FRICTION FACTOR FOR WATER FLOW THROUGH PACKED BEDS OF SPHERICAL AND NON-SPHERICAL PARTICLES

Transcription

1 SCIENTIFIC PAPER FRICTION FACTOR FOR WATER FLOW THROUGH PACKED BEDS OF SPHERICAL AND NON-SPHERICAL PARTICLES Tatjana Kaluđerović Radoičić 1*, Nevenka Bošković-Vragolović 1, Radmila Garić-Grulović, Mihal Đuriš and Željko Grbavčić 1 1 Faculty o Technology and Metallurgy, University o Belgrade, Karnegijeva 4, Belgrade, Serbia, IChTM - Deartment or Catalysis and Chemical Engineering, University o Belgrade, Njegoseva 1, Belgrade, Serbia ceived vised Acceted * Corresonding author. Tel.: ; ax: address: tanjak@tm.bg.ac.rs (T. Kaluđerović Radoičić).

2 Abstract The aim o this work was the exerimental evaluation o dierent riction actor correlations or water low through acked beds o sherical and non-sherical articles at ambient temerature. The exeriments were erormed by measuring the ressure dro across the bed. Packed beds made o monosized glass sherical articles o seven dierent diameters were used, as well as beds made o 16 ractions o quartz iltration sand obtained by sieving (olydiserse non-sherical articles). The range o bed voidages was , while the range o bed article ynolds numbers was rom 0.3 to 86 or sherical articles and rom 0.1 to 50 or non-sherical articles. The obtained results were comared using a number o available literature correlations. In order to imrove the correlation results or sherical articles, a new simle equation was roosed in the orm o Ergun`s equation, with modiied coeicients. The new correlation had mean absolute deviation between exerimental and calculated values o ressure dro o 9.04%. For non-sherical quartz iltration sand articles the best it was obtained using Ergun`s equation, with mean absolute deviation o 10.36%. Surace-volume diameter (d SV ) necessary or correlating the data or iltration sand articles was calculated based on correlations or d V =(d m ) and =(d m ) rom our revious aer [7]. Keywords: ressure dro, acked bed, sherical articles, quartz iltration sand, nonsherical articles Highlights Exerimental evaluation o ressure dro correlations in acked beds was conducted Pressure dro across beds o sherical and non-sherical articles was measured Sherical glass articles and quartz iltration sand were used as acking material Correlations in the orm o Ergun equation gave the best results The coeicients in Ergun equation are system-seciic

3 3 Introduction Packed beds o articles ermit a widesread means o contact between luid and solid hases and are used in many dierent industrial rocesses. Some examles o their alication include iltration rocesses, ion-exchange, catalytic reactions, heat transer, gas scrubbing, grain drying and others. The shae and size o articles that make u the bed are chosen or the characteristics o the seciic rocess. The article size and shae always aim at high rocess eectiveness, so a wide range o articles are used. In some alications, like in down-low granular ilters, olydiserse natural materials are used as articulate hase. When natural materials are used, the shae o the articles is irregular and their size alls into some granulometric interval. Dierently shaed articles ack with dierent degrees o bed voidage, which results in dierent ressure dro across the bed. The ressure dro through the acked bed is one o the most imortant arameters to be known or the adequate design o the rocess as well as or the estimation o the caital and oerating costs and sizing the ums or ans required to orce the luid through the bed. The ressure gradient through acked beds has been studied extensively and a large number o correlations were roosed [1-17]. The most widely used equation or ressure dro calculation was roosed by Ergun [1]: P ( 1 ) - = 150 H 3 ( 1 ) U U 3 d d (1) The riction actor introduced by Ergun is: P d - H U 3 1 () According to Eqs. (1) and (), Ergun`s equation or riction actor is: (3)

