Turbulent Heat Transfer Modellng n a Vessel Strred by a Ptched Blade Turbne Impeller BARBARA ZAKRZEWSKA, ZDZISŁAW JAWORSKI Chemcal Engneerng Faculty Szczecn Unversty of Technology Al. Pastó 42, 70-065 Szczecn POLAND aorsk@ps.pl zakrzeska@ps.pl http://.pmg.ps.pl Abstract: - The CFD modellng of the turbulent heat transfer n a strred vessel has been presented. Four turbulence models,.e. the standard and optmzed Chen-Km k-ε and also the standard and shear-stress transport k-ω models along th to near-all regon models ere used n the modellng. The predcted local values of the heat transfer coeffcent favourably compared th the expermental data from lterature. Key-Words: - CFD modellng, heat transfer, turbulent flo, strred tank, PBT mpeller, turbulence models. 1 Introducton Numercal modellng of heat transfer th the use of CFD codes has been dely appled n process engneerng. Hoever, only fe papers ere publshed on that modellng n strred vessels. Kamnoyama et al. [1] and Kuncecz et al. [2] obtaned a good agreement beteen the predcted and the expermental results n the lamnar flo regme. Hoever, n the turbulent flo regme ths agreement as poor [3, 4]. In the earler publcaton [5], concernng the CFD modellng n the Rushton turbne strred vessel, to possble reasons of the underpredcton of the heat transfer coeffcent ere gven. One of them as lo accuracy of the turbulent momentum transfer smulaton n the vessel, especally n terms of the turbulence characterstcs n the all et. In ths paper a heat transfer modellng methodology as proposed, hch has led to better agreement beteen the CFD and expermental data, beng close to the expermental error level. The same methodology as used n the present nvestgatons. 2 Modellng methodology The computatonal flud dynamcs (CFD) method as appled to the heat transfer modellng n a acketed strred tank. The vessel th a dshed bottom had the dameter, T = 58m, and as equpped th a ptched blade turbne mpeller and four standard baffles. The vessel as flled th strred flud to the heght H = T and the flud had physcal propertes smlar to those of ater. The four bladed mpeller had the dameter D = 83T and as stuated axally at the off-bottom dstance, C = 33H, and had the rotatonal speed, N, resultng n Re = 3.4x10 4. The vessel geometry and operatng condtons ere dentcal to those used n the experments publshed by Post [6]. The standard Reynolds averaged Naver-Stokes (RANS) equatons (Eq. (1)) along th the contnuty (Eq. (2)) and energy transport (Eq. (3)) equatons ere numercally solved th the help of the Fluent 6.0 code. The source term S T on the left-hand sde of Eq. (3) represents the vscous and turbulent energy dsspaton functon. It s usually omtted, except for the systems hen the large vscosty and the large velocty gradents appear. The values of the turbulent thermal conductvty, λ t, n Eq. (3) ere calculated from turbulent Prandtl number (Eq. 4), hch as assumed to be equal to 5. ( ρv ) ( ρv v ) t v + = 0 ( ρc T) = ( µ + µ ) t v p + ρf (1) p T ( c p v T) + ρ = ( λ + λ t ) + ST (3) t x µ Pr c t t = p (4) λt Based on the prevous numercal nvestgatons [5], four selected turbulence models ere used n order to close the equaton set. The follong models ere tested: the standard [7] and optmzed Chen-Km [8] k-ε models and also the standard k-ω [9] and the shear-stress transport k-ω (k-ω SST) [10]. The standard logarthmc all functons ere appled to descrbe the boundary flo at the vessel all. In addton, the standard k-ε (2)
model along th the enhanced all treatment as used n the CFD smulatons. Good predcton of the flo feld and turbulence ntensty, especally near the all regon s very mportant for the numercal heat transfer modellng. The effect of both the numercal grd densty and also the turbulence model on the turbulent momentum transfer modellng n the all et regon as reported elsehere [11, 12] for the standard strred tank equpped th ptched blade turbne mpeller and four baffles. The effect of the turbulence models from the k-ω famly for the same geometry as also nvestgated [13]. It as concluded that the flo feld as predcted qute ell, hereas the turbulence knetc energy as underpredcted, hen the 100K cells grd and the standard and optmzed Chen-Km k-ε models ere used. The k-ω models gave a loer accuracy of the predcted mean velocty