RELATIVE PARTICLE TO FLUID VELOCITY IN TURBULENT FLUID

Transcription

1 RELATIVE PARTICLE TO FLUID VELOCITY IN TURBULENT FLUID Ing. Vít Pešava Supervsor: prof. Ing. Pavel Dtl, DrSc. Czech Techncal Unversty n Prague, Faculty of Mechancal Engneerng, Department of Process Engneerng, Techncká 4, Prague 6, emal: Vt.Pesava@fs.cvut.cz Abstract The settlng partcle velocty n a turbulent feld affects many transfer phenomena such as offbottom partcle suspenson, partcle break up and attrton, mass and heat transfer at dssoluton and crystallzaton, polymerzaton, fludzaton, bochemcal reactons, heterogeneous chemcal reactons, etc. Most of these operatons occur n turbulently agtated tanks or n vertcals tubes so the determnaton of the relatve partcle to flud velocty s of crucal mportance. Currently, most of exstng approaches use the settlng partcle velocty n a stll lqud for desgnng processes. But the settlng partcle velocty n a stll lqud and the settlng partcle velocty n a turbulent feld may dffer sgnfcantly from each other. The goal of ths work s to develop a methodology for calculatng the settlng partcle velocty n a turbulent feld. The model of Dtl and Skrvanek [1] s used n ths methodology. Keywords Settlng partcle, lqud, turbulence, Matlab 1. Introducton Desgnng of process equpment demands an accurate determnng of the relatve partcle to flud velocty. Expermental nvestgaton of a turbulent feld nfluence on the settlng partcle velocty s a common way of solvng ths objectve. The measured relatve partcle to flud velocty s usually reported as a functon of the Stokes number or Kolmogoroff mcro length scale. Stokes number s defned as a rato between the partcle relaxaton tme and turbulent ntegral scale. An alternatve method s calculatng the relatve partcle to flud velocty usng the moton equaton of a partcle n flud. Ths method s utlsed n ths work. The nput values are measured tme dependences of local axal component of turbulent fluctuatons. The results of calculatons are subsequently compared wth LDA/PDA measurements of flud and partcle veloctes n an agtated vessel. Unfortunately a consderable uncertanty of the relatve velocty determned durng these measurements can occur. The relatve velocty s calculated as a dfference between partcle velocty and flud velocty. The uncertantes of these both ndvdual measurements have a sgnfcant mpact on the uncertanty of the determned relatve velocty. 2. Theoretcal Methods of measurng the hndered sedmentaton can be dvded nto two groups. The frst group comprses drect methods of measurements of veloctes usng PIV, LDA/PDA or hgh speed vdeo camera. The second group comprses ndrect methods of measurng veloctes

2 usng partcle dstrbuton measurements, partcle suspenson a mass transfer measurements. The measured values are usually depcted usng Stokes number or the rate of partcle sze and the ntegral length scale of turbulence. 2.1 Drect methods Šedvý [2] measured the dfference between lqud and partcle veloctes n a mxed vessel usng the LDA/PDA method. He observed a decrease of a settlng velocty caused by turbulence for glass beads and ron partcles wth szes 1.85 mm and 0.33 mm respectvely. Ths decrease was proportonal to the relatve sze of a partcle d/t and a rate of denstes Δρ/ρ. Yang and Shy [3] reported an ncrease n settlng velocty n a turbulent feld generated by two oscllatng grds. Heavy tungsten and glass partcles n an aqueous nearsotropc turbulence were nvestgated. The maxmum ncrease was observed as Stokes number was near unty. Doroodch et al. [4] nvestgated the mpact of turbulence on the drag coeffcent for partcles of dfferent sze and densty usng a hgh speed vdeo camera. Turbulence was generated by two oscllatng grds. Nylon and Teflon sphercal partcles wth szes from 2.38 to 7.94 mm were examned. They reported settlng velocty decreases wth ncreasng partcle sze. From the pont where the partcle sze was equal to the ntegral length scale of turbulence settlng velocty ncreased wth ncreasng partcle sze. A reducton of settlng velocty n a turbulent feld was reported by these authors. Ghatage et al. [5] measured hndered settlng slp velocty of steel partcles n a soldlqud fludzed bed. The drag coeffcent was ncreased by turbulence for all expermental condtons. 2.2 Indrect methods Magell et al. [6] nvestgated sold dstrbuton for sold-lqud suspensons n tanks strred by multple Rushton turbnes. Sold concentratons were measured usng an optcal technque and Peclet number calculated from expermental data. Settlng velocty was subsequently calculated from Peclet number and dsperson coeffcent. For larger partcles the retardaton of settlng veloctes sank down to 40 percent of settlng velocty n a stll lqud. But for tny partcles (less than 10λ) settlng velocty remaned equal. Brucato et al. [7] determned settlng velocty and drag coeffcent n Couette flow usng resdence tme technque. Partcles used n the experment were glass beads µm and slca partcles µm. They reported settlng velocty decreases wth ncreasng turbulence ntensty. An nfluence of partcle sze and turbulence ntensty on partcle drag was found. Partcle drag was ether unaffected or ncreased by free stream turbulence. A new correlaton for the estmaton of partcle drag coeffcents was proposed. 3 CD CD 0 4 d = CD 0 λ Sold concentratons of glass and plastc partcles of dfferent szes n tanks strred by multple mpellers were nvestgated by both Nocentn et al. [8] and Pnell et al. [9]. Partcle settlng veloctes were subsequently determned from Peclet number defnton. Retardaton of settlng velocty was measured n agreement wth prevous correlaton: U ( 16 / 1) 0. 6 / U = 0.4 tanh d + S t λ

