# Experimental Results and Models for SolidLiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids.

код для вставкиСкачатьDev. Chem. Eng. Mineral Process. 12(3/4), pp. 403-426, 2004. Experimental Results and Models for Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids M. Aghajad3,H. Muller-Steinhagen2*and M. Jamialahmadi3 I Dept of Chemical and Process Engineering, University of Surrey, Guildford, Surrey, UK Institute for Thermodynamics and Thermal Engineering, University of Stuttgart, Stuttgart, Germany Petroleum University of Technology, Ahwaz, Iran Fluidization technology relies almost solely on fluid/particle interaction wherein the liquid phase may exhibit Newtonian or non-Newtonian behavior. The steady motion of particles and the velocity-voidage relationship are the most important design parameters f o r fluidization, providing the basis for the prediction of heat and mass transfer coeflcients and information on hydrodynamic conditions. A summary of the literature on particle settling velocity, minimum fluidization velocity and velocityvoidage relationship is supplemented by new experimental results, which extend the range of investigated solid and liquid phase physical properties. Correlations for particle settling velocity and velocity-voidage relationship are developed and ver$ed against experimental data. Introduction Solidlliquid fluidized beds are used throughout the process industries for catalytic cracking, hydrometallurgical operations, crystallization and sedimentation. In processes where severe fouling of the heat transfer surfaces is expected, installation of a fluidized bed heat exchanger where the solid phase is cylindrical stainless steel or tantalum particles is also recommended. Over the years, fluidized beds have been the subject of intensive research, covering both fundamental and applied aspects. As a result, considerable progress has been made towards understanding the detailed hydrodynamics as well as in the development of sound methodologies for the design of fluidized bed systems. Most of these research efforts have been focused on low viscosity liquids with Newtonian behavior, in which the solid phase consists of spherical glass particles. Previous investigations on the hydrodynamic behavior of these systems have been documented and discussed, most recently by Jamialahmadi and Muller-Steinhagen [ 11. * Author for correspondence (hms@itw.uni-stuttgart.de). 403 M.Aghajani, H. Muller-Steinhagen and M.Jamialahmadi In recent years, solidliquid fluidized beds are also finding increasing application in the treatment of aqueous wastes, heavy oil cracking, polymerization, biological oxidation and fermentation. Here, the liquid phase is viscous with non-Newtonian behavior. Only limited information is available on the hydrodynamic and thermal performance of fluidized beds with non-Newtonian liquids. Fluidized beds involving high viscosity liquids and non-spherical heavy particles have been the subject of even less investigations. It is generally agreed that the terminal settling velocity of the particles, the minimum fluidization velocity and its corresponding bed voidage and the velocityvoidage relationship are the most important parameters of fluidization technology. Considerable empirical progress has been made on the understanding of these parameters and in establishing the velocity-voidage relationship in fluidized beds involving low viscosity Newtonian liquids. Correlations have been proposed, which may be found in several papers, e.g. Davidson et al. [2], and in a book recently published by Chhabra [3]. For non-Newtonian liquids, no such information is available to design engineers. For t h s reason, more detailed experimental investigation and analyses of fluidized beds are necessary. The aim of the present investigation was, therefore, to measure the hydrodynamic parameters over wide ranges of particle size, density and shape using liquids with Newtonian and nonNewtonian behavior. The predictions of various published correlations are compared with these experimental data, and new correlations and methodologies are presented for prediction of the terminal settling velocity of the particles, the minimum fluidization velocity and its corresponding voidage, and the velocity-voidage relationship in beds which are fluidized with Newtonian and non-Newtonian liquids. Experimental Equipment and Procedure Particle settling velocity apparatus Experiments to determine terminal falling velocities of different spherical and cylindrical particles are performed in vertical columns of different diameter ranging