CHAPTER-2 LITERATURE REVIEW. 2.1 Effect of Solids concentration on suspension settling

Transcription

1 CHAPTER-2 LITERATURE REVIEW PREAMBLE This chater is the discussion of the revious work by other researchers. At the beginning of the chater earlier studies on the effects of solids concentration and wall effects on the settling behaviour of susensions are discussed. The drag- coefficient articles Reynolds number relationshis roosed by earlier workers are resented and their recommendations are discussed. Next art of the chater includes discussion on rheological roerties of the susensions, the rheological behaviours and different rheological models. Further effects of system arameters like solids concentration, article size and temerature on susension viscosity are discussed. Different disersants are also discussed. The chater concludes with the discussion on alications of artificial neural networks in various fields of chemical engineering. 2.1 Effect of Solids concentration on susension settling When the concentration of solids is high, the distance between articles becomes so small that the articles come into contact with one another. As a result, the settling rocess occurs differently and Stokes law fails to describe the settling of articles beyond infinite dilution. Hindered settling is a term used to describe behavior at higher concentrations where sedimentation rates are largely related to concentration rather than to article size [1]. In hindered settling articles are crowded and surrounding articles interfere with the motion of individual articles, tyically creating a slowermoving mixture than would normally be exected. Uward dislacement of the liquid is also more rominent under hindered settling conditions. Settling rates in multiarticle systems are influenced by article interaction effects, which consist of collision among articles and non-hydrodynamic interactions. In the concentrated susensions of significant solid volume fraction (f), as the susension settles, it relaces equivalent volume of liquid. If susensions settle with their settling velocity (u m ) under the action of gravity, the dislaced liquid moves 8

2 uward. The liquid velocity (u w ) deends on the settling velocity of solids. According to continuity, the relative motion between solid articles and liquid can be resented as follows (Equation 2.1): fu + (1 - f) u = 0 (2.1) m w u = u - u (2.2) m w Where, u is the relative velocity between solids articles and the disersing medium and is the ertinent settling velocity as given below in Equation 2.3: u m u = (1 -f) (2.3) For sedimentation henomena, dimensional analysis suggests that, u u = f [(1 - f),(re )] (2.4) Where, u is the free falling velocity of the article alone in the same fluid and the container as the sedimentation of uniform sheres and Re is the article Reynolds number. Richardson and Zaki [2] extensively studied sedimentation of solid-liquid susensions of sherical articles; they investigated the deendency of the settling velocity on the solids volume fraction. Their results were summarized with the relationshi known worldwide as the Richardson-Zaki equation. Their relationshi (Equation 2.5) is a ower law of the void ratio (1-f). u = u (1 - f) n (2.5) The arameter n was found to be a function of the flow regime, exressed by the article Reynolds number Re, and of the article to column diameter ratio. Richardson and Zaki [2] have given the following relationshis (Equations 2.6 to 2.9) for the exonent; n as a function of article diameter (d ) to cylinder diameter (D) ratio and article Reynolds number Re : d n = ( ) for Re < 0.2 D (2.6) d n = ( ) for 0.2 < Re < 1 D (2.7) -0.1 n = Re for1< Re < 500 (2.8) n = < Re < 7000 (2.9) 9

3 When the ratio of the article to the container diameter is small, n has a value of about 4.6 [3], value of n between 4.65 and 5.1 was reorted by Barnea and Mizrahi [4], Caes [5] obtained the reduced values of n between 3 and 3.5. Turian et al [6] have suggested the alternative scheme of maintaining the value of n to be near to Recently Koo [7] reorted the value of n as 5.5 for settling of mono-disersed susensions. When Particles of mixed sizes settle, inter-article interactions are further comounded as the faster settling, larger article aroach and overtake the slower, smaller articles leaving then in their wake of trailing dislaced liquid. Garside and Al Dibouni [8], Cheng [9] and Baldock [10] confirmed the decreasing behaviour of n with the article Reynolds number. 2.2 Wall Effects on susension settling When articles settle in a quiescent liquid in a given container, the resence of the container wall at a finite distance from the settling article exerts retarding effect on it. This henomenon due to resence of solid boundary at a finite distance is known as wall effect. Wall effect is commonly reorted as a function of article to container diameter ratio (d /D). The wall effect becomes significant at higher values of d /D. Garside and Al-Dibouni [8] and Selim et al [11] examined wall effects on sedimentation articles. Settling of single sherical article and multisized article mixture in Newtonian fluid was exhaustively investigated. It was found that the correction for wall effects gives measurably better correlation of free falling velocity. For single article terminal velocity Garside and Al-Dibouni [8] recommended the following wall effect correlations: u éæ d öæ d öù = êç ç 1- ú for Re < 0.2 u ëè D øè D øû u d = æ ç ö for 0.2 < Re < 10 u è D ø é ù ê ú u 1 = ê ú for10 Re /2 u ê d ú < < ê æ ö 1- ú ê ç ë è D ø úû (2.10) (2.11) (2.12) 10

