Terminal Velocity of Solid Spheres Falling in Newtonian and non Newtonian Liquids

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1 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No Terminal Velocity of Solid Spheres Falling in Newtonian and non Newtonian Liquids VASSILIOS C. KELESSIIS Assistant Professor Technical University of Crete Abstract The prediction of the terminal velocity of solid spheres falling in non-newtonian shear thinning stagnant liquids is required in many industrial applications. Various correlations have been proposed but an accurate and comprehensive equation covering the whole range of ynolds numbers is not yet available. Furthermore, there is scarcity of much needed experimental data to aid in the derivation of the optimal correlation. Many investigators opt for using correlations derived for Newtonian liquids. However, there is a great variety of such correlations and not much is known about the optimal one or about the differences among them. The scope of this work was to propose an equation for predicting the terminal velocity of solid spheres falling in non-newtonian shear thinning stagnant liquids. The equation is firstly chosen by comparing the various proposed correlations using three indicators. It is then compared to experimental data taken using solid spheres falling through water solutions of Carboxylmethylcellulose of various concentrations and Newtonian fluids. The equation proposed compares favorably both to the experimental data presented and to experimental data reported by other investigators. 1. INTROUCTION Knowledge of the terminal velocity of solids falling in liquids is required in many industrial applications. Typical examples include hydraulic transport slurry systems for coal and ore transportation, thickeners, mineral processing, solid liquid mixing, fluidization equipment, drilling for oil and gas, geothermal drilling. In many of these processes it is the hindered falling velocity that is of interest, hindered by the presence of walls or by other particles [1]. Many observations show that the hindered falling velocity is proportional to the free (terminal) falling velocity of the solid particles. For this reason, there has been a great interest in predicting the free falling velocity of solid particles in liquids. Previous experimental and theoretical approaches have been reviewed as in [2]. The type of movement of a single solid sphere falling in Newtonian and non-newtonian liquids is well known; after a short acceleration time, it will fall at its terminal falling velocity V. For an unbounded liquid, V can be calculated from the knowledge of the liquid and solid physical properties and from the drag coefficient, C, defined as, C so that 4 gd (1) 3 2 V 4 gd V 3 C 0.5 Numerous attempts have been made to establish theoretical relationships of the terminal falling velocity of solid spheres but the theoretical and semi-theoretical expressions are normally only valid for < 1. For higher, resort must be made to experimental and empirical relationships. 2. CORRELATIONS FOR THE TERMINAL VELOCITY IN NEWTONIAN FLUIS There are over 50 correlations published for predicting a solid s terminal falling velocity with most of the work covering the case of Newtonian fluids. The standard approach is through an empirical expression relating the drag coefficient C to the particle ynolds number. The estimation of the velocity V is then done via a trial and-error procedure. Experimental data and theoretical analysis shows that for very low, C A while for very large, C B where Α, Β are constants. Past attempts were aiming at deriving correlations covering the entire range, for example, 0,01 < < Literature review reveals that the most comprehensive equation set for predicting C from for Newtonian (2)

2 44 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No 1-2 0, Table 1: Correlation set for rag Coefficient [3]. Πίνακας 1: Συσχετίσεις για το Συντελεστή Οπισθέλκουσας Δύναμης [3]. Range log Equations C ,881 0,82w 0, C log 1 0,7133 0, w 24 2 log C 1,6435 1,1242w 0, w ,2*10 4 1,2*10 4,4*10 4 4,4*10 3,38* log10 C 2,4571 2,5558w 0,9295w 0, 1049w log10 C 1,9181 0,6370w 0, 0636w log10 C 4,3390 1,5809w 0, 1546w with w log w 2 3 fluids has been published in [3] and the equations covering the range 0,01 < < 3,38*10 5 are shown in Table 1. The correlation was established from a critical review of available experimental data, and consists of six polynomial equations with 18 fitted constants for this ynolds range. While the predictions were quite accurate, efforts continued for the development of simpler equations with less fitted constants. Such expressions would be welcome in situations when the prediction of the terminal velocity is required several times (e.g. for