fiom 2.5 to 10 cm, and variable height (see Figure 1). All tests were performed in a random mode according to the following procedure. Initially, the columns were cleaned and the test liquid introduced. The system was then left for about 12 hours to allow trapped air bubbles to escape and homogenous conditions to be established throughout the system. The test particles were soaked in the test liquid for about 6 hours before they were introduced below the surface of the liquid, as close to the center of the column as possible. The terminal settling velocity of each particle was measured by timing its descent over a pre-marked distance using an electronic timer and a video camera. Each terminal settling velocity represents an average of ten repeat measurements. In the case of distilled water which has a low viscosity, the particle Reynolds number of heavy particles was above 1000, and hence sinking occurred very quickly and measurements were difficult and subject to error. To minimize the error, in this case each value of terminal falling velocity represents an average of at least twenty repeat measurements. Some measurements were repeated later to check the reproducibility of the experiments, which proved to be excellent. 404 SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Figure 1. Schematic diagram of experimentalfluidized bed system. Fluidized bed system A schematic diagram of the fluidized bed system is shown in Figure 1. The setup consists of fluidized bed column, pump, supply tank, liquid flow meter, control valves, AP cell and U-tube manometer with mercury under water as the manometric liquid. The column consists of calming, test and expansion sections. A stainless steel screen fitted above the calming section of the bed supports the solid particles. The liquid passing through the column and the return lines is discharged into the supply tank. The liquid temperature between the inlet and outlet of the test section was measured for determination of the liquid viscosity and density. A personal computer was used for data acquisition. Various types of spherical and cylindrical particles were used as the solid phase in this investigation. The physical properties of the particles are listed in Table 1. The minimum liquid fluidization velocity was measured from the intersection of pressure drop-velocity plots in fixed and fluidized bed regimes. For measurement of liquid velocity at very low volumetric flow rates, liquid from the column was passed through a graduated cylindrical column before returning to the supply tank. No significant difference was found when the minimum liquid fluidization velocity was determined with increasing or decreasing liquid velocity. Bed voidage is defined as: 405 M. Aghajani, H. Muller-Steinhagen and M. famialahmadi For a given mass of particles, therefore, only the bed height has to be measured. To start the experiment, about 0.3 kg of particles are fed into the column and the minimum height of the bed is recorded as the static bed height. Liquid is admitted into the system and the average bed expansion height is recorded up to the limit when the particles are at the point of being carried away from the system. Expanded bed height data were obtained using both increasing and decreasing liquid flow rates to investigate the possibility of a hysteresis effect. Static bed voidages were also measured separately for each particle type, in a column of the same diameter but of smaller height, by measuring the volume of water required to fill the void space. The results of these measurements are also included in Table 1. - Table 1. Physical properties of the solid particles. Cylindrical Spherical /Aluminum Aluminum Brass 1I Stainless Steel Tantalum Glass Glass Glass Lead Lead Carbon Steel Carbon Steel Stainless Steel 2x3 mm 3x3mm :;: I I PP [kg 1 m31 2.62 3.43 4 3 3.7 2600 2600 8500 7900 7900 17600 2700 2700 2700 11350 11350 7800 7800 8100 Y ESB [-I [-I 0.86 0.87 0.87 0.87 0.87 0.87 0.40 0.4 1 0.41 0.4 1 0.40 0.41 6 9 9 9 4 16 1 1 1 1 1 1 1 1 0.39 0.39 0.40 0.39 0.40 0.40 0.39 0.40 3.14 7.1 12.6 6.6 12.6 12.6 7.1 10.7 A* [m21 -- - I I Test liquids In order to cover a wide range of particle Reynolds numbers, a series of aqueous solutions of sugar and carboxymethylcellulose (CMC) were used as Newtonian and non-Newtonian liquids. The concentration of sugar and CMC was varied from 0 to 60 wt% and 0 to 1 wt% respectively. Rheology of these solutions was determined on a Carrimed 50 controlled stress rheometer equipped with a cone and plate assembly. The temperature was controlled by a Peltier element situated in the