4 where d U (1 ) (4) 4 For sherical articles, d is the diameter o the articles that constitute the acked bed, while or non-sherical articles d is usually taken to be the surace-volume diameter d sv [18, 19]. By analogy, all o the correlations roosed or sherical articles can be used or non-sherical articles using surace-volume diameter. Note that article shericity is deined as: d SV dv where d V reresents the volume diameter o the article. (5) The other literature correlations or riction actor in acked beds o sherical and nonsherical articles are shown in Table 1. Note that some authors used dierent orms or riction actor and ynolds number with resect to Erguns deinitions o and Table 1. Table 1, as shown in The overview o the ressure dro correlations or sherical articles shown in Table 1 is given in our revious aer [0], together with the exerimental data or the riction actor or air low through acked beds o sherical glass articles at ambient and elevated temeratures. The main conclusion o this study was that the overall best it o all our exerimental data is given by Ergun`s [1] correlation, with mean absolute deviation o 10.90% [0]. Some authors roosed riction actor correlations or non-sherical articles which included article shericity directly into the equations (Table 1) [14-17]. Allen et al. [14] reviewed the use o dierent correlations or ressure dro through acked beds o sherical and non-sherical articles. They have shown that the article shae, arrangement, acking method as well as surace roughness inluence the ressure dro signiicantly. The authors also conducted ressure dro measurements in air-articles system, using randomly acked beds o smooth and rough glass sheres, wooden cubes, wooden cylinders, acorns (ellisoids), mixed smooth sheres o dierent sizes and rounded and crushed rock with equivalent diameters rom 10.5 to 4.4 mm. The range o numbers was Based on their exerimental work, they roosed the correlation or non-sherical articles reresented by Eq.(18), Table 1.

5 Nemec and Levec [15] investigated single-hase low through acked bed reactors in 5 the range o numbers o 10< <500 with dense and loose acking o dierent uniormly sized sherical and non-sherical articles (glass and Al O 3 sheres, Al O 3 cylinders and rings as well as Ni-Mo trilobes and quadralobes) in the size range o mm. The luid used in their exeriments was nitrogen at 10 bars. The authors concluded that Ergun`s equation reresents a good aroximation o the luid low through the acked bed o sherical articles in the investigated range, while it under redicts the ressure dro over non-sherical articles under the same conditions. For non-sherical articles they roosed a new correlation which includes the articles shericity, given by Eq.(19), Table 1. Singh et al. [16] investigated the ressure dro o air low through acked bed solar energy storage system having large sized elements o dierent shaes (sheres and cubes) with d V = mm in the range o V numbers rom 157 to 674 and shericities rom 0.55 to 1. The correlation the authors roosed is given by Eq. (0), Table 1. Ozahi et al. [17] investigated the ressure dro in air-article systems, with article size 6 19 mm, and the range o shericity o 0.55 to 1. The range o V numbers in their exeriments was The correlation or riction actor they roosed is given by Eq. (1), Table 1. Several authors 9-13, 1 investigated the inluence o D c /d on ressure dro in acked beds o articles with a general conclusion that the eect o D c /d is negligible or D c /d >10. In addition, some recent studies investigated the ossibility to extend the use o Ergun`s equation to the olydiserse articles systems taking into account the article size distribution [, 3]. Flow through orous media was also studied in the ield o hydrology and or dierent alications in civil engineering. The exerimental results obtained were used to validate the semi-emirical relations or non-darcy low [4, 5, 6]. The resent study was conducted in order to investigate the otimal choice o riction actor correlation or calculating the ressure dro or water low through acked beds o sherical and non-sherical articles. The exerimental evaluation o literature correlations was conducted by measuring the ressure dro across acked beds o dierent acking and or dierent water low rates. The sherical articles used were monosized glass beads, while the non-sherical material used was olydiserse quartz iltration sand. The values o the equivalent diameter and the shericity o the quartz sand articles needed or the calculations were

