VELOCITY PROFILE MODELING FOR NON-ISOTHERMAL FLOWS INSIDE A CIRCULAR TUBE

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1 VELOCITY PROFILE MODELING FOR NON-ISOTHERMAL FLOWS INSIDE A CIRCULAR TUBE J. A. Gutieez a, A. A. Mendibuu a, I. Ávila a, and Edga Paz Peéz b a São Paulo Univesity (UNESP), School of Engineeing, Enegy Depatent, Guaatinguetá, SP, CEP , Bazil jodan.aao.gutieez7@gail.co b Catholic Univesity the Angels of Chibote Laboatoy of Fluid Mechanics - Peú ABRACT This eseach poposes a new ethod to establish the velocity field and the diensionless velocity pofile fo Newtonian and non-newtonian flows inside a cicula tube. Seveal studies developed egading diffeent fluid types (such as potency law fluid, Bingha and Heschel-Bulkley, aong othes) obseved that a ational o iational polynoial was used fo the dependent velocity field vaiable. Thus, a ational polynoial was established as a stating point fo this eseach as the dependent velocity field vaiable. Diensionless velocity pofiles obtained fo the poposed fluid-dynaics odel wee epeientally copaed only with diensionless velocity pofiles fo non-isotheal Newtonian flows of glyceol, in cooling as well as heating. On the othe hand, it was possible to calculate that RMS eos found using elative diensionless velocity data obtained fo the poposed fluid-dynaics odel ceates vey sall eos, which ae copaable to RMS eos found using data obtained fo application of a nueical ethod. Finally, the poposed fluid-dynaics odel was validated with a diensionless velocity pofile obtained fo the flow of a cooling pocess, esulting in the validity of the poposed odel. Received: July 9, 2017 Revised: Novebe 1, 2017 Accepted: Decebe 23, 2017 Keywods: heology, velocity pofile, non-isotheal fluids, cicula tube NOMENCLATURE Supescipts A T Coss-sectional aea to the pipe, 2 e Radial paaete e θ Angula paaete e Positional paaete e t Theodynaic paaete e T Tie, h F obj Conve objective function Q Quantity giving the deviation fo isoviscous behaviou Radial coodinate, R Tube adius, S T Tansvesal section V Velocity field, /s V Velocity on the pipe, /s V el Relative diensionless velocity V Aveage velocity, /s V Velocity of odel od Geek sybols Relative velocity coefficient Density, kg/ 3 Subscipts ep Epeiental Aial diection to the tube INTRODUCTION Reseach egading the heology of Newtonian and non-newtonian flows inside channels of diffeent tansvesal sections and cicula section tubes is cuently vey developed (Choi et al., 2016). Thee is a cetain assuance fo the study of Newtonian flows. Howeve, eseaches fo non-newtonian flows ae still being developed. A heological odel is initially chose fo study of non-newtonian flows (Igens, 2014), with a stating point being the elation of sheaing tension to sheaing ate (velocity pofile), fo diffeent heological odels, such as potency law, Bingha, Heschel Bulkley, aong othes (López- Caanza, Jenny, Noua, 2012; Peiinho et al., 2005). All of these odels ae esticted to bode conditions of flow inside the channel o tube. Maybe the ost coon heological odel is the potency law odel (Güzel, Figaad, Matinez, 2009; Hanley, Conin, Byne, 2013), and the geneal odel would be the Heschel Bulkley (Ancey, Bates, 2017; Bentad et al., 2017). A consequence of choosing a good heological odel is a good ageeent between the pofiles fo epeiental velocity and analytical o siulated velocity. The Dutch enginee Piete Bateld Kwant was one of the fist eseaches to wok with non-newtonian flows. In the theoetical eseach by Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p

2 Kwant, Zwaneveld and Dijksta (Kwant, Zwaneveld, Dijksta, 1973b) fo non-isotheal flows, the potency law odel was eployed to study velocity pofiles using two ethodologies: an appoiate ethod (includes the ass consevation equation and oentu equation) and a nueical ethod (includes pevious equations, plus the enegy equation). Late on, eseach conducted by Kwant, Fieens and Van De Lee (Kwant, Fieens, Van De Lee, 1973a) poduced epeiental esults to validate theoetical esults fo the wok of Kwant, Zwaneveld and Dijksta (Kwant, Zwaneveld, Dijksta, 1973b). Noally, seveal eseach studies use the copute fluid dynaics (CFD) fo flow odeling (Matins et al., 2014, 2016; Wang, Zhang, Wang, 