A review on modeling of hydraulic separators

Transcription

1 Novel Applied Researches in Geo-Science and Mining 2(2), Novel Applied Researches in Geo-Science and Mining Journal Homepage: A review on modeling of hydraulic separators Dariush Azizi, Ali Entezari Zarandi Department of Chemical Engineering, Université Laval, 1065 Avenue de la Médecine, Québec, Québec address: dariush.azizi.1@ulaval.ca Received 17 April 2016, Accepted 10 May 2016 K E Y W O R D S Hydraulic Separator Modeling Particle Separation Hindered Settling Cite this Paper: Azizi, D., Entezari Zarandi, A., 2016, A review on modeling of hydraulic separators, novel applied researches in geo-science and mining, 2(2), A B S T R A C T In this article, modeling of hydraulic separator as a gravity separator in mineral processing industry was taken into consideration. In this regard, crucial issues towards understanding minerals separation within the separator were deeply regarded. All parameters which could contribute to particle separation were comprehensively discussed. Besides, various proposed models concerning particles separation by the separator were illustrated as well as new assumptions in order of improvement the previous published models were introduced through which a new model can be developed 2017 elmdb.com. All rights reserved. 1- Introduction Since the mineral processing industry requires spending money and time very economically, modeling and simulation of processes have been always regarded prominently in this filed. The nature of the separation process within the hydraulic separator is extremely dependent on differences between slip velocities of particles in hindered settling situation. Therefore, the core hypothesis in modeling and simulation of the equipment process should be generated considering the particle slip velocity. This separator and its process have been already studied by many researchers. Lee (1989) developed a slip velocity model for hydraulic separator by modifying Concha and Almandra's empirical hindered-settling equation (1979). Galvin et al. (1999) proposed a slip velocity model based on Richardson and Zakin's hindered-settling equation (1954). They Modified the Richardson and Zaki's empirical model using the pressure drop inside the hydraulic separator. Honaker et al. (2000) used dynamic population mass balance to model separation process in hydraulic separator. They integrated slip velocity model into population mass balance to make process simulation. It should be mentioned that they utilized an empirical model suggested by Brauer et al. (1973) as well. Kim (2002) employed Lee's model (1989) to describe separation process and studied effect of operating parameters on the process. Sarkar et al. (2010) compared the number of empirical slip velocity models (Masliyah, 1979; Patwardhan and Tien, 1985; Van Der Wielen et al., 1996; Galvin et al., 1999b) to find more applicable one for hydraulic separator. They concluded that Galvin's model is better than other to predict particle segregation. Bazin et al. (2012) simulated hydraulic separator operation to contrast effect of operating parameters and feed characteristics with each other. Nonetheless, researchers have endeavored to find the best models and simulation methods in their works, in all investigations the suggested models have been obtained empirically. In other words, the main problem in the previous investigations is that they all have provided empirical slip velocity models. 13

2 In this situation, it seems that a number of vital operating and physical parameters related to the process have been missed and hence not to be factored in. Thus their models are only applicable in narrow range of operating conditions. Unfortunately, the vast majority of simulation programs dedicated to the mineral processing industry are based on these empirical models. As the simulators have been dependent on empirical models having imperfections, they cannot take into consideration the aforementioned influential parameters. Therefore, they cannot provide adequate analysis and good identification concerning operation optimization and flowsheet development. To solve these problems, the endeavor in this research report has been directed towards modeling and simulation of hydraulic separators regarding phenomenological issues of particle separation process. For this purpose, the slip velocity model must not be specific to a given plant or a given ore, therefore it has to be phenomenological rather than empirical. Indeed, the model has to be developed based on the physical principles so that it takes into account all crucial process variables i.e. operating condition, physical parameters and ore properties. With regard to above illustration, this report will be organized in some sections. Firstly, the first section will focus on hydraulic separator operation. In this section, operational condition, the process, influential variables, governing principles and the previous models will be comprehensively discussed. In the second section, methodology for the model development and generating the simulator will be provided. In this part, developing the slip velocity model based on physical principles and new assumptions will be debated. Also, different elements of the simulator will be reasoned and illustrated to generate initial structure of the flexible simulator. Afterwards, work plan will be given by which the project should be performed. Finally, conclusion of the report will summarize highlights