when resolving timedependent solid liquid flows). Simpler correlations have been proposed [4, 5], in the form C 24 B C 1 A* E 1 / Using 408 available data points (and with E 1) the final form of the equation in [5] is ,6459 C 1 0,186 (3) 2,6*10 0, ,95/ Even simpler expressions have been used [6, 7, 8]. In [6] reference is made to empirical expressions for predicting the drag coefficient, C, of varying degrees of complexity and accuracy and with a reasonable compromise, the equation proposed, attributed to allavalle, is given by 2 4,8 C 0,63 (4) In [7] a three part expression is proposed, C ; 3, C ; (5) 1/ , C 0,44; 500 while in [8] a two part expression is used, 0,6 C 18,5 ; 0,1 500; (6) C 5 0,44; *10 3. CORRELATIONS FOR THE TERMINAL VELOCITY IN NON-NEWTONIAN FLUIS In contrast to the vast work on Newtonian fluids there is less work related to the terminal velocity of solid spheres falling through non-newtonian fluids. Processes of interest include drilling applications, hydraulic conveying of non-newtonian slurries, mineral processing, waste water processing. A major hurdle with non-newtonian fluids is defining the proper rheology of the fluid for the appropriate shear rate. Most of the work has covered shear thinning fluids described by the power law equation for the shear rate shear stress relationship although there is also work covering Bingham plastics [9, 10, 11]. The most common approach taken by previous investigators is through the use of standard Newtonian relationships (C - ) but using a modified (non Newtonian or generalized) ynolds number, which for power law fluids is defined as 2n n V d gn (7) K This definition stems from the standard definition of the ynolds number, but using an apparent viscosity, μ α, i.e. Vd gn (8) a For a power law fluid, the apparent viscosity is

3 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No K a n K n1 The appropriate average shear rate for the case of motion of solid spheres in liquids over the entire particle surface is assumed as [12] V (10) d Combination of (9) and (10) gives n1 V a K (11) d Finally, substitution of (11) in (8) gives equation (7) for the generalized ynolds number. In experiments reported in [12] five different CMC solutions and one guar-gum solution was used, covering the range of flow behavior index n = [0,55 0,84] and flow consistency index of K =[0,03 4,22 P a *s n ]. The terminal falling velocities of spheres were measured in different cylindrical columns. The solids were glass beads (1.2 to 25 mm in diameter) and steel balls (0,8 to 20 mm in diameter), thus covering a range of gn = [0,1 200]. The data correlated very well with the Newtonian curve defined by, 0, 1 0,15 ; 0, C (12) with = gn. The agreement between experiment and the Newtonian curve was extremely good for gn > 0,1. At lower ynolds, the drag coefficient for the non-newtonian fluids was lower than for the Newtonian fluids. Measurements of terminal velocities of glass beads and steel balls falling through suspensions of kaolinite and titanium dioxide in water solutions of glycerol have been reported [13]. Testing various correlations proposed either for Newtonian or for non-newtonian fluids it was found that the Newtonian curve, with = gn, proposed elsewhere [14], 0,31 0,06 C 2,25 0,36 (13) ,45 yielded quite accurate results for the non-newtonian fluids tested. The relative error on the velocity prediction was around 9.3%. ported measurements of terminal velocity of single particles in non-newtonian fluids [15] were compared with correlation proposed for Newtonian fluids [16] given as, C 0,4 10 (14) assuming that the same equation can be extended to non-newtonian fluids with = gn, as in the previous study. asonable agreement was found suggesting that (9) the assumption made was acceptable for most engineering purposes. Measurements of terminal falling velocities in a still liquid were also reported [17] and compared with the predicted drag coefficient using various expressions available in the literature for pseudoplastic fluids. The agreement between the experimental and theoretical values was fair for < 0,2, and acceptable in the intermediate regime. From the above discussion it is apparent that the majority of the investigators conclude that the use of Newtonian correlations for non-newtonian fluids is justified, with an engineering accuracy, provided that the apparent viscosity is used. However, there is still a question as to which is the most proper correlation proposed for Newtonian fluids, as there are so many in the literature. In what follows a comparison will be made of the various available correlations aiming at finding the most suitable for use. The experimental data from this work will be then presented and compared both to experimental data from other investigators and to the predictions from the best available correlation. 