Cheng, N. S., and Law, A. W. K. (2003). Exponential formula for computing effective viscosity. Powder Technology. 129(1-3),

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1 THIS PAPER SHOULD BE CITED AS Cheng, N. S., and Law, A. W. K. (2003). Exonential ormula or comuting eective viscosity. Powder Technology. 129(1-3), EXPONENTIAL FORMULA FOR COMPUTING EFFECTIVE VISCOSITY Nian-Sheng Cheng 1 and Adrian Wing-Keung Law 2 ABSTRACT: An exonential model is roosed in this aer or evaluating the eective viscosity o a article-luid mixture. First, the theoretical consideration is restricted to the dilute condition without the eects o dynamic article interactions and luid turbulence. This leads to a ower series exressed in terms o article concentration, which can be viewed as an extension o the Einstein s ormula. The derived exression is then modiied by including indirectly eects o inter-article collisions, luid turbulence and random motion o the articles, which can cause the eective viscosity to be increased signiicantly with increasing article raction in the mixture. Finally comarisons are made between the resent study and various theoretical and emirical results available in the literature and satisactory agreement is observed. Keywords: eective viscosity, concentration o articles, susension, two-hase low, article interaction, dilute susension INTRODUCTION The henomenon o solid articles susended in luid is o key interest in hydraulic engineering such as sediment transort in rivers and slurry transort in ielines. Similar two-hase susensions can also be ound in industrial alications such as the rocessing o cement, comosite materials and oodstus. As an imortant macroscoic roerty o these susensions, the eective viscosity is usually required in the analysis o the related transort rocesses. 1 Assistant Proessor, School o Civ. and Struct. Engrg., Nanyang Technol. Univ., Nanyang Ave., Singaore, cnscheng@ntu.edu.sg 2 Associate Proessor, School o Civ. and Struct. Engrg., Nanyang Technol. Univ., Nanyang Ave., Singaore, cwklaw@ntu.edu.sg 1

2 It is well known that the earliest theoretical work on the eective viscosity was due to Einstein (1906) whose derivation led to the eective viscosity to be linearly related to the article concentration as ollows: μr = φ (1) where μ r = μ m /μ = relative viscosity deined as the ratio o the eective viscosity o the article-luid mixture, μ m, to the viscosity o luid, μ, and φ = volumetric concentration o the articles. This exression is exact when the viscous eect is dominant so that the creeing low equations can be alied at the article level. Since the inluence o article interaction is not considered, (1) is only alicable to susensions with low article concentrations, say, less than 2% (Acrivos 1995). Following Einstein s work, numerous exressions have been roosed to extend the range o validity to higher concentrations. They are either theoretical exansions o (1) to higher order in φ, or emirical exressions that were obtained based on exerimental data. The theoretical exansions are usually exressed in the orm o a ower series, 2 3 μ r = 1+ k1φ + k2φ + k3φ + (2) where k 1, k 2, k 3 = coeicients. Evaluation o these coeicients other than k 1 requires the ormulation o article interactions, which is rather diicult. This is one o the reasons that only the theoretical values o the coeicients related to lower order in φ, like k 2 and k 3, can be ound in the literature so ar. Even or these lower order coeicients, the calculations are available only or idealized cases when the article arrangement or its statistic roerties are simle. An examle or determining the k 2 -value was resented by Batchelor and Green (1972), who reorted that the coeicient k 2 was equal to 7.6 or a susension undergoing a ure straining motion. It should be noted however that Batchelor and Green introduced a rough interolation in their derivation, which led to an inaccuracy in the k 2 -value obtained. A re-calculation under the same condition was conducted by Kim and Karrila (1991), leading to the k 2 -value to be changed to Other k 2 -values have been reviewed by Hael and Brenner (1986). Furthermore, Thomas and Muthukumar (1991b) ound the third order coeicient, k 3, to be 6.40 by alying the multile scattering theory to the evaluation o the hydrodynamic interaction o three sheres. Following these theoretical attemts, it can be exected that more comlicated conigurations o the article arrangements will lead to urther diiculties in the mathematical ormulation. On the other hand, articles susended in luid should in act be distributed in a random manner so that their arrangement cannot be itted to a simle coniguration or described by a simle distribution unction. Thereore, although an idealized article arrangement may enable a theoretical evaluation o the k-coeicients in higher order, the alicability o the eective viscosity so derived is limited in ractice. Comarisons with exerimental results show that the existing theoretical exressions are only alicable to those conditions with low concentration u to 10% (Barnes et al. 1989). In addition to the theoretical exressions, various emirical relationshis have been roosed to evaluate the eective viscosity o a susension with higher concentration. An early survey o such relationshis made by Rutgers (1962) showed that 2