plate. As expected, the sugar solutions exhibited a constant shear viscosity whereas the CMC solutions displayed varying levels of pseudoplastic behavior. An examination of the steady shear stress-shear rate data suggested that the two-parameter power law fluid 406 SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids model provides an adequate representation of their pseudoplastic behavior. For steady shear, the power law is written as: r = kZ;" (2) where the best values of k and n were estimated using a nonlinear regression approach. The resulting values, with the density of each solution, are given in Table 2. It has been assumed that the average shear rate over the entire particle surface is d d P .With this definition, the apparent viscosity is given by: \"PI Table 2. Physical properties of Newtonian and non-Newtonian solutions. CMC solutions (25°C) Sugar solutions (25°C) wt% 20 40 60 PI Power Law Model : r = k y" [ kg/m3 ] P [ Pa. s ] wt% 1070 1150 1300 0.001714 0.005359 0.044410 0.2 0.4 0.6 0.8 1 PI [kg/m3] 998.3 998.3 998.3 998.3 998.3 k [Pa.s"] 0.0697 0.2084 0.3413 0.5756 2.538 n [-] 0.7468 0.6953 0.6882 0.6729 0.5519 i Results and Discussion Particle terminal velocity corrected for wall effect Particle settling velocity is essential for the prediction of bed voidage, and heat and mass transfer coefficients. The substantial number of investigations on this topic reflects the importance of particle settling velocity in solid-liquid systems (e.g. Lali et al. [4]; Chhabra [5]). The previous findings, particularly with respect to the influence of particle shape, density and the behavior of the liquid phase on the terminal velocity, are often of a contradictory nature. To clarify these uncertainties, extensive new experimental results are presented in t h s work for regions where the published data are insufficient. It is well known that the walls of the column exert an additional retardation effect on a settling solid particle. The extent of this wall effect is usually quantified by introducing a wall factor defined as: 407 M. Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi Terminal settling velocity in the presence of wall effect =-u, = Terminal settling velocity in the abscence of wall effect u, (4) It is obvious that f has a positive value less than 1. Figure 2 display representative results illustrating the relationshp between the measured terminal velocity and the settling column diameter for 3x3 mm aluminum particles in solutions with various physical properties. In all cases, the variation is almost linear. Therefore, it can easily be extrapolated to Dh+ m to estimate the corresponding terminal fall velocity of the particles in the absence of any wall effect. 0.35 1I c 1 - 0.3 1 Aluminum particle 1 d, = 3.43 mm v) 2 0.25 .-d ............ ?*.. U -=> 0 0.2 .- (0 0 € 0 0.15 c) 0 U .- 5 0.1 ................ ............ ................ ................ L 0.05 0 10 20 30 40 50 l/Dh [l/ml Figure 2. Variation of measured settling velocity of an aluminum particle with column diameter in various solutions. The extracted wall effect results are plotted as a f i c t i o n of d,,/Dh in Figure 3. The results conform (with an absolute mean average error of less than 6%) to the following equation proposed by Richardson and Zaki [6]: );( ; Log,, - =- 408 SolidILiquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids 1-00 - Richardson and Zaki [6] 0.60 1 0.00 0.05 1 1 1 1 1 1 1 1 1 0.10 1 0.15 1 1 1 1 0.20 Diameter ratio, dJDh Figure 3. Variation ofthe wall factor with diameter ratio. Correlation of particle free fall velocity data The forces acting on a particle moving with its terminal velocity through a fluid are in dynamic equilibrium Namely, the effective weight (the gravitational force minus buoyancy force) is equal to the drag force. Thus for the free terminal velocity yields: Equation (6) may be written in terms of fiee fall particle Reynolds number: Equation (6) shows that the terminal velocity of a particle is inversely proportional to the drag coefficient, Co. Theoretically, the drag coefficient can be obtained from the solution of the equation of momentum for the system. in the absence of the inertial terms it yields: c, =- 24 Re p m 409 M.Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi As the particle Reynolds number increases, the inertial terms become increasingly significant in the momentum equation and no analfical solutions are possible under these conditions. Therefore, almost all drag coefficients reported for higher Reynolds number have been obtained from experiments. These results are generally presented in graphical form as a complex function of the flow conditions. Most of this work has been reviewed and critically evaluated by several investigators (e.g. Clift et al. [7]; Khan and Richardson [8]). Clift et al. [7] and Lydersen [9] fitted these curves to a series of straight lines for calculating the value of CD for a given value of Re, embracing the complete standard drag curve. Description of these equations and conditions for which their application has been recommended are summarized in Table 3. For a given value of Reynolds number, calculation of CD using one of these correlations is a straightforward matter. Unfortunately, the form of these correlations is not convenient for the calculation of the free settling velocity for a given solidliquid system as the unknown velocity appears in both Repmand CD. This difficulty is overcome by writing Equation (7) in terms of the Arclumedes number, which is not a finction of the terminal velocity: 3 gP, (Pp - PI ) d ; A r = - C D R e 2Pm = 4 P2 (9) Equation (9) shows that the particle free fall Reynolds number is a function of Archimedes number only, and can be better presented in the form: Re, = F (Ar) (10) Several attempts have been made to establish this functionality between the Archimedes number and the particle Reynolds number. Most of the work is documented and critically evaluated by Khan and Richardson [8]. These authors proposed the following correlation based on a large body of experimental data extracted from the literature: Hartman et al. [15] proposed the following explicit relation for the prediction of the free fall velocity of a spherical particle in an infinite medium, which avoids the iterative solution of Equation (7) and the equations for CD: .&I, Repa = P(C)+log,o m where P(C)= ((0.0017795C - 0.0573)C + 1.0315)C - 1.26222 R(C)= 0.99947+ 0.01853sin(1.848C -3.14) and C = Log,,Ar 410 (12) (13) (14) Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Table 3. Recommended correlations for drag coeflcients. Re, < 0.01 C , = 2(I+ 0. I3 I5Ref 82-o 0.01 <Re,520 Re, C, = "(I +0.1935Re; ,OS) 20 5 Re, 5 260 Re, log c, = 1.6435- 1.12420+0.155802 IogC, = -2.4571+2.55580-0.92950~ 260 < Re, +0.1O49w3 < 1500 1500<Re,< I.2x104 IOgCD = -1.91 81 + 0.6370- 0.0636W2 1 . 2 ~ 1<Re$ 0~ 4.4~10~ log c, = -4.339 4 . 4 ~ 1 <Reps 0 ~ 3.38~10' + I ,5809W - 0. I 546 O 2 CD = 29.78- 5.30 3 . 3 8 1~0' <Re, 5 4x 10' CD = 0.10 - 0.49 4x10' <Re,< lo6 C, =0.19-- 8x104 lo6< Re, Re, Chhabra [ 131 R ~ ,= a Arb For non-Newtonian liquid; n = Rate index of power law model b = -0'954 0.16 Table 4 shows the percentage of absolute error, from a comparison between measured and calculated particle settling velocities, for the most frequently recommended correlations and models for Newtonian and non-Newtonian solutions. The best agreement for Newtonian solutions is obtained with the method suggested by Hartman et al. [ 141, followed by the standard equation of Lydersen [9]. 41 I M. Aghajani, H. Miiller-Steinhagen and M. Jamialahmadi Table 4. Absolute relative error (96) of correlations for the prediction of urn. Khan and Richardson [8] Few attempts have been made to establish the functional dependence of Archimedes number on particle Reynolds number for non-Newtonian solutions. Chhabra [13]presented a model for non-Newtonian fluids in terms of flow behavior index, which is also listed in Table 3. The best agreement for non-Newtonian fluids is obtained with the correlation of Khan and Richardson [8], as shown in Table 4. However, the variation between the predictions of the various correlations is quite considerable. Improved values of the constants used in Equation (1 1) are determined by non-linear regression analysis using all available data: Re,, = 0.334Ar0.654 (15) For non-Newtonian solutions, the apparent viscosity (pa) must be used in both Rep, and Ar. Therefore, Equation (15) is implicit with respect to the fiee particle terminal settling velocity (u,) and must be solved in parallel with Equation (3). Equation (15 ) predicts the fiee falling velocity of particles in non-Newtonian solutions with an absolute mean average error of less than 10%. Minimum fruidization vefocity The incipient or minimum fluidization point represents the transition between the fixed and fluidized states. It is readily recognized that the minimum fluidization velocity is one of the main design variables in such applications. Several investigators have studied this situation and developed correlations that can be used for Newtonian fluids. Most of these correlations have been compiled and critically evaluated by Couderc [ 151 who concluded that the minimum fluidization velocity for Newtonian liquids can be predicted with an average accuracy of about 15 to 20%, and larger errors may be encountered for non-spherical particles. In contrast, very little information is available on fluidized bed systems involving non-Newtonian fluids. The limited amount of work which is available in this area has been documented by Chhabra [3]. 