4. ISCUSSION ON THE USE OF VARIOUS CORRELATIONS In order to choose the best available correlation, it is very beneficial to compare the various correlations and determine whether there is any large difference among them. Such a comparison is performed here below. The procedure was as follows: for a given set of values of input variables, ρ, ρ s, d, μ, μ α, the variables V,, C are determined from each of the proposed correlations by trial and error, and are then compared to the derived results using four methods. Firstly, a graphical method is used, by plotting C values taken from the various correlations, a commonly followed practice, which however, may be questionable since it is a log log plot and small differences on the graph may indeed be large differences in the falling velocities. To avoid this uncertainty the RMS value is also used, given as the differences of the drag coefficient predicted in [3] and the drag coefficient of the other correlations, since the correlation in [3] is recognized as the most accurate correlation. This RMS value is, as in [4], QCd RMS _ C ; (15) N Q C C C 2 log 10 C log 10 where C C is the drag coefficient predicted from [3] and C i is the drag coefficient predicted from the other stated correlations. The ultimate comparison test, however, is made when the variable of interest is compared, that is, the terminal falling i

4 46 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No 1-2 velocity. For this, two indicators are used, the RMS error in velocity and the relative difference in velocity. The RMS error in velocity is defined as QV RMS _ V ; Q 2 V VC Vi (16) N with V C the velocity predicted from [3] and V i the velocity predicted from the other stated correlations. Lastly, the relative difference in velocity is defined as in [13], 1 _ / * N V VC Vi V (17) C N The results, covering the range 0,1 < < 1,4*10 5 and using 126 computed data points are presented below Clift Heider Felice Cherem oron Lali Machac Miura 10 C 1 0,1 0, , gn Figure 1: Log Log graph of C - from various correlations; Clift [3], Heider [5], Felice [6], Cherem [7], oron [8], Lali [12], Machac [13], Miura [15]. Σχήμα 1: Λογαριθμικό γράφημα C - για τις διάφορες συσχετίσεις. In Figure 1 the correlations in terms of the - C relationship are compared in a log - log graph. It is evident that the match is good for almost all correlations. Only the one used in [8] deviates from the others for < 2. Of course this seemingly good match should be expected given the large range covered and plotted. In Figure 2 a comparison is made between the velocities of the various correlations and the velocity predicted in [3]. A relative good match is seen between the velocities predicted from the stated simpler correlations and the velocity predicted by the more complex equation set [3], particularly for terminal velocity values less than 1 m/s. In Table 2 the different indicators, RMS_C, RMS_V and δ_v are compared, computed as stated above. Table 2: Comparison of various error parameters among different correlations. Πίνακας 2: Σύγκριση διαφόρων παραμέτρων λάθους για τις διαφορετικές συσχετίσεις. f. 5 f. 6 f. 7 f. 8 f. 12 f. 13 f. 15 RMS_C (-) 0,0145 0,0840 0,0408 0,1673 0,0262 0,0319 0,0505 RMS_V (m/s) 0,0154 0,1090 0,0506 0,0590 0,0484 0,0482 0,0806 _V (%) 1,30 8,45 3,38 12,80 2,48 3,06 5,23

5 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No All three indicators show different behavior for the various correlations. Except for the correlation used by [8], which shows larger deviations, the other correlations compare favorably with the most accurate but cumbersome correlation [3]. It is also apparent that the most accurate is the one proposed in [5] giving an RMS value for the difference in logarithms of drag coefficient between their proposal and that of [3] of only 0,0145, an RMS value for the difference in velocity of 0,0154 and a relative velocity error of only 1,3% for the 126 data points covering almost the entire range of interest. The equation proposed in [12] appears to be a good second choice. Furthermore, this analysis indicates that the RMS_C indicator is as good an indicator as the RMS_V and particularly the relative error in velocity δ_v. 3,0 2,5 Velocity (m/s) 2,0 1,5 1,0 0,5 0,0 0,0 0,5 1,0 1,5 2,0 2,5 3,0 Velocity, Clift (m/s) 'Heider' 'Felice' 'Cherem' 'oron' 'Lali' 'Machac' 'Miura' Figure 2: Comparison of terminal velocities predicted from stated correlations versus velocity predicted by [3] Heider [5], Felice [6], Cherem [7], oron [8], Lali [12], Machac [13], Miura [15]. Σχήμα 2: Σύγκριση προβλέψεων οριακών ταχυτήτων από τις αναφερόμενες συσχετίσεις με ταχύτητες πρόβλεψης από [3]. The above comparison showed