and turbulence knetc energy than the standard k-ε model. The fne grd (340K cells) as reected, because the values of the nondmensonal dstance from the tank all, y +, ere belo 20. Ths meant that recommendatons for usng all functons ere not fulflled. In the present study to numercal grds of dfferent densty ere employed. The number of grd cells as ether about 100K or 340K. It as decded to use the fne grd, because the mpeller nvestgated n ths paper dd not have standard dameter (D = 83T) and generated bgger turbulence than the standard mpeller. The values of y + ere checked after the smulatons. propertes ere defned n the Fluent code. The tank all, ncludng the dshed bottom, as dvded nto 4 quarters. One of them played the role of the heatng all. The opposte one as the coolng all. The mposed all temperatures ere equal to 308 and 288 K for the heatng and coolng alls, respectvely. To other quarters ere assumed adabatc and had the same temperature as the strred flud before the heat transfer process. Other boundary condtons ere the same for all the tank all quarters. The arrangement of the heatng, coolng and neutral alls s shon n Fg. 1. The segregated solver as employed to carry out the steady-state smulatons. The numercal computaton of the Reynolds equatons (1) along th the contnuty equaton (2) and th the selected turbulence model as the frst stage of the CFD smulatons. The flo feld and the turbulence quanttes ere obtaned. Next, the energy transport equaton (3) as solved, untl the normalzed sum of resduals decreased belo 10-8. The local values of the heat transfer rate,, ere obtaned as the result of the heat transfer smulatons and the local heat transfer coeffcent, h, as calculated from Eq. (5). The dfference beteen the temperature of the heatng (coolng) all, T, and the flud bulk temperature, T bulk, as the drvng temperature dfference. h = (T T bulk ) (5) neutral all, T n = 298K heatng all, T = 308K coolng all, T = 288K Fg. 1. Heat transfer boundary condtons The MxSm 1.7 preprocessor as used to generate to unstructured, hexahedral grds of dfferent densty. In order to smulate the acton of the ptched blade turbne mpeller, the opton of the multple frames of reference as used. The no-slp condtons at all fludsold boundares and the symmetry condtons at the free surface of the strred lqud ere chosen. All the heat transfer boundary condtons as ell as the flud 3 Results The contours of the computed heat transfer rate,, for all the turbulence models and grd denstes used are shon n Fg. 2. The hghest values of ere obtaned for the standard k-ε model th enhanced all treatment. Slghtly loer values of ere acheved for the other turbulence models tested. These values ere close to each other n the vertcal part of the heatng (coolng) alls. Large dscrepances beteen the predcted values of the heat transfer rate ere obtaned for the k-ε and the k-ω models n the bottom of the tank. The average values ere equal to 43000 and 33000 W/m 2, for the models from the k-ε and k-ω famles, respectvely. The local values of hgher than 39000 W/m 2 n the bottom of the tank ere shon as a hte places n the contours of n Fg. 2. The strred flud temperature, averaged n the hole tank volume as equal to 298.00 for all tested cases. The maxmum dfferences beteen the local and the average temperature ere equal to ± %. Therefore, n the calculatons of the local heat transfer coeffcent, the
Fne grd ~ 340K cells + WF + EWT Optmzed Chen-Km k-ε Standard k-ω k-ω SST 3.90 3.52 3.14 2.76 2.38 2.00 1.62 1.24 6 8 0 10 [W/m 2 ] 4 Coarse grd ~ 100Kcells + WF + EWT Optmzed Chen-Km k-ε Standard k-ω k-ω SST Fg. 2. Contours of the heat transfer rate at the heatng all for all turbulence models tested. WF all functons, EWT enhanced all treatment averaged value of temperature as used as the bulk flud temperature, T bulk n Eq. (5). The same results ere obtaned n prevous nvestgatons [5] hen the heat transfer modellng n a standard Rushton turbne strred tank as carred out usng the same thermal boundary condtons. The predcted local values of the heat transfer coeffcent, h, calculated from Eq. (5) ere compared th the expermental [6]. The comparson as performed at nne axal postons and for the angular poston of θ = 45 beteen the adacent baffles. The axal profles of the h coeffcent for all the