4 Lali et al [12] studied settling of sherical articles of glass and stainless steel in aqueous solution of CMC in the concentration range of 0.2 to 2 weight %, the settling velocities were measured in nine different cylinders in the range of mm diameter. It was found that the wall effect factor is function of both the article to container diameter ratio and the article Reynolds number. Di Felice and Parodi [13] have demonstrated that the container wall diameter influences the falling velocity only for dilute systems, which is in contrast with a widely used, established emirical relationshis. In creeing flow regime Francis [14] roosed the following relationshi for the wall effects: u u é d ù ê 1- ú = ê D ú ê 0.475d ú ê 1- ë D úû 4 (2.13) This relationshi is simler than those roosed by Clift et al [15]. Di Felis [16] roosed the following equations (2.14) and (2.15) for wall effects correction for the terminal settling velocity in viscous and inertial flow regimes resectively: Vt U u u é d ù ê 1- ú = ê D ú ê 0.33d ú ê 1- ë D úû é d ù ê 1- ú = ê D ú ê 0.33d ú ê 1- ë D úû (2.14) (2.15) 2.3 Drag coefficient article Reynolds number relationshi The C D -Re relationshi (Equation 2.16) was first roosed by Stokes [18] ; he obtained the solution to a general equation of motion for a single shere settling in a fluid by assuming the total drag as viscous drag. The relationshi is the well known Stokes law, alicable for Re < C D = (2.16) Re 11

5 Gilbert et al [19] roosed a three arameter equation (Equation 2.3.2) to correlate the drag coefficient and the article Reynolds number. They introduced a third arameter in addition to the two arameters given by Stokes. C = X Re Y + Z (2.17) D A five arameter model (Equation 2.18) suggested by Clift and Gauvin [15] incororated the corrections to the Stokes law; these additional arameters were obtained as a function of article Reynolds number. C 24 (1 Re B C = + A ) + (2.18) Re 1 + D Re D E Tourton and Levensiel [20] introduced two equations (Equations 2.19 and 2.20) to correlate the drag coefficient and article Reynolds number for falling sheres. These two equations are similar to the original three arameter model by Gilbert et al [19], and five arameter model by Clift and Gauvin [15]. C = 27.2 Re for Re < 2 10 (2.19) D = + + for Re < (2.20) CD ( Re ) Re Re Another five arameter equation obtained by Khan and Richardson [21] is as given below: C - - ( ) 3.45 = 2.25Re Re for Re < 3 10 (2.21) D Turton and Levensiel [20] tested their exerimental data with three arameter equation (Equation 2.19) and five arameter equation (Equation 2.20), as shown in the following Figures 2.1 and 2.2. The results show that the three arameter equation redict the drag coefficients value well for low article Reynolds number where as the redictions show significant error for the higher value of article Reynolds number. The five-arameter equation, on the other hand, is caable of showing a minimum C D value, and it can be seen from Figure 2.2 that this equation correlates the data extremely well throughout the whole article Reynolds number range. Equation 2.20 has the added advantage of converging to the Stokes equation at low article Reynolds number; it is a better correlation for rediction of the drag coefficient values in the sub critical regime (Re < 2 x 10 5 ). 12

6 Figure 2.1: Comarison between exerimental and redicted values of drag coefficient of sheres using three arameter model [20] A four arameter general drag coefficient-reynolds number relationshi roosed by Haider and Levensiel [22] is given below (Equation 2.22) C D 24 ( Re B C = A + ) + Re D 1+ Re (2.22) The exerimental data used to find the best values of the four arameters was same as used by Turton and Levensiel [20]. The final equation obtained for the drag coefficient is as given below (Equation 2.23): C D = ( Re ) + Re Re (2.23) The drag coefficient values redicted by this equation were comared with the exerimental values and earlier redictions by Turton and Levensiel [20] with the three arameter equation (Equation 2.21) and five arameter equation (Equation 2.22) as discussed earlier. The drag coefficient values calculated by four arameter model as deicted in Figure 2.3 were found more accurate. 13

7 Figure 2.2: Comarison between exerimental and redicted values of drag coefficient of sheres using five arameter model [20] An exlicit equation for article settling velocities in solid-liquid systems was resented by Zigrang and Sylvester [23]. This equation can be used to redict the settling velocity of articles in liquid-solid susensions; this correlation gives the terminal velocity of a single sherical article in a liquid as given below (Equation 2.24): u * 3/2 2 ( d* ) = (2.24) d * Where, d * and u * are dimensionless diameter and dimensionless terminal velocity of shere resectively. d * æ 3 2 ö = ç C Re 4 D è ø 1 3 (2.25) u * æ 4 Re ö = ç è 3 CD ø 1 3 (2.26) 14

8 Figure 2.3: Comarison between exerimental and redicted values of drag coefficient of sheres using four arameter model [22] Turton and Clark [24] roosed the following correlation (Equation 2.27) to reresent a very simle and accurate way to evaluate the terminal velocity of a shere of known dimensions and density in a known fluid. u é ù ê ú ê 1 ú = ê ú ê æ 18 ö æ 0.321ö + ú 2 ê ç ç ëè d* ø è d ú * ø û * (2.27) This correlation cannot redict the velocity of a article driven through a fluid by a force other than its own weight without further maniulation. There are number relationshis available in the literature for the drag coefficient as a function of article Reynolds number; some of them are reorted in the following Table

9 Table 2. 1: Drag coefficient- Reynolds number relationshis Author(s) Equation Range of Alicability Perry and 24 Re < 0.3 Chilton C D = Re [25] < Re < <1000 C D = 0.6 Re C D = < Re < Re < 100 C D = Re (1 - ex( Re ) Re E Re < 3 x 10 5 C D = Re Cheng [9] ( ) Flemmer and Banks [26] Where, E = 0.261Re Re (log Re ) 10 2 Saha et al [27] 18.5 C D = Re < Re < 1000 Choncha and Almendra [28] Madhav and Chhabra [29] Brown and C D æ 9.06 ö = ç Re 0.5 è ø Re C D = + Re ( ) C -1 Re Re ( ) D Lawler [30] 2 1 < Re < < Re < Re < 2 x 10 5 = Re + ) Flow curve modeling In order to describe the flow curves mathematically, several rheological models have been develoed. Mechanistic aroaches are, however, difficult to aly because of the diversity of the roerties resonsible for the rheological behaviour. More commonly, emirical models have been used to describe the characteristics of the flow behaviour. Several emirical equations relating shear stress to the shear rate have been develoed to model the basic flow curve shaes. The criteria for the suitability of the model are: i. It should be simle with minimum number of indeendent constants 16