that the correlation proposed in [5] performed better, but the correlation used in [12] also performed acceptably. Furthermore, the correlation in [12] has also been tested in non-newtonian fluids. Hence, both of these correlations will be used to determine whether they can also describe Newtonian and non Newtonian data collected from this work. 5. EXPERIMENTAL ETERMINATION OF SOLI VELOCITIES FALLING IN NEWTONIAN & NON NEWTONIAN LIQUIS ata has been collected for the falling velocities of single spheres in liquids using glass beads of different diameters. Through careful examination for the sphericity of the beads using a micrometer, five beads were chosen with corresponding diameters d 1, d 2, d 3, d 4, d 5. The density was determined by first measuring the mass of each bead on an electronic scale, giving masses m 1, m 2, m 3, m 4, m 5 and dividing by the volume of each sphere. The data are shown in Table 3. The liquids used were water and CMC solutions in water of 0,5%, 0,7%, 0,9% and 1,1% w/v concentrations. The solutions were prepared in batches of about 9 l by adding the necessary amounts of CMC in tap water, stirring continuously and letting the mixture age for 24 hrs. The solution was agitated prior to each set of measurements and a small sample was taken just before the experiments to determine the rheological properties using a coaxial rotating viscometer (Grace M3500a) having a continuously varying rotational speed of 0, RPM. The liquid density is the water density at the experiment temperature. All measurements were done at room temperature of 21 0 C. Rheometric data for the non-newtonian liquids were fitted to the power law to determine the power law parameters shown

6 48 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No 1-2 in Table 3. The 0,5% CMC solution exhibited Newtonian behavior while the other solutions exhibited power law behavior. The terminal velocity data was obtained in a cylindrical column of length of 1 m and a diameter of 0,1 m filled with the appropriate liquid. The particles were carefully dropped in the center of the column. The time of travel, t i, was determined with an electronic stop watch (accuracy 0,01 s) when the sphere passed between two fixed points, 0,10 m below the top of the column and 0,20 m above the bottom of Table 3: Experimental results of terminal falling velocities. Πίνακας 3: Πειραματικά δεδομένα οριακών ταχυτήτων καθίζησης. the column, so as to avoid acceleration and end effects. Thus, the total effective travel length was L = 0,60 m. Each sphere was dropped 5 to 10 times and the velocities were calculated from V i = L / t i. From the measurements, an average velocity and a standard deviation are computed for each combination sphere solution. Using the average velocities so determined and the solid liquid properties, the parameters of interest, C,, gn are estimated. The results are summarized in Table 3. Solid Liquid Measurements Calculations s d Soln. K n V s.dev C or gn 3 ( g / cm ) (cm) n ( Pa * s ) () ( cm / s) ( cm / s) () () 2,609 0,303 Water 0,001 1,0 34,72 0,64 0,69 190,1 2,572 0,345 36,89 0,92 0,61 566,7 2,727 0,205 27,71 0,58 0,51 671,9 2,449 0,230 29,29 1,01 0, ,6 2,260 0,115 16,56 0,11 0, ,5 2,609 0,303 0,5 % 0, ,34 0,37 1,17 120,2, 0 2,572 0,345 26,33 0,53 1,03 154,4 2,727 0,205 15,85 0,14 1,85 55,2 2,449 0,230 17,29 0,20 1,46 67,6 2,260 0,115 7,99 0,07 3,57 15,6 2,609 0,303 0,7 % 0,0165 0, ,50 0,14 1,68 50,9 2,572 0,345 21,96 0,32 1,47 64,2 2,727 0,205 12,75 0,12 2,85 22,1 2,449 0,230 14,03 0,18 2,21 27,3 2,260 0,115 5,98 0,06 6,94 7,3 2,609 0,303 0,9 % 0,0353 0, ,92 0,11 2,52 22,7 2,572 0,345 18,25 0,07 2,13 29,6 2,727 0,205 10,08 0,10 4,56 9,6 2,449 0,230 11,19 0,05 3,48 12,0 2,260 0,115 4,40 0,02 12,77 2,9 2,609 0,303 1,1 % 0,2648 0,7529 6,64 0,09 14,46 1,6 2,572 0,345 8,02 0,08 11,03 2,3 2,727 0,205 3,61 0,01 35,53 0,6 2,449 0,230 4,09 0,02 26,06 0,7 2,260 0,115 1,19 0,01 174,57 0,1 ata were not corrected for wall effect as explained in the Appendix. 6. ATA AN PREICTION COMPARISON The Newtonian as well as the non-newtonian data is plotted in Figure 3. ata from other investigators are also plotted, both non-newtonian data [15, 17] and Newtonian data [6, 18]. The predictions of C of the two similar correlations, equations (3) and (12), from the measured or gn are also shown in Figure 3. The lines corresponding to the two predictions coincide; hence no real difference is expected from the two correlations. Both Newtonian and non-newtonian fall around the two curves, with the non-newtonian data falling slightly below the curves and the Newtonian data falling slightly above. This is true both for data from this work (a set of 25 pairs) and for data from other investigators (a set of 48 pairs) for a total number of 73 pairs.

