Numerical Simulation of Heat Transfer during Microwave Heating of Magnetite

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1 ISIJ Iteratioal, Vol. 51 (011), No. 6, pp Numerical Simulatio of Heat Trasfer durig Microwave Heatig of Magetite Zhiwei PENG, 1) Jia-Yag HWANG, 1) Matthew ANDRIESE, 1) Waye BELL, 1) Xiaodi HUANG 1) ad Xili WANG ) 1) Departmet of Materials Sciece ad Egieerig, Michiga Techological Uiversity, Houghto, MI 49931, USA. jhwag@mtu.edu, zpeg@mtu.edu, mdadrie@mtu.edu, wmbell@mtu.edu, xihuag@mtu.edu ) School of Techology, Michiga Techological Uiversity, Houghto, MI 49931, USA. xilwag@mtu.edu (Received o Jauary 4, 011; accepted o February 8, 011) Numerical simulatio of heat trasfer durig the microwave heatig process of a oe-dimesioal (1-D) magetite slab subjected to covective, radiative boudary coditios was performed. The goverig equatios represetig the heatig process i the slab were discretized usig a explicit fiite-differece approach, ad a computer code was developed to predict the temperature distributios iside the slab. The heat geeratio from microwave irradiatio domiates the iitial temperature rise i the heatig ad the heat radiatio heavily affects the temperature distributio, givig rise to a temperature peak i the predicted temperature profile. As heatig cotiues, the temperature peak migrates iward. The microwave power level is crucial to obtai a high temperature icrease rate i the iitial heatig period (i.e. < 60 s for magetite). Microwave heatig at 915 MHz exhibits better heatig homogeeity tha 450 MHz due to larger microwave peetratio depth. To miimize/avoid temperature o-uiformity durig the microwave heatig the optimizatio of the object dimesio should be cosidered. KEY WORDS: heat trasfer; microwave processig; magetite; heatig homogeeity. 1. Itroductio Microwave heatig has gaied popularity i various applicatios ivolvig iro ad steel makig. 1 5) The distiguishig characteristic of this techique is attributed to its special heatig behaviors. It delivers heat istatly throughout the materials with volumetric heat geeratio resultig i a faster heatig rate tha covetioal heatig. Eergy savig ad less processig time are thus easy to achieve. Although microwave has show its superiority i materials heatig, a major drawback kow as o-uiform temperature distributio iside materials has also bee observed by may researchers. 6,7) To address the problem, accurate temperature determiatio iside the materials uder microwave irradiatio is quite ecessary. However, exact temperature measuremet i microwave heatig has bee idetified as a hard work sice most commo temperature measuremet tools like thermocouple ad pyrometer may ot provide precise measuremet data. The iteractio betwee thermocouple ad microwave lowers the accuracy of data measured while the complexity of emissivity cosideratios required to properly apply optical pyrometry heavily limits its extesive applicatio. 8) I compariso with direct temperature measuremet, temperature predictio by aalytical ad umerical methods seems to offer a promisig solutio to this problem. Both aalytical ad umerical methods are required to solve the heat trasfer differetial equatio coupled with Maxwell s equatios, but the former is foud to be much more difficult sice the heat geeratio from microwave heatig ad complex boudary coditios icludig covective ad radiative heat trasfer eed to be cosidered simultaeously to obtai closed-form mathematical solutio. 9 11) Coversely, umerical modelig has bee proved as a efficiet ad accurate method to predict the temperature of materials durig microwave heatig i the past 0 years. 1 16) Most of those works focused o the utilizatio of microwave i the field of food processig where oly heat diffusio ad/or covectio were cosidered. Meawhile, the variatios of dielectric properties of materials durig the heatig were geerally igored due to relatively low temperature rage ivestigated (geerally < 10 C). It is obvious that the same assumptio caot be applied at high temperature where heat radiatio becomes quite strog ad the dielectric properties may chage dramatically. Therefore, to accurately simulate the heat trasfer for high temperature microwave processig of materials, radiatio effect ad the temperature depedecy of dielectric properties of materials have to be cosidered. The aim of this study is to moitor the heat trasfer process i microwave heatig by predictig the temperature distributio iside a 1-D magetite slab usig explicit fiite-differece approach with full cosideratio of heat diffusio, covectio ad radiatio effect as well as temperature depedeces of thermophysical properties ad dielectric properties.. Modelig A 1-D object of homogeeous solid havig dimesio of L (Fig. 1) heated with microwaves was cosidered. Microwave eergy was assumed to be of uiform itesity ad 011 ISIJ 884

2 ISIJ Iteratioal, Vol. 51 (011), No. 6 Fig. 1. parallel polarizatio, impigig o both faces of the object. It was delivered i a trasverse electric ad magetic (TEM) mode at 915/ 450 MHz ad the microwave dissipatio i the object followed the Lambert s law (a satisfactory approximate alterative to Maxwell s equatios applied i microwave heatig provided o obvious stadig wave patter forms i materials). 17,18) Sice the same eergy was delivered ito both sides of the object, givig rise to a temperature distributio with mirror symmetry; thus oly oehalf of the object eeded to be cosidered. The mathematical aalysis pertiet to microwave heatig process was based o Fourier s law of heat coductio. The shrikage or deformatio of the object durig the heatig was assumed to be egligible ad the surroudig air temperature was cosidered as a costat. The mathematical heat trasfer equatio goverig the microwave heatig process i 1-D (x directio) slab object was give as: 1)... (1) = 1 k + k T Px ( ) + t ρcp x x ρcp x ρcp where T, ρ, c p, k, are the temperature, desity, specific heat capacity, ad thermal coductivity, respectively; P(x) is heat geeratio term by microwave absorptio. Accordig to Lambert s law, P(x) ca be expressed i terms of microwave power flux (P 0) ad peetratio depth (D p) as follows:... () The followig iitial ad boudary coditios were proposed: t = 0 T = T 0 0 x L... (3) x = 0 k t > 0... (4) x = 0 x = L k = ( )+ ( + ) 4 ( + ) 4 ht T εσ T T x t > 0... (5) where t, T 0, h, T, ε, ad σ are the time, iitial temperature, heat trasfer coefficiet, evirometal temperature, emissivity, ad Stefa-Boltzma costat, respectively. 3. Methodology Depictio of slab geometry. Px P 0 ( ) = D e p ( L x)/ D p The method used i this study was the explicit fiitedifferece approximatio, where the goverig equatios were trasformed ito differece equatios by dividig the domai of solutio to a grid of poits i the form of mesh ad the derivatives were expressed alog each mesh poit, referred as a ode. The spatial domai [0, L] was divided ito m sectios, each of legth Δx = L/m. Meawhile, the time domai [0, t] was divided ito segmets, each of duratio Δt = t/. The idex i represets the mesh poits i the x directio, startig with i = 0 beig oe boudary (slab ceter) ad edig at i = m (slab surface). Specifically, the followig differece equatios were used:... (6)... (7)... (8) To evaluate the coductivity spatial derivative i Eq. (1), the followig equatio was applied: k... (9) = ki+ 1 ki 1 x Δx By substitutig above differece equatios ito the heat trasfer equatio ad the iitial ad boudary coditios, the temperature of the sample at a give time could be determied. The solutio was foud by developig a computer code i a Mathematica 7.0 program. 4. Results ad Discussio + 1 = Ti Ti t Δt = Ti Ti x Δx The material cosidered i the simulatio is magetite derived from magetite cocetrate i Tilde Mie, Michiga. The thermophysical properties of the material ad modelig parameters are tabulated i Table ) 4.1. Heatig Time The temperature profiles for differet heatig time periods ragig from 1 s to 60 s (at 915 MHz) are show i Fig.. The highest temperatures iside the object are aroud 36 C, 1 C, 94 C, ad 767 C for 1 s, 10 s, 30 s, ad 60 s, respectively. Temperature i the object icreases rapidly with time due to the icrease of thermal eergy trasformed from the microwave irradiatio. Cotiued microwave heatig creates o-uiform temperature distributio i the slab. The temperature of slab ceter (L = 0 m) stays colder (37 C) after heatig for 60 s, givig a idicatio that the thermal ruaway may occur durig the microwave heatig. Additioally, the surface of the object (L = 0. m, L/Δx = 400) is foud to be the positio with the highest temperature i the iitial periods (~ 1 s). Loger heatig (> 60 s) leads to a temperature peak, which migrates iward with time, as represeted i Fig. 3. It is maily attributed to the effects of microwave heat geeratio ad thermal radiatio. I the iitial heatig, the thermal cotributio from microwave geeratio domiates the temperature rise i the sample ad weak thermal radiatio effect could be expected due to relatively low temperature of the object. As the heatig cotiues, the temperature of object icreases cosiderably, leadig to a high radiatio effect. Thus, a obvious temper T Ti Ti 1 Ti = x Δx ( ) ISIJ

ISIJ Iteratioal, Vol. 51 (011), No. 6 Table 1. Thermophysical properties ad modelig parameters used i the simulatio. Parameter Value Uits k 3.8558 1.37 10 3 T * W/K m c p 611.84+1.

3 ISIJ Iteratioal, Vol. 51 (011), No. 6 Table 1. Thermophysical properties ad modelig parameters used i the simulatio. Parameter Value Uits k T * W/K m c p T ** J/kg C ρ 800 *** Kg/m 3 α ( T) / ( T) m /s D p (915 MHz) T T T T T T 6*** m D p ( 450 MHz) T T m T T T T 6*** h 10 W/m C ε 0.96 **** Noe T 0 5 C T 5 C * Value calculated based o the data reported i Ref. 19. ** Value calculated based o the data reported i Ref. 0. *** Values take from Ref. 1. **** Value take from Ref.. Fig. 4. Temperature depedeces of magetite thermal diffusivity (α) ad microwave peetratio depth (D p). Fig.. Temperature distributios i magetite slab for differet microwave heatig periods at 915 MHz: a 1 s, b 10 s, c 30 s, ad d 60 s. Power: 1 MW/m ; Dimesio (L): 0. m. Fig. 5. Temperature distributios i magetite slab uder differet microwave heatig powers at 915 MHz: a 0.5 MW/m, b 1MW/m, c MW/m, ad d 4 MW/m. Heatig time: 60 s; Dimesio (L): 0. m. may ceramic materials), their cotributios are quite small, especially the heat diffusio. The heat diffusivity (α i Table 1 ad Fig. 4) is foud to be i the order of 10 6 m /s ad decreases with icreasig temperature. Fig. 3. Temperature distributios i magetite slab for differet microwave heatig periods at 915 MHz: a 60 s, b 300 s, c 600 s, ad d 1 00 s. Power: 1 MW/m ; Dimesio (L): 0. m. ature peak is formed iside the object after relatively log heatig time. Note that heat diffusio ad covectio also cotribute to the heat trasfer i microwave processig of materials. But for the magetite i this study (actually, for 4.. Heatig Power The temperature profiles for differet microwave powers (P 0 ) i the rage of 0.5 MW/m to 4 MW/m are give i Fig. 5. The temperature of the object icreases with icreasig microwave power. The highest temperatures after microwave heatig for 60 s are 78 C, 767 C, C, ad 1143 C for 0.5 MW/m, 1 MW/m, MW/m, ad 4 MW/m, respectively. It demostrates that a suitable power applied i microwave heatig is crucial to obtai high heatig rate i short time. Moreover, it is iterestig to ote that these highest temperatures locate at differet mesh positios (L/Δx): 396, 396, 390, ad 385, respectively. It shows the temperature peak shifts to the ceter of the object with icreasig 011 ISIJ 886

4 ISIJ Iteratioal, Vol. 51 (011), No. 6 power as the cotributio of microwave heat geeratio from higher power to temperature icrease becomes eve cosiderable with compariso to heat covectio ad diffusio. Owig to slower heat diffusio ad strog heat radiatio to eviromet at high temperature the peak migrates iward to keep heat balace betwee the object ad surroudig Microwave Frequecy I microwave processig of materials, the dissipatio of microwave power i materials highly relies o the microwave frequecy. It is kow that two frequecies, 915 MHz ad 450 MHz, are commoly assiged for idustrial ad domestic applicatios. To evaluate the effect of microwave frequecy o temperature distributio i magetite, the temperature profiles i the object for differet microwave heatig periods at frequecy of 450 MHz are show i Fig. 6 to compare with 915 MHz i Fig.. The compariso idicates there is egligible temperature differece betwee 915 MHz ad 450 MHz i the iitial heatig periods (1 s). As heatig time exteds to 60 s, the maximum temperature of the object at 450 MHz is foud to be much higher tha that at 915 MHz (996 C ad 767 C, respectively) ad the heatig rate is cosistet with the experimetal data reported i literature. 4,5,3) The heatig rate differece betwee two frequecies is attributed to the differet microwave wavelegths ad microwave absorptio properties (permittivity ad permeability) of the material at 915 MHz ad 450 MHz. Their effects o the heatig ca be idicated by the chage of microwave peetratio depth (D p) i materials: 1) D { λ π ε μ ε μ ε μ = + ( ) + 0 p r r r r r r / 1 1 / ( εr μr ) + ( εr μr ) + ( μ r ε r )... (10) where λ 0 is the microwave wavelegth i free space; ε r ad ε r are the real ad imagiary parts of complex relative permittivity, respectively; μ r ad μ r are the real ad imagiary parts of complex relative permeability, respectively. I this simulatio, the temperature depedeces of microwave peetratio depths at 915 MHz ad 450 MHz were determied via cavity perturbatio techique, as show i Table 1 ad Fig. 4. The microwave peetratio depth at 450 MHz is foud to be much smaller tha that at 915 MHz below 500 C, maily due to their differet microwave wavelegths. At higher temperature, the microwave peetratio depth is also greatly affected by the permittivity ad permeability. The magetite permittivity icreases with temperature, while the permeability decreases apparetly aroud Curie poit. Note that the magitude of permittivity is much larger tha that of permeability. Thus, the chage of permittivity domiates the variatio of microwave peetratio depth i magetite at high temperatures. The small microwave peetratio depth at 450 MHz results i a quick temperature icrease i short time (e.g. < 60 s). This idicates, uder the same coditios (power, heatig time, object dimesio, etc.), most of microwave eergy at 450 MHz would dissipate i the area closer to surface tha that at 915 MHz. As heatig cotiues, the temperature of object icreases ad the radiatio effect at the surface of object becomes quite strog. The differece of the highest temperatures betwee 915 MHz ad 450 MHz decreases, as show i Fig. 3 ad Fig. 7. The highest temperatures at 915 MHz after microwave heatig for 60 s, 300 s, 600 s, ad 1 00 s are 767 C, C, C ad 118 C, respectively. At 450 MHz, the couterparts are 996 C, 1154 C, 1 15 C, ad 1 85 C, respectively. Furthermore, owig to more eergy is located i the sectio close to surface, the temperature iside the object at 450 MHz is much lower tha that at 915 MHz. I other words, i the heatig time rage studied, temperature distributio at 915 MHz is more uiform tha 450 MHz. Hece, 915 MHz is more suitable for large scale microwave heatig of magetite where maximum temperature uiformity is demaded Object Dimesio Volumetric heatig is kow as a mai advatage of microwave processig of materials due to the propagatio behaviors of microwave. 4) However, this superiority also depeds o the object dimesio, as demostrated i Fig. 8. It shows the temperature distributios for slab with differet dimesios (L = 0. m, 0.15 m, 0.1 m, ad 0.05 m, respectively) after microwave heatig for 60 s at 450 MHz. As the dimesio decreases, the temperature homogeeity i the object is improved. The temperature peak magitude remais almost costat while its positio moves close to the ceter of the object. The object with dimesio (L) of 0.05 m uder microwave irradiatio exhibits better temper- Fig. 6. Temperature distributios i magetite slab for differet microwave heatig periods at 450 MHz: a 1 s, b 10 s, c 30 s, ad d 60 s. Power: 1 MW/m ; Dimesio (L): 0. m. Fig. 7. Temperature distributios i magetite slab for differet microwave heatig periods at 450 MHz: a 60 s, b 300 s, c 600 s, ad d 1 00 s. Power: 1 MW/m ; Dimesio (L): 0. m ISIJ

5 ISIJ Iteratioal, Vol. 51 (011), No. 6 Fig. 8. ature distributio tha the others. This could be clearly demostrated by the temperature icrease at the slab ceter with decreasig dimesio. The temperature at the slab ceter icreases from 5 C to 153 C as the dimesio decreases from 0. m to 0.05 m. This idicates a optimal dimesio of the material is required to obtai the miimum temperature o-uiformity ad high heatig performace. Also, it should be oted that further reductio of dimesio size (e.g. L = 0.0 m) would result i apparet stadig wave patter, which may dramatically worse the heatig uiformity. 17,5) 5. Coclusios Temperature distributios i magetite slab with differet dimesios (L) at 450 MHz: a 0. m, b 0.15 m, c 0.1 m, ad d 0.05 m. Heatig time: 60 s; Power: 1 MW/m. From the results obtaied from umerical simulatio of the heat trasfer of oe-dimesioal magetite slab uder microwave irradiatio with cosideratio of coductio, covectio, ad radiatio effect, the followig coclusios ca be draw: (1) The temperature distributio iside the object is o-uiform. () I the iitial periods, the thermal cotributio from microwave geeratio domiates the temperature rise. As the heatig cotiues, the temperature of object icreases cosiderably, resultig i a apparet radiatio effect. (3) Microwave heat geeratio ad heat radiatio from sample surface to eviromet lead to a temperature peak i the temperature profile, which migrates iward with time. (4) A optimal microwave power is required to obtai a high temperature icrease rate for magetite i short time. (5) Microwave heatig at 915 MHz exhibits better heatig homogeeity tha 450 MHz i large scale microwave heatig of magetite. (6) A reasoable dimesio of material is importat for miimizig temperature o-uiformity durig the microwave heatig. Ackowledgmets The authors wish to express their gratitude to the Michiga Public Service Commissio, U.P. Steel, ad the Uited States Departmet of Eergy (DOE) for fiacial support. Nomeclature c p : specific heat capacity (J/kg C) D p : peetratio depth (m) h: heat trasfer coefficiet (W/m C) i: idex of mesh poit alog x directio (dimesioless) k: thermal coductivity (W/K m) L: half slab width (m) m: umber of mesh grid (dimesioless) : umber of time grid (dimesioless) P 0: microwave power flux at the surface (MW/m ) P(x): heat geeratio (MW/m 3 ) t: time (s) Δt : time step (s) T: temperature ( C) T 0: iitial temperature ( C) T : evirometal temperature ( C) x: positio (m) Δx: space step (m) α : thermal diffusivity (m /s) ρ: desity (kg/m 3 ) ε : emissivity (dimesioless) σ : Stefa-Boltzma costat ( kg s 3 K 4 ) ε r : real part of complex relative permittivity (dimesioless) ε r : imagiary part of complex relative permittivity (dimesioless) μ r : real part of complex relative permeability (dimesioless) μ r : imagiary part of complex relative permeability (dimesioless) λ 0 : microwave wavelegth i free space (m) REFERENCES 1) N. Yoshikawa, E. Ishizuka, K. M ashiko, Y. Che ad S. Taiguchi: ISIJ It., 47 (007), 53. ) D. Malmberg, P. Hahli ad E. Nilsso: ISIJ It., 47 (007), ) Y. Makio: ISIJ It., 47 (007), ) Y. Wag: PhD Dissertatio, Michiga Techological Uiversity, Michiga, (005), ) J. Y. Hwag, X. Huag, S. Qu, Y. Wag, S. Shi ad G. Caeba: EPD Cog. Proc., TMS Publicatios, Warredale, PA, (006), 19. 6) Y. C. Ho ad K. L. Yam: J. Food Process Pres., 16 (199), ) R. Vadivambal ad D. S. Jayas: Food Bioprocess Techol., 3 (010), ) E. Pert, Y. Carmel, A. Birboim, T. Oloruyolemi, D. Gersho, J. Calame, I. Lloyd ad O. Wilso: J. Am. Ceram. Soc., 84 (001), ) J. Dolade ad A. Datta: J. Microwave Power E. E., 8 (1993), ) G. Fleischma: J. Food Eg., 7 (1996), ) G. Fleischma: J. Food Eg., 40 (1999), 91. 1) L. A. Campaoe ad N. E. Zaritzky: J. Food Eg., 69 (005), ) D. Aciero, A. Barba ad M. d Amore: Heat Mass Trasfer, 40 (004), ) Y. E. Li, R. C. Aatheswara ad V. M. Puri: J. Food Eg., 5 (1995), ) L. Zhou, V. M. Puri ad R. C. Aatheswara: J. Food Eg., 5 (1995), ) K. G. Ayappa, H. T. Davis, E. A. Davis ad J. Gordo: AIChE J., 37 (1991), ) S. Chatterjee, T. Basak ad S. Das: J. Food Eg., 79 (007), ) K. G. Ayappa ad H. T. Davis: Chem. Eg. Sci., 46 (1991), ) J. Molgaard ad W. W. Smeltzer: J. Appl. Phys., 4 (1971), ) E. Westrum Jr ad F. Grovold: J. Chem. Thermody., 1 (1969), ) Z. Peg, J. Y. Hwag, J. Mouris, R. Hutcheo ad X. Huag: ISIJ It., 50 (010), ) A. L. Sprague, T. L. Roush, R. T. Dows ad K. Righter: Icarus, 143 (000), ) S. L. McGill, J. W. Walkiewicz ad G. A. Smyres: Mat. Res. Symp. Proc., Materials Reserch Society, Warredale, PA, (1988), 47. 4) Y. Jiag, Y. Zhu ad G. Cheg: Cryst. Growth Des., 6 (006), ) H. W. Yag ad S. Guasekara: J. Food Eg., 64 (004), ISIJ 888

Mark Lundstrom Spring SOLUTIONS: ECE 305 Homework: Week 5. Mark Lundstrom Purdue University

Mark Lundstrom Spring SOLUTIONS: ECE 305 Homework: Week 5. Mark Lundstrom Purdue University Mark udstrom Sprig 2015 SOUTIONS: ECE 305 Homework: Week 5 Mark udstrom Purdue Uiversity The followig problems cocer the Miority Carrier Diffusio Equatio (MCDE) for electros: Δ t = D Δ + G For all the