6 6 obtained by using the correlations or volume diameter and shericity as a unction o mean sieve diameter roosed in our revious aer [7]. These correlations were derived or olydiserse ractions o quartz iltration sand with sieve diameters in the interval d m =0.359 to.415 mm [7]: d V d () d m (3) where d V and d m are in mm. m Exerimental aaratus The exeriments were erormed in the water-article system schematically shown in Fig. 1. The acked bed column () was used or ressure dro measurements. It was equied with a distributor and the calming section (e) in order to ensure the uniorm low o water through the bed. The uwards water low was induced using a um (b) and the low rate was measured using an electromagnetic low meter (d). The acked bed bulk temerature was measured using the temerature indicator (TI). The ressure dro in acked beds o dierent articles was measured using iezometers (h). The exeriments were erormed with two tyes o articles: glass sherical articles and olydiserse quartz iltration sand non-sherical articles, as shown in Fig.. The luid used was deaerated water at a nearly constant temerature o 0 C. In each run, water temerature was recorded and water density and viscosity were calculated. The article characteristics and range o the exerimental conditions are summarized in Table. Seven kinds o mono-sized sherical glass articles were used. The exeriments with glass sherical articles were conducted in two cylindrical columns: irst column o the diameter o 40 mm was used or mm articles and the second column o the diameter o 6 mm or mm articles. The ratio o the column diameter to the article diameter (geometric asect ratio) in the exeriments was between 10.0 and As the wall eects were not studied exerimentally in this aer, it should be noted that the literature on the range o geometric asect ratio in which the wall eects are negligible is somewhat divided. Generally, the wall eects are considered negligible at D c /d ratios less than 10, but there are some researchers who ound wall eects to be signiicant at D c /d ratios as high as 15-0 [10,1]. As in the case o some o our exeriments the D c /d ratios were in that range, the

7 existence o wall eects can not be excluded. The measurements o the ressure dro in the beds o non-sherical articles were conducted using 16 ractions o quartz iltration sand obtained rom the comany Kaolin - Valjevo. The raw material was irst washed by luidization to eliminate ine dust, then dried and sieved through a number o standard sieves with sieve oenings ranging rom.830 to 0.97 mm. The obtained ractions had the sieve diameters in the interval o dm = (ds,n + ds,n+1)/ = to.415 mm and the ratio between the two successive sieve sizes dr =ds,n/ds,n + 1 was in the interval 1.13 to 1.715, where ds,n is the size o the oening o the sieve through which the article had assed and ds,n+1 is the size o the oening o the sieve on which the article was retained, as shown in Table. A total o 3 runs were conducted (7 with sherical articles and 16 with non-sherical articles) and a total o 75 data oints were collected (511 or sherical and 14 or nonsherical articles). Bed article ynolds number, 7 varied between 0.3 and 86 or sherical articles and between 0.1 and 50 or non-sherical articles. All o the water suericial velocities used in the exeriments were below the minimum luidization velocity or the resective articles. The velocities were in the range o U mf, where U mf reresents minimum luidization velocity. Fig. 1. Fig.. Table. sults and discussion The results o the riction actor vs. obtained by the exerimental measurements o ressure dro are shown in Figs. 3 and 4 or sherical and non-sherical articles, resectively. The comarison between the exerimental results and the selected literature correlations is given in Table 3 and in Figs. 5 and 6. The literature correlations or sherical articles shown in Table 1 were tested or all the exerimental data. The suracevolume diameter and the shericity needed or the calculations or non-sherical articles were obtained using the correlations rom our revious aer [7] or quartz iltration sand (Eqs. () and (3)). For quartz iltration sand beds, the literature correlations seciically deined or non-sherical articles were also tested: Allen et al. [14], Nemec and Levec [15],

8 8 Singh et al. [16] and Ozahi et al. [17] correlations. Fig. 3. Fig. 4. Table 3. The mean absolute deviation between the measured values o the ressure gradient and the values obtained rom the literature correlations were calculated according to the ollowing equation: N 1 ( P / H ) calc ( P / H ) measured (4) N ( P / H 1 ) measured where N is the number o data oints, (ΔP/H) calc and (ΔP/H) measured are the calculated and the measured ressure gradients. As can be seen rom Table 3, the best it o our exerimental data or ressure dro in beds o sherical articles was obtained using Cheng [9] correlation, with mean absolute deviation o 10.89%. The correlations o Macdonald et al. [] and Montillet et al. [4] also gave very good results in itting our exerimental data with mean errors o 1.18 and 13.13%, resectively. A number o other correlations tested gave results with mean absolute deviation in the range o 16-0%. It should be noted that Hicks [6] correlation gave the mean error o 50.19% or sherical articles and 84.34% or non-sherical articles. The reason or such a large error is that the range o numbers in this aer was below 183, while the seciied range o alicability o Hicks correlation is >500. The correlations with mean absolute deviation between exerimental and correlated ressure dro less than 0% or beds o sherical articles are shown in Fig. 5. The correlations shown in Fig. 5 were calculated or D c /d = 5 and bed orosity o = 0.40 (the mean values in our exeriments) in order to be able to show the correlations with direct deendence on as lines. Fig. 5. Comared to the data o our revious aer [0], in which air-sherical articles system was investigated, the results are in the similar range or Macdonald et al. [] (1%) and Cheng [9] (11 and 1%) correlations, which gave very good results at ambient temerature both or air-articles and water-articles systems. On the other hand, the correlations o Ergun [1], Tallmadge [7], ichelt [11], Gibilaro et al. [3], Einseld and