7 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No TUC-water TUC-0,5% TUC-0,7% TUC-0,9% TUC-1,1% Miura Pinelli Felice Hartman eqn.3 eqn.12 C 1 Hence, both correlations can predict with good accuracy the drag coefficient and predictions compare well with data from this work as well as data from other investigators, both for Newtonian and non- Newtonian fluids. 7. CONCLUSIONS 0,1 0, The foregoing analysis has indicated that the terminal falling velocity of solid spheres through stagnant non- Newtonian shear thinning fluids can be predicted with engineering accuracy from various proposed correlations for Newtonian fluids. This can be achieved provided the apparent viscosity of the fluid is used, evaluated at a shear rate given as the ratio of the falling velocity to the sphere diameter., gn Figure 3: Comparison of predictions with experimental data; Miura [15], Felice [6], Hartman [18], Pinelli [17] TUC = data from this work. Σχήμα 3: Σύγκριση προβλέψεων με πειραματικά δεδομένα. In addition, the RMS_C value, computed and shown in Table 4 for all experimental data, indicates that the errors from both equations are around It is also evident that the RMS_C indicator is as good a deviation indicator as the RMS in velocity and the deviation in velocity. Table 4: RMS error on drag coefficient between experimental data and predictions. Πίνακας 4: Λάθος RMS στο συντελεστή οπισθέλκουσας δύναμης μεταξύ πειραματικών δεδομένων και προβλέψεων. Equation (3) Equation (12) RMS_C (-) 0,094 0,092 From the many simpler correlations (with respect to the cumbersome correlation set [3]) that have been proposed to date for predicting the terminal falling velocity of solid spheres through stagnant Newtonian fluids, the equations proposed in [5], equation (3), and in [12], equation (12), give the best approximation. This conclusion is evident since all three indicators used, the RMS error on C, the RMS error on velocity and the percent deviation in velocity, are minimum. In addition it was shown that all three indicators are very similar. Experimental data have been gathered, both for Newtonian and shear thinning non-newtonian fluids, regarding the terminal falling velocity of solid spheres (glass beads). For the non-newtonian data, the generalized ynolds number covered a range of 0,1 to 64, while for Newtonian data the ynolds number range was from 0,1 to

8 50 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No The experimental data were compared to experimental data for both Newtonian and non-newtonian fluids from other investigators, with ynolds number ranges of 0,1 to 928 and 0,4 to 770 respectively. The comparison yielded very similar C behavior and all data fall along the same Newtonian curve, thus justifying the use of Newtonian curves. Equations (3) and (12) gave overall RMS errors, on the predicted versus measured drag coefficient, of ~0,09. Hence either equation can be used with good engineering accuracy, for the ranges of ynolds numbers listed above, for the prediction of terminal velocity of solid spheres falling through Newtonian and non-newtonian shear thinning fluids. ACKNOWLEGEMENTS The author would like to thank Mr. Mpandeli for collecting the experimental data. Special thanks to Professor Kawase and Professor Magelli for the kind provision of their experimental data in the proper form for the analysis given in this