3 2.3 Dtl and Skrvanek model The model of Dtl and Skrvanek [1] s used n ths work. The equlbrum of forces actng on a partcle s utlsed n ths model. Fluctuatons of local axal veloctes are added for determnng of the drag force actng on a partcle. Accordng to [1] the basc equaton for partcle moton n lqud velocty feld s as follows: d Re v ν ρl Re = Ar 18 Re ( Re ) * dt ω d 2 ρs Re (1) where dmensonless number s defned n [1] as: ρ 2 S ω d D = (2) ρ ν L and Re = Re v Re u (3) The local turbulent velocty s n [1] smplfed by a snusod so the values of fluctuatng Re u can be expressed as Re u d = a sn(2π ω t) ν (4) The value of velocty ampltude a and the frequency ω for the correspondng turbulent flow must be determned expermentally. The rato X between the partcle sedmentaton velocty hndered by turbulence and the termnal velocty u sed can be expressed as: X = ( Re Re ) ( v u) v Rev sed u = u sed (5) The termnal sedmentaton velocty u sed s calculated n MATLAB from Equaton 6 usng the Newton's method. 4 Ar 24 Re Re ( Re ) = 0 Re (6) Equaton (1) was solved n MATLAB usng Runge-Kutta method (ODE45). 3. Calculatons The veloctes were newly also expressed usng the Fourer transform n ths work. The measured records of real veloctes obtaned by LDA [10] were frstly re-sampled to a

4 constant tme step by sample and hold method [11]. Ths treatment s a necessary condton for the Fourer transform to work properly. Then the measured tme dependences were decomposed nto the frequency doman usng the Fourer transform. The calculated coeffcents of the Fourer seres were n the next step transformed usng the equatons (7-9) nto parameters sutable for utlzaton n the computatonal model n MATLAB. The computatonal model was created usng [12]. 2 2 a = Re( FT) + Im( FT) (7) ω = T (8) Im( FT) ϕ = Re( FT) (9) These parameters are a, ω and ϕ n the equaton (10). FT s the Fourer transform of measured data. Fnally the recomposed tme dependency of veloctes s a sum of snes created usng the parameters. Re u d = a sn(2π ω t + ϕ ) ν (10) The process data nputs for calculatons are lsted n Table 1. Ampltudes and frequences for modelng a smplfed sne lqud velocty are taken from [1]. The measured data obtaned by LDA for calculatons wth a real flud velocty were chosen accordng to correspondng revolutons of the mpeller. LDA measurements were performed on a tank T=0.3 m n dameter agtated by a ptched sx-blade turbne at mpeller speeds 450 and 600 rpm. Iron and glassy partcles are consdered n an aqueous suspenson wth densty kg.m -3 and vscosty x10-6 m 2.s -1. Table 1. - Input process data Sold phase densty Materal [kg.m -3 ]. Ampltude [m.s -1 ]. Frequency [s -1 ]. Revolutons of mpeller [mn -1 ]. Glass Glass Iron Iron Results The results are plotted as a dependences of X on D or X on a partcle dameter for dfferent materals and revolutons. A comparson of results calculated n Matlab and Excel [1] for glassy partcles at revolutons 450 rpm and 600 rpm are depcted n Fgure 1. Comparsons of results of calculatons wth a smplfed sne lqud velocty and the real lqud velocty performed n MATLAB are depcted n Fgures 2 and 3.

5 Fg. 1. X as a functon of D for glassy beads at 450 rpm and 600 rpm.comparson of Excel and Matlab results Fg. 2. X as a functon of a partcle dameter for glassy beads at 450 rpm and 600 rpm. Comparson of results wth smplfed and real veloctes.