3 large discreancies exist among the dierent relationshis. Acrivos (1995) claimed that the discreancies could artially be attributed to the viscometric measurements that were normally subjected to shear-induced article diusion. Table 1 shows some tyical examles o the existing emirical ormulas, where φ max = maximum article concentration and μ in = intrinsic viscosity. It is noted that or those relationshis where the so-called maximum concentration is included as a arameter, the eective viscosity aroaches ininity when the concentration is equal to the maximum value. This may not be hysically reasonable. Strictly seaking, there are only two extreme conditions that are meaningul or the eective viscosity. The irst condition is a susension without articles, imlying that the eective viscosity is the same as the luid viscosity. The other is a susension without luid, which would then theoretically behave as a solid with ininite viscosity. On the other hand, only or articles with regular shaes can the maximum concentration be mathematically determined or a given acking arrangement. For examle, the maximum concentration can reach 1.0 or cubes acked ace-to-ace and 0.74 or sheres with the closest hexagonal acking arrangement. With irregular articles, the maximum value varies markedly even though the article size is uniorm. As a matter o act, the maximum concentration and intrinsic viscosity were oten used as two emirical arameters in itting exerimental data to an emirical ormula (e.g., Leighton and Acrivos 1987, Barnes et al. 1989). Other than the traditional aroaches mentioned above, Fan and Boccaccini (1996) studied the eective viscosity using toological transormation. Their results showed that the eective viscosity increases with increasing continuity o the solid articles. This suggests the existence o two limiting cases. One is the minimum eective viscosity when the articles are comletely discontinuous, and the other is the maximum eective viscosity when the articles are comletely continuous. Their results should be viewed as qualitative since a number o toological arameters were deined or characterizing the microstructure o the two-hase system, o which some are not exerimentally deterministic or even not hysically clear. This study starts with a derivation o the eective viscosity or the dilute condition without dynamic article interaction. The exression o the eective viscosity so obtained recovers the Einstein's ormula. The exression is urther modiied to be alicable or high article concentrations. Comarisons o the resent study with other emirical relationshis are inally rovided. DERIVATION Eective Viscosity or Dilute Conditions The ollowing aroach illustrates an iterative concet, based on the Einstein s ormula, that is simle in imlementation yet retains the basic hysical reresentation o the situation. The objective is to derive the eective viscosity in the dilute condition but with the article concentration beyond the range o validity o the Einstein ormula. Thereore, the inter-article collision and eects o the luid turbulence and article random motion are assumed to be insigniicant. 3

4 Consider a susension made o clean luid with uniormly distributed articles, whose volumes are v and v, resectively. First, divide v into n sub-volumes, i.e. v = Δ v,1 + Δv,2 + Δv,3 + + Δv,n (3) With this division, a series o slightly dierent susension samles can be constructed with rogressively increasing article concentration, as sketched in Fig. 1. It is noted that although the article volume varies in the dierent samles, the luid volume remains constant. For simlicity, all the article volume increments can be set to be equal, i.e., Δv,i = Δv or i = 1, 2,, n. Hence, the samle volumes are v + Δv, v + 2Δv,, v + nδv, resectively. The eective viscosities o the samles are denoted as μ m,1, μ m,2,, μ m,n, resectively. Their relationshi can be concetualized as ollows. Consider the (i-1)th samle with the eective viscosity, μ m,i-1. I an additional volume o articles o Δv is urther disersed in this samle, it will then be changed to the ith samle with the eective viscosity increased to μ m,i. Since Δv is arbitrary, it can be chosen in such a manner that Δv /(v +i Δv ). or i = 1, 2,, n, is small within the range o validity o the Eintein s equation. With these considerations, it is exected that (1) is alicable to the i th susension in the orm μ m, i = μ m, i 1 ( φ r, i ) (4) where φ r,i = i th relative volumetric concentration deined as Δv φ r i = (5), v + iδv Substituting (5) into (4) yields Δv μ m, i = μm i (6), 1 v + iδv Note that μ m,0 = μ. Alying (6) reeatedly rom i = 1 to i = n, we can get n 2.5Δv μ m, n = μ 1+ (7) i = 1 v + iδv Furthermore, since Δv = v /n and v /(v + v ) = φ, (7) can be changed to n 2.5φ μ m n = μ 1+ (8), i = 1 n(1 φ ) + iφ I n is large, the eective viscosity o the susension with the volumetric article concentration φ can be exressed as n 2.5φ (9) μm = μ lim 1+ n i = 1 n(1 φ ) + iφ Using the Gamma unction and its recursive roerty, (9) can be converted to 4