of the report Hydraulic separator operation Hydraulic separator is a well-known gravity separator which has been widely used for particle separation based on density and size (Wills, 2006; Gupta and Yan, 2006). This separator is a big column which its bottom looks the same as a coin through which underflow leaves the equipment. Materials which should be classified are fed on top of equipment. Water is added to solid feed in order to create the pulp with suitable solid percent. In this situation, the particles will be better distributed throughout the equipment. It should be mentioned that solid percent of feed plays very important role in process so that changes in this parameter can have significant effect on the performance of equipment. The feed is distributed on the top of the classifier and next particles settle down. Feed rate has to be controlled for which some controller (for example PID controller) should be used during the process. Observation of flow rate should be also considered for underflow and overflow to control separation process. Teeter water enters above the cone (upper part of unit) to provide the upward fluid flow to generate the influential forces for segregation between particles. Figure 1 represents the operation and structure of hydraulic separator (Sarkar et al., 2008; Das et al., 2009). 2- Principles of hydraulic separator The first section of this report introduces hydraulic separator completely. In this section, principles of hydraulic separator which consist of operation condition, influential parameters, and physical matters related to the process and proposed models for hydraulic separator will be discussed. This section has been provided, because the equipment characteristics and preceding works have to be put into perspective to make a coherent foundation regarding the project objectives. Figure 1: Schematic diagram of hydraulic separator (Das et al., 2009). There are some influential parameters related to the hydraulic separator process which should be introduced at this time. They are categorized based on their function and situation, named physical and operating parameters. The operating parameters are directly related to operating condition so that they can be changed during the process. These parameters consist of feed rate, teeter water rate, solid percent, bed-pressure, rate of underflow and 14

3 particle distribution. In regard to operating parameters, Feed rate is so substantial in separation process. A low feed rate prevents the formation of an adequate fluidized bed. In contrary, a high feed rate can make excessive compacting of the fluidized bed. Teeter water should be considered as significant factor as well. A little change in teeter water rate can cause sharp effect on cut point. Solid percent and bed-pressure have to be exactly controlled to make sufficient and suitable voidage within the unit. Voidage is defined as distances between particles in suspension environments (Honaker et al., 2003; Young, 1999). In the case of physical parameters, height and diameter of unit should be taken into consideration. If height of unit is higher or lower than optimum condition, particle cannot be separated very well. For instance, in very high column, tiny particles cannot move to overflow acting teeter water force and consequently they will be trapped in closed circuit in the hydraulic separator. On the other hand, in very short unit, the cut point will be changed so that teeter water force will push some coarse and dense particles into overflow. In general, higher hydraulic separators have better performance compared with shorter ones mainly because particles have more time to be affected by various forces for segregation. The diameter can also have significant impact on solid percent and voidage in hydraulic separator. If hydraulic separator has small diameter, the solid concentration will be increased in fluidized bed. In this situation, voidage will be reduced and particles can be affected by drag force more and more. Hence particle separation will be done more by drag force instead of gravity force (Honaker et al., 2003; Young, 1999). Every particle within hydraulic separators moves down with a particular velocity which is named the slip velocity. With regard to separation process, it should be illustrated that particles separation takes place through competition between the slip velocity of particles and the velocity of the teeter water (Sarkar et al., 2008; Das et al., 2009). Particles which have a slip velocity equal to the teeter water velocity have equal chances of settling or being transported upward by water. However, if the slip velocity of a particle is greater than the teeter water velocity, the particle settles downwards and reports to the underflow. Otherwise it is carried to the overflow (Bazin et al., 2011; Galvin et al., 1999b) Hindered settling Since there is hindered settling condition inside hydraulic separators, this phenomenon should be clarified to make better understanding about the process. Hindered settling term is used for a suspension environment in which impact of neighboring particles on each other is so crucial (Honaker et al., 2003). In hindered settling condition, inter-particle distances have close relationship with the slip velocity of particles. A change in distances between particles can cause a change in the particle slip velocity. In fact, inter-particle distance and nature of settling are directly related to concentration of particles in