paper. NOMENCLATURE C drag coefficient (-) C C drag coefficient from correlations [3] (-) C i drag coefficient from stated correlations (-) d particle diameter (L) diameter of test column (L) f wall velocity correction factor (-) f 0 f wall velocity correction factor for low wall velocity correction factor for high g acceleration of gravity (L/T 2 ) K consistency index of fluid (M/LT n ) L length of measurement column (L) m mass of each sphere (M) n flow behavior index (-) N number of measurements (-) ynolds number based on particle diameter, Vd gn generalized ynolds number, 2n n V d K (-) (-) (-) (-) t i time travel of sphere (T) V solid terminal velocity (L/T) V C V i solid terminal velocity predicted by correlations from [3] solid terminal velocity predicted by stated correlations (L/T) (L/T) V m solid terminal velocity in columns (L/T) Greek letters Δρ density difference, (ρ s ρ) / ρ (-) ρ s solid density (M/L 3 ) ρ liquid density (M/L 3 ) τ shear stress (M/LT 2 ) shear rate (T -1 ) μ liquid viscosity (M/LT) μ a apparent liquid viscosity (M/LT) REFERENCES 1. Kelessidis, V. C. and Mpandelis, G., 2003, Flow Patterns and Minimum Suspension Velocity for Efficient Cuttings Transport in Horizontal and eviated Wells in Coild-Tubing rilling, SPE paper presented at the 2003 SPE/ICoTA Coiled Tubing Conference, Ηοuston, TX, 8 9 April. 2. Chhabra, R. P., 1993, Bubbles, rops and Particles in non- Newtonian Fluids, CRC Press, Boca Raton, Fl. 3. Clift, R., Grace, J. R. and Weber, M. E., 1978, Bubbles, rop and Particles, Academic Press, New York. 4. Turton, R. and Levenspiel, O., 1986, A Short Note on the rag Correlation for Spheres, Powder Techn., 47, Heider A. and Levespiel, O., 1989, rag Coefficient and Terminal Velocity of Spherical and Nonspherical Particles, Powder Techn., 58, Felice, R., 1999, The Sedimentation Velocity of ilute Suspensions of Nearly Monosized Spheres, Int. J. Mult. Flow, 25, Cheremisinoff, N., P. & Gupta, R., 1983, Handbook of Fluids in Motion, Ann Arbor Science, Michigan. 8. oron, P., Garnica,. and Barnea,., 1987, Slurry Flow in Horizontal Pipes: Experimental and Modeling, Int. J. Mult. Flow, Vol. 13 (4), Valentik, L. and Whitmore, R. L., 1965, The Terminal Velocity of Spheres in Bingham Plastics, Brit. J. Appl. Phys., 16, Saha, G., Purohit, N. K., and Mitra, A. K., 1992, Spherical particle terminal falling velocity and drag in Bingham liquids, Intern. J. of Min. Proc., 36 (3-4), Briscoe, B. J., Glaese, M., Luckham, P. F. and n, S., 1992, The Falling of Spheres through Bingham Fluids, Colloids and Surfaces, 64 (1), Lali, A. M., Khare, A. S., Joshi, J.. B. and Migam, K.. P., 1989, Behavior of Solid Particles in Viscous non-newtonian Solutions: Falling Velocity, Wall Effects and Bed Expansion in Solid - Liquid Fluidized Beds, Powder Technol., 57, Machac, I., Ulbrichova, I., Elson, T. P., and Cheesman,. J., 1995, Fall of Spherical Particles through Non Newtonian Suspensions, Chem. Engr. Science, 50 (20),