2013). The CFD odel eploys bode conditions, the ass consevation equation, the oentu equation, aong othes. Advanceents egading the study of non-newtonian flows is divese, such as the study of the tube inclination effect ove the eoval dynaics of a viscous-plastic fluid by a Newtonian fluid (Alba, Figaad, 2016), the behavio of a Heschel Bulkley fluid laye when it is suddenly inclined and subjected to gavitational foces (Ancey, Bates, 2017), the effects of the velocity pofile in the entance of a cooling channel ove flow (Ki et al., 2016), the study of the velocity pofiles (befoe and afte a poous zone) of a tubulent flow in a staight channel (Choi et al., 2016), o the theal conditioning in the tube wall (Betsche, Knippe, Wetzel, 2016; Tu et al., 2015; Weigand, Abdeloula, 2014). Wang et al. (2017) pefoed one inteesting study, in which a heological odel was not used fo odeling the velocity pofile of a heteogeneous flow. These authos established a ational polynoial in the vaiable depending on the velocity field in ode to odel the velocity pofile. Such ational polynoial has 10 tes, which ae based on space coodinates, tube diaete, paticle size, ice faction, and aveage flow velocity. The idea to establish a ational polynoial to a depending vaiable fo the velocity field was also eployed in anothe study (Aao, Henández, Olivencia, 2015). This cuent eseach consists on odeling diensionless velocity pofiles fo laina Newtonian flows in which the tube wall tepeatue was constant fo each epeient. In ode to do that, a genealization was eployed egading the polynoial epession poposed in the peviously entioned eseach (Aao, Henández, Olivencia, 2015) fo the dependent velocity field vaiable. This polynoial epession was esticted by appopiate bode conditions as those used in pevious eseaches that used heological odels. MATERIALS AND METHODS In ode to define the velocity field dependent vaiable (consideing only the velocity aial coponent) of a flow fo any fluid inside a cicula section tube, a ational polynoial was eployed, which was uch siple than the one eployed by Wang et al. (2017). V e,e θ,e,e t,e 1 t,,e 2 t,e j T n i Vi e θ,e,e t,e 1 t,,e 2 t,e j T e i1 t1 t2 t j T V e,e,e,,e,e (1) The velocity field dependent vaiable is defined on Eq. (1), whee e, e θ and e, ae the independent vaiables, which depends on intenal adius, the angle, and position egading the beginning of the tube, espectively. They ae independent vaiables that elate cylindical coodinates. The independent vaiable is an aveage theodynaic paaete (o a paaete that elates a theodynaic popety in an iplicit anne) evaluated in a cetain tansvesal section (S T ) to the tube o it is chaacteistic fo all the tube (fo eaple: constant tepeatue o constant heat flow). Finally, the e T independent vaiable is the tie vaiable. Soe consideations wee foulated, which ae pesented as follows: i) The tube though which the fluid flows is intenally sooth and it is placed on a hoizontal position. ii) The flow unde study is in coplete hydodynaic developent. iii) A value of n=3 was eployed to evaluate the velocity field dependent vaiable. iv) The flow egien is stationay. v) The velocity field is independent fo the e θ vaiable. vi) Due to denotative siplicity, only one theodynaic paaete was eployed and naed as e t (but soe change in such consideation can be odified in futue studies, without any poble). vii) The following notations wee established by denotative siplicity: e,e,et e,,t e,et e,t e t j (2) (3) The intenal adius and the tube tansvesal cicle aea ae naed as R and A T, espectively. e (4) R 88 Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p

