prodtion term and are onerned of a violation of Clasis ineqality. Al Nimir et al. [6, 7] disssed an overshoot and eqilibrim entropy prodtion. Haji Sheik et al. [8] pointed ot some anomalies in Eq. (). Tzo [9] has fond Eq. () to be admissible within the framework of the seond law of thermodynamis. Sharma [9-5] has presented losed form analytial soltions for different geometries within the bonds of the seond law of thermodynamis; the damped wave eqation was derived by aonting for aeleration of the moleles in the Stokes-Einstein formlation, demonstrating that the overshoot disappeared sing the physially reasonable initial ondition. Final ondition in time was sed and bonded soltions withot violating the seond law of thermodynamis were presented. Antaki [6] has disssed some analytial soltions for onvetive bondary ondition. Analytial soltions are derived in this stdy for the damped wave ondtion eqation in three dimensions (D). The soltions are in the form of Bessel omposite fntion of the third order. The derived spatio-temporal temperatre profiles an be sed to gage the ampaity risks in PCB interonnetions.. THEOY The defet region is modeled as a spherial shell with radis. The effets of the defets an be in any diretion. The transient temperatre field srronding the srge needs to be represented sing spherial oordinates. Spherial oordinates an be sed when events in arbitrary diretion beomes important. Spherial oordinates are sed when the ross-setional area along the path of heat ondtion hanges sh as in onial objets. The governing eqation for temperatre, T ( K) an be obtained as follows; an energy balane on the spherial shell at a distane r from the origin an be written. hen ombined with the damped wave diffsion and relaxation eqation, it beomes Let T T α T r r () t t r r r T T t Ts T ; ; r Eq. () is made dimensionless by sing the variables defined in Eq. (), then Eq. () beomes The time and spae onditions an be written as r α r, (5), (6) >,, (7), (8) These bondary onditions represent the onstant wall temperatre srfae bondary ondition in an infinite medim. The infinite spae assmption lends the soltion with fewer terms and is physially reasonable. The two time onditions, i.e., one at initial time and the other after infinite time, has been assmed as a first approximation. Sbstitting / into Eq. (), we an get The damping term an be removed by the sbstittion exp(-n) and is defined in Eq. (). As shown in the preeding setions for n ½, Eq. (9) beomes; () () (9) - 7 -

3 Jornal of Basi and Applied Physis Feb., ol. ss., PP () Let The term / an be negleted for a large, bease at large distanes from the point of distrbane, the effet annot be seen to an appreiable extent. an be expeted to be small for a large r as exp (-/)/r. For a large, Eq. () an be modified as follows. Now eah of the terms in Eq. () in terms of and is transformed in terms of, 8 () () 6 () Comparing Eq. () with the generalized Bessel eqation, the soltion is; a ; b ; ; d -/6; s ½ The order p of the soltion is then p and s d. The generalized Bessel eqation and forms of Bessel soltion are given in []. n the following eqation (), is the order of the fntion, is the modified Bessel fntion of the first kind and seond order and K is the modified Bessel fntion of the seond kind and seond order. ( ) ( ) K () here an be seen to be zero as is finite and not infinitely large at. At, the veloity of heat times, the time ordinate t beomes the spae ordinate r. This an be seen at the point where the distrbane wold have travelled at the speed of heat. ( ) e (5) here an be eliminated between the above eqation and the eqation from the bondary ondition. The bondary ondition is given by Eq. (7). ( ) e (6) Ths for >

4 Jornal of Basi and Applied Physis Feb., ol. ss., PP (7) For >, J (8) On examining Eq. (8), it an be seen that the Bessel fntion of the seond order and first kind will go to zero at some vale of. The first root of the Bessel fntion ors when ½( - ) / 5.56 (9) How 5.66 was obtained is disssed in details by graph in ef. []. t has something to do with the zeros of the fntion onsidered. n this ase, it s the first root of the Bessel fntion. Or () hen an exterior point in the infinite sphere is onsidered, a lag time an be allated prior to when there is no heat transfer to that point. After the lag time, there exist two regimes. One is desribed by Eq. (8) and the third regime is desribed by Eq. (7). Ths, lag sqrt( p ) Fig. Three regimes of dimensionless temperatre at an exterior point from the defet All the three dimensions of the spherial oordinates are onsidered. The /r sbstittion is sed and the spatio-temporal temperatre in the infinite sphere is derived as follows.

5 Jornal of Basi and Applied Physis Feb., ol. ss., PP The governing eqation for the temperatre is obtained when the energy balane eqation and the onstittive damped wave diffsion and relaxation eqation are ombined. The eqation is made dimensionless by sing the sbstittions in Eq. (). Then the governing eqation in three dimensions in spherial oordinates an be written as; ot sin () Sbstitting / into Eq. (), we an get Sin ot () The damping term an be removed by a wexp(-n) sbstittion. As shown in the preeding setions for n ½, Eq. () beomes ot sin () For small by Taylor approximation as shown in [], sin () Let, (5) Sin, Then, Sin (6) Eq. () then beomes for a large, (7) Consider the transformation, ( - - ) As shown in Eq. (), the derivatives in Eq. (7) in for independent variables beome onverted into one independent variable (), and Eq. (7) beomes 8 (8)

6 Jornal of Basi and Applied Physis Feb., ol. ss., PP Or, 6 9 (9) Comparing Eq. (9) with the generalized Bessel eqation given in [], the soltion is a 9/; b ; ; d -/6; s ½ Then the order p of the soltion is p 7/ ( ) ( ) () here an be seen to be zero as is finite and not infinitely large at. An approximate soltion an be obtained by eliminating between the above eqation and the eqation from the bondary ondition. The eqation from the bondary ondition an be written as; ( ) e () Dividing Eq. () by Eq. () () For small, J () n the reeping heat transfer limit, Eq. (7) an be approximated as; () Creeping limit is sitable for problems in spherial geometry. The harateristis of transient temperatre lose to the sphere an be expeted to be ompletely different from that far from the sphere. Creeping flow assmptions have been made sessflly in flid mehanis in the flow with sphere problems. By analogy between momentm transfer and heat transfer, Eq. (7) is approximated as Eq. (). After the transformation, the PDE with variables is onverted to a Bessel eqation with one variable:

7 Jornal of Basi and Applied Physis Feb., ol. ss., PP (5) The order of the Bessel soltion for Eq. (5) an be allated by omparing Eq. (5) with the generalized Bessel eqation and the soltion is when a, b,, d -/6, s ½, the order p of the soltion is ( ) ( ) K (6) an be seen to be zero as is finite and not infinitely large at. The bondary ondition is given in Eq. [7]. An approximate soltion an be obtained by eliminating between the above eqation and the eqation from the bondary ondition. The eqation from the bondary ondition an be written as; ( ) e (7) Dividing Eq. (6) by Eq. (7), (8) For small, J (9) For small, J () The soltion is in terms of a Bessel omposite fntion of the third order and first kind for a small and a modified Bessel omposite fntion of the third order and first kind for a large. The first root of the Bessel fntion of the third order is allated by sing 7 terms of the series expansion of the Bessel fntion in a Pentim miroproessor sing a Miorsoft Spreadsheet p to for deimal plaes. The root is fond to be 6.8.

8 Jornal of Basi and Applied Physis Feb., ol. ss., PP. 6- ½( - ) / 6.8 () Or () hen an exterior point in the infinite sphere is onsidered, a lag time an be allated prior to when there is no heat transfer to that point. After the lag time, there exist two regimes. One is desribed by Eq. () and the third regime is desribed by Eq. (9). Ths, lag p p p 6.88 (). CONCLUSONS Analytial soltions are presented to evalate the ampaity risks in PCB interonnetions. Earlier stdies have sed nmerial soltions. n [], onjgate gradient nmerial analysis method is sed for the non-forier heat ondtion eqation. n this stdy, analytial soltions are presented for the damped wave ondtion and relaxation non Forier model. Method of relativisti transformation is sed. Soltions in one dimension and in three dimensions are given. The orders of the Bessel omposite fntions are: (i) 7/ for the dimensional ase; (ii) for the three dimensional ase and (iii) for the ase with the reeping limit assmption.three regimes an be seen in eah soltion: (i) inertia lag regime; (ii) rising regime haraterized by Bessel omposite fntion and; (iii) rising regime haraterized by modified Bessel omposite fntion in spae and time. The final ondition is made to obtain physially realisti soltions. These soltions are within the framework permitted by the seond law of thermodynamis. Other reports with violating the seond law and overshoot orrene an be seen de to the se of nrealisti initial amlation ondition []. The orrene of soltion with spatio-temporal symmetry an have impliations in the speial theory of relativity proposed by Sir Albert Einstein. Model an be sed to prevent potential damage to predetermined exitation. EFEENCES []. J. Mrgan and S. Krishnan, System and method to analyze and determine ampaity risks on PCB interonnetions, US Patent , Dell Prodts, ond ok, T. [] K.. Sharma, On the soltion of damped wave ondtion and relaxation eqation in a semi-infinite medim sbjet to onstant wall flx, nt. Jornal of Heat and Mass Transfer, vol. 5, pp. 6-6, 8. [] C. Cattaneo, A form of heat ondtion whih eliminates the paradox of instantaneos propagation, Comptes ends, vol. 7, pp. -, 958. [] C. Cattaneo, Slla Codzione del Calone, Atti. Sem. Fis. Univ. Moderna,, 8, 98. [5] P. ernotte, Les paradoxes de la theorie ontine de l eqation de la haler, C.. Hebd. Sean. Aad. Si., vol. 6(), pp. 5-55, 958. [6] D. D. Joseph and L. Preziosi, Heat waves, eviews of Modern Physis, vol. 6, pp. -7, 989. [7] D. D. Joseph and L. Preziosi, Addendm to heat waves, eviews of Modern Physis, vol. 6, 75-9, 99. [8] M. N. Ozisik and D. Y. Tzo, On the wave theory of heat ondtion, ASME J of Heat Transfer, vol. 6, pp , 99. [9] D. Y. Tzo, Maro to mirosale heat transfer: the lagging behavior, CC Press, New York, 996. [] K.. Sharma, Damped wave transport and relaxation, Elsevier, Amsterdam, 5. [] K. Mitra, S. Kmar, A. edavarz and M. K. Moallemi, Experimental evidene of hyperboli heat ondtion in proessed meat, Jornal of Heat Transfer, vol. 7, , 995. [] Y. Taitel, On the paraboli, hyperboli and disrete formlation of heat ondtion eqation, nternational Jornal of Heat and Mass Transfer, vol. 5(), pp. 69-7, 97. [] C. Bai and A. S. Lavine, On hyperboli heat ondtion and seond law of thermodynamis, Jornal of Heat Transfer, vol. 7 (), pp. 56-6, 995. [] E. Zanhini, Hyperboli heat ondtion theories and non-dereasing entropy, Phy. eview B-Condensed Matter and Material Physis, vol. 6(), pp , 999. [5] A. Barletta and E. Zanhini, Thermal-wave heat ondtion in a solid ylinder whih ndergoes a hange of bondary temperatre, Heat and Mass Transfer/ arema-nd Stoffebertragng, vol. (), pp. 85-9, 997. [6] M. A. Al-Nimr and O. M. Haddad, The dal-phase-lag heat ondtion model in thin slab nder a) Fltating Thermal Distrbane, Heat Transfer Engineering, vol. (5), pp. 7-5,. [7] M. A. Al-Nimr and M. Naji, The hyperboli heat ondtion eqation in an anisotropi material, nt. J of Thermophysis, vol., pp. 8-87,. [8] A. M. Haji-Sheik,. J. Minkowyz, and E. M. Sparrow, J. Heat Transfer, vol. (), pp. 7-9,. [9] K.. Sharma, s overshoot a mathematial artefat, JP Jornal of Heat and Mass Transfer, vol. 7(), pp. 5-6,. [] K.. Sharma, On analytial soltions dring damped wave ondtion and relaxation in a finite slab sbjet to the onvetive bondary ondition, nternational Jornal of Thermophysis, vol. (), pp. -,. [] K.. Sharma, Damped wave ondtion and relaxation in ylindrial and spherial oordinates, Jornal of Thermophysis and Heat Transfer, vol. (), pp ,

9 Jornal of Basi and Applied Physis Feb., ol. ss., PP. 6- [] K.. Sharma, Manifestation of aeleration dring transient heat ondtion, Jornal of Thermophysis and Heat Transfer, vol. (), pp , 6. [] K.. Sharma, A forth mode of heat transfer alled damped wave ondtion, nd Annal Convention of Chemists Meeting, Santiniketan, ndia, Febrary, 6. [] K.. Sharma and. Trton, Mesosopi approah to orrelate srfae heat transfer oeffiients with pressre fltations in gas-solid flidized beds, Powder Tehnology, vol. 99(), pp. 9-8, 998. [5] K.. Sharma, On the derivation of an expression for relaxation time from stokes-einstein relation, rd a) ACS National Meeting, Chiago, L, Marh 7, Amerian Chemial Soiety, ashington, DC. [6] P. J. Antaki, Soltion for non-forier dal phase lag heat ondtion in a semi-infinite slab with srfae heat flx, nternational Jornal of Heat and Mass Transfer, vol. (), pp. 5-58,