9 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No Kahn, A. R. and Richardson, J. F., 1987, The sistance to Motion of a Solid Sphere in a Fluid, Chem. Engr. Comm., 62, Miura, H., Takahashi, T., Ichikawa, J., and Kawase, Y., 2001, Bed Expansion in Liquid Solid Two-Phase Fluidized Beds with Newtonian and Non-Newtonian Fluids over the Wide Range of ynolds Numbers, Powder Techn., 117, Molerus, O., 1993, Principles of Flow in isperse Systems, Chapman & Hall, Berlin. 17. Pinelli,. and Magelli, F., 2001, Solids Falling Velocity and istribution in Slurry actors with ilute Pseudoplastic Suspensions, Ind. Eng. Chem. s., 40, Hartman, M., Havlin, V., Trnka, O. and Carsky, M., 1989, Predicting the Free-Fall Velocities of Spheres, Chem. Engr. Sci., 44 (8), Felice, R., Gibilaro, L. G. and Foscolo, P. U., 1995, On the Hindered Falling Velocity of Spheres in the Inertial Flow gime, Chem. Engr. Sci., 50 (18), Fidleris, V., and Whitmore, R., L., 1961, Experimental etermination of the Wall Effect for Spheres Falling Axially in Cylindrical Vessels, British J. of Applied Physics, 12, Chhabra, R. P. and Richardson, J. F., 1999, Non-Newtonian Flow in the Process Industries, Butterwoth-Heinemann, Oxford. APPENIX WALL EFFECT The retarding effect of bounding walls is assessed with great difficulty, particularly for non-newtonian liquids. From an extensive study of wall effect for non-newtonian fluids [12], a comprehensive equation was proposed, but with an average error of ~13%. The correction factor is a function of d / and of the non-newtonian parameters. In the case of this investigation, d / 0,03. Using the Newtonian correlation [19], for 500 < < 2*10 5, the error introduced in the measurements of this study is ~ 0,1%. In addition, using the correction from [19] for d / < 0,1 and > 100, the wall correction factor, f, defined in (A-1) is ~1. For the laminar region ( < 100), the equation proposed in [19] is, Vm f V 1 d / 1 0,475( d / ) 4 A-1 Using this correction a wall correction factor of f = 0,88 is predicted. An equation proposed for pseudoplastic fluids [20] covering the entire range is given by, / f 1 / f 1 / f 1 / f 2 1 / 3 1 1,3 * 1 gn 0 A-2 where f 0 = 1 1,6 (d /) is the wall correction factor in the low ynolds regime, and f = 1 3 (d / ) 3,5 is the wall correction factor in the high ynolds regime. The correction factor for the case of this investigation is computed for various ynolds numbers and the results are shown in the Figure A-1. correction factor (-) 1,00 0,99 0,98 0,97 0,96 0,95 0,94 0,93 0,92 0,91 0,90 0,0 1,0 2,0 3,0 4,0 sphere diameter (mm) =0,1 =1,0 =10 =100 =300 Figure A-1: Terminal velocity wall correction factor. Σχήμα Α-1: Συντελεστής διόρθωσης τοιχωμάτων για οριακή ταχύτητα. The maximum correction is of the order of 5,5%. In view of the above stated uncertainties and since the critical parameter, d /, is very small in this case, no correction for wall effect is made, since it is very small and it would be more accurate not to correct for wall effect than to apply a more uncertain correction factor. Vassilios C. Kelessidis Assistant Professor, Technical University of Crete, Mineral sources Engineering epartment.

10 52 Τεχν. Χρον. Επιστ. Έκδ. ΤΕΕ, V, τεύχ , Tech. Chron. Sci. J. TCG, V, No 1-2 Εκτεταμένη περίληψη7 Οριακή Ταχύτητα Στερεών Σφαιρών που Καθιζάνουν σε Νευτώνεια και μη Νευτώνεια Υγρά ΒΑΣΙΛΕΙΟΣ Χ. ΚΕΛΕΣΙΔΗΣ Επίκουρος Καθηγητής Πολυτεχνείου Κρήτης Περίληψη Η πρόβλεψη της οριακής ταχύτητας καθίζησης στερεών σφαιρών σε Νευτώνεια και μη Νευτώνεια υγρά απαιτείται σε πολλές βιομηχανικές εφαρμογές. Πολλές εξισώσεις πρόβλεψης έχουν προταθεί στο παρελθόν, αλλά δεν υπάρχει έως σήμερα μία εξίσωση που να προβλέπει με ακρίβεια την ταχύτητα καθίζησης και να καλύπτει όλο το εύρος των αριθμών ynolds. Επιπρόσθετα, δεν υπάρχουν