9 9 Schnitzlein [10] and Zhavoronkov et al. [1] erormed better in air-articles system, while the correlation o Montillet et al. [4] erormed better in water-articles systems. In order to imrove the correlation results or sherical articles, a new simle equation is roosed in the orm o Ergun`s equation, with modiied coeicient: (5) The mean absolute deviation between the values calculated rom Eq. (5) and the exerimental data is 9.04%. The new correlation is shown in comarison to the exerimental data in Fig. 3. For non-sherical articles (quartz iltration sand), the best it o the exerimental data was obtained using Ergun`s equation [1], with mean absolute deviation o 10.36%. The correlations o ichelt [11], Tallmadge [7] and Einseld and Schnitzlein [10] also gave very good results in itting the exerimental data with mean errors o 10.64%, 11.19% and 11.31%, resectively. The correlations with mean absolute deviation o ressure dro less than 0% or quartz iltration sand acked beds are shown in Fig. 6. The correlations shown in Fig. 6 were calculated or D c /d =73 and bed orosity o = 0.40 (the mean values in our exeriments) in order to be able to show the correlations with direct deendence on as lines. Fig. 6. The correlations secially derived or non-sherical articles gave very oor results in correlating our exerimental data. The mean absolute deviations or these correlations were in the range o 59.40% to 85.%. The reason or this is that the correlations o Allen et al. [14], Sing et al. [16] and Ozahi et al. [17] were derived rom exeriments in systems in which the articles were o large diameters (6 0 mm equivalent diameters) and the numbers were larger than 75, while in our system, the numbers o non-sherical articles were smaller than 5. Allen et al. [14] and Singh et al. [16] correlations were derived or acked bed solar energy storage systems with article materials being wooden cubes, crushed rock, concrete and masonry bricks. On the other hand, the correlation o Nemec and Levec [15] was derived or the articles in the range o mm, but in the high ressure system (5 0 bar). The very oor erormance o the correlations derived or the non-sherical articles in our quartz sand acked beds emhasizes the act that the riction actor in acked beds strongly deends on other variables besides the shericity o the articles. It can be concluded that the

10 10 riction actor is very system seciic and that it deends on article shae and size, voidage, arrangement, acking method as well as surace roughness. It is interesting to note that almost all o the correlations that gave good results in correlating our exerimental data both or sherical and non-sherical articles were in the orm o Ergun`s equation, i.e. reresented a modiication o this equation with dierent coeicients. From this act it can be concluded that the orm o Ergun`s equation with two added terms describing viscous and inertial eects is adequate or reresenting the riction actor in acked beds. However, the coeicients in the equation are very system-seciic and care should be taken when choosing the adequate equation or ressure dro calculation or the seciic urose. The correlation chosen or the seciic system should be obtained rom the data in the similar range o exerimental conditions as the system it is intended to be alied to. Conclusions The resent study was conducted in order to investigate the otimal choice o riction actor correlation or water low through acked bed o articles. The best it o exerimental data or sherical articles was obtained using Cheng [9] correlation (mean absolute deviation o 10.89%). Ergun`s equation [1] gave better results in correlating the data or non-sherical articles with mean absolute deviation o 10.36% comared to 19.8% or sherical glass articles. Ergun`s equation was modiied in order to imrove the it or sherical articles and a new correlation was roosed. The mean absolute deviation between the exerimental data and the roosed correlation is 9.04%. The correlations secially derived or non-sherical articles gave very oor results in correlating our exerimental data robably because they were derived or systems with much larger articles acked in a dierent arrangement. Most o the correlations that gave good results in correlating exerimental data both or sherical and or non-sherical articles were in the orm o Ergun`s equation with modiied coeicients, thus showing that the orm o Ergun`s equation with two added terms describing viscous and inertial eects is adequate or reresenting the riction actor in acked beds. However, the coeicients in the equation are very system-seciic.