5 n 1 φ (10) Γ Γ n + 1 φ φ μ r = lim n Γ n 1 φ + Γ + 1 n 3.5 φ φ Furthermore, using the ollowing relationshi (Abramowitz and Stegun 1970) Γ ( x + a) a b lim = lim x (11) x ( x b) x Γ + where a, b = arameters, (10) can inally be simliied to n 1 φ (12) μ r = lim n = ( 1 φ ) = φ n φ φ where φ = 1- φ = volumetric raction o luid. The above derivation is based the iterative concet without any emiricism, hence the resulting ormula (12) should be considered as analytically vigorous and a direct extension o (1). The ormula can urther be exanded in a ower series as μ r = 1 + φ + φ + φ + φ + φ + (13) This gives k 2 = 4.38 and k 3 = 6.56, which are very close to the coeicients comuted by Thomas and Muthukumar (1991a, 1991b), as discussed reviously. As the dynamic eects o articles and luid are not included, (12) can only be alied to dilute susensions. I the article concentration is increased, inter-article collisions and random motion o articles are unavoidable, and thus the eective viscosity is also increased. This henomenon can be illustrated eectively by introducing dierent stress comonents or luid and solid hases, resectively, as detailed in Aendix I. Theoretically, the eective viscosity subjected to the dynamic eects in the non-dilute conditions can be obtained i the dierent stresses included in (24) can be ormulated. Unortunately, this is almost imossible as the inormation related to the article collision and kinetic stress is still very limited in the literature. Alternatively, an exonential model is roosed in the ollowing to extend (12) or the case o higher article concentrations. 5

6 Exonential Model There are a large number o measurements available in the literature or the eective viscosity. It is known that a considerable discreancy exists in the measured viscosities at higher article ractions reorted by dierent investigators, although oten each o their own data set showed little scatter (Acrivos 1995, Fan and Boccaccini 1996). The discreancy may deend on the roerty o articles, shear rate and mechanism o the viscometer. For a seciic susension, however, the relative viscosity generally varies with increasing luid raction described as ollows. First, the relative viscosity declines raidly with increasing luid raction, φ, or small φ -values. This is then ollowed by a gentle reduction i φ is urther increased. For the limiting condition o φ = 1, the relative viscosity would then be equal to one. Such variations can be illustrated in Fig. 2, or examle, by an emirical relationshi which was obtained or a susension comrised o 46 μm olystyrene sheres (Leighton and Acrivos 1987). Also suerimosed in the igure is (12), which serves as an asymtote to the measurements or very high φ -values but deviates rom the emirical relationshi, as exected, i φ is reduced. Fig. 2 shows that the sloe o the trendline itted to the exerimental data, when lotted on the logarithmic scale, always decreases with increasing luid raction, aroaching 2.5 or the dilute condition. Mathematically, as a irst aroximation, this can be exressed as d ln μr 2. 5 = (14) β d lnφ φ where β = exonent. Integration o (14) with resect to φ leads to 2.5 μ r = α ex (15) β βφ Since μ r = 1 or φ = 1, 2.5 α = ex (16) β Substituting (16) into (15) yields (17) μ r = ex 1 β β φ or 2.5 ( ) 1 (18) μ r = ex 1 β β 1 φ Fig. 2 shows a amily o curves lotted according to (17). It is noted that with increasing β-values, the μ r - φ relationshi deviates gradually rom μ r = φ -2.5, and or the same luid raction, μ r increases with increasing β-values. COMPARISON WITH PREVIOUS STUDIES 6

7 With dierent β-values, (18) can be used to reresent various emirical relationshis included in Table 1, which were derived largely based on exerimental data. For examle, the ormula given by Mooney (1951) or φ max = 0.74 and that reorted by Leighton and Acrivos (1987) can be well aroximated by (18) with β = 2; while Barnes (1989) exression or the high shear condition is close to (18) or β = More comarisons are given in Fig. 3, where the relative viscosity is lotted against the concentration o articles, and the corresonding β-values are ound varying rom 0.95 to 3.9. Similar to (12), (18) can also be exanded as a ower series with β as a arameter: μr = 1 + φ + + β φ + + β + β φ (19) β + β + β φ β + β + β + β φ Clearly, or β = 0, (19) reduces to (13). Comaring (19) with (2) yields that all the k- coeicients excet or k 1 = 2.5 are unctions o β. The comuted results included in Table 2 show that k 2 = 5.63 ~ 9.38, k 3 = ~ 30.73, k 4 = ~ and k 5 = ~ i β varies rom 1 to 4. Also included in Table 2 are various values o the coeicients rom k 1 to k 5, which were obtained theoretically or emirically by revious researchers. The coeicients or the ormula reorted in Thomas (1965) were calculated by exanding his ormula into a series, while those or Ward's ormula (see Gra 1977) by setting k 1 = 2.5. Table 2 shows that there are marked dierences among the various values or each coeicient, including those obtained in the resent study. However, it is interesting to note that the comuted coeicients o k 2 to k 5 or β = 2 are very close to those reorted by Ward (1955, see Gra 1977) who suggested the ollowing exression μ r = 1 + kφ + ( kφ ) + ( kφ ) + ( kφ ) + ( kφ ) +... (20) to be itted to the exerimental data or the concentration u to 35%, where k = 2.5 or sheres. CONCLUSIONS The objective o this study was not to develo another emirical ormula to reresent the existing exerimental data on to o other dozens o emirical relationshis available in the literature. Rather, we demonstrate an aroach to calculate the eective viscosity or a article susension that is simle to imlement yet maintains the underlying hysical reresentation o the situation. The results obtained indicate that it is not necessary to involve the so-called maximum article concentration, which is closely related to the randomness o article arking, in comuting the eective viscosity. For the condition without dynamic article interactions and turbulence eects, the derived ormula serves as an extension o the well-known Einstein s exression. This ormula, 7

8 ater urther modiication, is able to match well with the emirical relationshis reorted reviously. The analytical results obtained can also be given in the orm o ower series or comarison uroses. 8

9 APPENDIX I. STRESS ANALYSIS The bulk stress in a two-hase susension is comosed o two comonents, one due to the luid hase and the other due to the article hase. Simly, it can be exressed as τ m = φ τ + φ τ (21) where τ = luid hase stress and τ = article hase stress. Furthermore, it is known that the luid hase stress can be decomosed as v t τ = τ + τ (22) where τ v t = luid viscous stress and τ = luid turbulence stress. In comarison, the article hase stress can be divided into a static comonent, which is only associated with the stationary existence o the articles in the luid, and a dynamic comonent, which is related to the relative motion among the articles or article interaction. The relative motion causes velocity luctuations o the articles as well as inter-article collisions. For convenience, the article hase stress can also be divided into three comonents as ollows (Hwang and Shen 1989) c k τ = τ + τ + τ (23) c where τ = article collision stress, τ k = article kinetic stress and τ = article resence stress. The collision stress is the rate o momentum transer due to the collisions between the articles. The kinetic stress is the rate o momentum transer caused by the random motion o the articles, and thus can also be called the article turbulence stress. The article resence stress is not due to the relative motion among the articles. Rather, it is a result o the existence o the articles in the luid. The volume raction o the articles alters the local low attern o the luid and thus aects the bulk rheological roerty. Substituting (22) and (23) into (21), one gets v t c k τ m = φ ( τ + τ ) + φ ( τ + τ + τ ) (24) Eq. (24) shows that or a susension with a seciied concentration o articles, the stress minimizes i the contributions by the luid turbulence, inter-article collision and article random motion can be neglected, i.e., τ t, τ c and τ k are aroximately equal to zero. This condition in eect is imlied in deriving (1) or the case o dilute susensions (Hwang and Shen 1989), where the article resence stress can simly be obtained by integrating the Stokes stress. With an increase in the article raction, article interactions become signiicant and at least τ c and τ k cannot be ignored, thus the stress increases. For the same strain rate, the higher the stress, the larger the eective viscosity o the susension. Theoretically, the eective viscosity can be derived by considering the various stress comonents included in (24). However, since general inormation related to the collision and kinetic stresses is still not available, diiculties arise in order to urther ormulate the dierent stress comonents. 9