suspension environment. When concentration is increased, the average distances between two neighboring particle is decreased. Consequently, the nature of settling is changed from free settling to hindered settling (Sarkar et al., 2008; Richardson and Zaki, 1954). Free settling condition occurs in clear water in which effect of particle size on settling is dominated more than density. By contrast, effect of particle density on slip velocity and particle settling is more considerable in hindered settling condition (Mirza et al., 1979; Davies et al., 1968; Liver et al., 1961). Free settling and hindered settling of particles have been depicted in figure 2. Figure 2: The difference between free settling in clear fluid and hindered settling in sediment-fluid mixture of a sediment particle (Pal et al., 2013) Effective forces When particles settle down in suspension, they will be influenced by various forces. These forces can make particle segregation and particle movements throughout the unit. A general function has been proposed to find the all effective forces during the sedimentation. Based on this general function, these forces are directly dependent on some parameters which are given in equation (1) (Richardson and Zaki, 1954): R=f (μ, ρ f, d p, U, ε) (1) Where, R is total resistive force, μ is dynamic viscosity of fluid, ρf is density of the fluid, dp is particle diameter, U is the particle terminal settling velocity and ε is voidage. The sum of different forces acting on each particle must be equal with its weight (FG). In a clear 15

4 fluid, the total force on a sediment particle can be separated into drag force (FD) and buoyant force (FB). Relationship between the forces in an equilibrium position is (Pal et al., 2013; Gupta and Sathiyamoorth, 1999): F G=F D+F B (2) The expression of FG, FD and FB can be written as follows (Pal et al., 2013; Gupta and Sathiyamoorth, 1999): F G = 1 6 πd p 3 ρ p g (3) F B = 1 6 πd p 3 ρ f g (4) F D = 1 2 C Dρ f U 2 πd p 2 4 (5) Where CD is the dimensionless drag coefficient, ρ p and ρ f are mass density of sediment particle and mass density of clear fluid respectively and dp is sediment particle diameter, U is the terminal settling velocity of sediment particle in clear fluid. Concerning drag force, "the drag coefficient is defined as the ratio of the force on the particle and the fluid dynamic pressure caused by the fluid times the area projected by the particles" (Yang, 2003). This parameter can be directly calculated by some equations which are dependent on particle's Reynolds number (Gupta and Sathiymoorth, 1999) Zones Hydraulic separator domain can be divided into separated zones respecting the process and equipment features. These features are comprised of fluid regime, size and density distribution of particles inside the unit, settling condition and operating parameters. These zones are named overflow collection zone, upper intermediate zone, feed zone, lower intermediate zone, thickening zone, and underflow collection zone (Honaker et al., 2000). The zones are depicted in figure 3. Figure 3: Schematic of the characteristic zones within a hydraulic separator (Honaker et al., 2000). Each zone has particular properties by which the zones can differ from each other concerning their effects on the separation process. For instance, since the overflow collection zone is in relation to overflow stream, there are many tiny and light particles in this zone. The particles settling in the overflow collection zone may be either free or hindered settling. This is dependent on the superficial feed and teeter water velocity. The main feature of upper intermediate zone is that this zone transfers tiny particles to upper zone. The fluid regime and particle distribution are very similar to the previous zone. The next zone is feed zone in which feed is entered into the unit. In this zone, fluid regime is so turbulent and the initial particle segregation occurs due to differential acceleration. In lower intermediate zone, there is significant particles concentration. Also, Fluidized bed is formed in this zone which plays vital role in particle separation. In lower part of this zone, teeter water flow is entered into the equipment which causes particle separation. The zones located under the teeter water stream are thickening and underflow collection zones. The particles which have higher velocity than teeter water stream will transfer to these zones in order for particles to move to underflow. Hence the main part of coarse and dense particle is found in these zones (Honaker et al., 2000) Effective parameters of the process Knowing all the effective parameters of the process is a requisite to develop the phenomenological slip velocity model. To describe separation process in hydraulic separators, some research works have been performed. In these works, researches have taken into account some physical and operating parameters of the process 16