11 11 Acknowledgment Financial suort o the Serbian Ministry o Education and Science (Project ON170) is grateully acknowledged.

12 1 Nomenclature a A b c d m d d s d SV d s,n d s,n+1 d V D c emirical coeicient in Eq. (6), dimensionless area o a article emirical coeicient in Eq. (6), dimensionless emirical coeicient in Eq. (6), dimensionless article sieve diameter acked bed article diameter sieve oening article surace-volume diameter the size o the oening o the sieve through which the article had assed the size o the oening o the sieve on which the article was retained article volume diameter column diameter A riction actor deined according to Eq. (18) o riction actor deined according to Eq. (1) S riction actor deined according to Eq. (0) riction actor deined according to Eq. () modiied riction actor deined according to Eq. (7) H P P P bed height ressure bed ressure dro, Pa dimensionless bed ressure dro, Eq.(1) =( Ud )/μ article ynolds number =( Ud )/(μ(1-ε)) bed article ynolds number V article ynolds number deined according to Eq. (0) V bed article ynolds number deined according to Eq. (1) A bed article ynolds number deined according to Eq. (18) U U mf V suericial luid velocity minimum luidization velocity (suericial) volume o a article

13 13 Greek letters voidage μ luid viscosity ρ ρ σ Ψ luid density article density mean absolute deviation article shericity, dimensionless Subscrits luid mf minimum luidization article

14 14 REFERENCES 1. S.S. Ergun, Chem. Eng. Prog. 48 (195) F. Macdonald, M. S. El-Sayed, K. Mow, F. A. L. Dullien, Ind. Eng. Chem. Fundam. 18 (1979) L. G. Gibilaro, R. Di Felice, S. P. Waldram, Chem. Eng. Sci. 40 (1985) A. Montillet, E. Akkari, J. Comiti, Chem. Eng. Process. 46 (007) H. Watanabe, Inte. J.Eng.Fluid Mech. (1989) R.E. Hicks, Ind. Eng. Chem. Fundam. 9 (1970) J.A. Tallmadge, AIChE J. 16 (1970) J. Lee, K. Ogawa, J. Chem. Eng. Jn. 7 (1994) N.S. Cheng, Powder Technol. 10 (011) B. Eiseld, K. Schnitzlein, Chem. Eng. Sci. 56 (001) W. ichelt, Chem. Ing. Tech. 44 (197) N.M. Zhavoronkov, M.E. Aerov, N.N. Umnik, J. Phys. Chem. 3 (1949) (in Russian). 13. R.C. Raichura, Ex. Heat Transer 1 (1999) K.G. Allen, T.W. von Backström, D.G. Kröger, Powder Technol. 46 (013) D. Nemec, J. Levec, Chem. Eng. Sci. 60 (005) R.Singh, R.P. Saini, J.S. Saini, Sol. Energy 80 (006) E.Ozahi, M.Y.Gundogdu, M.Ö. Carinlioglu, Adv. Powder Technol. 19 (008) D. Geldart, Powder Technol. 60 (1990) W.-C. Yang, in Handbook o Fluidization and Fluid-article Systems, W.-C. Yang, Marcel Dekker, New York, (003). 0. R. Pešić, T. Kaluđerović Radoičić, N. Bošković-Vragolović, Z. Arsenijević, Ž. Grbavčić, Chem. Ind. Chem. Eng. Q., 014, DOI:10.98/CICEQ P. 1. R. Di Felice, L.G. Gibilaro, Chem. Eng. Sci. 59 (004) M. Mayerhoer, J. Govaerts, N. Parmentier, H. Jeanmart, L. Helsen, Powder Technol. 05 (011) A. Luckos, J.R. Bunt, Fuel 90 (011) M. Sedghi-Asl, R. Hassan, J. Hydrol. Eng. 49 (011) K. N. Moutsooulos, I. N. Paasyros, V. A. Tsihrintzis. J. Hydrol. 374 (009) M. B. Salahi, M. Sedghi-Asl, M. Parvizi, J. Hydrol. Eng. (015) T. Kaluđerović Radoičić, M.Đuriš, R.Garić-Grulović, Z.Arsenijević, Ž.Grbavčić, Powder Technol. 54 (014)