10 ii. iii. Its constants should be easily determined and The constants should have some hysical significance Regression methods can be used to fit the equations to the rheological data. The fit of these equations can be then comared using statistical methods. Models with smallest number of coefficients are referred because they are usually simle to use. The following sections describe some of the models commonly used to reresent the rheological roerties of the susensions. Included are Newton s law of viscosity, three models can be used to describe the shear thinning behaviour and three models that describe lastic behaviour. The shear thinning roerties are often characterized by the ower law model, the Cross model and the Cerreau model. Models that describe lastic flow roerties must have yield stress term such as the Bingham lastic model, the Herschel Bulkley model and the Casson Model Newton s law of viscosity The flow behaviour of Newtonian fluids can be described by Newton s Law of Viscosity (Equation 2.28) du t = m æ ö ç (2.28) è dr ø Where, m is the Newtonian viscosity of the fluid. It has been used to describe the rheology of non-interacting sheres in Newtonian liquids [18, 31]. Above a critical solids concentration, susensions exhibit non-newtonian roerties The ower law model æ du ö t = k ç è dr ø n (2.29) This model (Equation 2.29) is also referred as Ostwld-De Waele model. This equation can be used to model all three basic viscous flow behaviours. The coefficients k and n are referred to as the fluid consistency index and flow index, resectively [32]. A high k value imlies that the susensions have a high viscosity. The deviation of n from unity is a measure of the non-newtonian behaviour. Clearly, at n equal to unity, the equation becomes Newton s law of viscosity and k becomes m. For n is less than unity, the model describes seudolastic flow and, for n greater than one it describes the dilatant flow. This model has been used to describe many solid-liquid susensions [6]. 17

11 2.4.3 The Cross model The Cross model (Equation 2.30) was develoed base on the assumtion that the seudolastic flow is due to the rocess of formation and ruture of linkages in chains of articles. This rocess is believed to occur in aggregated systems where links between articles result in the formation of chain-like aggregates which ruture due to Brownian motion and shearing [33-34]. -m æ du ö m = m + a ç (2.30) è dr ø Where, m is the aarent viscosity at infinite shear rate. A value of m equal to 2/3 was found to be suitable for many seudolastic systems which reduce the number of coefficients in the equation to 2 [33]. The Cross model has been alied to susensions that exhibit seudolastic behaviour [35] The Carreau model The Carreau model (Equation 2.31) was develoed to describe the seudolastic roerties of olymer solutions and melts. -s 2 æ æ du ö ö m = m 1+ ç A (2.31) ç è dr ø è ø It is based on molecular network theory which describes the non-newtonian flow with resect to the creation and loss of segments. The rate of creation and loss of these segments is function of shear rate which results in the non-newtonian resonse to shear [36-38] The Bingham lastic model The Bingham lastic model (Equation 2.32) describes the simlest tye of viscolastic flow. The model characterizes the ideal case in which a comlete structure breakdown occurs once a yield stress has been exceeded. Once the yield stress is exceeded a linear relationshi exists between the shear stress and shear rate [39]. du t = t 0 + h æ ö ç è dr ø (2.32) Where, t 0 is the yield stress and h is the lastic viscosity. The Bingham model is widely used because of its simlicity. Many susensions including red mud, kaolinite, clays, drilling mud and yeast susensions have been modeled with equation [6, 35, 40-41]. 18

12 2.4.6 The Herschel Bulkley Model The Herschel Bulkley model (Equation 2.33) is the combination of the Oswald-De Waele ower law model and the Bingham lastic model. As the yield stress becomes small the equation takes form of the ower law model and as the index, n, aroaches unity, it becomes Bingham lastic model. t n æ du ö = t 0 + k ç è dr ø (2.33) This simle, versatile and ractical model has been widely used to characterize both dilute and concentrated susensions [6, 35] The Casson model 1/2 1/2 du = æ ö o + ç t t h è dr ø (2.34) The Casson model (Equation 2.34) is a simle two arameter model that has a hysical basis and is derived from structural arguments. This model is caable of fitting low shear rate curvature and therefore it estimates the yield stress comare well to the values obtained exerimentally [6]. 2.5 Effect of Solids Concentration on Susension Viscosity Several studies have shown that the viscosity of susensions increase in an exonential manner with solids concentration and becomes infinite at the maximum acking fraction. It is well known that the earliest theoretical work on the effective viscosity was due to Einstein [18] whose derivation led to the effective viscosity to be linearly related to the article concentration as follows (Equation 2.35): Where, hr = f (2.35) hr = µm/µf = relative viscosity of the susension µm = effective viscosity of the article-fluid mixture µf = viscosity of fluid, and f = solid volume fraction of the susension This exression is exact when the viscous effect is dominant so that the creeing flow equations can be alied at the article level. This equation is based on the viscous 19