turbulence models and grd denstes are shon n Fg. 3. A good agreement beteen the predcted and expermental values of h, th dscrepances of about 8% (fne grd) and 12% (coarse grd) as obtaned for the axal postons = to 5. Near to the free surface of the strred lqud the dfference n the h values from CFD and experment as about 35% for the fne grd and 28% for the coarse grd. Smlarly, n the loer part of the vessel, belo the mpeller level, the h values ere underpredcted, especally by the standard k-ω model hen the dscrepances ere equal to 44%. For the other cases tested, they ere about 35% beng roughly the same for the to tested grds. 4 Conclusons CFD modellng of the turbulent heat transfer n strred tank equpped th the non-standard ptch blade turbne mpeller and four flat baffles as carred out. The follong conclusons ere dran from the analyss of the modellng results: 1. The best agreement beteen the predcted and expermental [6] values of the local heat transfer coeffcent, h, as obtaned for the fne grd and standard k-ε turbulence model along th the enhanced all treatment. The dfferences ere equal to 12.2%. 2. The smulated values of h ere underpredcted near to the free surface of the strred lqud and to the tank bottom n all the tested cases. 3. The same methodology used n the present and prevous [5] nvestgatons resulted n a good
340K cells Post, 1983 k-epslon + WF k-epslon + EWT optmzed Chen-Km Post, 1983 k-epslon + WF k-omega k-omega SST 100K cells Fg. 3. Axal profles of heat transfer coeffcent for all turbulence models and grd denstes. WF all functons, EWT enhanced all treatment agreement beteen the CFD and expermental values of h for the standard and non-standard geometry of the strred tank. ACKNOWLEDGEMENTS The State Commttee for Scentfc Research s gratefully acknoledged for fnancal support thn the grant No. 4 T09C 003 22 and thanks are also due to Fluent Inc. for usng the MxSm softare. References: [1] M. Kamnoyama, M. Watanabe, K. Nsh, M. Kamano, Numercal smulaton of local heat transfer coeffcents n strred vessel th mpeller for hghly vscous fluds, Journal of Chemcal Engneerng of Japan, Vol. 32 No. 1, 1999, pp. 23-30. [2] Cz. Kuncecz., A. Nedzelska, M. Petrzykosk, Formulaton and soluton of a heat transfer model for rbbon mpellers (n Polsh), In Ŝ ynera Chemczna Procesoa, Vol. 25, No. 3/2, 2004, pp. 1225-1230. [3] Z. Jaorsk, Numercal studes on acket heat transfer n strred tanks, CHISA Congress, Prague, 31 August 2000, G7 Symposum on CFD, 2000. [4] A.J. Mannng, R. Mann, M. Colley, CFD model of the flo from a heated mpeller n a strred vessel, In: IChemE Symp. Ser., 140, Flud Mxng 5 Conference, 4-5 July 1996, Bradford, UK, pp. 95-105. [5] B. Zakrzeska, Z. Jaorsk, CFD modellng of turbulent acket heat transfer n a Ruston turbne
strred vessel, Chemcal Engneerng and Technology, Vol. 27, No. 3, 2004, pp. 237-242. [6] T. A. Post, Geometrcal nfluences on local and total mass and heat transfer n an agtated tank. PhD thess. Sss Federal Insttute of Technology, Zurch, 1983. [7] B. E. Launder, D. B. Spaldng, The numercal computaton of turbulent flos, Computer Methods n Appled Mechancs and Engneerng, Vol. 3, 1974, pp. 269-289. [8] M. Jenne, M. Reuss, A crtcal assessment on the use of k-ε turbulence models for smulaton of the turbulent lqud flo nduced by a Rushton-turbne n baffled strred-tank reactors, Chemcal Engneerng Scence, Vol. 54, 1999, pp. 3921-3941. [9] D.C. Wlcox, Reassessment of the scale-determnng equaton for advanced turbulence models, Amercan Insttuton of Aeronautcs and Astronautcs Journal, Vol. 26, No. 11, 1988, pp. 1299-1310. [10] F.R. Menter, To-equaton eddy-vscosty turbulence models for engneerng applcatons, Amercan Insttuton of Aeronautcs and Astronautcs Journal, Vol. 32, No. 8, 1999, pp. 1598-1605. [11] Z. Jaorsk Z., B. Zakrzeska, Modellng of the turbulent all et generated by a ptched blade turbne mpeller. The effect of turbulence model, Trans IChemE Part A, Vol. 80, 2002, pp. 846-854. [12] B. Zakrzeska, Z. Jaorsk, Modellng of the turbulent all et generated by a ptched blade turbne mpeller. The effect of numercal grd densty (n Polsh), In Ŝ ynera Aparatura Chemczna, Vol. 4S, 2002, pp. 149-150. [13] B. Zakrzeska, Numercal heat transfer modellng n strred tanks. PhD thess. Szczecn Unversty of Technology, Poland, 2003.