10 APPENDIX II. REFERENCES 1. Abramowitz, M., and Stegun, I. A. (1970). Handbook o Mathematical Functions. Dover Publications, New York. 2. Acrivos, A. (1995). "Bingham award lecture 1994: Shear-induced article diusion in concentrated susensions o noncolloidal articles." Journal o Rheology, 39(5), Barnes, H. A., Hutton, J. F., and Walters, K. (1989). An Introduction to Rheology. Elsevier Science Publishers B. V., Amsterdam, The Netherlands. 4. Batchelor, G. K., and Green, J. T. (1972). "The determination o the bulk stress in a susension o sherical articles to order c 2." J. Fluid Mech., 56, Einstein, A. (1906). "Eine neue bestimmung der molekuldimensionen." Ann. Physik, 19, Fan, Z., and Boccaccini, A. R. (1996). "A new aroach to the eective viscosity o susensions." J. Materials Sci., 31, Gra, W. H. (1971). Hydraulics o Sediment Transort. McGraw-Hill Book Comany, New York. 8. Hael, J., and Brenner, H. (1986). Low Reynolds Number Hydrodynamics. Martinus Nijho Publishers, Dordrecht, The Netherlands. 9. Hwang, G. J., and Shen, H. H. (1989). "Modeling the solid hase stress in a luidsolid mixture." Int. J. Multihase Flow, 15(2), Kim, S., and Karrila, S. J. (1991). Microhydrodynamics: Princiles and Selected Alications. Butterworth-Heinemann, Boston. 11. Leighton, D., and Acrivos, A. (1987). "The shear-induced migration o articles in concentrated susensions." Journal o Fluid Mechanics, 181, Metzner, A. B. (1985). Rheology o susensions in olymetric liquids. Journal o Rheology, 29, Mooney, J. (1951). "The viscosity o a concentrated susension o sherical articles." J. Colloid Sci., 6, Rutgers, J. (1962). "Relative viscosity o susensions o rigid sheres in Newtonian Liqiud." Reol. Acta, 2, Thomas, C. U., and Muthukumar, M. (1991a). "Convergence o multile scattering series or two-body hydrodynamic eects on shear viscosity o susensions o sheres." J. Chemical Physics, 94(6), Thomas, C. U., and Muthukumar, M. (1991b). "Three-body hydrodynamic eect on viscosity o susensions o sheres." J. Chemical Physics, 94(7), Thomas, D. G. (1965). "Transort characteristics o susension: VIII. A note on the viscosity o Newtonian susensions o uniorm sherical articles." J. Colloid Sci., 20, APPENDIX III. NOTATION The ollowing symbols are used in this aer a = arameter; b = arameter; k = coeicient; 10

11 k i = coeicient (i = 1, 2, 3, ); n = total number o susension samles; τ = luid hase stress; t τ = luid turbulence stress; v τ = luid viscous stress; τ m = bulk stress tensor; τ = article hase stress; c τ = article collision stress; k τ = article kinetic stress; τ = article resence stress; v = volume o luid; v = volume o articles; α = coeicient; β = exonent; μ = viscosity o luid; μ in = intrinsic viscosity; μ m = eective viscosity o article-luid mixture; μ m,i = eective viscosity o the ith susension samle (i = 1, 2,, n); μ r = relative viscosity = μ m /μ ; φ = volumetric raction o luid; φ = volumetric concentration o articles; φ max = maximum volumetric concentration; φ r,i = ith relative volumetric concentration; = increment o volume o articles (i = 1, 2,, n). Δv,i 11

12 Table 1. Emirical Formulas or Eective Viscosity Researchers μ m μ μ in φ max Mooney (1951) 2.5φ 0.52~ ex φ / φ max Thomas (1965) φ φ ex( 16.6φ ) 1 Metzner (1985) φ 1 φ max Leighton and 2 0.5μinφ Acrivos (1987) / φ φ max Barnes et al. μinφ max φ 2.71~ 0.63~ (1989) φ max 12

13 Table 2. Values o Coeicients included in Eq. (2) Study k 1 k 2 k 3 k 4 k 5 Theoretical Einstein (1906) 2.5 Batchelor and Green (1972) Kim and Karrila (1991) Thomas and Muthukumar (1991) Emirical Ward (1955, see Gra 1977) Thomas (1965) Present β = (Eq. 19) β = β = β = β =

14 Cations or Figures Fig. 1. Sketch o Susension Series Fig. 2. Eective Viscosity Decreases with Increasing Fluid Fractions Fig. 3. Variations o Eective Viscosity with β as a Parameter Fig. 4. Comarisons with Previous Emirical Relationshis 14

15 Samle 1 2 i-1 i n μ m,i = μ m,1 μ m,2 μ m,i-1 μ m,i μ m,n φ r,i = v Δv + Δv v Δv Δv + 2Δv v + ( i 1) Δv v + iδv v + nδv Δv Δv Fig. 1. Sketch o Susension Series 15

16 100 μ r 10 Leighton and Acrivos (1987) Eq. (12) μ r = φ φ Fig. 2. Eective Viscosity Decreases with Increasing Fluid Fractions 16

17 100 μ r 10 1 β = Eq. (12) φ Fig. 3. Variations o Eective Viscosity with β as a Parameter 17

18 Present study 80 φ max Mooney(1951, φ max =0.74) = 70 φ max Mooney (1951, φ max =0.52) = μr Thomas (1965) Metzner (1985) Leighton and Acrivos (1986) Barnes et al (1989, zero shear) Barnes et al (1989, high shear) β= φ Fig. 4. Comarisons with Previous Emirical Relationshis 18