412 Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids In this present study, the minimum fluidization velocity (ud) was obtained by plotting pressure drop gradient versus liquid phase Reynolds number for fixed and fluidized regimes. The transition point is designated as the minimum fluidization velocity, and the corresponding bed voidage is denoted by E&. Thus for Rep-Red, the pressure drop across the bed remains constant as illustrated in Figure 4 for typical experiments with cylindrical particles. All other results obtained in this study conform to this behavior. I00 -E . a" 25 90 80 70 Y a E 60 20 10 0 0 200 400 600 800 1000 Particle Reynolds number, Re, Figure 4. Typical variation ofpressure drop with particle Reynolds number for cylindrical particles. Most attempts to develop models for the estimation of minimum fluidization velocity are based on the fact that at the point of incipient fluidization, the pressure drop for a fixed bed is equal to the apparent buoyancy weight of solid particles. The most widely used correlation for fixed beds is that of Ergun [ 161: M.Aghajani, H. Muller-Steinhagenand M.Jarnialahrnadi At incipient fluidization, this pressure drop is equal to: AP -=(I - Em/ 1( P , - P I )g L Combining Equations (1 6 ) and (1 7) yields: 1.75 -Re$+ 3 150( 1- zm/ ) YEmf v/ 2 3 Em/ Remf- Ar = 0 Equation (1 8) is a dimensionless equation for the minimum fluidization velocity. The main problem with the solution of this equation is that ~dis unknown. Based on experimental observations, Wen and Yu [ 171 suggested that: 1 = 14 3 Y Em/ - and Y ' 2 = 11 3 Em/ (19) With these approximations the solution of Equation (18) yields: Re,/ = /(33.7)* + 0.0408Ar - 33.7 (20) Comparison of all experimental data for Newtonian fluids with Reynolds numbers calculated according to Equation (20) shows an absolute mean average error of about 7%. This is a good indication that Equation (20) is suitable for the prediction of the minimum fluidization velocity of solidliquid fluidized beds with Newtonian behavior. Chhabra [3]compiled all the available correlations for the prediction of u d for systems with a non-Newtonian nature. Table 5 shows a comparison between measured and calculated minimum fluidization velocities for these correlations. Evidently, none of these correlations seems to predict the experimental data for nonNewtonian solutions satisfactorily. The present investigation shows that Equation (20) predicts the experimental data for non-Newtonian solutions with an absolute mean average error of lo%, if the apparent viscosity is used in the Red and Ar numbers. Table 5. Performance of correlationsfor predicting minimum fluidization velocity. Kumar and Upadhyay [2 11 Kawase and Ulbrecht [22] 414 Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Velocitpvoidage relationship Theoretical and empirical correlations available for the prediction of design parameters, such as heat and mass transfer coefficients, are strong functions of the bed voidage. Therefore, accurate knowledge of t h s relationship is crucial for the reliable estimation of transfer coefficients. Considerable progress has been made in establishing the velocity-voidage relationship in fluidized bed systems, and several correlations have been proposed for its prediction. Jamialahmadi and MiillerSteinhagen [25] compiled the published correlations and conditions for which their application has been recommended. Most of these correlations are empirical and apply only over a restricted range of Reynolds number, for specific particles, or for Newtonian fluids. Furthermore, the prediction of bed voidage requires the use of iterative solutions for most of these correlations. Development of a new bed-voidage correlation When a liquid flows upwards through a bed of particles at low velocity (u < ud), a fixed bed exists where the solid particles rest on top of each other and on the bottom of the column. In this regime, the height and, therefore, the voidage of the bed remains constant at fixed bed voidage while the pressure drop increases with liquid velocity. If the velocity of the fluid is sufficiently high (u > ud), the solid particles will be freely supported in the liquid to create a