13 energy dissiated by the flow around the shere, relating the relative viscosity of the solids content. With increasing solids concentration, there are a greater number of article interactions: these are resonsible for the exonential increase in viscosity. These interactions result in energy dissiation that can be described in terms hydrodynamic, and aggregation effects. These effects are also resonsible for the increase in non-newtonian nature of the susensions that is found at higher concentrations. Following Einstein s work, numerous exressions have been roosed to extend the range of validity to higher concentrations. They are either theoretical exansions of Einstein formula to higher order in f, or emirical exressions that were obtained based on exerimental data [42-44]. The theoretical exansions are usually exressed in the form of a ower series as given below (Equation 2.36): 2 3 hr = 1 + k1f + k2f + k3f +... (2.36) Where k 1, k 2, k 3 are the coefficients. Evaluation of these coefficients other than k 1 requires the formulation of article interactions, which is rather difficult. This is one of the reasons that only the theoretical values of the coefficients related to lower order in f, like k 2 and k 3, can be found in the literature so far. Even for these lower order coefficients, the calculations are available only for idealized cases when the article arrangement or its statistic roerties are simle. An examle for determining the k 2-value was resented by Batchelor and Green [45] who reorted that the coefficient k 2 was equal to 7.6 for 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 [46], leading to the k 2-value to be changed to Other k 2-values have been reviewed by Elic and Phan-Thien [47]. Furthermore, Thomas and Muthukumar [48] found the third order coefficient, k 3, to be 6.40 by alying the multile scattering theories to the evaluation of the hydrodynamic interaction of three sheres. Following these theoretical attemts, it can be exected that more comlicated configurations of the article arrangements will lead to further difficulties in the mathematical formulation. On the other hand, articles susended in fluid should in fact be distributed in a random manner so that their arrangement cannot be fitted to a 20

14 simle configuration or described by a simle distribution function. Therefore, although an idealized article arrangement may enable a theoretical evaluation of the k-coefficients in higher order, the alicability of the effective viscosity so derived is limited in ractice. When aggregation effects exist, the aggregates may be considered to be the susended units in the susension. This result in an increase in effective solids content as the aggregate will tra liquid [49]. The size of aggregates deends on the tye of attractive forces, the shearing conditions and solids concentration in the susensions [50]. Many of these effects have been considered in the evaluation of the coefficients of the general ower formula [51]. This model, which is extension of the Einstein s equation alies only to moderately concentrated susensions and does not cover the entire solids concentration range [43, 51]. In addition to the theoretical exressions, various emirical relationshis have been roosed to evaluate the effective viscosity of a susension with higher concentration. An early survey of such relationshis made by Rutgers [51] showed that large discreancies exist among the different relationshis. Table 2.2 shows some tyical examles of the existing emirical formulas, where fm is the maximum article concentration and h is the intrinsic viscosity. It is noted that for those relationshis where the so-called maximum concentration is included as a arameter, the effective viscosity aroaches infinity 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 meaningful for the effective viscosity. The first condition is a susension without articles, imlying that the effective viscosity is the same as the fluid viscosity. The other is a susension without fluid, which would then theoretically behave as a solid with infinite viscosity. On the other hand, only for articles with regular shaes can the maximum concentration be mathematically determined for a given acking arrangement. For examle, the maximum concentration can reach 1.0 for cubes acked face-to-face and 0.74 for sheres with the closest hexagonal acking arrangement. With irregular articles, the maximum value varies markedly even though the article size is uniform. As a matter of fact, the maximum concentration and intrinsic viscosity were often used as two emirical arameters in fitting exerimental data to an emirical formula [52-53]. 21

15 Eilers [18] used the maximum solids content as an uer limit in relating relative viscosity to the solids fraction for bitumen emulsions. Chong et al [54] modified Eiler s equation to correlate the relative viscosity to the solids content for oly-diserse susensions. Mooney [55] considered the crowding effects of mono-diserse shere to obtain his equation. He also showed how his aroach could be extended to bimodal and oly-diserse susensions. Krieger and Dougherty [56] also used article crowding, in similar manner to derive their equation. Ting and Luebber [57] considered the effects of liquid-solid density ratio, article size distribution and article shae on acking arrangement to derive an emirical relation between the relative viscosity and some function of these arameters for varying solids contents. 22

16 Table 2.2: Formulas for effective viscosity Einstein s Equation [18] Eiler Equation [18] -2 æ f ö hr = ç 1- è fm ø Leighton and Acrivos s Equation [52] 2 æ ö ç 0.5hf hr = ç 1+ f 1 ç - f è m ø Chong et al s Equation [54] 2 æ f ö ç f m hr = ç f 1 ç - f è m ø Mooney s Equation [55] æ ö ç 2.5f hr = exç f 1 ç - f è m ø Krieger Dowarty s Equation [56] f - m hr = [1- ] hf fm Liu s Equation [58] n h = [a( f - f)] Metzner s Equation [59] 2 æ ö ç f hr = ç f 1 ç - f è m ø Quemada s Equation [60] f -2 m hr = [1- ] f f hr = ( f) r m m 2.6 Effect of Temerature on Susension Viscosity Liquid viscosity decreases with the increase in temerature. The trend of decreasing viscosity at elevated temeratures occurs due to increased kinetic energy of the articles romoting the breakage of intermolecular bond between adjacent layers which results in decrease in viscosity of the susension. For the limestone slurry of 70 weight % solids concentration, He et al [61] investigated influence of temerature on the viscosity. They found that, the slurry viscosity decreases with increase in the temerature from 13 to 55 0 C. Reduction in the viscosity from the range of to 23

17 Pas to the range of 0.01 to Pas as for the increase in temerature from 13 to 55 0 C was reorted. Mikulasek et al [62] shown the decrease in viscosity of the titanium oxide susensions of 10 to 30 volume % concentration over the range of temerature from 20 to 50 0 C. Similar results were observed by Roh et al [63] for coal water susensions for the temerature range of 15 to 50 0 C and Senaati et al [64] for limestone susensions in the temerature range of 30 to 60 0 C. The decrease in the susension viscosity may be the result of decrease in the interarticle forces between articles with temerature. The decreased value of the viscosity with increased temerature confirms to fundamental roerties of any viscous materials as for all normal liquids [65]. Work of different researchers [66-69] shown that the susension viscosity also decreases with the increase in temerature. Senaati et al [64] resented a simle Arhenious tye equation for the temerature deendence of susension viscosity. An identical relationshi was confirmed by Li et al [70] for temerature deendence of silicon carbide susensions, they observed increase in viscosity as temerature of the slurries increased from 20 to 60 0 C. The increase in the viscosity was observed because the susensions induce flocculation trend, which might be enhanced at higher temerature. 2.7 Effect of Particle Size on Susension Viscosity It is exected that the effects of article size on susension viscosity varies deending on the range of article size where these effects are studied because the actual interarticle forces will vary with the size of articles. Renehan et al [71] found that susensions of coal articles in a non-newtonian susension of bentonite clay exhibited both increase and decrease in viscosity with increasing article size. Exerimental results from Hasan et al [72] show that for high concentrations there is a decrease in viscosity with increasing article size, whereas at lower concentrations there is an increase with increasing article size. The size of articles in a susension has been shown strong influence on its rheological roerties. In articular it has been shown that susensions with very large (higher than 100 mm) or very small (less than 10 mm) articles have a higher viscosity than susension containing immediately sized articles i.e mm [71]. These differences in the viscosities can be 24

18 exlained by contributions of the hydrodynamic and aggregation effects. These effects contribute to the viscosity to a greater or lesser degree deending on the size of the articles. It is difficult to determine the effect of article size on the viscosity of susensions since arameters like shae and charge, which characterize the system may vary with article size. Most authors do not secify recisely the accuracy of the size distribution, or rovide much detail as to exerimental conditions. Therefore the disarity in observed results is not unexected. Some investigators have reorted an increase in viscosity with the decrease in the article size [73-76]. This increase in viscosity has been attributed to the aggregation effects that dominate fine article behaviour [43, 70, 76-79]. Inter-article forces of attraction and reulsion are resonsible for these effects which dominate the movement of colloidal size articles [43, 71]. The magnitude of these factors may be large enough to influence the movement of course articles. Rehenan et al [71] showed that the movement of coal articles as course as 200 mm were influenced by surface charge related effects, that were large enough to contribute to the rheological roerties. Sweeney and Geckler [74] suggested that the contribution from electroviscous effects to the rheological roerties of fine article susension can be very significant. They showed that the viscosity of a susension of glass sheres in anionic susending fluid, where electroviscous effects should be large, was greater than the viscosity of the same shere susension in non-ionic fluid, in which electroviscous effects should be small. For susensions of glass and olystyrene sheres, Parkinson et al [73] Saunders [78] exlained the increase in non-newtonian behaviour with decreasing article size, by contribution of aggregation effects. Inter-article forces of attraction dominate over inertial forces for small articles which make them more suscetible to aggregation than course ones. With increasing shear rate the attachments break resulting in more disersed susension. As the article becomes disersed the aarent viscosity decreased resulting in the shear thinning behaviour [75]. Since this disersion with shearing is not instantaneous, at a given shear rate the aarent viscosity will decay with time resulting in the thixotroic characteristics of such susensions. The structure that forms as a result of the aggregation of the fine article contributes to the yield stress. Below a critical solids concentration the structure may not be continuous 25

19 and therefore no yield stress may be resent [80]. The effects of article size are more ronounced at high solids concentrations than at low ones. The magnitude of the hydrodynamic effects is roortional to secific surface area of the articles. Since the secific surface area increases with decreasing article size, the contribution of the hydrodynamic effects also increases. Therefore the increase in viscosity of susensions resulting from decreasing article sizes can be exlained by the greater amount of hydrodynamic energy dissiation [78]. According to Hasan et al. [72] increased article interaction for smaller articles leads to increase in the susension viscosity. The formation of aggregates or flocks with decreasing article size due to the increasing significance of inter-article forces at smaller article sizes increases the susension viscosity. Kawatra and Eisele [81] observed an increase in viscosity with reduction in article size for constant solids concentration. This is due to increased surface area, which binds u water molecules and thus increases the effective solids concentration. The results of several investigations [71, 75, 77] have shown that the increase in viscosity occurs as article size increases. This result has been exlained by the energy dissiated due to hysical article interactions. Since coarse articles have greater inertia than fine ones they will collide rather than sli ast each other. The hysical collisions dissiate energy through friction and loss of translational and rotational momentum. Thomas [43] and Hasan et al [72] suggest that at low concentrations article interactions are minimal and viscosity may increase with article size. Mangesana at al [82] shown that viscosity increases with article size at a fixed concentration for any given shear rate. The reason suggested for this increase was that articles of greater size ossess greater inertia such that on interaction, the articles are momentarily retarded and then accelerated. In both these stages their inertia affects the amount of energy required. This dissiation of energy is what may aear as extra viscosity. The contributions of the hydrodynamic, aggregation and electroviscous effects clearly deend on the size of the articles. In order to minimize the viscosity, it seems that an otimum article size exists that is not too large and not too small. 26

20 2.8 Disersing Agents Disersing agent can be classified into: (i) Inorganic comounds, and (ii) olymeric disersants [83]. While inorganic disersants are rimarily to control the charge density at the solid/liquid interface, the olymeric comounds also rovide steric hindrance. Susensions of aggregated articles can exhibit yield stress and tyically have shear thinning flow roerties [84]. Disersing the articles results in more Newtonian flow behaviour and reduced aarent viscosity. It should be noted, however, that by increasing electrostatic reulsion, electroviscous forces can also become imortant resulting in greater aarent viscosities. Organic reagents tyically used in iron ore rocessing are olysaccharides such as starch, guar gum and carboxyl methyl cellulose which are used to flocculate iron ore [85-86]. The effectiveness of these regents deends on the nature of the reagents, the article surface chemistry, the tye of dissolved ions and the H. Tyical inorganic disersing agents are sodium olyhoshates and sodium silicates [84]. These disersing agents are widely used in rocessing of iron ores. 2.9 Artificial Neural Networks (ANN) ANNs are massively connected arallel distributed rocessor made u of simle rocessing units, which has a ability for storing exerimental knowledge and making it available for use. ANNs are well known for their utility as universal aroximator of comlex nonlinear relationshis between rocess variables and roduct quality roerties. ANNs have the ability to learn from inut data and are very useful for the rediction of comlex high-dimensional data. ANN methods have a broad range of alications in agriculture [87], weather forecasting [88], water resources engineering [89-90] finance, medicine, robotics, etc. As in other fields of science, ANN methods have become oular in chemical engineering too, soon after their develoment. Artificial neural networks have been successfully used for dynamic modeling and control of chemical rocesses and fault diagnosis [91], in the catalytic reactor modeling and design of solid catalysts [92] and for modeling the kinetics of a chemical reaction [93]. The alicability of ANN modeling in emulsion liquid membranes [94] and in the rediction/estimation of the vaor liquid equilibrium data [95-96] has also been investigated. Melkon Tatlier et al [97] used ANN model for the rediction of the molar 27

21 comosition of the zeolitess. They estimated the quantities of Si and H 2 O in the zeolites by using ANN. They reorted the amounts of deviation from the exerimental results for the Si and H 2 O contents to be equal to about 10 and 20%, resectively. Bhat and McAvoy [98] alied ANN to dynamic modelling for a H-controlled CSTR. The redicted H values when comared to two other aroaches demonstrated that ANN could redict more accurately than conventional method. Willis et al [99] discussed the alication of ANN as both inferential estimator and redictive controller. The results demonstrated that ANN could accurately redict the rocess outut and significantly imroved the control scheme. Pollard et al [100] utilized backroagation trained ANN for rocess identification. They concluded that ANN was articularly useful when the inut-outut maing was unknown since ANN was able to accurately reresent nonlinear behaviour in a black box manner. Rajasimman et al [101] used ANN for modeling of an inverse fluidized bed bioreactor. Initial substrate concentration and the chemical oxygen demand (COD) values for a given retention time were redicted by using ANN. Predictions of these arameters were obtained with desired accuracy. Shaikh and Al-Dahhan [102] comared the selected literature correlation and the ANN model for the rediction of gas holdu in bubble column reactor. The standard deviation of the ANN results for different column arameters was within 14% which was still less than the other models. They further suggested the use of ANN for scale u of he bubble column reactors. Sablani et al [103] develoed ANN model caable of exlicitly calculating the friction factor for a non Newtonian fluid over a wide range of Reynolds number. Among different architectures studies they found the maximum average mean error to be about 3% for the laminar and turbulent regions. Fadare and Ofidhe [104] develoed artificial neural network for the rediction of friction factor in ie flow. The roosed model found accurate for rediction of factor using relative roughness and Reynolds number as inut arameters. ANNs are good at attern recognition, trend rediction, modeling, control, signal filtering, noise reduction, image analysis, classification, and evaluation. In fact, the uses for neural networks are so numerous and diverse that these alications may seem to have nothing in common. However, they all share the ability to make associations between known inuts and oututs by observing a large number of examles. ANNs have been successfully alied in various fields of chemical 28

22 engineering which include, modeling of multihase reactors [105], Process control [91, ], Catalysis [110], Estimation of concentration of comonents in a mixture [ ], calculation of heat transfer coefficient [114], modeling of um-mixer characteristics [114], Membrane searations [110], Reactor modeling [105, 116], waste water treatment [117]. The success of the ANN deends on the selection of the rocess variables, the quality of the data and the tye of algorithm used. The backroagation [ ] is the most well known and widely among all tyes of ANN algorithms. The back roagation is the multilayer feed forward network with different transfer functions in an artificial neuron and a owerful learning rule. The learning rule is known as backroagation, which is kind of error minimization technique with backward error roagation. ANN is attractive due to its information rocessing characteristic such as nonlinearity, high arallelism, fault tolerance as well as caability to generalize and handle imrecise information [118, 120]. These characteristics have made ANN suitable for solving a variety of roblems Critical Araisal The two hase systems of solid-liquid susensions are inherently unstable. Sufficient amount of literature is available on settling behaviours of different tyes of susensions for different ranges of article Reynolds numbers. Among the relationshis for the single article terminal settling velocity in concentrated susensions, Richardson and Zaki s equation is confirmed and recommended by several theoretical and exerimental studies. From the available literature it is well understood that, the hindered settling coefficient n varies from susension to susension. No unique value of n is available for all susensions. It varies from 3 to 5.1 for the susensions of different tyes of solids. In the literature various drag coefficient- article Reynolds number (C D -Re ) relationshis are available. The available relationshis contain two to five constants. No settling data for the hindered settling coefficient n of magnetite ore susensions is available in the literature. Because of the unstable nature of these two hase systems the, available C D -Re relationshis are erroneous if alied to all tyes of susensions. Hence there is good scoe for studying settling behaviour of magnetite ore susensions and to obtain the useful data on the n values. From the exerimental 29

23 data it would be ossible to develo exerimental C D -Re relationshi for magnetite ore susensions. Rheological characteristics of the susensions deend on various arameters. Particle size and concentration of solids significantly influence the shear stress-shear rate relationshi and the relative viscosity of the susensions. The effect of solids concentration on the susension rheology is not the same for all tyes of susensions. It deends on the nature and tye of the solid articles and their surface characteristics. Studies by Some susensions show the increase in non Newtonian behaviour with the increase in solids concentration, whereas the decrease in non Newtonian behaviour of the susensions also has been resented in some studies. Same susension also can show different rheological behaviours at different solids concentrations and shear rates. Particle size effect on susension rheology and relative viscosity was studies by many workers. The earlier studies show increase in non Newtonian behaviour and relative viscosity with the increase in article size of the solids in susensions as well as decrease in non Newtonian behaviour and relative viscosity with increased article size. Well established emirical equations are available in the literature to describe the flow curves of the susensions, but the susensions of different tyes of solids articles can not be resented by a same equation. Even for same susensions different equations were suggested by a few workers. For magnetite ore susensions, to describe the shear stress- shear rate relationshi, equations such as ower law, Bingham lastic and Casson equation were roosed by earlier workers. For very dilute susensions, the relative viscosity can be redicted by Einstein s formula. For concentrated susensions, numbers of equation are available in literature, but any of these equations is not suitable for all tyes of susensions. Different equations are suitable for different tye of susensions. No such relationshi for magnetite ore susensions was found in the literature. Hence there is good scoe studying rheological behaviour of the magnetite ore susensions, to investigate the effects of article size and solids concentration on the susensions rheology and to find out a suitable equation to describe the flow curves. It would be worth to know the effect of article size on relative viscosity of magnetite ore susensions and to find out a suitable better fitting equation for the deendence of relative viscosity on the solids concentration in the susension. Further there is good 30

24 otential to develo artificial neural network models to describe the flow curves for magnetite ore susensions and the relative viscosity-solids volume fraction relationshis for the same. References 1. Rushton, A. S. Ward and R. G. Holdich, Solid-Liquid Filtration and Searation Technology, Wiley-VCH Verlag GmbH, Weinheim, Germany, (1996) 2. J F Richardson and W N Zaki., The Sedimentation of a Susension of Uniform Sheres under Condition of Viscous Flow, Chemical Engineering Science, 3, (1954), S Mirza and J F Richardson, Sedimentation of Susensions of Particles of Two or More Sizes, Chemical Engineering Science, 34, (1979), E. Barnea, J. Mizrahi, Generalized Aroach to the Fluid Dynamics of Particulate Systems Part 1: General Correlation for Fluidization and Sedimentation in Solid Multiarticle Systems, The Chemical Engineering Journal, 5, (1973), E. Caes, Particle Agglomeration and the Value of the Exonent n in the Richardson-Zaki Equation, Powder Technology, 10, (1074), R. M. Turian, T. W. Ma, F. L. G. Hsu and D. J. Sung, Characterization, settling and rheology of concentrated fine articulate mineral slurries Powder Technology, 93, (1997), Sangkyun Koo, Estimation of hindered settling velocity of susensions, Journal of Industrial and Engineering Chemistry, 15, (2009), Garside and M.R. Al-Dibouni, Velocity-voidage relationshis for fluidization and sedimentation in solid-liquid systems, Industrial and Engineering Chemistry Process Design and Develoment, 16, (1977), Nian-Sheng Cheng, Comarison of formulas for drag coefficient and settling velocity of sherical articles, Powder Technology, 189, (2009), T.E. Baldock,, M.R. Tomkinsa, P. Nielsena and M.G. Hughes, Settling velocity of sediments at high concentrations, Coastal Engineering, 51, (2004), M.S. Selim, A.C. Kothari and R.M. Turian, Sedimentation of multi-sized articles in concentrated susensions, A.I.Ch.E.Journal, 29, (1983),

25 12. M. Lali, A. S. Khare and J. B. Joshi, Behaviour of Solid Particles in Viscous Non-Newtonian Solutions: Settling Velocity, Wall Effects and Bed Exansion in Solid-Liquid Fluidized Beds, Powder Technology, 57, (1989), Di Felice, R. and Parodi, E., Wall effects on the sedimentation velocity of susensions in viscous flow, AIChE Journal, 42, (1996), Alfred W Francis, Wall Effects in Falling Ball Method for Viscosity, Journal of Alied Physics, 4, (1933), Clift R, Grace J. R. and Weber M. E., Bubbles, Dros and Particles, Academic Press New York (1978) 16. R Di Felice, A Relationshi for the Wall Effect on the Settling Velocity of a Shere at Any Flow Regime, International Journal of Multihase Flow, 22 (3), (1996), Edward J. Was, John P Kenny and Ramesh Gandhi, Solid-liquid Flow Slurry Pieline Transortation-Bulk Materials Handling, Trans Tech Publications Cranfield, UK (1977) 18. Sumer M. Peker Serife S. Helvaci, Solid Liquid Two Phase Flow, Elsevier Amsterdam, The Netherlands (2008) 19. M. Gilbert, L. Davis and D. Altman, Velocity lag of articles in linearly accelerated combustion gases, Jet Proulsion 25, (1955), R. Turton and 0. Levensiel, A Short Note on the Drag Correlation for Sheres, Powder Technology, 47, (1986), A.R. Khan and J.F. Richardson, The Resistance to Motion of a Solid Shere in a Fluid, Chemical Engineering Communications, 62(1 6), (1987) 22. Haider and 0. Levensiel, Drag Coefficient and Terminal Velocity of Sherical and Nonsherical Particles Powder Technology, 58, (1989), J. Zigrang and N. D. Sylvester, An exlicit equation for article settling velocities in solid-liquid systems, AIChE Journal, 27(6), (1981), R. Turton and N. N. Clark, An exlicit relationshi to redict sherical article terminal velocity, Powder Technology, 53, (1987), R. H. Perry and C. H. Chilton, Chemical engineers' handbook, McGraw- Hill, New York, USA (1973) 26. R.L.C. Flemmer and C.L. Banks, On the drag coefficient of a shere, Powder Technology, 48(3), (1986),

26 27. G. Saha, N.K. Purohit and A.K. Mitra, Sherical article terminal settling velocity and drag in Bingham liquids, International Journal of Mineral Processing, 36 (3-4), (1992), F. Concha and E. R. Almendra, Settling velocities of articulate systems, 1. Settling velocities of individual sherical articles, International Journal of Mineral Processing, 5 (4), (1979), G. Venu Madhav and R. P. Chhabra, Drag on non-sherical articles in viscous fluids, International Journal of Mineral Processing, 43 (1-2), (1995), P.P. Brown and D.F. Lawler, Shere drag and settling velocity revisited, Journal of Environmental Engineering, 129 (3), (2003), Harris J., Rheology and non-newtonian flow, Longman, New York, USA (1977) 32. Bird, R.B., Stewart, W.E. and Lightfoot, E.N., Transort Phenomena, John Wiley & Sons, New York, USA (2001) 33. Malcom M Cross, Rheology of non-newtonian fluids: a new flow equation for seudolastic systems, Journal of Colloid Science, 20, (1965), , 34. Malcom M Cross, Rheology of viscoelastic fluids: elasticity determination from tangential stress management, Journal of Colloid and Interface Science, 21, (1968), Mewis J and Saull A J B, Rheology of concentrated disersions, Advances in Colloid l Interface Science, 6, (1976), Carreau P. J., Rheological equations from molecular network theories, Transactions of Society of Rheology, 16(1), (1972), Carreau P. J. and De Kee D., Review of some useful rheological equations, Canadian Journal of Chemical Engineering, 57, (1979a), Carreau P. J. and De Kee D., Analysis of viscous behaviour of olymeric solutions, Canadian Journal of Chemical Engineering, 57, (1979b), Howard A Barnes, A Handbook of Elementary Rheology, Cambrian Printers, Llanbadarn Road, Aberystwyth, UK (2000) 40. Q.D. Nguyen and D.V. Boger, Alication of rheology to solving tailings disosal roblems, International Journal of Mineral Processing, 54, (1998), S K Nicol and R J Hunter, Some Rheological and electrokinetic roerties of kaolinite susensions, Australian Journal of Chemistry, 23(11), (1970), 33

27 Cheng, N. S., and Law, A. W. K., Exonential formula for comuting effective viscosity, Powder Technology, 129(1-3), (2003), Thomas D. G. "Transort characteristics of susension: VIII. A note on the viscosity of Newtonian susensions of uniform sherical articles." Journal of Colloid Science., 20, (1965), G. S. Khodakov, On susension rheology, Theoretical Foundations of Chemical Engineering, 38(4), (2004), Batchelor, G. K., and Green, J. T., The determination of the bulk stress in a susension of sherical articles to order c 2, Journal of Fluid Mechanics., 56, (1972), Kim, S., and Karrila, S. J., Microhydrodynamics: rinciles and selected alications, Butterworth-Heinemann, Boston, USA (1991) 47. V Ilic and N Phan-Thien, Viscosity of concentrated susensions of sheres, Rheological acta., 33, (1994), Thomas, C. U., and Muthukumar, M., Convergence of multile scattering series for two-body hydrodynamic effects on shear viscosity of susensions of sheres, Journal of Chemical Physics, 94(6), (1991), T. B. Lewis and L. E. Nielsen, Viscosity of disersed and aggregated Susensions of sheres, Journal of Rheology, 12, (1968), Bruce A. Firth and Robert J. Hunter, Flow roerties of coagulated colloidal susensions : I. Energy dissiation in the flow units, Journal of Colloid and Interface Science, 57(2), (1976), Rutgers, J. Relative viscosity of susensions of rigid sheres in Newtonian Liqiud, Reologica Acta, 2, (1962), David Leightont and Andreas Acrivos, Viscous resusension, Chemical Engineering Science, 41(6), (1986), Barnes, H. A., Shear-Thickening (``Dilatancy'') in susensions of nonaggregating solid articles disersed in Newtonian liquids, Journal of Rheology, 33(2), (1989), Chong, J.S., Christiansen, E.B. & Baer, A.D., Rheology of concentrated susensions, Journal of Alied Polymer Science, 15, (1971), Mooney, M, The viscosity of concentrated susension of sherical articles, Journal of Colloid Science, 6, (1951), Irvin M. Krieger and Thomas J. Dougherty, A mechanism for non- 34