15 15 Figure cations Fig. 1 Schematic diagram o the exerimental system (a-reservoir; b-um; c-valve; d- electromagnetic low meter; e-calming section; -column; g-overlow; h-iezometers; TItemerature indicator) Fig. Some o the articles used in the exeriments Fig. 3. vs. or sherical articles Fig. 4. vs. or non-sherical articles Fig. 5. Comarison o exerimental data o vs. articles with chosen correlations or sherical Fig. 6. Comarison o exerimental data o vs. with chosen correlations or nonsherical articles

16 Tables Table 1 Some imortant literature correlations or riction actor in acked beds erence Friction actor Eq. number range Sherical articles Ergun [1] (3) 1< < Macdonald et al. [] (6) - Gibilaro et al. [3] 18 (1 ) Note: 3 /( 1 ) (7) - Montillet et al. [4] Kuerten, re. in [5] Hicks 6 Tallmadge 7 Table 1 - continued 0. 1 D c = a d a=0.061 (<0.39), a=0.050 ((>.39); For (D c /d )>50 term (D c /d ) 0. =. 5 (1 ) 1 6 = (1 ) (10) = (1 ) 4. (1 ) 1/6 = 3 3 (8) 10< < D c /d (9) 0.1< < < <610 4 (11) 0.1< <10 5

17 17 erence Friction actor Eq. number range Lee and Ogawa (1 ) n = where n (1) 1< <10 5 Cheng [9] AM B M c D d M M d D D A c c M d D D.. B c c / (13a) (13) - Eiseld and Schnitzlein [10] 154 M M B M by Eq.(13a), ) D / d (. B c (14) 0.01< <

18 Table 1 - continued erence Friction actor Eq. number range ichelt [11] 150 M M B M by Eq.(13a), B. 5 d / D c (15) - 18 Zhavoronkov et al. [1] A 1. B A B 1 D / d ( 1 ) c 1 (16) - Raichura [13] AM B M M by Eq.(13a) 103 A 6 ( 1 ) 80 ( d M 1. 8 B ( d / Dc ) M 1 / Dc ) (17) -

19 Table 1 - continued erence Friction actor Eq. number range Non-sherical articles 3 P 1 V a b A = < < 3000 H c U A 1 A A Coeicients a, b and c deend on article tye and acking structure. Allen et al.[14] U Note: V V d (18) SV A 4, (1 ) A A 6 Nemec and Levec [15] Singh et al. [16] Ozahi et al. [17] 3 3 A, A / 4/3 (19) S Note: P d - V H U UdV V, v ex[ (log ) ] 3/ (1 ), / (1 ) * 3 P d V 76 = 1.76 O - L 1 V Note: * 1 P P / UdV U, V, (1 ) V S V /, o (0) (1) 10< < < V < < V <7773

20 Table. Particle characteristics and the range o the exerimental conditions d or d m mm d s,n+1 d s,n D c mm mm mm kg/m 3 cm/s Sherical articles Non-sherical articles U

21 1 Table 3. Comarison o exerimental data or riction actor with dierent correlations rom the literature σ (%) erence Sherical articles Non-sherical articles Ergun [1] a Macdonald et al.[] a Gibilaro et al. [3] a Montillet et al. [4] a Kuerten, re. in [5] a Hicks [6] a Tallmadge [7] a Lee and Ogawa [8] a Cheng [9] a Eiseld and Schnitzlein [10] a Raichura [13] a ichelt [11] a Zhavoronkov et al.[1] a Allen et al.[14] b Nemec and Levec [15] 63.5 c Singh et al.[16] 85. Ozahi et al.[17] This aer (Eq.(5)) 9.04 a-using d SV ; b-using coeicients or crushed rock, d SV =4.4 mm; c-using coeicients or cylindrical articles

22 Figures Figure 1.

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24 Figure 3. 4

25 Figure 4. 5

26 Figure 5. 6

27 Figure 6. 7