fluidized bed. In fluidized beds, the height of the bed increases while the pressure drop balances the buoyant weight of the fluidized particles and remains constant. Thus it can be written: ) = cons tan t W ( U , E - gSE (21) Differentiating Equation (2 1) gives: aAP dAP = -du au aAP + a(E - d(& - E S B ) = 0 Pressure drop is a function of bed voidage and liquid velocity and can be expressed as: The velocity dependency may be justified on theoretical grounds with the exponent x talung values of 1 and 2 for laminar conditions where Stokes' Law is applicable and for turbulent conditions where Newton's law is valid, respectively. On the other hand, the voidage dependency is chosen quite arbitrarily. The partial derivatives in Equation (22) can be evaluated from Equation (23). Differentiating Equation (23) partially with respect to velocity and keeping the bed voidage constant produces: 415 M. Aghajani, H. Miiller-Steinhagenand M. Jamialahrnadi Similarly: Substituting the values given by Equations (24) and (25) into Equation (22), and after some algebraic manipulation and rearrangement, results in: Equation (26) is linear and subject to the condition that, at u = u,, the bed voidage is unity. Its solution in the range of umf S u Iu, yields: where the exponent z is known as the fluidization index, and is equal to: 7 Y = -X Rearranging Equation (27) gives the following explicit expression for the bed voidage: It is well known that the hydrodynamic behavior of fluidized beds is different and independent from that of the corresponding fixed bed. However, this is only correct when fluidization is fully developed and the superficial liquid velocity is considerably larger than the minimum fluidization velocity. For low superficial velocity close to the minimum fluidization velocity (u,f), the hydrodynamic behavior of fluidized beds is close to that of a static bed. The results of this investigation show that there is a smooth transition between fixed and fluidized beds. This is confirmed by several other investigators, e.g. Foscolo et al. [26] who stated that all the empirical evidence indicates a smooth transition from fixed to fluidized bed behavior. Therefore, in the range of low superficial velocities close to the minimum fluidization velocity, the fluidized bed voidage in Equation (29) approaches the constant static bed voidage. Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Prediction of static bed voidage (ssB) The static bed voidage (ESB ) in Equation (29) can be calculated from the correlation of Fan and Thinakaran [27]: €SB = 0'15' + 0.365 ; Dh 2 2.033 dP Equation (30) generally under-predicts the static bed voidage. Considering that in solidtliquid fluidization the diameter of particles is relatively large (e.g. d, >2 mm), the interaction forces between particles, and between particles and walls, are not taken into account in the following equations for predicting static bed voidage. At the expense of a slight loss in accuracy this is a reasonable assumption. Improved values of the constants used in Equation (30) are determined for spherical and cylindrical particles by regression using all available data. For spherical particles, the static bed voidage is obtained from: and for cylindrical particles: 0.15 Equations (31) and (32) predict the static bed voidage for spherical and cylindrical particles with an absolute mean average error of less than 3%. Prediction of thefluidization index (zj When the particles are fluidized with non-Newtonian CMC solutions, the behavior of the bed was similar to that observed for Newtonian solutions. Figure 5 shows typical plots of Equation (27) for various particles fluidized with Newtonian and nonNewtonian solutions. The results obtained with the remaining solidliquid combinations also conform to this behavior, and no noticeable differences were observed. For each experimental run, the value of z has been determined by performing regression analysis on velocitylvoidage data. Examination of the calculated values suggests that: z=p($, Re,,) (33) 417 M. Aghajani, H.Miiller-Steinhagen and M. Jamialahmadi 1 0.9 0.8 0.7 0.6 -A 5 0.5 0.4 \ 1” 0.3 0.2 0.1 0.2 0.1 0.3 0.4 (E-ESB)/(~-ESB) 0.5 0.6 0.7 0.8 0.9 1 1-1 Figure 5. Typical plot of Equation (27) for Newtonian and non-Newtonian solutions. The effect of the wall is considered when the terminal velocity is corrected for wall effect using Equation ( 5 ) from Richardson and Zaki [ 6 ] . Hence, Equation (33) reduces to: z = F(Re,) (34) The calculated fluidization index (z) for various particle sizes, types and shapes, and fluids with Newtonian and non-Newtonian nature, plus all the z values which are recalculated from the reported experimental data by Richardson and Zaki [ 6 ] , are plotted as a function of particle terminal Reynolds number (Re,, ) in Figure 6. The general shape of the curve is similar to that observed for the variation of the Richardson and Zaki exponent, n, with Re,,. At low and high particle terminal Reynolds numbers (Re,, < 0.2 and Re,, > 500), the fluidization index (z) is almost independent of particle Reynolds number. Between these boundaries, the value of z decreases graduaIly as Re,, increases. This functionality can be well represented by the following equation: 0.65(2 + 0.5Rei:’) Z= 418 (1 + 0.5 Re’:; ) (35) Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Equation (35) is independent of the nature of the fluids and it can equally well be used for Newtonian and non-Newtonian solutions. The nature of the solutions is taken into consideration by using the apparent viscosity in the calculation of parameters such as the Archimedes number and the particle Reynolds number. 1.6 1.4 4 0 - I .2 L N J P) 1 ‘El .- ‘C 0 0.8 cp 0.6 E 0.4 : Newtonian solution 0: non - Newtonian solution 0.2 PI = 998-1 300 Kdm’ p = 0.001- 0.05 Pa. s Solid physical properties p, = 998 Kg/m’ pp= 2600 - 17600 Kg/m’ p = 0.01 - 0.55 pa. d,= 1 - 5 mm 0 0.0001 0.001 0.01 0.1 1 10 Particle Reynolds number, Rep, 100 1000 10000 [-I Figure 6. Variation ofjluidization index with particle terminal Reynolds number for Newtonian and non-Newtonian solutions. Comparison with experimental data The predictions from Equations (29) and (35) for 3x3 mm brass particles and fluids with Newtonian and non-Newtonian behavior are shown in Figures 7a and 7b, respectively. The calculated trends are in excellent agreement with the experimental results of all investigators. Furthermore, for low liquid velocity, Equation (29) approaches the static bed voidage (ESB ) as a limiting condition. The applicability of the presented model for Newtonian and non-Newtonian (shear-thinning power law) fluids is demonstrated in Figures 8 and 9, where the experimental data of different investigators and also the data measured in this investigation are compared with those predicted from Equation (29). Table 6 shows the content of the database used for this comparison. 419 M.Aghajani, H. Muller-Steinhagen and M.Jamialahmadi 1 0.9 -- 0.8 6 0.7 I 0 M I e0 > 2 0.6 0.5 G 0.3 0 0.1 0.2 0.3 0.5 0.4 0.6 0.7 0.8 Superficial liquid velocity, us Ids] Figure 7a. Comparison of measured and predicted bed voidages for Newtonian solutions. 1 A 9/ / 0.9 -- 0.8 w 6 0.7 4 = 3.43 mm I pp = 8500 Kg/m' M .- U 0 r z 0.6 . Solutions Experiment Predicted l.Owt%CMC ...._......... 0.8wt% CMC A -..0.6wPhCMC A 0.4 wt%CMC 0 0.5 (j n?wP/nrMr - ------ 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 Superficial liquid velocity, us Im/s] Figure 76. Comparison of measured and predicted bed voidages for non-Newtonian solutions. 420 Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids 0.4 0.5 0.6 0.7 E ( experimental 0.8 0.9 1 ) Figure 8. Comparison of measured bed voidages with values calculated from Equation (29). 0.4 0.5 0.6 0.17 0.8 0.9 1 E ( experimental ) Figure 9. Comparison of measured bed voidages with values calculated from Equation (29)for Newtonian and non-Newtonian solutions. 421 M. Aghajani, H. Muller-Steinhagen and M. Jamialahmadi Table 6. Summaly of database usedfor comparison with the present model. Liquid Pure water Pure water Richardson and "aki L 1 30 1 0.5,1,0.3,6.4 1 Pure water 2745,1060,7740 Pure water Pure water Loeffler and Ruth Pure water ~ Pure water Pure water Garside and Al- Pure water Pure water Pure water Jamialahmadi et Pure water Present work Present work 464 I 422 same as Table 1 1 same as Table 1 same as Table 1 1 sameas Table I Pure water and Sugar solutions same as Table2 CMC solutions same as Table 2 Ten of the most frequently recommended correlations from the literature have been compared with the experimental data. The results of thls comparison are summarized in Table 7 in terms of average relative error and the standard deviation of prediction. The model developed in the present investigation clearly out-performs all other correlations. This table also indicates whether correlations tend to under-predict "- " or over-predict the measurements. Correlations with "- - " or "+ + '' have a high tendency to under-predict or over-predict the measurements, and for correlations with Y " no clear tendency was found. I'+" 422 Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids Table 7. Average relative error of values predicted