6 Fg. 3. X as a functon of a partcle dameter for ron spheres at 450 rpm and 600 rpm. Comparson of results wth smplfed and real veloctes. 5. Conclusons The followng conclusons can be drawn: True descrpton of local lqud turbulent velocty requres at least 8192 terms and evaluaton program was constructed n MATLAB For both cases namely snusodal velocty and real turbulent velocty the program calculatng relatve partcle to lqud velocty expressed as X was programmed and verfed. The dfference between results prevously obtaned n Excel and newly n MATLAB was found excellent. Comparson of X vs. d obtaned for snusodal and real lqud veloctes have a smlar course, however the curves are postponed and results wth a snusodal velocty take lower values. The expermentally obtaned values for X usng PDA [2] are not n contrast wth calculated ones. From ths follows that smplfcaton of lqud velocty by a snusod s stll an open problem for next research.

7 Acknowledgement Ths research has been supported by the Grant Agency of Czech Republc under grant No. P Ths research has been supported by the Grant Agency of Czech Republc under grant No. SGS14/061/OHK2/1T/12. Symbols Ar Archmedes number (1) a ampltude of the fluctuatng flud velocty component (m s -1 ) C D drag coeffcent (1) D shaft dameter (m) D dmensonless number defned by Eq. (2) (1) d partcle dameter (m) g acceleraton due to gravty (m s -2 ) L shaft length (m) n revolutons of mpeller (s -1 ) Rev sed partcle Reynolds number at sedmentaton by ts termnal velocty n a stll flud (1) Re u local flud Reynolds number defned by Eq. (4) and (10) (1) Re v partcle Reynolds number defned by Eq. (1) (1) Re dfference between the partcle and flud Reynolds numbers - see Eq. (3) (1) T tank dameter (m) t tme (s) u local flud velocty (m s -1 ) v partcle local velocty (m s -1 ) X the rato between the sedmentaton velocty n a turbulent feld and the termnal velocty n a stll lqud, -see Eq. (5) (1) Greek letters φ phase shft (1) λ Kolmogoroff mcroscale (m) ν knematc vscosty (m 2 s -1 ) ρ L flud densty (kg m -3 ) ρ L partcle densty (kg m -3 ) ω frequency (s -1 ) References [1] Dtl, P. - Skrvanek, J.: Sedmentaton hndered by a turbulent snusodal velocty feld. In XXI Ogólnopolska Konferencja Inżyner Chemcznej Procesowej - Materały konferencyjne. Szczecn: Zachodnopomorsk Unwersytet Technologczny w Szczecne, 2013, p ISBN [2] Šedvy, V. - Dtl, P. - Reger, F. - Severa, M.: Dmensonless Flow Characterstcs n Mxed Suspenson Obtaned by LDA/PDA. In: Flud Mxng 6. Rugby: Insttuton of Chemcal Engneers p [3] Yang, T.S., Shy, S.S.: The settlng velocty of heavy partcles n an aqueous near-sotropc turbulence. Physcs of Fluds, 15, , 2003.

8 [4] Doroodch, E., Evans, G. M., Schwarz, M. P., et al.: Influence of turbulence ntensty on partcle drag coeffcents. Chemcal Engneerng Journal, 135, , [5] Ghatage, S.V., Sathe, M.J., Doroodch, E., Josh, J.B., Evans, G.M.: Effect of turbulence on partcle and bubble slp velocty. Chem. Eng. Sc., 100, , [6] Magell, F., Fajner, D., Nocentn, M., Pasqual, G.: Sold dstrbuton n vessels strred wth multple mpellers. Chem. Eng. Sc., 45(3), , [7] Brucato, A., Grsaf, F., Montante, G.: Partcle drag coeffcents n turbulent fluds, Chemcal Engneerng Scence, 53, ,1998. [8] Nocentn, M., Pnell, D., Magell, F.: Dsperson coeffcent and settlng velocty of the solds n agtated slurry reactors strred wth multple rushton turbnes. Chem. Eng. Sc., 57, , [9] Pnell, D., Montante, G., Magell, F.: Dsperson coeffcents and settlng veloctes of solds n slurry vessels strred wth dfferent types of multple mpellers. Chem. Eng. Sc., 59(15), , [10] Pešava, V. - Dtl, P.: Určení parametrů fluktuačních rychlostí turbulence v míchané nádobě. In Procesní technka Praha: České vysoké učení techncké v Praze, Fakulta strojní, 2013, s ISBN [11] Benedct, L.H., Nobach, H., Tropea, C.: Estmaton of turbulent velocty spectra from laser Doppler data, MEASUREMENT SCIENCE & TECHNOLOGY, 2000, vol. 11, Issue: 8, p [12] Zaplatílek, K., Donar, B.: MATLAB: začínáme se sgnály. 1. vyd. Praha: BEN - techncká lteratura, s. ISBN