3 The e independent vaiable was defined in Eq. (4), as 0 R. Definition of e and e t independent vaiables will depend on the study which will be chosen as a efeence. It is ipotant to notice that it was not established if the flow egien is laina, tansitoy o tubulent, since it was not defined if the fluid is incopessible due to the dependence of the behavio egading the theodynaic paaete along the flow. D,,t and D,t ae non-epty sets in which thei eleents ae (e,,t ) and (e,t ), espectively. The siplified velocity field (by pevious consideations) and the flow density (ρ) wee established as eal functions defined in the D,,t doain. V e,,t 3 2 3,t 2,t 1,t,t V e e V e e V e e V e (5) Equation (5) shows a siplified epession of the velocity field dependent vaiable. ρ, e,t V ρ e,,t V e,,t dat V e,,t da T e,t (6) V e,,t dat (7) AT On the othe hand, it was necessay to establish flow aveage density (ρ, ) and flow aveage velocity ( V ) as eal functions defined in the D,t doain. Dependent vaiables fo both functions ae defined on Eq. (6) and (7), espectively. e V e,t αe,t V,t (8) A elative velocity coefficient was defined and identified as α. It is pesented on Eq. (8). Soe conditions wee established, which seved to estict the V (e,,t ) dependent vaiable. These conditions ae pesented as below: V e e,,t e 0 t 0 (9) V 1,e,e 0 (10) ρ e,,t V e,,t dat (11) ρ, e,t V e,t AT In Eqs. (9) - (11) the conditions of: Condition of aiu velocity in the tube ais, Non-slippey condition in the tube intenal wall and Mass consevation condition, espectively, ae pesented. 3 2,t α e 5e 6e 1 V e,,t V e,t e 10e (12) A final epession fo the velocity field dependent vaiable was obtained by using Eq. (9) (11) to estict V (e,,t ), and it is pesented on Eq. (12). e V,,t Vel e,,t V e,t (13) Finally, the epession fo elative diensionless velocity pofile ( V ) is shown on Eq. el (13). Fo laina isotheal Newtonian flows, α(e,t )=2 is eployed, ceating a known epession fo the velocity diensionless pofile equivalent to V (e el,,t )=2-2 e 2. The RMS eo (oot ean squae) was eployed in ode to evaluate the pedictability of the elative diensionless velocity pofile egading eseach epeiental data that ight be used (Devoe, Bek, 2012). V 2 i 1 ep V i odi (14) RMS The atheatical epession fo RMS eo, applied in this study, is pesented on Eq. (14). Whee is the nube of epeiental data and V is the ep epeiental velocity o the epeiental elative diensionless velocity povided by a cetain adius. The V velocity is obtained by the poposed odel od fo Eq. (12) o Eq. (13). This depends on data povided by studies used to validate the poposed odel. The epeiental elative diensionless velocity was defined as the epeiental velocity divided by the aveage velocity. It is clea that Eq. (12) needs to be used to validate the odel, in case epeiental velocities ae povided. Thus, the dependent vaiable can be epesented as V (e,,t ) by V (e, V (e,t ), V (e,t )). Due to denotative siplicity easons, the dependent vaiable Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p

4 was consideed to be V (e,,t ) by V (e, V, V ). Fo the case in which epeiental velocities ae povided, a ethodology was developed to allow calculation of oe suitable velocity values, V and V. ep 2 F V,V V V V,V (15) obj,1 i i i1 The coesponding conve objective function was established in Eq. (15). Thus, this objective function has to satisfy the following conditions: obj,1 V F V,V Fobj,1 V,V V 0 0 (16) (17) Once V and V velocity values ae obtained fo diffeent epeients, values fo elative velocity coefficient can be calculated (α). In case epeiental elative diensionless velocities ae povided, a siila ethod to the pevious one can be applied, changing the objective function and the conditions of the fist deivative. α 2 ep i (18) Fobj,2 V V α i i1 The coesponding conve objective function was established in Eq. (18). Thus, this objective function has to satisfy the following condition: F obj,2 α α 0 (19) A behavio fo the elative velocity coefficient can be established with these ethods, fo any position and theal condition in the tube wall unde study. An appopiate epession can be attibuted to the elative velocity coefficient by using a linea egession odel (Kleijnen, 2015). RESULTS AND DISCUSSION In ode to validate the poposed fluid-dynaics odel, epeiental data wee eployed fo Kwant (Kwant, 1971) and Kwant et al. (Kwant, Fieens, Van De Lee, 1973a), the woking fluid being glyceol in both studies. In addition, data egading elative diensionless velocities obtained though the nueical ethod poposed by Kwant et al. (1973b) wee copaed with epeiental elative diensionless velocities. The of data egading epeiental elative diensionless velocities and fo the nueical solution wee deived fo scaling velocity pofile figues. Kwant et al. (1973a) studied velocity pofiles fo Newtonian