by published models as compared to experimental data. Newtonian liquids Average Standard relative deviation rrror(%) (%) Prediction - Non -Newtonian liquids Average relative 'rror (%, itandam reviatioi 3redictio (%) 3 e 19.98 5.61 f 15.12 1 I .87 9.51 4.75 f 16.81 25.79 8.22 -- 6.52 1.50 11.16 8.74 - All liquids Average Standard relative deviation ?rror(%) (%) 17.56 5.08 13.45 13.08 5.46 15.14 12.96 20.61 9.57 f 18.21 14.52 12.79 8.36 -- 8.61 10.45 10.89 5.62 13.48 12.44 - 50.93 55.74 ++ 34.83 36.93 12.66 9.57 f 10.36 8.21 9.59 7.47 f 9.90 18.09 15.09 f 6.95 4.44 f f f t 1 6.64 - -423 M. Aghajani, H. Muller-Steinhagen and M. Jamialahmadi Bed voidage at minimum fluidization velocity Bed voidage at minimum fluidization velocity is one of the important factors to be considered in the design of fluidized bed systems. The main problem with the prediction of minimum fluidization velocity is that Equation (1 8) is highly sensitive to the value of ~pllrandthe shape factor, w. For spheres, Wen and Yu [33] selected a value for ~pllrof 0.42, and Bamea and Mednick [43] used a value of 0.415. Theoretically it should be possible to determine the bed voidage at minimum fluidization velocity by using one of the velocity-voidage relationships for fluidized bed systems. Equation (29) is general and includes the effect of various operational and geometrical parameters, and the nature of the fluids, on the velocity-voidage relationship. Bed voidage at minimum fluidization velocity predicted from Equation (29) for various solidliquid combinations is: The minimum fluidization velocity ( u d ) can be calculated from Equation (20), and E~~ from Equations (31) and (32) for spherical and cylindrical particles, respectively. It is worthwhile to note that Equation (18) can be used as a cross-check because the predicted upllr and E& should conform to this equality. The prediction of Equation (36) is verified against experimental data for various particle sizes, types and shapes, and for fluids with different natures. The absolute mean average errors of 6% illustrate the excellent applicability of this model for the prediction of bed voidage at minimum fluidization velocity. Conclusions An experimental and theoretical investigation of particle settling velocity and velocity-voidage relationship in solid/liquid fluidized bed systems has been undertaken using distilled water and sugar solutions as Newtonian fluids, and CMC solution as non-Newtonian fluids, and a variety of solid particles as solid phase. New and simple correlations for the prediction of particle settling velocity and bed voidage for both Newtonian and non-Newtonian fluids are presented. The present experimental data as well as a database containing a large number of published bed voidages over a wide range of operational parameters, and liquid and solid phase physical properties, are compared with the predictions from various correlations in the literature. The best prediction is obtained using the correlations recommended in the present investigation. Nomenclature A A, Ar 424 cross-sectional area of the column, m2 projected area of particle, m2 Archimedes number, (= gdP3 ( P p -Pi) PI / PI3 CD d, d,, Dh drag coefficient particle diameter, m equivalent particle, m hydraulic diameter of fluidized bed, m Solid/Liquid Fluidized Beds Involving Newtonian and Non-Newtonian Liquids wall factor acceleration due to gravity, Ids2 k viscosity coefficient in power law model, Pa. S" k constant L bed height, m M mass of particles, kg n Richardson and Zaki exponent n rate index in power law model Remf minimum fluidization, (= PI UmfdP 1PI) particle Reynolds number, (= PI us dP 1 PI) particle terminal Reynolds number in an infinite fluid, (= PI u,dP / PI) superficial liquid velocity, d s minimum fluidization velocity, m/S particle terminal velocity corrected for wall effect, m/s particle terminal velocity in an infinite fluid, m/s v z volume, m3 fluidization index Greek symbols q~ shape factor E bed voidage y shear rate, s-' p dynamic viscosity, kg/m. s pa apparent viscosity, kg/m. s p density, kg/m3 T shear stress, Pa Subscripts and superscripts a apparent D drag h hydrodynamics 1 liquid mf minimum fluidization p particle RZ Richardson and Zaki s superficial velocity SB static bed t terminal velocity T total 00 infinity References I. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Jarnialahmadi, M., and Miiller-Steinhagen. H. 2000. 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Received: 24 April 2003; Accepted after revision: 11 September 2003. 426

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