πολλά διαθέσιμα και απαιτούμενα πειραματικά δεδομένα που θα υποβοηθούσαν στην ανάπτυξη της κατάλληλης συσχέτισης. Πολλοί ερευνητές προτείνουν τη χρήση εξισώσεων που ισχύουν για Νευτώνεια ρευστά. Είναι, όμως, γεγονός ότι υπάρχουν πολλές τέτοιες συσχετίσεις και δεν είναι γνωστό ποια από τις συσχετίσεις είναι η βέλτιστη, όπως, επίσης, και δεν έχουν αναφερθεί στοιχεία για τις διαφορές μεταξύ των συσχετίσεων αυτών. Ο στόχος της εργασίας αυτής είναι να προτείνει μία εξίσωση για την πρόβλεψη της ταχύτητας καθίζησης στερεών σφαιρών σε μη Νευτώνεια ψευδοπλαστικά στατικά ρευστά. Η εξίσωση αυτή επιλέγεται από τις ήδη προταθείσες συσχετίσεις ως η βέλτιστη, με τη χρήση τριών κριτηρίων. Ακολούθως, συγκρίνονται οι προβλέψεις με πειραματικά δεδομένα ταχυτήτων καθίζησης στερεών σφαιρών σε Νευτώνεια και μη Νευτώνεια υγρά. Η προτεινόμενη εξίσωση προβλέπει το συντελεστή οπισθέλκουσας δύναμης με καλή ακρίβεια, όταν συγκρίνεται με τα πειραματικά δεδομένα της εργασίας αυτής αλλά και με δημοσιευθέντα πειραματικά δεδομένων άλλων ερευνητών. 1. ΕΙΣΑΓΩΓΗ Η γνώση της ταχύτητας καθίζησης στερεών, που καθιζάνουν σε υγρά, απαιτείται σε πολλές βιομηχανικές εφαρμογές, όπως για παράδειγμα, στην υδραυλική μεταφορά γαιανθράκων ή ορυκτών, σε παχυντές, κατά την επεξεργασία ορυκτών, στην ανάμιξη στερεών υγρών, σε μηχανήματα ρεύστωσης, σε γεωτρήσεις πετρελαίου, φυσικού αερίου και γεωτρήσεις γεωθερμίας. Στις περισσότερες των ανωτέρω περιπτώσεων, αυτό που ενδιαφέρει πραγματικά είναι η «αναχαιτιζόμενη» ταχύτητα καθίζησης, αναχαιτιζόμενη λόγω τοιχωμάτων αγωγού ή λόγω άλλων σωματιδίων [1]. Παρατηρήσεις μας οδηγούν στο συμπέρασμα ότι η αναχαιτιζόμενη ταχύτητα είναι ανάλογη της «ελεύθερης» (οριακής) ταχύtητας καθίζησης. Ανασκόπηση των διάφορων συσχετίσεων, που έχουν προταθεί, έχει αναφερθεί στην [2]. Η ταχύτητα καθίζησης προκύπτει από τον ορισμό του συντελεστή οπισθέλκουσας δύναμης (εξίσωση 1) και τελικά προκύπτει η εξίσωση (2). Έχουν γίνει πολλές προσπάθειες στο παρελθόν για τη θεωρητική πρόβλεψη της ταχύτητας καθίζησης, αλλά οι προσεγγίσεις είναι συνήθως ικανοποιητικές για < 1. Για μεγαλύτερους αριθμούς, γίνεται συνήθως προσφυγή σε πειραματικές και εμπειρικές συσχετίσεις. 2. ΣΥΣΧΕΤΙΣΕΙΣ ΓΙΑ ΤΗΝ ΟΡΙΑΚΗ ΤΑΧΥΤΗΤΑ ΣΕ ΝΕΥΤΩΝΕΙΑ ΡΕΥΣΤΑ Υπάρχουν πλέον των 50 συσχετίσεων, που έχουν αναφερθεί στο παρελθόν, που συσχετίζουν τον C με τον βασισμένο στη διάμετρο του στερεού σωματιδίου. Πειραματικές μετρήσεις και θεωρητικές προσεγγίσεις δίδουν τον C αντιστρόφως ανάλογο του για χαμηλούς αριθμούς και ίσο με μία σταθερά για μεγάλους αριθμούς. Η ανάπτυξη συσχετίσεων εστιάζεται στην κάλυψη εύρους αριθμού, 0,01 < < Οι πλέον ολοκληρωμένες συσχετίσεις έχουν αναφερθεί στην [3] που καλύπτουν ένα εύρος 0,01 < < 3,35*10 5 και δίδονται στον Πίνακα 1. Για το εύρος των αριθμών υπάρχουν έξι πολυωνυμικές εξισώσεις με 18 υπολογισθείσες σταθερές. Έχει αναγνωρισθεί ότι οι εκτιμήσεις βάσει των συσχετίσεων αυτών είναι αρκετά ακριβείς, αλλά υπάρχει συνεχής αναζήτηση για την ανάπτυξη περισσότερο απλών εξισώσεων με λιγότερες σταθερές. Τέτοιες εξισώσεις θα ήταν ιδιαίτερα ελκυστικές σε περιπτώσεις όπου απαιτείται συνεχής υπολογισμός των ταχυτήτων καθίζησης, για παράδειγμα κατά την επίλυση προβλημάτων ροής στερεών υγρών μη μόνιμης κατάστασης. Μία τέτοια συσχέτιση αναφέρεται ως εξίσωση (3), από την [5], ενώ ακόμη πιο απλούστερες εξισώσεις είναι οι εξισώσεις (4), (5) και (6) από [6], [7] και [8] αντίστοιχα.