laina flows in a tube whee wall tepeatue is constant, but diffeent in each epeient that is pefoed. Epeiental and nueical elative diensionless velocities ae a function of +, Q and Re paaetes, which ae the diensionless aial position, iso-viscosity standad paaete egading the flow, and the Reynolds nube evaluated at the aveage flow tepeatue, espectively. Fo oe efeences egading these paaetes, the study by Kwant et al. (1973a) can be evised. The cuent study consideed that the positional paaete ( + ) does not ceate significant changes in the elative diensionless velocity pofile. This assuption was consideed due to the fact that seveal velocity pofiles ae not pesent fo the sae tube theal condition. In addition, the Reynolds nube ( Re ) was not selected as a theodynaic paaete since all flows that geneate epeiental and nueical elative diensionless velocity data ae chaacteized by a laina egien, and also have a Reynolds nube in the ange of 0.24 to 55 (Kwant, 1971), which was consideed vey low to poduce significant changes in the diensionless velocity pofile. Due to pevious assuptions, the Q theodynaic paaete was consideed the ost influential in epeients conducted, since it quantifies heat tansfeence between the flow and the tube, chaacteizing the heat tansfe phenoenon on heating o cooling pocesses. It is then possible to establish that e t =Q. The Q value is equivalent to zeo in case the flow and the tube have the sae tepeatue. el V e,q α Q 5e 6e 1 10e 10e (20) The poposed fluid-dynaic odel to deteine elative diensionless velocity pofiles is pesented on Eq. (20). Figue 1 shows the diensionless velocity pofile fo the flow of a cooling pocess with Q=-2.06 (Kwant, 1971). Fo this case, one α=3.053 was deteined. The RMS eo fo data obtained fo the poposed ethod and the nueical ethod was equivalent to and , espectively. Figue 2 shows a diensionless velocity pofile fo the flow of a heating pocess with Q=1.49 (Kwant, Fieens, Van De Lee, 1973a). In this case, one α=1.577 was deteined. The RMS eo fo data obtained fo the poposed odel and the nueical ethod was equivalent to and , espectively. 90 Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p

5 Figue 3 shows the diensionless velocity pofile fo the flow of a heating pocess with Q=2.34 (Kwant, 1971). In this case, one α=1.298 was deteined. The RMS eo fo data obtained fo the poposed odel and the nueical ethod was equivalent to and , espectively. With diffeent calculated α values and Q values used in pevious epeients, a linea egession odel was developed with α=2 fo Q=0, in ode to calculate the elative velocity coefficient value (α). 2 αq0.0432q Q (21) Figue 1. Diensionless velocity pofile fo Q= This linea egession odel pesented on Eq. (21) have a deteination coefficient of R 2 = Though the use of the linea egession odel to calculate the elative velocity coefficient (α) fo isotheal laina Newtonian flows, a value of α= was obtained. Thus, this velocity coefficient value is vey close to the α=2 value, which is calculated theoetically. In ode to validate the poposed fluid-dynaics odel fo non-isotheal Newtonian flows, an epeiental diensionless velocity pofile was evaluated fo Q=-1.35 (Kwant, Fieens, Van De Lee, 1973a). A good ageeent was obseved between the diensionless velocity pofile (obtained fo the poposed fluid-dynaics odel) egading epeiental data. This evaluation was pesented on Fig. 4. Figue 2. Diensionless velocity pofile fo Q=1.49. Figue 4. Diensionless velocity pofile fo Q= Figue 3. Diensionless velocity pofile fo Q=2.34. Though the use of Eq. (21), a value of α=2.655 was calculated. The RMS eo fo data obtained fo the poposed odel (using Eq. (21) to calculate the elative velocity coefficient) was equivalent to , and the RMS eo fo data obtained though the nueical ethod was equivalent to Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p

6 CONCLUSIONS It can be concluded fo the esults that diensionless velocity pofiles obtained fo the poposed fluid-dynaics odel efficiently calculate epeiental diensionless velocities fo nonisotheal Newtonian flows. Diensionless velocity pofiles obtained though the use of the fluiddynaics odel ae bette to odel diensionless velocity pofiles in cooling pocesses, in copaison to diensionless velocity pofiles in heating pocesses, fo which heating is even highe in the flow. Due to the fleibility to which the vaiables ight depend, o how the elative velocity coefficient (α) can be defined, it is possible to eploy the poposed fluid-dynaics odel in futue studies to obtain velocity pofiles fo non-newtonian flows. REFERENCES Alba, K., and Figaad, I. A., 2016, Dynaics of the Reoval of Viscoplastic Fluids fo Inclined Pipes, Jounal of Non-Newtonian Fluid Mechanics, Vol. 229, pp Aao, J., Henández, O., and Olivencia, J., 2015, Cálculo del Capo de Velocidad de un Flujo Laina de Agua al Inteio de una Tubeía, Enfiándose con el Medio Abiente Después del Copleto Desaollo Hidodináico, Agoindustial Science, Vol. 5, pp (in Spanish) Ancey, C., and Bates, B. M., 2017, Stokes Thid Poble fo Heschel Bulkley Fluids, Jounal of Non-Newtonian Fluid Mechanics, Vol. 243, pp Bentad, H., Esael, A., Noua, C., Lefeve, A., and Ait-Messaoudene, N., 2017, Enegy Gowth in Hagen Poiseuille Flow of Heschel Bulkley Fluid, Jounal of Non-Newtonian Fluid Mechanics, Vol. 241, pp Betsche, D., Knippe, P., and Wetzel, T., 2016, Epeiental Investigation on Heat Tansfe in Laina, Tansitional and Tubulent Cicula Pipe Flow, Intenational Jounal of Heat and Mass Tansfe, Vol. 95, pp Choi, M. K., Cho, M. K., Lee, H. W, Jung, H., and Lee J. W., 2016, Genealized Equation fo the Design of a Baffle to Geneate Abitay Flow Velocity Pofiles, Jounal of Wind Engineeing and Industial Aeodynaics, Vol. 149, pp Devoe, J. L., and Bek, K. N., 2012, Moden Matheatical Statistics with Applications, Spinge Book. Güzel, B., Figaad, I., and Matinez, D. M., 2009, Pedicting Laina-Tubulent Tansition in Poiseuille Pipe Flow fo Non-Newtonian Fluids, Cheical Engineeing Science, Vol. 64, pp Hanley, K. J., Conin, K., and Byne, E. P., 2013, Dispesion in Paticle Velocity Resulting fo Rando Motion though a Spatially-Vaying Fluid Velocity Field in a Pipe, Powde Technology, Vol. 245, pp Igens, F., 2014, Rheology and Non-Newtonian Fluids, Spinge Book. Ki, D. H., Lee, B. J., Pak, J. S., Kwak, J. S., and Chung, J. T., 2016, Effects of inlet Velocity Pofile on Flow and Heat Tansfe in the Entance Region of a Ribbed Channel, Intenational Jounal of Heat and Mass Tansfe, Vol. 92, pp Kleijnen, J. P. C., 2015, Design and Analysis of Siulation Epeients, Spinge Book. Kwant, P. B., 1971, Non-Isotheal Laina Flow, Doctoal Thesis, Delft Univesity of Technology. Kwant, P. B., Fieens, R. H. E., and Van de lee, A., 1973a, Non-Isotheal Laina Pipe Flow - II. Epeiental, Cheical Engineeing Science, Vol. 28, pp Kwant, P. B., Zwaneveld, A., and Dijksta, F. C., 1973b, Non-Isotheal Laina Pipe Flow - I. Theoetical, Cheical Engineeing Science, Vol. 28, pp López-caanza, S. N., Jenny, M., and Noua, C., 2012, Pipe Flow of Shea-Thinning Fluids, Coptes Rendus - Mecanique, Vol. 340, pp Matins, N. M. C., Caiço, N. J. G., Raos H. M., and Covas, D. I. C., 2014, Velocity-Distibution in Pessuized Pipe Flow Using CFD: Accuacy and Mesh Analysis, Coputes and Fluids, Vol. 105, pp Matins, N. M. C., Soaes, A. K., Raos, H. M., and Covas, D. I. C., 2016, CFD Modeling of Tansient Flow in Pessuized Pipes, Coputes and Fluids, Vol. 126, pp Peiinho, J., Noua, C., and Desauby, C., Théon, B., 2005, Laina Tansitional and Tubulent Flow of Yield Stess Fluid in a Pipe, Jounal of Non-Newtonian Fluid Mechanics, Vol. 128, pp Tu, W., Tang, Y., Hu, J., Wang, Q., and Lu, L., 2015, Heat Tansfe and Fiction Chaacteistics of Laina Flow though a Cicula Tube with Sall Pipe Insets, Intenational Jounal of Theal Sciences, Vol. 96, pp Wang, J., Wang, S., Zhang, T., and Battaglia, F., 2017, Matheatical and Epeiental Investigation on Pessue Dop of Heteogeneous Ice Sluy Flow in Hoizontal Pipes, Intenational Jounal of Heat and Mass Tansfe, Vol. 108, pp Wang, J., Zhang, T., and Wang, S., 2013, Heteogeneous Ice Sluy Flow and Concentation Distibution in Hoizontal Pipes, Intenational Jounal of Heat and Fluid Flow, Vol. 44, pp Weigand, B., and Abdeloula, M., 2014, Aial Heat Conduction Effects in the Entance Region of Laina Duct Flows: Coelations fo the Local Nusselt Nube, Intenational Counications in Heat and Mass Tansfe, Vol. 51, pp Engenhaia Téica (Theal Engineeing), Vol. 16 No. 2 Decebe 2017 p