Heat transfer analysis of canned food sterilization in a still retort

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1 Available online at Jounal of Food Engineeing xxx (28) xxx xxx Heat tansfe analysis of canned food steilization in a still etot A. Kannan *, P.Ch. Gouisanka Sandaka 1 Depatment of Chemical Engineeing, Indian Institute of Technology Madas, Chennai 6 36, India Received 6 Febuay 27; eceived in evised fom 3 Decembe 27; accepted 6 Febuay 28 Abstact Computational fluid dynamics (CFD) analyses povide insight on the natual convective pocesses occuing duing the steilization of canned liquid food. The definition and estimation of heat tansfe coefficients petaining to the tansient heat tansfe occuing in cylindical food cans is pesented. The heat tansfe coefficients defined on the basis of absolute mass flow aveaged and volume-aveaged tempeatues ae compaed. The contibutions to the oveall heat tansfe ate fom diffeent sufaces of the unifomly heated cylindical can ae evaluated. The influence of the tempeatue diffeence to viscosity atio on the peculia Nusselt numbe tends is discussed. Unified coelations ae povided fo the Nusselt numbe as functions of Fouie numbe, food can aspect atio and themal conductivity of the food medium in both the conduction and convection dominated heat tansfe egimes. Ó 28 Elsevie Ltd. All ights eseved. Keywods: Themal food steilization; Heat tansfe coefficients; Natual convection 1. Intoduction * Coesponding autho. Tel.: ; fax: addess: kannan@iitm.ac.in (A. Kannan). 1 Cognizant Technology Solutions, Chennai. Liquid food mateial is often steilized in still etots with steam flowing aound the suface of the can. This food is steilized mainly by natual convection when vigoous agitation is uled out. Such situations may aise duing in-package pasteuization of liquids in bottles. Futhe, agitation equipment may not be justified when liquid food is pocessed in low volumes. Food steilized in cans include in-packaged heat pasteuized bee, evapoated milk, thin soups and both, pueed vegetables, fuit and vegetable juices and vegetable soups (Datta and Teixeia, 1988; Kuma et al., 199). It is impotant to develop bette undestanding of the heat tansfe phenomena that ae associated with the themal steilization pocess fom the viewpoint of impoving its design and opeation. This wok focuses on the definition and diect estimation of heat tansfe coefficients obtained duing the steilization pocess fom CFD simulations. Unified Nusselt numbe coelations ae dawn fom CFD simulations on heat tansfe in food cans with diffeent aspect atios and food medium themal conductivities. 2. Backgound and scope Engelman and Sani (1983) studied the pasteuization of bee in glass bottles though finite element simulations. Datta and Teixeia (1988) pesented tansient flow and tempeatue pattens estimated fom numeical simulations of natual convective heat tansfe occuing in a unifomly heated can. Zechman and Pflug (1989) analyzed the influence of medium popeties and metal containe size on the location of slowest heating zone (SHZ) in canned media. Kuma et al. (199) numeically simulated the natual convection in canned thick viscous liquid food heated though the sidewall. Kuma and Bhattachaya (1991) simulated the steilization of pseudoplastic liquid food, which was heated fom all sides. Ghani et al. (1999) caied out tansient numeical simulations of natual convective heating pocess in canned food steilization and analyzed the slowest heating zone chaacteistics. Ghani et al. (22a) /$ - see font matte Ó 28 Elsevie Ltd. All ights eseved. doi:1.116/j.jfoodeng

2 2 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx Nomenclatue AARD aveage absolute elative deviation C p specific heat of the liquid in the can, J/ kg C C nc natual convection constant CMC caboxy-methyl cellulose D diamete of can, m E a activation enegy, kj kg 1 mol Fo Fouie numbe Fo 1 Fouie numbe duing tansient peiod g gavitational acceleation constant, m/s 2 G Gashof numbe G VP Gashof numbe fo a vetical plate h heat tansfe coefficient, W/m 2 C h M local heat tansfe coefficient based on absolute mass flow aveage tempeatue, W/m 2 C h V avg volume aveaged heat tansfe coefficient, W/ m 2 C h total specific total enthalpy, J/kg J zeo-ode Bessel function J 1 fist-ode Bessel function k themal conductivity of the liquid in the can, W/ m C k e effective themal conductivity of liquid food in the can, W/m C K fluid consistency index, Pa s n L height of the can o length of vetical plate, m n exponent to shea ate in dynamic viscosity expession Nu Nusselt numbe Nu c Nusselt numbe in conduction dominated egime Nu hc Nusselt numbe fo a hoizontal cylinde Nu max maximum Nusselt numbe Nu educed Nusselt numbe Nu S aspect atio scaled Nusselt numbe Nu S aspect atio scaled Nusselt numbe pedicted fom theoy Nu VP mean Nusselt numbe fo a vetical plate P Pandtl numbe P VP Pandtl numbe fo a vetical plate q w wall heat flux, W/m 2 C Q heat tansfe ate, W adial coodinate R adius of the can, m Ra Rayleigh numbe Ra d Rayleigh numbe based on diamete Ra d Rayleigh numbe calculated fo the initial time Ra educed Rayleigh numbe Ra max maximum Rayleigh numbe S aspect atio S M souce tem fo momentum S E souce tem fo enegy SHZ slowest heating zone t time of heating, s T tempeatue, C T initial tempeatue inside food can, C T wall tempeatue at the wall of the can, C ht M i absolute mass flow aveaged tempeatue, C ht i i initial volume-aveaged tempeatue, C ht M i axially aveaged absolute mass flow aveaged tempeatue, C ht V i volume aveage tempeatued, C V velocity vecto V velocity in the adial diection, m/s V z velocity in the axial diection, m/s z axial coodinate Z axial distance, m Geek lettes a T themal diffusivity, m 2 /s b themal expansivity, K 1 c ate of shea, s 1 c n nth oot of the zeo-ode Bessel function d Konecka delta l appaent viscosity, Pa s l o consistency index, Pa s n l ef chaacteistic viscosity, Pa s n volume aveaged viscosity, Pa s q density, kg/m 3 h azimuthal coodinate hdti volume aveaged tempeatue diffeence, C studied both expeimentally and numeically the themal deactivation of bacteia in food pouches. In anothe study, Ghani et al. (22b) pesented simulation and expeimental esults on the themal destuction of vitamin C in food pouches. Subsequently, Ghani et al. (22c) investigated themal steilization of caot oange soup in a hoizontally oiented cylindical can. Ghani et al. (23) investigated the themal steilization ate when the can containing viscous caot oange soup was otated at 1 pm. The combined natual and foced convection was obseved to split the slowest heating zone into two egions. Vama and Kannan (25, 26) investigated the effects of modifying the geomety of the conventional cylindical food can as well as the can s oientation on the ate of themal steilization of pseudoplastic food fom CFD simulations. A cone pointing upwads was found to heat faste than the cylinde of equal volume while the cone pointing downwad heated the slowest. Accuate measuement of tempeatues inside the can undegoing themal teatment is difficult because pobes may distub the tempeatue-velocity fields. Futhe, the slowest heating zone is not fixed at one location (Ghani

3 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx 3 et al., 22a). Fo this eason, mathematical modeling appoach has been extensively applied to pedict the tempeatue pattens in the themal steilization pocess (Teixeia et al., 1969; Naveh et al., 1983; Datta and Teixeia, 1987, 1988; Nicolai et al., 1998; Faid and Ghani, 24). Even in the elatively simple conductive heat tansfe study using tempeatue sensos, Maa and Romano (23) obseved that the senso pesence, location and size elative to the can dimensions influenced the estimated cold spot tempeatue evolution and steility values. They futhe noted that the steility time calculations should incopoate a safety facto when tempeatue sensos ae used especially in small cans. Ealie studies have mainly focused on tempeatuevelocity-concentation pofiles, location and movement of the slowest heating zone and effects of geomety modifications, motion and oientation. Howeve, fo a natual convection dominated heat tansfe poblem, knowledge of the fundamentally impotant heat tansfe coefficients is equied. In the liteatue, natual convective heat tansfe coefficients have been mainly ecoded fo flows past single infinitely long heated walls, paallel walls at diffeent tempeatues and inclined plates. In the case of heated cylindes o sphees, the flow domain extenal to the heated suface has been mainly studied. Rectangula cavities, concentic cylindes and sphees maintained at diffeent tempeatues have also been investigated (Dewitt and Incopea, 21). Relatively little wok appeas to have been caied out in the case of isothemal cylindical cans such as those encounteed in food steilization. Faid and Ghani (24) obseved that a coelation fo pedicting heat tansfe coefficient is not available fo liquids contained in vetical and hoizontal cans. These authos poposed a technique fo calculating steilization times. Using the simplified classical solution fo tansient heat tansfe in infinitely long cylindes, they attempted to pedict the slowest heating zone tempeatue. The slowest heating zone tempeatue was used in the definition of the dimensionless tempeatue. An effective themal conductivity (k e ) was used in the themal diffusivity tem to facto in the contibution fom natual convection. The heat tansfe coefficient coelation developed by Catton (1978) fo paallel vetical walls that wee maintained at diffeent tempeatues was applied. The authos justified this appoach on the basis that the natual convection induced ciculation olls wee simila to those obseved in cylindical cans. Howeve, in the cylindical can situation, while the oute wall tempeatue is constant, the axis tempeatue is vaying fom top to bottom. In a ecent wok, Matynenko and Khamtsov (25) eviewed Nusselt numbe coelations fo natual convection in cylindical enclosues. These authos summaized tansient two dimensional heat tansfe esults inside a completely filled hoizontal cylinde at aspect atios exceeding 2.5. Coelations wee poposed fo Nusselt numbe as functions of Rayleigh numbe and Fouie numbe in the tansient peiod. The involved natue of these coelations testifies to the complex natue of the phenomena occuing inside the cylinde. Howeve, this wok does not indicate the methodology though which the heat tansfe coefficients wee estimated. In the pesent wok, the methodology fo estimating the heat tansfe coefficient diectly fom the CFD data is illustated. The tempeatue diving foce used in the heat tansfe coefficient has to be popely defined and two of these definitions ae compaed. The contibutions to the oveall heat tansfe ate fom the diffeent sufaces of the food can ae compaed. Unified coelations ae dawn fo the Nusselt numbe fo diffeent can aspect atios and food medium themal conductivities. This study focuses on the heating cycle involving steam. In a ecent study, Ghani (26) has also analyzed the cooling cycle of the food steilization pocess using CFD. 3. Details of food systems and can geomety The pseudo plastic fluid involving.85% w/w CMC solution in wate was taken as the model system. The popeties of this system given in Table 1 wee based on the study of Kuma and Bhattachaya (1991). Steffe et al. (1986) suggested that this model to be applicable fo tomato puee, caot puee, geen bean puee, applesauce, apicot and banana puees, which ae egulaly canned and peseved usually by heating. Fo validating the CFD calculations, expeiments wee also conducted. Commecial CMC (LR gade) was pocued fom Sdfine-Chem Limited, Mumbai and a.85% w/w solution was pepaed. This expeimental system was chaacteized by the Anton Paa cone and plate Rheomete (Physica MCR 31) of the high polyme-engineeing laboatoy of IIT Madas. The powe law paametes fo this solution ae also included in Table 1 (Neehaika, 27). Othe listed popeties ae taken to be the same fo both systems. Fo aiding the development of coelations, a ange of themal conductivities ( W/m C) and food can aspect atios ( ) wee consideed in this study. Two standad cylindical cans (Can Manufactues Institute, Washington) wee used in the analysis. The cans wee Table 1 Popeties of the model and expeimental food system System popeties Value/expession Units Viscosity (l) l ¼ l o exp nea _c n 1 Pa s Consistency index (l ) RT.2232 (model system) Pa s n (expeimental system) Activation enegy (E a ) J/g mol Paamete fo shea ate, n.57 (model system).82 (expeimental system) Specific heat (C p ) 41 J/kg C Themal conductivity (k).7 W/m C Density (q ef ) 95 kg/m 3 Coefficient of volume expansion (b).2 C 1

4 4 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx Table 2 Details of food cans used in the CFD simulations Can label D (m) L (m) S ( ) labeled on the basis of thei aspect atios. The can labeled as Regula has been commonly used in pevious studies (Kuma and Bhattachaya, 1991; Ghani et al., 1999; Vama and Kannan, 25, 26). The simulations based on this can ae used to illustate the analysis of heat tansfe coefficients in Section 5. Futhe, a non-standad can dimension labeled Long was also included in the analysis fo studying the effects of paametes. The details of the cans used in the simulations ae given in Table Expeimental pocedue Mesh volume 1 1 (m 3 ) Minimum Maximum Numbe of hexahedal elements used fo simulations Regula , Medium ,5 Long ,6 Expeiments wee caied out with the egula can (Neehaika, 27). Openings wee dilled into the top suface of the food can and Teflon plugs wee inseted into these holes to fit the 3 mm diamete themocouples vetically. A themocouple was used to measue the tempeatue at the midpoint of the can. The food can was housed within a cylindical chambe. The cylindical chambe was made of stainless steel (gade 34) with an intenal ing suppot to hold the food can vetically. The can was held above the chambe base to ensue that the steam flowed feely on all sides aound the can. A flanged aangement was welded to the cylindical body of the chambe so that the lid could be tightly bolted. A ubbe gasket was used between the flanges to ensue that the chambe was leak poof. A themocouple was used to monito the steam tempeatue in the chambe duing the couse of the expeiments. Afte the top flange was bolted to the cylinde base, the steam line was activated. At the stat of the un, the steam outlet pot at the top was patially opened and subsequently contolled with mino adjustments to maintain steam tempeatue at 121 C. Afte the tempeatues in the can eached constant values, the steam line was deactivated and the condensate dained off the chambe. The epoducibility of the midpoint tempeatue measuements wee found to be within ±3%. 5. Govening equations and numeical solution methodology The commecial CFD package CFX v5.7, which is based on the finite volume method, was used to solve the govening tanspot equations fo the defined geomety and associated bounday conditions. The domain was defined in the global co-odinate fame. A non-staggeed collocated gid is used. The continuity equation is modified in an extension to the scheme adopted by Rhie and Chow (1983). This enables ovecoming of checkeboad oscillations in pessue and velocity associated with collocation. The linea sets of equations that aise afte discetization ae solved in a coupled fashion, theeby avoiding segegation. This method ensues obustness, efficiency and simplicity at the cost of high stoage of all the coefficients Tanspot equations The genealized tanspot equations that wee solved in CFX (23) ae (a) Continuity equation oq þðqv Þ¼ ot ð1þ (b) Momentum equation oqv þðqv V Þ ¼ð pd þ lðv þðv Þ t ÞÞ þ S M ot (c) Enegy equation oqh total Dp ot Dt þ ð qvh totalþ ¼ðkT ÞþS E In Eq. (3) h total is defined as the specific total enthalpy. In CFX, the momentum souce tem S M is defined by S M ¼ðq q ef Þg ð4þ Applying the Boussinesq appoximation, the body foce tem (S M ) hence becomes S M ¼ q ef bðt T ef Þg ð5þ When this fom is used, the pessue in the momentum equation excludes the hydostatic contibution fom q ef (CFX, 23). As thee ae no intenal souces of enegy, S E is taken to be zeo. A mateial temed CMC was ceated in the pe-pocesso libay and its popeties wee specified in accodance with Table 1. It is assumed that the flow involves an incompessible high viscous liquid with negligible viscous dissipation effects. Based on the popeties defined, the softwae uses an inbuilt mechanism fo geneating enthalpy vaiations with tempeatue Bounday and initial conditions Ghani et al. (1999) suggest that simulations based on unifom heating on all sides of the can with constant wall tempeatue ae appopiate fo a geneal pupose. All walls of the enclosue ae kept at 121 C with the cylinde placed in an upight position. At the cuved wall of the can, no slip conditions apply with specified tempeatues T ¼ T wall ¼ 121 C; V ¼ m=s; V z ¼ m=s fo 6 z 6 L at ¼ R ð2þ ð3þ

5 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx 5 At the top and bottom walls of the containe, again no slip conditions apply with specified tempeatues T ¼ T wall ¼ 121 C; V ¼ m=s; V z ¼ m=s fo 6 6 R at z ¼ andl Initially the fluid is at est and at a unifom initial tempeatue T ¼ 4 C; V ¼ m=s; V z ¼ m=s fo 6 6 R and 6 z 6 L The condensing steam is assumed to maintain a commonly applied constant tempeatue of 121 C at all boundaies. This tempeatue is assumed to apply at the liquid boundaies owing to the vey small themal esistance of the can mateial. Futhe, the tempeatue at the liquid boundaies is assumed to each this tempeatue fom the initial conditions without any lag Mesh and time step details CFD simulations, especially those involving tansients, ae time consuming even with faste pocessos. Exploiting the axisymmety, only a segment of the cylinde was used. The stuctued mesh option fom CFX Build v 5.6 was applied. To esolve the apid changes in velocities and tempeatues nea the walls, fine ectangula mesh was applied close to the walls, which gadually gave way to inceasingly coase ones towads the coe. Fo the cylindes used in the study, the smallest element size was aound.3 mm, while the lagest element size was nea 1 mm in the adial diection. Along the axial diection, the mesh sizes wee between 1 and 2 mm. The total numbe of hexahedal elements in the domain and mesh volumes fo each case is included in Table 2. The tansient uns wee caied out by adopting the step size-time elationship povided by Kuma and Bhattachaya (1991). Upon doubling the time step sizes, the esults did not diffe significantly. Howeve, the ealie fine time step stategy was chosen. The esults fom the stuctued meshes used hee compaed well with the ealie epoted values fo midpoint tempeatues of Kuma and Bhattachaya (1991) as well as fom Vama and Kannan (25). The unstuctued mesh scheme was used by Vama and Kannan (25, 26) in thei pevious studies. A coase stuctued mesh scheme was also investigated in the pesent wok to study the effect of mesh sizes. Fo instance, a coase mesh scheme was tested fo the egula cylinde with 25, hexahedal elements as against 55, hexahedal elements used in the fine mesh scheme. The element sizes wee oughly 1.9 times highe along the adial diection with the minimum and maximum mesh sizes being.6 mm and 2.3 mm. The esults wee not found to be in significant vaiance. Howeve, the efined mesh scheme whose details wee shown in Table 2 was used in this wok. Finally, the axisymmety condition was also veified by simulating ove the entie cylinde. No significant azimuthal flow o vaiations in tempeatues wee encounteed. Detailed esults of mesh independency studies ae povided elsewhee (Gouisanka, 26). The simulations wee executed in the Intel Pentium (R) 4, 3.6 GHz and 1 GB RAM with Windows XP platfom. The CPU time fo the egula cylinde case fo instance was estimated at 1.2 h Solution methodology The tansient calculations wee caied out using the obust and bounded fist-ode backwad Eule scheme. CFX has the optional schemes of fist ode upwind diffeence and numeical advection with specified blend facto. The blend facto may be vaied between and 1 to opt between the fist- and second-ode schemes in ode to contol numeical diffusion. Howeve, these schemes ae not obust and non-physical oveshoots and undeshoots may occu. Hence, the high-esolution scheme option, which maintains the blend facto as close as possible to unity without violating the boundedness pinciple, was chosen. In the pesent simulations, non-physical oveshoots o undeshoots in the solution wee not encounteed. Convegence citeion was fixed at esiduals oot mean squae (RMS) value lowe than 1 4. The low Gashof numbes encounteed duing the entie couse of simulations justified lamina flow conditions. 6. Results and discussion 6.1. Calculation of wall heat flux At epesentative time steps, heat fluxes at the walls (sidewall, top and bottom sufaces) wee detemined using the CFX post pocesso. The wall heat flux at any given height on the cuved suface is elated to the tempeatue gadient by the following equation: q w ¼ k ot o ð6þ ¼R Tempeatue depends on both and z and inceases towads the cuved wall, i.e. with inceasing adial coodinate. The heat tansfe coefficient is defined as follows: q w ¼ hðt w T bulk Þ ð7þ 6.2. Estimation of suitable bulk tempeatue Bid et al. (22) obseve that cae should be taken to note the definitions of the heat tansfe coefficients when using teatises and handbooks. Many liteatue souces do not clealy define the heat tansfe coefficients. To obtain the heat tansfe coefficient fom the heat flux (Eq. (7)), it is necessay fist to define the efeence o bulk tempeatue (T bulk ). Fo flow past submeged bodies, the bulk fluid tempeatue may be set as that of the infinite suoundings. Fo fluid flow though a conduit, cup mean

6 6 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx mixing tempeatues ae often used. Howeve, fo the natual convection engendeed flow within the confines of a food can, the fluid flow is ciculatoy. Howeve, the cup mean mixing tempeatue would not apply at a given coss section because the net flow acoss the coss section would be zeo. Hence, it is possible to define a bulk tempeatue in tems of absolute mass flow aveaged tempeatue (ht M i) as given below R R ht M i¼ jvj zt d R R jvj z d ð8þ The heat flux fom the vetical cuved wall was estimated. Hoizontal slice planes, acoss which the flow occus, ae constucted at diffeent locations along the cuved wall. Nine equidistant slice planes ae used and these ae located at millimete intevals along the cuved wall of the egula cylinde. The influence of the top and bottom walls would be minimal especially fo long cylindes. Howeve, a simila pocedue, by constucting vetical planes, may not be adopted fo estimating the local heat tansfe coefficients at the top and bottom sufaces of the can. Fo this case, the method is faught with uncetainties associated with heat tansfe contibutions not only fom the top and bottom walls but fom the cuved wall as well. The tempeatue field would be influenced by heat flux fom the cuved wall as well as fom the top and bottom sufaces. This effect would be significant especially when the diamete of the cylinde is small elative to the height, i.e. fo high aspect atios. It would be then incoect to calculate the local heat tansfe coefficient based only on the heat flux contibution fom the bottom o top suface. Hence, a moe convenient definition of the heat tansfe coefficient is necessay and it will be inteesting to compae its pedictions with the ealie poposed method. It is also possible to base the mean heat tansfe coefficient on the volume-aveaged tempeatue in the domain. The volume-aveaged tempeatue is the tempeatue that would be obtained if the entie contents of the can wee to be mixed. The volume-aveaged tempeatue is defined as shown below RRR T d dhdz ht V V i¼ RRRV d dhdz ð9þ The heat tansfe coefficients esults based on both these methods ae pesented and compaed in the next section Heat tansfe coefficient based on the absolute mass flow aveaged tempeatue The local heat tansfe coefficient (h M ) was estimated using Eq. (8) afte constucting the hoizontal slice planes as discussed in Section 5.3. Its vaiation along the axial diection is plotted fo diffeent time steps as illustated in Fig. 1. In this figue, it may be seen that each time step the heat tansfe coefficient is nealy a constant along the vetical diection. Thee is a small incease nea the top h (W/m 2o C) Z (m) Fig. 1. Vaiation of the local heat tansfe coefficients at diffeent times (in seconds). and bottom walls due to heat tansfe effects fom these sufaces. The decline in heat tansfe coefficients with time may be attibuted to leveling of tempeatue gadients in the domain. This in tun was caused by conduction and natual convective mixing. The heat tansfe coefficient vaied fom an initial value of nealy 5 W/m 2 C to only 4 W/m 2 C at the final stages of the heating pocess. The axially aveaged heat tansfe coefficient based on the absolute mass flow aveaged tempeatue ðh M avg Þ is defined by R L h M avg ¼ hm dz ð1þ L This aveaged heat tansfe coefficient is shown in Fig. 2 as a function of time Heat tansfe coefficient based on the volume aveaged tempeatue The method outlined ealie equied a tedious pocedue of evaluating heat tansfe fluxes at diffeent locations and calculating the absolute mass flow aveaged tempeatue at each plane. The advantage is that it enables the pediction of local heat tansfe coefficients, which howeve in the pesent case do not vay significantly along the axial diection (Fig. 1). A moe convenient method would be to evaluate the total heat flux into the cylinde at any given time instant. The aveage heat tansfe coefficient may then be estimated in tems of the volume-aveaged tempeatue accoding to q V avg ¼ hv avg ðt wall ht V iþ ð11þ The volume aveaged heat tansfe coefficient ðh V avgþ is compaed with the absolute mass flow aveaged heat tansfe coefficient ðh M avgþ in Fig. 2. It is inteesting to note fom this figue that volume aveaged heat tansfe coefficients ae in easonable ageement with those estimated ealie based on the absolute mass flow aveaged tempeatues. The eason fo this effect is analyzed next.

7 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx h (W/m 2o C) qw (W/m 2 ) time (s) M h avg V h avg time (s) bottom top cuved wall total Fig. 2. Compaison of aveage heat tansfe coefficients based on aveage absolute mass flow aveage and volume aveaged tempeatues. Fig. 3. Vaiation of heat flux with time at the top, bottom and cuved sufaces and thei total contibution. The aveage heat tansfe flux fom the cuved walls (q avg ) is based on the aveage of the poduct of the heat tansfe coefficient and the tempeatue diving foce. Expessing the heat flux at any given time instant in tems of ht M i and aveaging along the vetical diection leads to the following expession fo the aveaged heat flux ðq M avg Þ q M avg ¼ h M ðt w ht M iþ ð12þ h M avg ðt w ht M iþ The ove ba is used to denote the aveaging along the axial diection to distinguish it with the ealie aveages epesented by the symbol h i. Usually the aveage of poduct of two quantities is not the same as the poduct of the aveages of the individual quantities. Howeve, in the pesent case appoximation to Eq. (12) was made possible since thee was not too much of a vaiation of the local heat tansfe coefficient (h M ) along the axial diection (Fig. 1). As the wall tempeatue is maintained constant, the above equation may be expessed as q M avg ¼ hm avg ðt w ht M iþ ð13þ This equation may be compaed with Eq. (11) whee the aveaged heat tansfe coefficient was based on volume aveage tempeatue. These two equations may now be used to analyze why the volume aveaged heat tansfe coefficient that consides the entie domain is close to the absolute mass flow aveaged heat tansfe coefficient that is only based on the cuved suface. The contibutions to the total heat flux fom the cuved, bottom and top sufaces wee analyzed. The suface goups option in CFX enables the estimation of heat tansfe fluxes eithe fom individual sufaces o fom thei combination. The esults ae shown in Fig. 3. Initially the contibutions to the heat flux fom the thee sufaces ae unifom. Howeve, with inceasing time, the heat flux fom the top wall suface declines apidly due to attainment of highe tempeatues inside the can at the uppe egions. The heat flux fom the bottom wall is even moe than the total heat flux. The total heat flux is an aveage value based on the sum of heat tansfe ates fom all the thee sufaces divided by the total aea. This coincides moe o less with the heat flux fom the cuved wall. While the heat flux fom the bottom wall alone appeas to dominate, it is to be noted that its effect would be lowe on the heat tansfe ates since the suface aea of the bottom suface is only 18.22% of the cuved suface (see Fig. 3). Fom Fig. 3 it is obseved that the total heat flux may be well epesented by the contibution fom the cuved suface. Hence q M avg will be close to qv avg. Fom Eqs. (11) and (13) it may be infeed that fo the heat tansfe coefficients h M avg and hv avgto be compaable, the axial aveage of mass flow aveaged tempeatue ðht M iþ should be close to the volume aveaged value (ht V i). As the azimuthal vaiation in the absolute mass flow aveaged tempeatue is negligible owing to axisymmety, aveaging this tempeatue along the axial diection of the can would lead to a value that is close to the volume aveaged tempeatue (ht V i). This has been veified as illustated in Fig. 4 ove the entie duation of the themal steilization pocess Estimation of heat tansfe ates fom the sufaces The heat tansfe ates fom the top, cuved and bottom sufaces wee compaed at diffeent time instants. Fom each of the suface goups ceated ealie and the conveged esults, the heat tansfe flux, aea and heat tansfe ate could be calculated. The heat tansfe ate contibutions fom the thee sufaces as well as the total heat tansfe ate ae depicted in Fig. 5 as a function of time. While the ate of enegy tansfe to the cylindical segment initially was aound 1 W, it declines by two odes of magnitude nea the end of the steilization pocess. Until the fist 8 seconds, the bottom and top sufaces contibuted almost equally to the total heat tansfe ate. Late howeve, the heat tansfe ates decline substantially, especially fom the top suface. Diffeences pesist between the two significant heat tansfe sufaces viz. the bottom suface

8 8 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx T ( o C) time (s) and the cuved sidewall with the latte dominating. In this context, any innovations to enhance the themal steilization pocess would equie concentating on the cuved sidewall of the cylinde athe than the top o bottom walls o to device means though which heat tansfe fom the top and bottom sufaces may be impoved. It is inteesting to note that Vama and Kannan (26) obseved that a cone pointing upwads (CNU) povided significant heat tansfe enhancement ove a cylinde and even moe so than a cone pointing downwad (CND). The CNU aangement would totally obviate the heat tansfe fom the top while the CND aangement gives pedominance to it theeby leading to poo pefomance Nusselt numbe analyses < T V > < T M > Fig. 4. Compaison between the volume aveage tempeatue and axial aveage of the absolute mass flow aveage tempeatue. Q (W) time (s) bottom top cuved wall suface goup Fig. 5. Compaison of heat tansfe ates fom the top, bottom and cuved sufaces of the food can and thei total contibution. The heat tansfe coefficients based on the volume-aveaged tempeatues wee expessed in the Nusselt numbe fom using cylinde length (L) as the chaacteistic dimension. Diffeent themal conductivities of the food medium wee also consideed in the analyses to investigate thei effect on the heat tansfe pocess. The Nusselt numbe (Nu) vaiation with Rayleigh numbe (Ra) was fist detemined with the latte defined as follows: Ra ¼ gbql3 hdt i ð14þ a T Plotting the Nusselt numbes diectly as a function of the Rayleigh numbes fo diffeent cylindes and themal conductivities led to peculia tends as shown in Fig. 6. Two Nusselt numbe values ae possible fo the same Rayleigh numbe in the egion whee the banches ovelap. Fo a given aspect atio (S), the Nusselt numbe cuve is located at lowe values of Rayleigh numbe at highe themal conductivities. Fo a given themal conductivity, the cuve is shifted towads highe Rayleigh numbe values at highe aspect atios. These imply that heat penetation into the can would take a longe time to complete at highe aspect atios and smalle themal conductivities. The Nusselt numbe tends wee analyzed next by focusing on the volume-aveaged quantities such as the tempeatue diving foce (hdti), fluid medium viscosity () and thei atio (hdti/). At lowe times, when compaed to hdti, deceases moe apidly leading to the initial incease in hdti/. Subsequently, hdti tends to zeo while attains a nealy constant value of 3.4 Pa s theeby poducing a declining tend in hdti/ atio. Hence a maximum is obseved in the plot of hdti/ with time. These ae illustated fo the egula cylinde case with food medium themal conductivity of.7 W/m CinFigs Othe cylindes exhibited simila tends. Fo a fixed can aspect atio and food medium themal conductivity, the maximum in hdti/ is a chaacteistic popety of the food medium as it gets heated gadually. This maximum coelated with the poduct of the themal conductivity and aspect atio as shown hdt i ¼ 8:22ðkSÞ :144 ð15þ max The coelation coefficient value fo this fit was.99. Nu (-) Ra (-) Regula (.7) Medium (.7) Medium (1.4) Long (.7) Long (.35) Regula (1.4) Fig. 6. Vaiation of Nusselt numbe with Rayleigh numbe at diffeent aspect atios. Food medium themal conductivities W/m C ae shown in paentheses.

9 <ΔT > ( o C) < > (Pa. s) time (s) Fig. 8. Vaiation of volume aveaged viscosity with time fo the egula cylinde with food themal conductivity of.7 W/m C. <ΔT>/ < > (Pa -1 s -1. o C) The ð hdt i Þ values fo each of the cans may be scaled by its coesponding maximum value to yield the educed ð hdt i Þ. with Fouie num- The empiical fit to vaiation of be Fo ¼ at is given by R 2 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx time (s) Fig. 7. Vaiation of volume aveaged tempeatue diving foce (hdti) with time fo the egula cylinde with food themal conductivity of.7 W/ m C. time (s) hdt i Fig. 9. Vaiation of with time fo the egula cylinde with food themal conductivity of.7 W/m C. hdt i hdt i 1:58 expð 11:15FoÞ ¼ ð16þ 1 þ 1:42 expð 74:84FoÞ The deviation (absolute) of the model pediction fom the CFD value was divided by the CFD value to give the absolute elative deviation. The aveage value of such deviations was epoted as the aveage absolute elative deviation (AARD). The AARD fo the pedictions fom Eq. (16) was estimated to be 13.7%. Howeve if the Fouie numbes coesponding to vey high heating times wee not consideed, the AARD educed to 7.7%. The citical Fouie numbe coesponding to the maxi- mum value of hdt i may be detemined by setting the fist deivative of Eq. (16) with espect to Fouie numbe to zeo. This leads to a citical Fouie numbe (Fo c ) of.28. Since heat tansfe data ae expessed commonly in tems of Rayleigh numbe athe than by hdt i, the Rayleigh numbe dependency on Fouie numbe is next detemined. Since the Rayleigh numbes coveed a wide ange of values ( ) as was shown in Fig. 6, it was also necessay to scale the Rayleigh numbe. Fo each can-themal conductivity combination, the maximum Rayleigh numbe (Ra max ) was obtained fom the coesponding value of hdt i. Hence, defining Ra max as max gbl 3 hdt i q max Ra max ¼ ð17þ a T the individual Rayleigh numbe values wee scaled to give the educed Rayleigh numbe as given in Ra ¼ Ra ð18þ Ra max The Fouie numbe values wee also scaled by dividing with the citical Fouie numbe to yield the educed Fouie numbe (Fo ). The educed Rayleigh numbe coelated Ra (-) Fo (-) Regula (.7) Medium (.7) Medium (1.4) Long (.7) Long (.35) Regula (1.4) coelation Fig. 1. Vaiation of educed Rayleigh numbe with educed Fouie numbe fo food cans of diffeent aspect atios and themal conductivities. The dashed line indicates the citical educed Fouie numbe.

10 1 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx with Fo as given in Eq. (19) and the fit with the data ae shown in Fig. 1. Ra ¼ 1:18 expð1:72fo Þ ð19þ 1 þ :75 expð2:2fo Þ The citical educed Fouie numbe is shown by the dashed line. The aveage absolute elative deviation (AARD) was estimated to be 13%. Howeve if the data coesponding to vey high heating times (Fo > 1) wee not consideed, the AARD educed to 7.5%. The Fouie numbe is the pi- may vaiable fo coelating hdt i and Nu. Howeve, in natual convection liteatue, the Nusselt numbe is coelated usually in tems of the Rayleigh numbe. As shown in Eq. (19), thee is an explicit elation between the educed Rayleigh numbe and educed Fouie numbes. The decision to use the educed Fouie numbe diectly o in combination with the educed Rayleigh numbe will be govened by the tends exhibited by the data in the diffeent heat tansfe egimes. These will be discussed next in the development of coelations fo Nusselt numbe. Two Nusselt numbe egimes, classified as sub citical and supecitical wee defined, depending on whethe the Fig. 11. Tempeatue contous (a) and steamline pattens (b) at nealy the same Rayleigh numbe but at diffeent times fo the egula cylinde with food medium themal conductivity of.7 W/m C.

11 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx 11 Fo was less than o geate than unity. This was necessay in view of the type of the dominant heat tansfe mechanism pevalent in these two egimes. The tempeatue contous and steamlines plots in the sub-citical and supecitical egimes fo appoximately the same Rayleigh numbes fo the egula cylinde case ae shown in Fig. 11a and b. It may be infeed fom these figues that in the subcitical egime, the heat tansfe is mainly by conduction with unifom Kenels of tempeatue zones. In the supecitical egime, the convection cuents contot the tempeatue pofiles Sub citical egime The Nusselt numbe data that wee defined in tems of the can length (L) as the chaacteistic dimension wee escaled by dividing with the aspect atio (S). The scaled Nusselt numbe (Nu S ) is defined as shown in Nu S ¼ Nu S ¼ h L k L ¼ h D ð2þ k D The scaled Nusselt numbe will lead to the chaacteistic dimension being diamete of the cylinde athe than its length. In the sub citical egime, the scaled Nusselt numbe data was coelated as follows: Nu S ¼ 8:92 Fo :45 ð21þ The coelation fit is easonable fo a wide ange of Fouie numbes, themal conductivities and aspect atios. In ode to facilitate compaison with the theoetical pediction, this coelation may be expessed in tems of actual Fouie numbe as follows: Nu S ¼ 1:78 ð22þ Fo :45 The plot in tems of the actual Fouie numbe is given in Fig. 12. The aveage absolute elative deviation (AARD) was estimated to be 4.7%. In the conduction dominated heat tansfe egime, the Nusselt numbe by definition becomes unity. Futhe, fom the heat conduction equation, we have ot ot ¼ a o 2 T T ð23þ o 2 Fom scaling analysis descibed by Bejan (1995), it may be shown that if d T, whee d T is the conduction laye thickness, the above equation will become DT t a T DT d 2 T This esults in the following elation fo d T p d T ffiffiffiffiffiffi a T t ð24þ ð25þ NuS (-) Using the definition of the Fouie numbe, this equation becomes d T D pffiffiffiffiffi Fo ð26þ 2 Hence using the conduction laye thickness as the tue chaacteistic dimension fo conduction-dominated situations, the definition fo Nusselt numbe becomes Nu C ¼ h d T k ¼ 1 Hence Nu C ¼ h D 2 pffiffiffiffiffi Fo ¼ 1 k ð27þ ð28þ Hence the scaled Nusselt numbe given by theoy ðnu S Þ is given by Nu S ¼ h D k ¼ p 2 ffiffiffiffiffi ð29þ Fo This simplified expession is shown in compaison with the actual coelation pediction (Eq. (22)) and CFD data in Fig. 12. While the tend is captued by the theoetical model (Eq. (29)), its pedictions ae howeve highe than the actual coelation (Eq. (22)). The eason may be that the adial popagation of the conductive heat flux may have been inhibited by the onset of axial convection cuents that cay away the heat in a diection pependicula to the diection of conductive heat tansfe. In any case fo Fo < 1, the conduction mechanism is the dominant if not the sole mechanism of heat tansfe. The discepancy in the exponent, which deviated fom.5, was futhe analyzed. It was obseved that a maginally bette coelation fit could be obtained by expessing the scaled Nusselt numbe as a Nu S ¼ Fo (-) Fo :5 bra Regula (.7) Medium (.7) Medium (1.4) Long (.7) Long (.35) Regula (1.4) Theoy Coelation Fig. 12. Coelation fo Nusselt numbe in the sub citical egime. ð3þ This coelation was poposed to account fo the effects of convection on the Fouie numbe exponent. Fo the ange of Rayleigh numbes studied, the modified exponent to Fo

12 12 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx (as well as Fo) anged only between.44 and.46. Hence, the simple fom given by Eqs. (21) and (22) was etained Supecitical egime The analysis in this egime is complicated since the simple conduction heat tansfe is no longe dominating. Natual convection cuents would influence the tempeatue pofile stongly as shown in Fig. 11. In addition, the conduction effects may be still impotant at supecitical Fouie numbes not much highe than unity. The vaiation of Nusselt numbe with the educed Fouie numbe in the supecitical egime is shown in Fig. 13. It may be infeed that moe than the themal conductivity, it is the aspect atio that is playing a key ole. It may be seen that fo long cylindes with twice the vaiation in themal conductivities, the Nusselt numbes ae nealy the same. Thee is a mino effect of themal conductivity especially befoe the peak. The tend suggests that lowe the themal conductivity value, highe the Nusselt numbe. Highe the aspect atio, highe is the Nusselt numbe. This figue also indicates, that the Nusselt numbe fist inceases befoe deceasing with inceasing Fouie numbe, a tend distinct enough not to be ignoed. The simple scaling of the Nusselt numbe, which unified the data in the sub citical egime, failed in the supecitical egime. The maximum in the Nusselt numbe was coelated to the aspect atio and themal conductivity as shown below Nu max ¼ 12:84 S:69 ð31þ k :18 The coelation coefficient value fo this fit was.98. Using this coelation, the Nusselt numbes in the supecitical egime wee educed by dividing with Nu max. The tend of the educed Nusselt numbe passing though a maximum may be modeled as a combination of two effects, of which one would vanish at highe Fouie numbes. In the modeling pocess, the educed Rayleigh numbe was also used since the tends showed a monotonic vaiation with this vaiable Nu (-) Fo (-) Regula (.7) Medium (.7) Medium (1.4) Long (.7) Long (.35) Regula (1.4) Fig. 13. Vaiation of Nusselt numbe with educed Fouie numbe in the supecitical egime. at highe Fouie numbes as would be expected in natual convective pocesses. Matynenko and Khamtsov (25) popose a simila appoach involving both Fouie and Rayleigh numbes fo tansient two dimensional heat tansfe inside a completely filled hoizontal cylinde at aspect atios exceeding 2.5. Thee ae both similaities and diffeences between thei wok and the pesent study. These authos defined a Fouie numbe ange ( Fo 1 ) in the tansient peiod, whee Fo 1 was coelated with the Rayleigh numbe at initial time (Ra d ) as follows: Fo 1 ¼ :4Ra :25 d ð32þ A function f was defined in the Pandtl numbe ange.7 25, as shown below f ðpþ ¼1 :21 expð :247PÞ ð33þ The Nusselt numbe was defined by :279f ðpþra :15 d Nu hc ¼ Fo½Fo :575 :486f ðpþra :25 d Š ð34þ The authos also identified a quasi steady state egime, whee the Rayleigh numbe (Ra d ) anged between 1 5 and 1 1 in which the Nusselt numbe could be coelated as follows: Nu hc ¼ :59Ra :25 d ð35þ Simila to the pesent study, the heat tansfe coefficients deceased apidly followed by a much slowe decease (Fig. 2). Howeve, thei coelations ae not applicable to the pesent case whee the cylinde alignment is vetical. The aspect atios would be dastically diffeent fo the hoizontal oientation of the cylindical food can. Futhe the Pandtl numbes encounteed in the pesent wok ( ) ae much highe than the ange specified fo thei coelations. Howeve, the coelations show that both Fouie numbes and Rayleigh numbes ae necessay to model the Nusselt numbe even though the Rayleigh numbe itself depends on the Fouie numbe. The simple natue of this dependency (Eq. (32)) in contast to the pesent case (Eq. (19)) also indicates that the fluids behavio in the two analyses is diffeent. In addition, the effect of aspect atio is not pesent in these coelations. Howeve, the Nusselt numbe is shown to be a monotonic function of Rayleigh numbe in the quasi steady state egime. In the poposed appoach, the Nusselt numbe may be modeled in tems of a convective tem, which is solely a function of the Rayleigh numbe and a conductive heat tansfe coection to it that is applicable at values close to the citical Fouie numbe. The conductive heat tansfe coection would vanish apidly at high values of supecitical Fouie numbes. The following coelation was fitted to the data Nu ¼ 1:5Ra :16 :2 Ra7:96 Fo :5 ð36þ

13 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx 13 The second tem becomes insignificant at highe Fouie numbes that coespond to lowe values of the educed Rayleigh numbe (see Fig. 15). Hence, it may be seen that at highe Fouie numbes, the Nusselt numbe simplifies to Nu ¼ 1:5Ra :16 ð37þ The coelation poduces a fit within ± 1% to the CFD numeical data Validation of tempeatues and heat flux data obtained fom CFD Numeical calculations fo lamina flow CFD simulations ae staightfowad, as the govening tanspot equations do not pose additional complications. This is in contast to tubulent flow whee additional model equations ae equied to povide closue (Sinivasan et al., 25). Hence, the eo may aise mainly fom numeical discetization of the time and space domains. The sufficiency of the discetization steps wee veified though mesh and time step independency studies. To the best of ou knowledge, infomation in fom of expeimental data and empiical coelations fo heat fluxes ae not available fo the tansient natual convection flow situations in enclosed containes and non-newtonian food systems encounteed in themal food steilization. As descibed in Section 4, expeiments wee conducted with a test system to validate the CFD tempeatue pedictions. The midpoint location was chosen fo measuement to avoid end and wall effects. The measued tempeatues wee found to be in easonable ageement with the CFD pedictions as shown in Fig. 14. Next, the mean Nusselt numbe values obtained fom CFD simulations of the actual model system was validated. If the vetical wall of the cylinde is appoximated as an isothemal plate, then an appopiate liteatue coelation fo Nusselt numbe may be used fo at least an ode of T ( o C) Time (s) Expeiment CFD Fig. 14. Compaison between expeimentally measued and CFD pedicted midpoint tempeatues. magnitude compaison. Fo natual convective heat tansfe into a non-newtonian fluid flowing steadily past an isothemal vetical suface the following coelation fo the mean Nusselt numbe has been suggested (Skelland, 1967; Chhaba and Richadson, 1999) Nu VP ¼ C nc ðg VP Þ 1 2ðnþ1Þ ðpvp 3nþ1 Þ n ð38þ whee the Gashof and Pandtl numbe definitions fo a vetical plate ae given as follows: G VP ¼ q2 L nþ2 ½gbðT w T ÞŠ 2 n ð39þ P VP ¼ qc P k K 2 2 1þn K 1 n L 1þn ðlgbðt w T ÞÞ 3ðn 1Þ 2ð1þnÞ q ð4þ In Eqs. (39) and (4), K efes to the fluid consistency index of the powe law fluid. It is taken as the poduct of l and the exponential tempeatue dependent tem given in Table 1 fo the model system. K was evaluated at the film tempeatue, which was defined as the aithmetic mean of the wall and bulk flow tempeatue. In the pesent analysis, the bulk flow tempeatue was chosen as the volumeaveaged tempeatue. The tempeatue T used in Eqs. (39) and (4) efe to the tempeatue outside the themal bounday laye. Fo flow past a flat plate, this tempeatue would be the bulk fluid tempeatue. Fo flow inside the cylindical can, T is taken as the volume aveaged tempeatue. The constant C nc in Eq. (38) was.636 fo the model system. The CFD based Nusselt numbes based on the volume aveaged heat tansfe coefficients ðh V avg Þ wee compaed with Eq. (38) fo the egula cylinde at two diffeent food medium themal conductivities (Fig. 15). It may be obseved that the coelation fit to CFD data is not good initially when conduction is the dominant mechanism. In the conduction-dominated sub citical egime, the Nusselt numbe tend followed the analytical expession (Eq. (29)) as discussed in Section 6.6. Howeve, a easonable fit may be obseved late when the convection effects ae established inside the can. The flat plate coelation will inceasingly apply fo food cans of lage adii and low food medium themal conductivities as shown in Fig. 15b Estimation of the volume aveaged tempeatues To pedict the total heat flux fom Eq. (11), the heat tansfe coefficient (h V ) and volume-aveaged tempeatue hti ae necessay. The heat tansfe coefficient in the sub citical and supecitical egimes may be estimated fom the Nusselt numbe coelations (Eqs. (21) and (36)). As suggested by one of the anonymous eviewes, the volume-aveaged tempeatue at any time instant may be estimated by integating the following lumped paamete model fo tansient heat tansfe qvc p dht i dt ¼ h V AðT w ht iþ ð41þ

14 14 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx a b Nu (-) Nu (-) Fo (-) CFD Coelation (Eq. 38) Fo (-) CFD Coelation (Eq. 38) Fig. 15. Compaison of CFD pedicted mean Nusselt numbes fo the egula cylinde with the coelation fo the flat plate (a) k =.7 W/m C (b) k =.35 W/m C Sub citical egime (Fo 6 1) Fom Eqs. (2) and (21), the heat tansfe coefficient (h V ) may be expessed as a function of Fo. The esulting expession may be integated to yield the following analytical expession fo the volume-aveaged tempeatue ht i¼t w ðt w T Þ exp :454 S þ :5 S Fo :55 ð42þ Afte substituting the values fo the wall tempeatue and initial tempeatue, the above equation gives the volumeaveaged tempeatue in Kelvin as ht i¼394:15 81 exp :454 S þ :5 S Fo :55 ð43þ When the volume-aveaged tempeatues fom this equation wee compaed with those pedicted fom CFD, the ageement was not too bad. A maximum tempeatue diffeence of about 2 C was obseved at Fo = 1. Howeve, when initiating the calculations in the supecitical egime (Fo > 1), this tempeatue offset would lead to futhe eos that cay ove when estimating hti at diffeent times in the supecitical egime. Hence, an altenative was sought to pedict tempeatues bette in the sub citical egime. As discussed in Section 6.6 and illustated in Fig. 11a, this egime is chaacteized by conduction dominated heat tansfe. Hence, the tempeatue in the domain may be estimated fom the analytical solution of the two-dimensional tansient heat conduction equation. Using the poduct solution method descibed in Holman (1997), the tempeatue T(,z) in the finite cylinde of length L and adius R may be obtained by taking the poduct of the solutions fo the infinite cylinde of adius R and the infinite flat plate of thickness L. The esulting expession is given by T T ¼ 1 8 X 1 X 1 ð 1Þ mþ1 T w T p ð2m 1Þ m¼1 n¼1 2m 1 J cos pz c n R " L c n J 1 ðc n Þ exp 2m 1 # 2 p 2 at exp L at c 2 n R 2 ð44þ whee c n is Bessel oot of the function J ðcþ n ¼ ð45þ whee J and J 1 ae Bessel functions of odes zeo and one espectively. The solutions to Eq. (45) ae given in standad mathematical handbooks o may be obtained fom MAT- LAB Ò calculations. These solutions apply fo the case when the extenal suface heat tansfe coefficient is vey high. This is justified in the pesent analyses as condensing steam is used as the heating medium. Eq. (44) may be volume aveaged using the fom of Eq. (9) to obtain the following elation ht i T ¼ 1 32 X 1 T w T p 2 X 1 1 m¼1 n¼1 ½ð2m 1Þc n Š 2 " # 2 p 2 at exp exp 2m 1 L at c 2 n R 2 ð46þ The volume-aveaged tempeatues in the sub citical egime wee pedicted using Eq. (46) and wee found to be in easonable ageement with the CFD pedictions (Table 3). Especially at Fo = 1, the diffeence between the volume-aveaged tempeatues pedicted by CFD simulations and Eq. (46) was about.5 C Supecitical egime (Fo >1) In this egime, convection has set in the cylinde and the volume-aveaged tempeatue may be obtained using the lumped paamete appoach. Using Nu = h V L/k, the above equation becomes dht i qvc p ¼ Nu k dt L AðT w ht iþ ð47þ In the supecitical egime, diect analytical integation is not possible because the Nusselt numbe is a function of Rayleigh numbe in addition to time (Eq. (36)). The Rayleigh numbe (Ra) in tun, is a function of the can dimensions, time (o equivalently Fo ) and the tempeatue diving foce (DT). These dependencies ae shown in Eqs. (17) (19). The Nusselt numbe (Nu) is obtained by multiplying its educed fom (Nu ) with Nu max using Eq. (31). Estimates of hti in the supecitical egime wee then

15 A. Kannan, P.Ch. Gouisanka Sandaka / Jounal of Food Engineeing xxx (28) xxx xxx 15 Table 3 Volume-aveaged tempeatue pedictions in the subcitical and supecitical egimes No. Time (s) Medium can (.7 W/m C) Fo ( ) hti CFD (K) hti model (K) The supecitical egime values ae shown in bold. obtained by numeically integating Eq. (47) with the ODE45 outine of MATLAB Ò. The initial value fo hti is povided at a time coesponding to Fo = 1 using the analytical conduction solution pediction Eq. (46). The pedictions based on the methods fo both egimes ae shown in Table 3. A good match between the volume-aveaged tempeatues obtained fom the lumped paamete model and CFD simulations may be obseved. 7. Summay and conclusions CFD analyses povide insight on the themal steilization pocess in elatively long food cans of diffeent aspect atios. Two diffeent methods wee used to define and estimate the heat tansfe coefficient. The volume aveaged tempeatue based definition of the heat tansfe coefficient is moe convenient to use elative to the absolute mass flow aveaged heat tansfe coefficient. The heat flux and heat tansfe ate contibutions fom the diffeent sufaces of the food can ae diffeent. The cuved suface, owing to its lage suface aea especially fo cylindes with high aspect atio, dominates the heat tansfe pocess. The Nusselt numbes exhibited a shap decease with time. The Nu Ra plot indicated banching at high Rayleigh numbes. This was attibuted to the atio of volume-aveaged tempeatue diving foce to viscosity goup exhibiting a maximum as it vaied with time. The citical Fouie numbe at which the hdti/ value attained maximum was estimated. A unified coelation was developed fo the educed Rayleigh numbe as a function of educed Fouie numbe. The Nusselt numbe seemed to be govened essentially by conduction based mechanism in the sub-citical Fouie numbe ange. Nusselt numbe analysis in the sub citical egime indicated that the aspect atio switched the chaacteistic dimension fom a length scale to a diamete scale. This is mainly due to conductive heat tansfe in the adial diection. Howeve, in the supecitical Fouie numbe ange, the Nusselt numbe was inceasingly govened by natual convection. Coelations wee developed fo the Nusselt numbe both in the sub citical and supecitical egions. The application of CFD techniques in pedicting tempeatue vaiation with time was validated expeimentally using a test system. The CFD deived Nusselt numbes wee also validated with an available liteatue coelation developed fo isothemal plates. A easonable match was obseved in the convection-dominated egime especially at low food medium themal conductivity. The volumeaveaged tempeatues wee estimated easonably fom the analytical solution in the conduction dominated sub citical egime and the lumped paamete model in the supecitical egime. The pesent wok may find applications in themal steilization optimal contol studies whee the enegy consumption is incopoated into the objective function. Pevious studies mainly apply optimization stategies on the much simple conduction model, which may not be ealistic fo convection heated liquid foods. The coelations developed may be used to estimate the heat tansfe fluxes as a function of time, which upon integation will lead to estimation of enegy consumption. Acknowledgement The simulations wee caied out fom the facilities given by the Cente fo Computational Fluid Dynamics, Indian Institute of Technology Madas, Chennai Refeences Bid, R.B., Stewat, W.E., Lightfoot, E.N., 22. Tanspot Phenomena, second ed. John Wiley, New Yok. Bejan, A., Convection Heat Tansfe, second ed. John Wiley, New Yok. Can Manufactues Institute, Washington. (< Catton, I., Natual convection in enclosues. In: Poceedings of the Sixth Intenational Heat Tansfe Confeence, vol. 6, Toonto, Canada, pp CFX, 23 Documentation, CFX Ansys Ltd., Didcot, Oxfodshie, UK. Chhaba, R.P., Richadson, J.F., Non-Newtonian Flow in the Pocess Industies. Buttewoth Heinemann, Oxfod. Datta, A.K., Teixeia, A.A., Numeical modeling of natual convection heating in canned liquid foods. Tansactions of Ameican Society of Agicultual Enginees 3 (5), Datta, A.K., Teixeia, A.A., Numeically pedicted tansient tempeatue and velocity pofile duing natual convection heating of canned liquid foods. Jounal of Food Science 53 (1), Dewitt, D.P., Incopea, F.P., 21. Fundamentals of Heat and Mass Tansfe, fifth ed. John Wiley, New Yok. Engelman, M., Sani, R.L., Finite element simulation of an inpackage pasteuization pocess. Numeical Heat Tansfe 6, Faid, M.M., Ghani, A.G.A., 24. A new computational technique fo the estimation of steilization time in canned food. Chemical Engineeing and Pocessing 43,

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