The influence of the way of modelling the radiative heat transfer on the temperature distribution in a charge heated inductively

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1 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) The nfluence of the way of modellng the radatve heat transfer on the temperature dstrbuton n a charge heated nductvely ROMAN PRZYŁCKI ADAM KACHEL JERZY BARGLIK Department of Electrotechnology Slesan nversty of Technology Krasńskego 8, 0-09 Katowce POLAND roman.przyluck@polsl.pl Abstract: The paper presents an analyss how the way of modellng radatve heat transfer nfluences on temperature dstrbuton n a charge heated by nducton. The calculatons were provded as coupled allowed couplng of electromagnetc and temperature felds. Two-dmensonal calculaton model was used. The results were obtaned from the experment consstng of several calculaton cases. They dffer each other n the ntensty of heatng and the way n whch the heat transfer was taken nto consderaton. There were modeled cases where multple reflectons phenomena was taken nto account, effectve emssvty was appled, free heat radaton to half space was analyzed and fnally radaton heat transfer was completely neglected. Depeng on the adopted models the maxmum temperature devaton n the heated charge was determned as 96 o C. Key-Words: Radatve heat transfer, nducton heatng. Introducton Inducton heatng seems to be more and more popular heatng method. The desgn of nducton heatng systems takes advantage of computer smulatons. A smulaton model of an nducton heatng process must take nto consderaton an analyss of electromagnetc and temperature felds, and n some cases also the felds of phase transton or stress / dsplacement. Apart from the number of the physcal felds calculated, correspong models dffer also n the appled smplfcatons. The paper analyses the nfluence that those varous models of radatve heat transfer have on the obtaned results. In the experment temperature dstrbuton was montored on two cross-sectons p and p (see Fg.). Calculatons were made by usng professonal software Flux D and the own sngle purpose program Trad prepared by the authors, whch allows the calculaton of radatve heat transfer wth takng nto account multple reflectons m. The dmensons of the nductor are the followng: heght of nductor profle h w = 0.0m, ts wdth w w = 0.0m, thckness of nductor s wall t w = 0.00m. The dmenson of the model n the drecton on axs z s 0.5m. Calculaton model A heater model chosen for the experment was one for flat charges wth the geometry shown n Fg.. The basc dmensons of the model are as follows: heght of the charge h c = 0.m, half of the wdth of the charge w c = 0,0m, ar gap between the charge and thermal nsulaton a g = 0.007m, thckness of thermal nsulaton w = 0.005m, heght of thermal nsulaton h = h w = Fg.. Calculaton model geometry ISSN: ISBN:

2 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) Lookng at the desgn, t s a typcal heatng system because the general gudelnes suggest that the mnmal rato of nductor length to charge length should be between. and. The charge was made from non-magnetc steel (ASTM 3), thermal nsulaton was a plate made of ceramc fbre, and the nductor was made from copper (Fg., Fg.). Thermal calculatons were conducted on the bass of Fourer-Krchhoff s equaton (3) [], [3]: T ρc + ( λ T ) = q (3) t ρ - densty, kg/m 3, c - specfc heat, J/(kg K), T - temperature, K, t - tme, s, λ - thermal conductvty, W/(mK), q - volumetrc densty of heat sources, W/m 3. Wth approprate boundary condtons. For all calculaton varants the followng boundary condtons (Fg.) were dentcal: dt fd,da,ae : λ = 0 () ab, cd, gh,h, jk, kl, mn,no,op : dt λ = α ( T Ta ) + εσ ( T T a ) (5) α = 0, T a = 0 for edges: ab, cd : ε = 0.6; mn, op : ε = 0.; gh, h, jk,kl,no : ε = 0.5 Fg.. Boundary condtons locaton bc, gl : dt λ = α ( T Ta ) (6) The analyss of electromagnetc feld was performed on the bass of Equaton ()[], []: jωγ A + A = γ V () µ A - magnetc vector potental, Vs/m, ω - angular frequency, rad/s, γ - conductvty, S/m, µ - magnetc permeablty, Vs/(Am), V - electrc potental of source, V. Wth approprate boundary condtons (). ae, ef, fd, da : A = 0 () α = 0, T a = 0 qr,rs,st, tq, ef : T = const (7) for edges: qr, rs, st, tq : T = 30 o C; ef : T = 0 o C. α - convecton heat transfer coeffcent, W/(m K), ε - emssvty, -, σ - Stefan-Boltzmann constant, W/(m K ), T a - ambent temperature, o C. Depeng on the varant, the boundary condtons descrbng heat transfer through radaton for edges bc, gl were changed. They are dscussed n detal n secton.. ISSN: ISBN:

3 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09). Models of radatve heat transfer The nducton heatng s often used as an ntroductory stage n the metal thermal treatment and forgng. The temperatures n the charge, depeng on the knd of the heated metal, may be dffer from 500 to 00 o C. It s wdely beleved that for so hgh temperatures radatve heat transfer must not be neglected [], [5], [6]. Some of the commercal programs that are used for a coupled analyss of electromagnetc and temperature felds, lke Opera, do not allow for radatve heat transfer n any way. Others, lke Flux, can model a boundary condton as n Equaton (5), whch represents radatve heat transfer to half-space. As can be seen n Fg., the charge s surrounded by a thermal nsulaton layer, whch should be taken nto account whle consderng radatve heat transfer. There are programs whch allow the assumpton of a boundary condton for radatve heat transfer between two surfaces (whch s suffcent for the adopted calculaton model (Fg.)), for example Fluent, but t cannot carry out a full analyss of electromagnetc feld. It can be concluded from the specfcatons, that program Ansys allows for radatve heat transfer by the method of multple reflectons, however, the authors have not as yet come to a possblty of usng ths program. Calculaton model presented n Fg. suggests that radatve heat transfer n ths system should be modeled takng nto account multple reflectons. A mathematcal model presented below s wrtten for a cylndrcal system (as more complex one because t takes nto consderaton self-rradance). A computer program Trad prepared on the bass of ths model makes possble calculatons for both flat and cylndrcal systems. The followng smplfyng assumptons were adopted n the experment: o The whole system s n thermal steady state (quas-steady state); o Both prmary and reflected radaton are of dffusve (Lambertan) type; o Radatng surfaces are curved (for cylndrcal system); o Radatng surfaces emt energy evenly through the whole spectral range (grey bodes). A smple formula for radaton energy balance, after takng nto account Lambert and Krchhoff laws, yelds a dfferental equaton system of (8) and (9) s obtaned [3], [7]. The soluton of the equatons system (8), (9) descrbes total radant extance emtted from the surfaces S and S (Fg. 3). Total radant extance descrbes effectve radaton of the surface, whch s a sum of prmary and reflected radatons. Equatons (0) and () descrbe the resultng rradance I and heat transfer Q for the surfaces S and S, respectvely. It must be underlned that for curved surfaces systems the reflected radaton s the result both mutual- and selfrradance. That s why Equaton (8) contans ntegraton over all surfaces. In the case of flat surfaces selfrradance does not occur, thus n Equaton (9) doman of ntegraton was reduced to the opposte surface. a) b) Fg. 3. System wth radatve heat transfer for a - flat confguraton, b - cylndrcal confguraton ( x ) = a σt + ( a ) k( x,x ~ ) ( x~ ) + k S I S ( x,x ) ( x ) ds ( x ) = aσt + ( a ) k( x, x ) ( x ) d S S (x ) = a (x ) aσt ~ ds + (8) (9) (0) ISSN: ISBN:

4 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) k pjq [ I (x ) σt ] Q ( x ) = a () p = = a p p σ T n k j j= q= jq p pjq + k p jq ( a ) jq p jq ( x, x ) j d S d S j () (3) to use t because edges bc, gl are parallel and usually located close to each other. Emssvty determned n ths way may be used n Equaton (5) descrbng a flux of energy emtted by the heated surface, but stll the problem of determnng the ambent temperature (T a ) remans unsolved. Another possble smplfcaton s an assumpton that the charge s not surrounded by a thermal nsulaton layer and emts energy freely to half-space. The nfluence of such smplfcatons on the obtned results s presented below n the secton 3. - total radant extance from -th surface, W/m, a - absorpton coeffcent a = ε, -, k (x,x j ) - optcal couplng functon (ntegral kernels), -, I - rradance on the -th surface, W/m, Q - radatve heat exchange, W/m, p - dscreet element of surface -th, -, n j - number of j-th surface elements, -, =, - ndex of surface, -. 3 Calculaton experment Calculaton varants dffer n the way the radatve heat transfer s modelled on the edges bc, gl, n assumed ambent temperature, and n the ntensty of heatng. The montored quantty was temperature dstrbuton obtaned on cross-sectons p and p (Fg. ). In a general case such a system of Fredholm ntegral equatons of the second knd that was obtaned cannot be solved by analytcal methods. In ths paper the soluton was obtaned wth the use of numercal teratve method wth a proper dscretzaton of surface, owng to whch equatons (8) and (9) were reduced to lnear equatons (). In order to mnmze numercal errors resultng from a dscrete dvson of the surface nto elements the kernel k(x,x j ) was averaged by pars of elements accorg to formula (3), where, depeng on the densty of surface dvson and the system geometry, the ntegral may be reduced do sum of a few terms. ε = + ε ε d () T λ = εσ ( T Ta ) (5) ε - emssvty of bc edge, -, ε - emssvty of gl edge, -. Another way of modellng of the radatve heat transfer may be usng of the effectve emssvty formulaton expressed by Equaton () [3]. It s possble Fg.. Cross-sectons on whch the temperature was montored The calculatons were conducted untl the temperature n the heated charge was around 000 o C for the varant n whch radatve heat transfer was modelled usng program Trad, that s for 36 seconds for ntensve heatng and 0 seconds for non-ntensve heatng. For ntensve heatng (nductor current 3000A) the followng varants were consdered: o MI- radatve heat transfer modelled wth the use of program Trad; o MI - radatve heat transfer modelled wth the use of effectve emssvty () and boundary condton (5). Ambent temperature T a = 0 o C; ISSN: ISBN:

5 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) o MI3 - radatve heat transfer modelled as free emsson to half-space, T a = 0 o C o MI radatve heat transfer neglected. Smlar calculaton varants were carred out for nonntensve heatng (nductor current 000A): o LI- radatve heat transfer modelled wth the use of program Trad; o LI - radatve heat transfer modelled wth the use of effectve emssvty () and boundary condton (5). Ambent temperature T a = 0 o C; o LIa - as n LI but assumed ambent temperature T a = 000 o C; o LIb - as n LI but assumed ambent temperature T a = 500 o C; o LI3 - radatve heat transfer modelled as free emsson to half-space, T a = 0 o C; o LI - radatve heat transfer neglected. maxmum temperature devaton s 9 o C (between varants MI and MI3). The temperature dstrbuton closest to the reference varant was obtaned for varant MI, that s for the model usng the effectve emssvty. Fgure 5 presents an example temperature dstrbuton for the whole heatng system for varant MI. In Fgure 6 an analogous dstrbuton s presented for varant MI. Fg.6. Temperature dstrbuton for varant MI, after 36 seconds of heatng Fg.7. Temperatures dstrbutons on secton p for ntensve heatng after 36 seconds of heatng Fg.5. Temperature dstrbuton for varant MI, after 36 seconds of heatng Comparng the two fgures t can be seen that only takng nto account multple reflectons allows modelng of temperature dstrbuton n the whole system (charge, thermal nsulaton, nductor). A smlar concluson can be reached on the bass of Fgure 7, where the dstrbuton of temperature on secton p l s presented for all varants of ntensve heatng. Maxmum temperature devaton n the charge on the cross-secton s 9 o C. In relaton to reference varant MI The nfluence of the model of radatve heat transfer on the temperature dstrbuton on the surface of the charge (secton p ) (Fg.9) s smlar. Maxmum temperature devaton n comparson wth the reference varant n ths case s 6 o C. From the dagram t can also be concluded that modelng of radatve heat transfer usng effectve emssvty produces the results closest to the ones obtaned wth the use of the model allowng for multple reflectons. Fgures 0 and present temperatures dstrbutons obtaned for non-ntensve heatng. As t could be ISSN: ISBN:

6 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) expected, the dfferences between temperatures are much more sgnfcant n ths case. presentng temperatures dstrbutons on secton p for non-ntensve heatng. The presented experment does not allow any general conclusons. Yet as t can be seen n Fgures 9 to, the sole change n ntensty of heatng causes that dfferent smplfed models of radatve heat transfer produce results closest to the referental results. Fg. 8. Temperatures dstrbutons on secton p lmted to the area of the charge for ntensve heatng after 36 seconds Fg.. Temperatures dstrbutons on secton p for nonntensve heatng after 0 seconds Fg. 9. Temperatures dstrbutons on secton p for ntensve heatng after 36 seconds Fg. 0. Temperatures dstrbutons on secton p lmted to the area of the charge for non-ntensve heatng after 0 seconds Maxmum temperature devaton n the area of the charge on cross-secton p l n relaton to reference varant s 6 o C. The smallest devaton can be observed n the case where the modelng of the radatve heat transfer was neglected. A smlar dependency can be seen n Fg. For ntensve heatng the best smplfcaton s to use effectve emssvty (varant MI), for non-ntensve heatng the results that are closest to the referental ones are obtaned n the model n whch radatve heat transfer s neglected (LI). It can be thus concluded that t s very dffcult to select a smplfed model of radatve heat transfer wthout adversely affectng the accuracy of the calculatons. Concluson The paper analyses the nfluence of dfferent models of radatve heat transfer on the temperature dstrbuton n the nducton-heated charge. Ten dfferent models were analysed. The model takng nto account multple reflectons was adopted as a referental varant (the authors program Trad was used). In the case of ntensve heatng, the temperature dstrbutons that were closest to the referental dstrbutons were obtaned for varant MI, n whch the heat transfer was modeled wth effectve emssvty. However, n the case of non-ntensve heatng the best results were obtaned n the varant where the transfer was neglected. Therefore, t can be concluded that smplfed models of radatve heat transfer cannot be used f the accuracy of calculatons s to be mantaned. Addtonally, only the model allowng for multple reflectons makes t possble to carry out an analyss of temperature feld n the whole heatng system consstng of the charge, thermal nsulaton and nductor. ISSN: ISBN:

7 Proceegs of the 7th IASME / WSEAS Internatonal Conference on HEAT TRANSFER, THERMAL ENGINEERING and ENVIRONMENT (HTE '09) Stll, usng Trad s not flawless. Frst of all, the calculaton tme s sgnfcantly longer, for example n the case under consderaton, one tme step wthout Trad lasted mnutes, whle the same step wth Trad took around 30 mnutes. Also, n the case when the heat flux exchanged between surfaces s substantal a queston of the calculaton stablty occurs. Ths problem could be partly solved by shortenng of the tme step. Acknowledgement The work s supported by the Mnstry of Scence and Hgher Educaton (3 T08C 06 30). References: [] Flux D ser s Gude, Cedrat, 006. [] J.Turowsk, Oblczena elektromagnetyczne elementów maszyn urządzeń elektrycznych, WNT, 98. [3] B.Stanszewsk, Wymana cepła, podstawy teoretyczne, PWN, 979. [] A.Kachel, R.Przyłuck, Smulaton of nducton heatng process wth radatve heat exchange, Journal of Achevements n Materals and Manufacturng Engneerng, Vol., May 007, pp [5] J.Barglk, M.Czerwńsk, M.Herng, M.Wesołowsk, Radaton n Modelng of Inducton Heatng Systems, IOS Press, 008, pp. 0-. [6] R.Przyłuck, A. Kachel, Model oblczenowy ndukcyjnego układu grzejnego z uwzglęenem radacyjnej wymany cepła, Przegląd Elektrotechnczny, 7/008, pp [7] P.Dutre, Global Illumnaton Compendum, Cornell nversty, 00. ISSN: ISBN: