Numerical heat transfer during Herschel Bulkley fluid natural convection by CVFEM

Transcription

1 Iteratioal Joural of Heat ad Techology ol. 36, No. 2, Jue, 208, pp Joural homepage: Numerical heat trasfer durig Herschel Bulkley fluid atural covectio by CFEM Diego A. asco *, Carlos Salias 2, Nelso Moraga 3, Roberto Lemus-Modaca 4 Departameto de Igeiería Mecáica. Uiversidad de Satiago de Chile. Av. Lib. Berardo O Higgis 3363, Satiago, Chile 2 Departameto de Igeiería Mecáica. Uiversidad del Bío-Bío. Av. Collao 202, Cocepció, Chile 3 Departameto de Igeiería Mecáica. Uiversidad de La Serea. Beavete 980, La Serea, Chile 4 Departameto de Ciecia de los Alimetos y Tecología Química, Uiversidad de Chile. Satos Dumot 964, Idepedecia, Satiago, Chile Correspodig Author diego.vascoc@usach.cl Received: 9 November 207 Accepted: 9 April 208 Keywords: free covectio, heat trasfer, o- Newtoia fluid, CFEM ABSTRACT Numerical predictio of heat trasfer by atural covectio of a Herschel-Bulkley o- Newtoia fluid iside a square cavity has bee computatioally aalyzed. Usteady 2D fluid mechaics ad heat trasfer were described i terms of the o-liear coupled cotiuity, mometum ad heat equatios. These equatios were solved by the cotrol volume fiite elemet method (CFEM) with Gauss-Seidel/System Over-Relaxatio couplig algorithm. The effect of the Ra, Pr, B ad the rheological behavior idex () o the o-newtoia fluid thermal ad mometum behavior were studied. The o-newtoia fluid flow was described by the rheological model of Herschel-Bulkley. Results for the streamlies ad isotherms alog the eclosure walls are preseted. It was foud that the effect of the Pr ad B is more importat whe the Ra is lower (0 3 ). I additio, the behavior idex had a sigificat effect o the CPU time for the differet studied cases.. INTRODUCTION Processes ivolvig shear thiig o-newtoia fluids are of great importace i several biological applicatios, chemical ad food idustry [-3]. For istace, thermal processig of pulps ad fruit juices is iterestig i food idustry. Several studies have demostrated that rheological properties of fruit juices are well described by shear-thiig No-Newtoia models such as Power-law [4-5], Hershel- Bulkley [6], Bigham ad Casso [7]. Most of foregoig metioed experimetal studies cosider the effect of solutes cocetratio, water cotet, ad temperature i the rheological parameters. Natural covectio ivolvig Newtoia ad o- Newtoia fluids has bee broadly ivestigated. Both experimetal ad umerical studies have bee made; hece bechmark reliable results are available for two ad threedimesioal domais [8-]. Nevertheless due to experimetal difficulties, to the best kowledge of the authors, oly umerical studies have bee coducted with No- Newtoia fluids i both 2D ad 3D domais [2-5]. The authors, Awar-Hossai ad Reddy-Gorla [6], studied the flow of a power law No-Newtoia fluid past a isothermal vertical slotted surface. They foud that as the behavior idex icrease, both the frictio factor ad the Nusselt umber icreased. Also with a power law No-Newtoia fluid, a study of the Rayleigh-Béard problem cofirmed the previous observatio respect the behavior idex of shear-thiig fluids (0<<). Nevertheless, for shear-thickeig fluids (>), the opposite effect was observed by Lamsaadi ad Naïmi [7]. A aalytical ad umerical study by the fiite differece method was performed i a differetially heated sided walls of a rectagular eclosure filled with a power law fluid [8]. It was foud that at high Pradtl umbers atural covectio is maily cotrolled by the flow behavior idex ad the Rayleigh umber. The same physical situatio was studied by ola et al. [9] ad Tura et al. [3] with a square eclosure filled with a yield stress fluid obeyig the Bigham model. I the former work, through domai discretizatio by low order fiite elemets, the yield stress effect o temperature ad velocity distributio was studied. The obtaied results show that at higher yield stress values the apparitio of rigid zoes modified the vortices positio ad the temperature distributio turs out to be cotrolled by thermal diffusio. Meawhile, i the latter work, the Fluet v.6 commercial software was used to solve the coupled coservatio equatio of mass, mometum, ad eergy. Respect to the rheological properties of the o-newtoia fluid, for high values of the Bigham umber, o sigificat flow is iduced withi the eclosure. It is kow that i most buoyacy-drive motios, the flow is slow due to moderate temperature gradiets [20]. Kumar ad Bhattacharya [2] predicted usteady atural covectio ad heat trasfer durig sterilizatio of a temperature depedet power-law No- Newtoia fluid usig the fiite volume method. The authors foud out that there was ot a fixed poit, which could be called the coldest poit at all times. Certaily, the rheological properties have a sigificat ifluece o the trasiet processes [22]. Dea et al. [23] preseted a wide review of results ad umerical methods cocerig with forced covectio for Bigham plastic fluids. The cotrol volume fiite-elemet method (CFEM) developed by Baliga ad Patakar [24] holds the topological characteristics of the fiite elemet method (FEM) ad the coservative properties of the fiite volume method (FM). The cotrol volumes are made up of several fiite elemets 575

2 ad each elemet cotributes to three differet cotrol volumes. The way how the cotrol volumes are built allows that the fluxes through the boudaries ca be calculated through iterpolatio techiques developed i FEM [25]. Thus, CFEM cosiders liear orthotropic variatios of the properties ad idepedet variables withi the fiite elemet, cosiderig the discrete variable cetered o the cotrol volume. The umerical approach leads to the formulatio of liear algebraic equatios systems that are solved through iterative ad direct methods like Gauss-Seidel with SOR ad Gauss Elimiatio [26]. The CFEM method has bee successfully used i applicatios with thermofluids [24], movig boudary problems [27-28], solid mechaics [25], ad fuctioally graded materials [29]. The mai objective of the preset work is to examie the atural covectio heat trasfer characteristics of a o- Newtoia fluid filled i a square cavity based o the Herschel-Bulkley fluid model ad solved by fiite elemet/cotrol volume method (CFEM), for differet Ra, B ad Pr umbers ad behavior idex. A compariso of the behavior of streamlies, pressures, velocity profiles, isotherms, Nu umbers i the cavity, caused by the variatio i the goverig parameters, is preseted. 2. PHYSICAL SITUATION The atural covectio of the o-newtoia fluid iside a square domai cosidered i this paper is show i Figure. The problem is assumed to be two-dimesioal. The dimesios of the eclosure are equal ad deoted by L. The left ad right vertical walls are differetially heated at isothermal temperatures of Tc ad Th, respectively, whereas the remaiig walls are cosidered adiabatic. Θ= U==0 Y X θ Y Y=L = 0 U==0 Figure. Physical 2D domai of atural covectio heat trasfer process g U==0 θ Y Y=0 = 0 L Θ=0 U==0 for the atural covectio of a No-Newtoia fluid i a twodimesioal square cavity are described by: Mass coservatio equatio: u + v = 0 () x y (ρu) t (ρv) t Mometum coservatio equatios: + u (ρu) x + u (ρv) x + v (ρu) p (η(γ ) u) + = 0 (2) y x + v (ρv) y Eergy coservatio equatio: (ρcpt) t + u (ρcpt) x (η(γ ) v) ρgβ(t T ref) + p y = 0 (3) + v (ρcpt) (K T) = 0 (4) y The o-dimesioal versio of the goverig equatios is achieved accordig to the followig dimesioless parameters: X = x L ; Y = y L θ = T T H pl ; P = ; U = u ; = T H T C η 0 gβ TL gβ TL Η = η η o ; Γ = L gβ TL v gβ TL (5) (6) γ (7) The o-dimesioal goverig equatios are the expressed as: Mass coservatio equatio: U + = 0 (8) X Y Ra Pr Ra Pr Mometum coservatio equatios: U U (U + ) (Η(Γ ) U) + P = 0 (9) X Y X (U + X ) (Η(Γ ) ) Ra Y Pr Eergy coservatio equatio: Ra Pr (U θ X θ + P Y = 0 (0) θ + ) ( θ) = 0 () Y The o-dimesioal parameters obtaied are: Ra = gβ(t H T c )L 3 η 0 α (2) 3. MATHEMATICAL MODEL 3.. Goverig equatios Pr = η 0Cp k 3.2. Herschel-Bulkley fluid model (3) The goverig equatios are formulated for a icompressible o-newtoia fluid with costat thermophysical properties. Desity for the gravitatioal term i the y-mometum equatio is assumed to have liear depedece of temperature chages accordig to the Boussiesq approximatio. Therefore, dimesioal equatios This model is characterized by showig o deformatio up to a certai level of stress, however above this yield stress the material flows readily [30]. The, Herschel-Bulkley model i tesorial form ca be expressed as: τ = ( τ y + kγ ) γ ; for τ > τ γ y (4a) 576

3 γ = 0; for τ τ y (4b) It could be oticed that for = the Herschel-Bulkley model is reduced to the Bigham model ad for τ y = 0 the Ostwald-De Waele power-law model is obtaied. Papaastasiou [3] proposed a alterative approach to overcome the issue that represets yield stress, avoidig the discotiuity i the flow curve due to the icorporatio of the yield criterio. The modificatio ivolves the icorporatio of a expoetial term, thereby permittig the use of oe equatio for the etire flow curve, before ad after yield. A modified Herschel-Bulkley model to approximate the rheological behavior is give by: B Η(Γ ) = Γ + [ exp( MΓ )] (5a) Γ B = τ y η 0 L gβ T M = m gβ T L (5b) (5c) I additio, the m parameter whe icreasig the Papaastasiou modified Herschel-Bulkley model is approached the Herschel-Bulkley model at low shear rates. The bi-viscosity models ad Papaastasiou s modificatio are empirical improvemets desiged primarily to afford a coveiet viscoplastic costitutive equatio for umerical simulatios [32] Boudary ad iitial coditios The goverig equatios are complemeted through the implemetatio of the boudary ad iitial coditios. Noslippery boudary coditios for the velocities at the walls are settled. Heat trasfer boudary coditios iclude uiform hot ad cold temperature o the vertical left ad right walls, respectively, alog adiabatic coditio o the other walls. I additio, the iitial coditio for velocities, pressure ad temperature iside the cavity correspods to ull values. Depedig o the trasported quatity, terms i equatio (7a) are modified accordig to Table. I the same way that fiite volume method, the startig-poit of this method is the itegratio of the geeralized trasport equatio (7b) over a cotrol volume with a cotour area S (=,, N). Table. Coefficiets of the geeralized trasport equatio (8) for each variable φ ρ Γ S U Ra Pr Ra Pr Η(Γ ) Η(Γ ) P X P Y + Ra Pr θ θ Ra Pr Applyig the Gree Theorem: d + ( v ) ds F d = 0 t ( ) S (8) where each is made up by partial cotributios of the fiite elemets e (e=,..,e) (Figure 2): = N = l (9a) with l = E e= e ad S = E e= e (9b) S e θ(0, Y) = ; θ(, Y) = 0 (6a) θ Y Y=0 = 0; θ Y Y= = 0 4. CFEM IMPLEMENTATION (6b) The goverig equatios could be represeted by a geeral expressio (7a), which cotais the trasiet, covective, diffusive ad source terms: φ t + φ (Γ φ) F = 0 (7a) This expressio could be represeted as well i terms of flux divergece (J ): φ t + (J ) F = 0 (7b) Figure 2. Local system of coordiates Y/Z defied i the ceter of each cotrol volume coformed by fiite elemets I the same way each S e is built by two segmets. For the cotrol volume C cetered o the local vertex (Figure 2), S l l k = S ka + S l kc, with ormal vectors ( a, c ) to the segmets betwee the medias of the FE (a, c) to the cetroid (G). The itegratio of the covective ad diffusive terms of the equatio (7a) requires a fuctio, which represets the distributio of ϕ over each FE. Geerally, a liear fuctio (20a) is adopted for the diffusio term ad a expoetial fuctio based o local coordiates (20b) for the covective term: φ(x, y) = Ax + By + C (20a) with: φ(z, ζ) = AZ + Bζ + C (20b) J = v φ Γ φ (7c) I both cases, the coefficiets A, B, ad C are defied 577

4 accordig to the odal values of the depedet variable φ ad the coordiates of the vertices of the FE (local odes,2 ad 3). The o-liear iterpolatio of the covective term requires first the defiitio of a local system of coordiates (Z,ζ) for each FE, where the Z-axis is parallel to the velocity vector v i with module U (Figure 2). The ew Z-values are obtaied through a expoetial fuctio of the origial coordiate values X ad the Peclet umber: Z = Γ ρu [exp (Pe(X X max ) X max X mi ) ] (2) The approach above alog with a implicit backward Euleria scheme for the trasiet term, allows the represetatio of the equatio (7a) i discrete terms for each ode p of the domai: a p φ p = b a b φ b + b p (22) The obtaied system of liear algebraic equatios builds up sparse matrices with ull values distributed radomly. To avoid uecessary operatios with ull values, the o-ull values of the sparse matrices ad its positios are stored i two vectors through the compressed row storage (CRS) scheme. The resultig systems of equatios are solved through the iterative method Gauss-Seidel with SOR (System Over- Relaxatio). For more details about the computatioal implemetatio see the work of Salias et al. [27]. The calculatios were accomplished by usig a persoal computer (Itel Core I7, 3.40GHz; 8 GB memory). The Itel Fortra compiler IFORT uder Liux platform was used for the compilatio of the Fortra 90 source codes. All variables were defied as double-precisio floatig-poit data types. The covergece criteria applied to stop the velocity ad temperature calculatios was based i a maximum value for the differece i the calculated value at two successive iteratios, defied as follows: k φ i,j,k φ k i,j,k 0 6 (23) 5. SCALING ANALYSIS Followig the same procedure performed by Tura et al. [3] for scalig aalysis of atural covectio of Bigham fluids, the effects of Ra, Pr, B ad o the Nusselt umber for a Herschel-Bulkley fluid are aalyzed. The wall heat flux q is scaled as: q~k T δ th ~h T (24) which gives rise to the followig expressio: Nu~ hl ~ L or Nu~ L g(pr, B, ) (25) k δ th δ where the thermal boudary layer thickess δ th is related to the hydrodyamic boudary layer thickess δ i the followig maer: δ δ th ~g(pr, B, ) (26) where g(pr,b,) is a fuctio of Pr, B ad behavior idex (). To estimate the hydrodyamic boudary layer thickess δ, a balace of iertial ad viscous forces i the vertical directio is cosidered: ρ [ v2 L ] ~ τ δ (27) where v is a characteristic velocity scale (v~ gβ TL). For Herschel-Bulkley fluids the shear stress τ ca be estimated as τ~τ y + μ ( v δ )2 which upo substitutio i Eq. (27) gives: ρ [ v2 L ] ~ [τ y + μ ( v δ )2 ] δ (28) Usig Eq. (27) ad (28), the hydrodyamic boudary layer thickess ca be estimated from the equatio: f(δ) = δ + [ L Ra Pr ] B δ B μ τ y v ~0 (29) Stoer ad Bulirsch [33] proposed a modificatio of the Newto method to estimate the first approximatio of a root of a polyomial accordig to: δ~δ 0 f (δ 0 )± (f (δ 0 )) 2 2f(δ 0 )f (δ 0 ) f (δ 0 ) (30) δ 0 is a proof ordiate value chose as (δ 0 = L Pr Ra B), a relatively close value to the miimum (δ mi = L + Pr Ra B) of the fuctio f(δ) with: After evaluatig δ 0 i the equatio (30) ad replacig δ i equatio (27), the hydrodyamic boudary layer thickess ca be estimated as: δ~ L Ω {2 B [( )B B2 ( Ra 2] ) 2 B HB Pr } (3) where B HB is the modified Bigham umber for a Herschel- Bulkley fluid. B HB = τ y μ (L v ) (32) This scalig gives rise to the followig expressio for the thermal boudary: δ th ~mi [ L, Nu ~max g(pr,b,) L Ω {2 B [( )B B2 BHB ( Ra Pr ) 2] Substitutio of Eq. (33) ito Eq. (25) yields: [.0, Ra Pr { 2 2 B+ 2 [( )B 2 +4 B2 ( Ra 2] B HB Pr ) 2 } g(pr, B, ) ] 2 }] (33) (34) The expressio (34) provides a useful isight ito the expected behavior of Nu i respose to variatios of Ra, Pr, B, ad. The scalig aalysis suggests that Nu is expected to decrease with icreasig B for give values of Ra ad, whereas Nu icreases with icreasig Ra for a give value of B ad. For costat Ra, Pr, ad B, the effect of o Nu is remarkable for shear-thiig fluids (0<<) meawhile for shear thickeig fluids the effect is egligible. For =, the same scalig expressio for Nu preseted by Tura et al. [3] 578

5 is obtaied: Ra ~max 2 Pr 2 Nu.0, f(pr, B) { 2 [ B+ 2 [B2 +4( Ra 2 2] Pr ) } ] (35) results, a mesh of 8 8 odes is used durig the calculatios with Ra=0 5, ad a mesh of 6 6 (Figure 3) odes is used. No-uiform meshes with a higher desity of elemets ear to the eclosure walls are used sice the larger velocity, ad temperature gradiets take place i these zoes. 6. RESULTS AND DISCUSSIONS 6.. Bechmark compariso ad mesh idepedecy study To verify the implemeted umerical code ad the pressure-velocity couplig algorithm (PRIME), the wellkow de ahl Davis [34] bechmark problem of Newtoia atural covectio process is replicated. The verificatio was performed for Rayleigh umbers ragig from 0 3 to 0 5 ad Pradtl umber correspodig to air, equal to Pr=0.7 (Table 2). It should be metioed here that because of the used referece value of velocity ( gβ TL), the obtaied velocity values must be multiplied by RaPr compare with the Bechmark results. For the fier used mesh (8 8), the relative errors do ot exceed % for the Nusselt ad the maximum velocity values of U(Y=0.5) ad (X=0.5). Table 2. Compariso of obtaied results with the Bechmark for Pr=0.7 Nu Nu max U max (Y=0.5) max (X=0.5) Ra= Bechmark Ra= Bechmark Ra= Bechmark Table 3. Compariso of corrected obtaied results for Bigham fluid results with the ola et al. [9] for Ra=0 5, Pr=.0 ad B=0.95 U max max Nu Nu max (Y=0.5) (X=0.5) ola et al (3) The mesh idepedecy study was performed for costat Rayleigh umber ad Pradtl umber values (Pr=.0; Ra=0 5 ) cosiderig a o-newtoia Bigham fluid (B=0.95). Accordig to the defiitio of the Bigham umber (5b), this situatio is equivalet to that made by ola et al. [9] for a costat yield stress (τ y=300 Pa). A compariso of the obtaied values for the Nusselt umber ad maximum values of U(Y=0.5) ad (X=0.5) is show i Table 3. For a o-uiform mesh of 8 8 odes, the absolute error does ot exceed 3%. The, accordig to this ad to the deviatio of the obtaied results respect to the bechmark Figure 3. No-uiform mesh used with 372 odes ad 7200 elemets for cases with Ra< Natural covectio heat trasfer behavior The atural covectio heat trasfer process of a Herschel- Bulkley fluid is studied by varyig the dimesioless parameters: Pr, Ra, B ad. Table 4 shows the Nusselt umbers obtaied for Ra=0 3, Ra=0 4 ad Ra=0 5 as a fuctio of the Bigham ad Pradtl umbers for a costat behavior idex (=0.7). Table 4. Nusselt umbers for a Herschel-Bulkley fluid (=0.7) B Pr/Ra Accordig to these results, the Nusselt umber is a strog fuctio of the Ra (see values i bold), while Nu is a weaker fuctio of the Pr (see values i cursive) ad B (see uderlied values). As predicted by the scalig aalysis, the Nu umber icreases with the Ra umber, ad it decreases as the Pr ad B umbers icreases. Whe the B umber rises the fluid has higher yield stress; therefore the covectio is depleted, ad heat trasfer by coductio becomes more importat. I Figure 4 ca be see how the umerical experimet coducted at a low Rayleigh umber (Ra=0 3 ). It allows observig the effect of the B ad Pr umbers i the diffusio heat trasfer mode. Whe the B ad Pr umbers are lower tha 0.0 ad 0., respectively, the effect of the covective heat trasfer starts to be oticeable sice the isotherms adopt a sigmoidal shape. Whe the Pr, therefore the Gr umber becomes lower, ad Bigham umbers become higher, coductio is the more importat heat trasfer mechaism, ad the isotherms are almost parallel to the isothermal surfaces, as expected. The distributio of isotherms i Figure 5 shows the importace of the buoyacy forces o the heat trasfer process whe the Rayleigh umber is icreased (Ra=0 4 ). I this set of results, it ca be see the effect of heat trasfer by coductio whe the Pradtl ad Bigham umbers icrease (Pr=0 ad B=.0), that is whe the fluid trasfers mometum at a higher rate tha eergy ad 579

6 fluid mechaics for a higher Rayleigh umber (Ra=04) are show. I geeral, the streamlies are similar to those observed for Ra=03. However, whe the Pr ad B umbers are icreased, the cocetric fluid trajectories appear more flatteed ad squared, respectively. The higher viscosity regios, demarcated i red i the same figure, that occupy a broader regio i the cavity are foud whe B=0. ad B=.0, for Pr=0 ad Pr=.0; respectively. However, the effect of coductio o heat trasfer becomes importat oly i the latter case (Figure 5). the yield stregth becomes higher tha the viscosity of the fluid, or whe the fluid teds to be a solid material. B/Pr B/Pr Figure 4. Effect of Pr ad B umbers o isotherms for Ra=03 ad =0.7 B/Pr Figure 6. Effect of Pr ad B umbers o streamlies for Ra=03 ad =0.7. The red zoes correspod to uyielded regios i the cavity 0. B/Pr Figure 5. Effect of Pr ad B umbers o isotherms for Ra=04 ad = I order to icrease the comprehesio of the behavior of the Herschel-Bulkley fluids, a aalysis of the behavior of the fluid as a fuctio of the dimesioless umbers Ra, Pr ad B is performed. Whe the Rayleigh umber is low (Ra=03), the streamlies are cocetric, adoptig a rouded shape ear to the ceter of the cavity, ad takig the form of the walls as they move away from the ceter (Figure 6). I this case, oly oe cetral vortex ca be see without the presece of secodary flows, which i two-dimesioal domais are evideced by the presece of smaller vortices ear to the lower corers of the cavity. The absece of these vortices is explaied by zoes of high viscosity (Η>0), where the yield value is ot exceeded (uyielded zoe demarcated i red i the same figure). Whe the Bigham umber icreases, from 0.0 to 0., the uyielded zoes appear close the ceter of the cavity, breakig the "atural covectio motor" ad promotig the heat coductio. It is evidet that the uyielded zoes have a symmetrical distributio aroud the ceter of the vortex whe the coductio is the more importat heat trasfer regime (Pr=0, B=0., ad Pr=.0, B=). I Figure 7, the streamlies that describe the behavior of the.0 Figure 7. Effect of Pr ad B umbers o streamlies for Ra=04 ad =0.7. The red zoes correspod to uyielded regios i the cavity A study of the vertical compoet of the velocity alog the axis X for Y = 0.5 (Figure 8) is added to the previous descriptive aalysis. Whe the Pradtl umber is icreased by oe order of magitude, from Pr=0. to Pr= for a costat Bigham umber, the velocity decreases by the same order of magitude. However, this depletio is more proouced whe the Rayleigh umber is lower (Ra=03), ad the Pradtl umber is higher (Pr=0). For Ra=04, whe the Pradtl 580

7 umber is icreased from 0. to.0, the velocity is decreased about oe order of magitude, but a further icrease from.0 to 0 does ot chage the velocity profile substatially. The Bigham umber has ot a importat effect o the velocity distributio,0.6whe Pr is costat. This observatio is more evidet whe the yield stregth becomes as importat as the viscous forces (B=), ad the umber Rayleigh is lower 3 (Ra=0 ) I Figure 0, the effect of the Pradtl umber o the Ycompoet of velocity profile alog the X-axis (Y=0.5) is show. The observed shape of the velocity profile is typical of the atural covectio processes with a high Grashof umber. Whe the Pradtl umber becomes higher the maximum velocity value close to the walls drops as well as the velocity gradiet, sice a decrease of the shear stress o the walls is expected. The Herschel-Bulkley model is characterized by a expoetial term called the behavior idex (). Whe = the Hershel-Bulkley models becomes the Bigham model. The effect of the behavior idex is studied i the rage of 0.7 to.5. Figure shows the distributio of isotherms withi the cavity for differet behavior idex values. From the qualitative poit of view, o sigificat differeces are observed. A deeper isight ito the effect of the behavior idex is give by observig the Nusselt umber. The aalysis of the variatio of the Nusselt umber with the behavior idex (Figure 2) cofirms the predictio made by the scalig aalysis. With Ra, Pr ad B as parameters, the icrease of the behavior idex causes a decrease of the effect of covectio i the heat trasfer process (lower Nusselt umber), beig this effect more importat whe the behavior is i the rage 0<< a) X 0.6 X 0.2 B=0.0; Pr=0. B=0.; Pr=0. B=; Pr= b) B=0.0; Pr= B=0.; Pr= B=; Pr= 0.6 XX.0.0 B=0.0; Pr=0 B=0.; Pr=0 Figure 8. ertical compoet of velocity alog the horizotal axis X (Y=0.5) for a) Ra=03 ad b) Ra=04 with =0.7 For a costat Bigham umber, the icreasig of the Rayleigh umber to Ra=05 (Pr=0.) makes the heat covectio more importat tha the heat coductio. I Figure 9, this effect is oticed by observig the sigmoidal shape of the isotherms. From the qualitative poit of view, the effect of the Pradtl umber o the isotherms distributio is ot importat, except i those poits where the velocity gradiet has a proouce chage, some of them are highlighted i Figure 9, especially those close to the wall at a lower temperature (X=L). The behavior of the streamlies predicts the existece of two vortices i the ceter of the cavity, slightly displaced whe Pr=, which ted to be flatteed ad fused whe the Pradtl umber rises. The areas of high viscosity (Η>0) appear i a regio of high itesity of the streamlies ad i the upper left ad lower right of the cavity =0.9 = =.2 =.5 Figure. Effect of the behavior idex () o streamlies ad isotherms for Ra=05 ad B= Nu Figure 9. Effect of the Pr umber o isotherm ad streamlies for Ra=05, B=0. ad =0.7 Figure 2. Nusselt umber variatio as a fuctio of the behavior idex () for Ra=05 ad B=0. The effect of the behavior idex i the streamlies is ot relevat from the qualitative poit of view (Figure ). The positios of the vortex i the ceter of the cavity are equally displaced from the cetral axis (Y=0.5). I the Figure 3, the effect of the behavior idex i the Y-velocity profile alog the axis X is show. A detailed view allows distiguishig the larger variatio of the velocity profile ear to the isothermal walls of the cavity. It is observed that there is o a sigle tred. Higher Y-velocities ear to the wall are observed whe < Figure 0. ertical compoet of velocity alog the horizotal axis X (Y=0.5) for Ra=05, B=0. ad =0.7 58

8 ad after certai critical positio (X 0.) the Y-velocities values are higher for >. The behavior idex has a importat effect o the computatioal time. For the smallest studied value (=0.7) the CPU time is higher (360 h) tha for = (270 h). For itermediate values <<.5 the CPU times are of the same order (270 h) whe =. If the behavior idex is higher (=2), the CPU time is sigificatly lower (00 h) tha for =. These differeces ca be explaied by the relative weight of the diffusive term with respect to the covective term i the Navier-Stokes equatio. Whe the strai rate is more sigificat tha oe ad <, the relative weight give to the diffusive term is lower, ad therefore the covective term becomes more importat. The opposite occurs whe > X Figure 3. Effect of the behavior idex () i the Y-velocity profile alog the horizotal axis (Y=0.5) for Ra=0 5 ad B=0. 7. CONCLUSIONS =0.7 =.0 =.5 =0.9 = =0.7 =0.9 =.0 = = X A developed code based o CFEM was implemeted to solve atural covectio heat trasfer i a cavity for a Herschel-Bulkley o-newtoia fluid. The performed scalig aalysis allowed a prelimiary idetificatio of the importace of Ra, Pr, B,, ad Nu umbers. The Ra umber had a sigificat effect o both heat trasfer ad fluid mechaics. The effect of the Pr ad B is more importat whe the Ra umber is low (0 3 ). Whe the Pr (0) ad B (.0) umbers are icreased, the heat trasfer becomes a diffusive process, because of the reduced effect of covectio ad the icreased ifluece of the yield stress, respectively. Also, the icreasig of the behavior idex causes a decrease i the Nu umber. This effect was more importat whe <. The aalysis of the vertical compoet of the velocity () alog the horizotal axis (X) allows idetifyig a chage i the tred of the velocity profiles. Close to the isothermal walls, decreases whe the behavior idex was icreased, ad this tred was reversed after a critical poit, located aroud X=0. ad X=0.95. The behavior idex had a effect o the CPU time. For oe of the studied cases (Ra=0 5, B=0., Pr=), whe the behavior idex is the lower (=0.7) the computatio time is 33% greater tha for the Bigham fluid (=). For itermediate values <<.5 the CPU time was of the same order of magitude tha whe the fluid was of the Bigham type (=). Whe the performace idex was larger, =2, the CPU time was much lower (63% over =). These differeces were explaied by the differet relative weight of the diffusio cocerig to the covective terms i the Navier-Stokes equatios. ACKNOWLEDGMENT This work was fuded by CONICYT/Chile uder Fodecyt Project REFERENCES [] Bares H. (999). The yield stress-a review-everythig flows? Joural of No-Newtoia Fluid Mechaics 8(-2): [2] Zhu H, Kim Y, De Kee D. (2005). No-Newtoia fluids with a yield stress. Joural of No-Newtoia Fluid Mechaics 29(3): [3] Tabilo-Muizaga G, Barbosa-Cáovas G. (2005). Rheology for the food idustry. Joural of Food Egieerig 67(-2): [4] Gratão A, Silveira Jr., Telis-Romero J. (2007). Lamiar flow of soursop juice through cocetric auli: Frictio factors ad rheology. Joural of Food Egieerig 78(4): [5] Trigilio-Tavares D, Alcatara M, Tadii C, Telis- Romero J. (2007). Rheological properties of froze cocetrated orage juice (FCOJ) as a fuctio of cocetratio ad subzero temperatures. Iteratioal Joural of Food Properties 0(4): [6] Telis-Romero J, Telis, Yamashita F. (999). Frictio factors ad rheological properties of orage juice. Joural of Food Egieerig 40(-2): [7] adrese S, Quadri M, de Souza J, Hotza D. (2009). Temperature effect o the rheological behavior of carrot juices. Joural of Food Egieerig 92(3): [8] Leog W, Hollads K, Bruger A. (998). O a physically-realizable bechmark problem i iteral atural covectio. Iteratioal Joural of Heat ad Mass Trasfer 4(23): [9] Leog W, Hollads K, Bruger A. (999). Experimetal Nusselt umbers for a cubical-cavity bechmark problem i atural covectio. Iteratioal Joural of Heat ad Mass Trasfer 4(): [0] Fusegi T, Hyu J, Kuwahara K, Farouk B. (99). A umerical study of three-dimesioal atural covectio i a differetially heated cubical eclosure. Iteratioal Joural of Heat ad Mass Trasfer 34(6):

9 [] We-Ha ST, Reui-Kuo LR-K. (20). Threedimesioal bifurcatios i a cubic cavity due to buoyacy-drive atural covectio. Iteratioal Joural of Heat ad Mass Trasfer 54(-3): [2] O Doova E, Taer R. (984). Numerical study of the Bigham squeeze film problem. Joural of No- Newtoia Fluid Mechaics 5(): [3] Tura O, Chakraborty N, Pool R. (200). Lamiar atural covectio of Bigham fluids i a square eclosure with differetially heated side walls. Joural of No-Newtoia Fluid Mechaics 65(5-6): [4] Khalifeh A, Clermot J. (2005). Numerical simulatios of o-isothermal three-dimesioal flows i a extruder by a fiite-volume method. Joural of No-Newtoia Fluid Mechaics 26(): [5] Barth W, Carey G. (2006). O a atural-covectio bechmark problem i o-newtoia fluids. Numerical Heat Trasfer, Part B: Fudametals 50(3): / [6] Awar-Hossai M, Reddy-Gorla R. (2009). Natural covectio flow of o-newtoia power-law fluid from a slotted vertical isothermal surface. Iteratioal Joural of Numerical Methods for Heat & Fluid Flow 9(7): [7] Lamsaadi M, Naïmi M. (2006). Natural covectio i a vertical rectagular cavity filled with a o-newtoia power law fluid ad subjected to a horizotal temperature gradiet. Numerical Heat Trasfer, Part A: Applicatios 49(0): [8] Lamsaadi M, Naïmi M, Hasaoui M. (2006). Natural covectio heat trasfer i shallow horizotal. Eergy Coversio ad Maagemet 47(5-6): [9] ola D, Boscardi L, LatchéJ. (2003). Lamiar usteady flows of Bigham fluids: a umerical strategy ad some bechmark results. Joural of Computatioal Physics 87(2): [20] Sigier D, alezuela-redo A. (2000). O the lamiar free covectio ad stability of grade fluids i eclosures. Iteratioal Joural of Heat ad Mass Trasfer 43(8): [2] Kumar A, Bhattacharya M. (99). Trasiet temperature ad velocities profiles i a caed o- Newtoia liquid food durig sterilizatio i a still-cook retort. Iteratioal Joural of Heat ad Mass Trasfer 34(4-5): [22] Kim G, Hyu J, Kwak H. (2003). Trasiet buoyat covectio of a power-law o-newtoia fluid i a eclosure. Iteratioal Joural of Heat ad Mass Trasfer 46(9): [23] Dea E, Glowiski R, Guidoboi G. (2007). O the umerical simulatio of Bigham visco-plastic flow: Old ad ew results. Joural of No-Newtoia Fluid Mechaics. 42-3): [24] Baliga B, Patakar S. (980). A ew fiite elemet formulatio for covectio diffusio problems. Numerical Heat Trasfer 3(4): [25] oller. (2009). Basic cotrol volume fiite elemet methods for fluid ad solids, IISC Press, Sigapore. [26] Silva J, De Moura, L. (200). A cotrol-volume fiiteelemet method (CFEM) for usteady, icompressible, viscous fluid flows. Numerical Heat Trasfer, Part B: Fudametals 40(): [27] Salias C, Chavez C, Gatica Y, Aaias R. (20). Simulatio of wood dryig stresses usig CFEM. Lati America Applied Research 4(): [28] Estacio K, Noato L, Magiavacchi N, Carey G. (2008). Combiig CFEM ad meshless frot trackig i Hele Shaw mold fillig simulatio. Iteratioal Jouural of Numerical Methods i Fluid. 56: [29] Charoesuk J, essakosol P. (200). A high order cotrol volume fiite elemet procedure for trasiet heat coductio aalysis of fuctioally graded materials. Heat ad Mass Trasfer 46(-2): /s [30] Herschel W, Bulkley R, (926). Kosistezmessuge vo Gummi-Bezollösuge. Colloid Polym Sci. 39: [3] Papaastasiou T. (987). Flows of materials with yield. Joural of Rheology 3(5): [32] Abdali S, Mitsoulis E, Markatos N. (992). Etry ad exit flows of Bigham fluids. Joural of Rheology 36: [33] Stoer J, Bulirsch R. (980). Itroductio to umerical aalysis, Spriger, New York, USA. [34] ahl Davis G. (983). Natural covectio of air i a square cavity: A bech mark umerical solutio. Iteratioal Joural for Numerical Methods i Fluids 3: NOMENCLATURE A, B, C B Cp F g Gr h J K k L m M Nu p P Pe Liear iterpolatio coefficiets Bigham umber, B= τ y/μ (L/(gβΔT)) 0.5 heat capacity, J kg - K - source term gravitatioal acceleratio, m s -2 Grashof umber, Gr = gβ TL 3 /η o 2 covective coefficiet, W m -2 K - iterfacial flux cosistecy idex, Pa s - thermal coductivity, W m - K - characteristic dimesioless legth expoetial growth parameter, s expoetial growth parameter behavior idex ormal uitary vector Nusselt umber, Nu = hl/k Pressure, Pa dimesioless pressure, P=pLη o/u o Peclet umber, Pe = ρul/µ 583

10 Pr q Ra S t T u, v U, x, y X, Y Z Greek symbols α β γ γ Γ δ Δ ζ η Η θ Pradtl umber, Pr = η ocp/k heat flux, W m -2 Rayleigh umber, Ra = gβ ΔT L 3 /η oα cotour area of a cotrol volume, m 2 time, s temperature, K velocity compoets, m s - dimesioless velocity compoets, U=u/u o; =v/u o volume of a cotrol volume, m 3 velocity vector legths, m dimesioless legths, X=x/L; Y=y/L local coordiate parallel to the velocity vector thermal diffusivity, m 2 s - thermal expasio coefficiet, K - deformatio rate, s - deformatio rate tesor dimesioless deformatio rate, Γ=L/u oγ boudary layer thickess, m differece betwee values local coordiate perpedicular to the velocity vector apparet viscosity, Pa s dimesioless apparet viscosity, H=η/η o dimesioless temperature, θ = (T-T C)/(T H- T C)) ρ τ y τ ϕ Subscripts a, b, c C e H k i, j max mi b p ref th o Superscripts E k l N desity, kg/m -3 yield stress, N m -2 shear stress tesor depedet variable, u, v, T vertices of a fiite elemet cold correspodig value of a fiite elemet hot cotour of a cotrol volume Cartesia coordiate compoet maximum value miimum value eighbor odes referece ode referece thermal referece value total umber of fiite elemets iteratio umber value of a cotrol volume value of a cotrol volume total umber of cotrol volumes 584