5 into their models which can be useful for model development. Kumar et al. (2011) used regression method to find relationship between operating parameters and cut size. They concluded that feed rate, pulp density and teeter water have important impact on separation process and also feed rate has significant effect on particles segregation. They also found teeter water and pulp density have impact on performance of hydraulic separator. Bazin et al. (2012) studied differences between effect of operating conditions and ore characteristics on hydraulic separators performance. They found most of the variations which observe in overflow and underflow are owing to the operating parameters instead of ore characteristics. Dey et al. (2012) studied performance of hydraulic separator and modeled the process empirically using experimental design and regression method. They quantified influence of the operating variables, namely, teeter water flow rate, bed pressure and feed rate. They also concluded that a high mass yield of the concentration is favored by a low bed pressure and low teeter water rate. In addition, in order to achieve a good grade of the product, a high bed pressure and high teeter water flow rate are required. According to above illustration, it can be concluded that operating parameters have been more considered to generate the slip velocity models. The parameters such as teeter water, ore characteristics, voidage, feed rate, solid percent and bed pressure play vital roles in separation process, hence they should be taken into consideration properly. If the slip velocity model is developed considering all of the aforementioned parameters, the model will be able to predict velocity of particles with high accuracy Modeling of the process The slip velocity models Generally, the particle slip velocity is dependent on three main physical phenomena which are named the terminal settling velocity, the inter-particle distance and the suspension density. The second and third cases are closely related to each other so that a change in one of them can have an inverse effect on the other (Sarkar et al., 2010, Pathwardan et al., 1985). These phenomena have to be fundamentally studied and considered to generate the phenomenological slip velocity model. Concerning these phenomena, a general model can be introduced to represent the slip velocity model. This general model has been given as follows: V =U F(ε) G(ρ) (6) Where V, U, F(ε) and G(ρ) are the slip velocity, the terminal settling velocity of the particle, the effect of neighboring particles and the effect of the suspension density respectively. Some slip velocity models have been developed regarding this model and idea. Relating to the general model, Richardson and Zaki (1954) proposed their particle slip velocity model for hindered settling condition. Their model can be employed to compute the particle slip velocity for only one kind of particle having specific size and density. It is a considerable imperfection of their model to describe particle settling in hydraulic separators, because there are various particles within the unit. On the other hand, their model takes into account all three mentioned phenomena which it confirms basic idea of their model. The Richardson and Zaki model is given as follows: V =U (1 φ) n i 1 (7) Here φ is the volume fraction of the species and ni is a constant which is directly related to Reynolds number. ni is Richardson and Zaki index. 1-φ is voidage (ε) which depends on distances between particles in suspension. Lockett and Al-Habbooby (1974) developed their model regarding the Richardson and Zaki model. They claimed that their model can be utilized for three different particles having specific size and density. Their model had been well structured respecting the phenomena. This model is provided in the following equation: V =U (1 φ i φ k φ l ) n i 1 (8) Where φ i, φ k and φ l are the volume fractions of three different types of particle. Nonetheless Lockett and Al-Habbooby (1974) stated their model can be applied for particles with different density and size, experimental results have rejected this idea outright (Kumar et al., 2011). This can be considered as drawback of their model. Masliyah (1979) proposed a new functional form of F(ε) and G(ρ) for multi-particle system which can be found as follows: F(ε)= ε n i 2 (9) G(ρ)= ( ρ p ρ sus ρ p ρ f ) (10) Whereρ p, ρ f and ρ sus are density of particle, density of fluid and density of suspension. 17

6 He obtained the slip velocity model based on steady-state momentum balance principle for a particle in suspension. Subsequently, He changed his model a little by taking into consideration combination of the particle hindrance, particle concentration and local voidage. Particle hindrance and influence of particle concentration are regarded based on equations (9) and (10) respectively. In this approach, He perceived the local voidage for which a cell model was applied. This cell model is considered regarding volume conservation of each phase (Sarkar et al., 2008; Masliyah, 1979). In his model, voidage has constant value which it reduces accuracy of the model. Van Der Wielen et al. (1996) introduced new functions for F(ε) and G(ρ). In their model, they took into account steady- state slip velocity correlation and the drag force originating from the fluidization of mono-dispersed particles. Their model takes into account voidage as a constant value, while voidage changes throughout the unit. In their model, the functions of F(ε) and G(ρ) can be found as follows: F(ε)= ε 0.79n i 1 (11) G(ρ)= ( ρ p ρ sus ρ p ρ f ) n i/4.8 (12) Galvin et al. (1999b) proposed one equation to define F(ε) and G(ρ). They considered dissipative pressure gradient due to the weight of the particle in the liquid (excluding the hydrostatic pressure gradient) as the main driving force to develop their model (Sarkar et al., 2010). Accordingly, F(ε) and G(ρ) can be computed based on a function of the density differences: F(ε) G(ρ)=( ρ p ρ sus ρ p ρ f ) n i 1 (13) By considering equations (9) and (16), the Galvin's model will be: V =U ( ρ p ρ sus ρ p ρ f ) n i 1 (14) It should be explained that use of slip velocity correlations presented in equation (14) is restricted to particles having a higher density than the suspending liquid. The effect of particle size is also not taken into account in formulating the dissipative pressure gradient. The other above models of slip velocity correlations is analytical in nature. Therefore, their use is not restricted to particles having a density higher than that of the liquid. In all models the wall effect, which can assume a significant role, is not considered and also particles are assumed to be spherical. Furthermore, particles interactions have not been considered when formulating the models (Sarkar et al., 2008). Furthermore, there are several slip velocity models which are not dependent on the general model given in equation (6) and the phenomena. In this case, Conchas and Almendra (1979) proposed a slip velocity model for spherical particles respecting volume fraction of particle. In this empirical model, the slip velocity can be determined based on dimensionless parameters relating to volume fraction. In fact, this model predicts the slip velocity of particles based on particle concentration of particles. It means that voidage plays crucial role in the model. This model has been given as follows: 1 V = d f 1 (φ )f 2 (φ)[{ f 2 1 (φ)d 3 2 2} 1] 2 (15) Where d is dimensionless diameter, f 1 (φ ) and f 2 (φ) are two functions which are bound up with the volume fraction and V is the slip velocity. Related to multi-particle systems, Shih et al. (1987) applied numerical methods to extend onedimensional hydrodynamic model. They tried to model particle settling considering various kinds of particle. They solved the momentum equations of solid and fluid using numerical methods for which a mesh of finite difference cells fixed has been employed. Their model had good agreement with experimental results. Lee (1989) developed new empirical model to describe particles separation using Conchas and Almendra's model (1979) for hydraulic separator. He claimed that Conchas and Almendra's model (1979) was not well formulated for high solid concentrations, where more particle-particle interactions occur. He utilized statistical analysis to make a few corrections and modifications in the equation (15). Their model is dependent volume concentration, particle properties and fluid characteristics. It seems reasonable to consider this model better than Conchas and Almendra's model, because this model takes into account more influential parameter concerning the particle slip velocity. Indeed, Lee Considered particle properties and fluid characteristics, while Conchas and Almendra did not take them into account. This model has been depicted as follows: 1 V = f d p ρ 1 (φ )[{ ( d3 2 ρ s ρ p ρ p g ) 1/2 f p 0.75μ 2 2 (φ )} 1] (16) 2 Kim (2002) utilized Lee' model (1989) in order to study particle separation and operating parameters in hydraulic separator. He studied particle separation dynamically considering convection diffusion equation and also he assessed influences of 18

7 some parameters such as solid feed rate, teeter water rate, feed solid composition, column height, and feed entrance place on the process. Honaker at el. (2000) employed a dynamic population balance equation to model hindered settling separation in hydraulic separator. They used an empirical slip velocity model to generate their population mass balance model. This empirical model suggested by Brauer et al. (1973) has two correction factors, namely, upward fluid flowing against the settling of a particle (k f ) and particles settling in a dense suspension (K c ) have been applied. They took into account these correction factors in order to consider effect of voidage and teeter water on prediction of the particle slip velocity. The Brauer's empirical model has been provided as follows: V= U k f K c (17) Besides, there are others theoretical slip velocity model which have been provided by some researchers. In these models, the effort was to explain hindered settling of particle respecting theoretical issues. In this case, Happel (1958) proposed a cell model for inter-particle interaction. He claimed that his model is applicable in different particle concentration rages. Pathwardan et al., (1985) also proposed a cell model in which the particle particle interaction in the suspension is represented by the average distance between two neighboring particles. They stated that their model has ability to predict the settling velocities of individual types of particles in a suspension containing mixtures of particles. Batchelor (1972) developed hydrodynamic models for the slip velocity of particle. He considered hydrodynamic interaction between particles to develop his model. Reed and Anderson (1980) developed their model considering hydrodynamic interaction among particles as well. They endeavored to represent that the settling velocity of a dispersion of spherical particles is depends on concentration. Garside and Al-Dibouni (1977) built up their model using particle Reynolds number, voidage and also terminal settling velocity. Barnea and Mizrahi (1973) proposed their models for particle slip velocity considering pressure drop Models comparison Various slip velocity models were discussed in the previous section. Researchers have been interested in comparing models in order to find the best model. Their works are extremely useful to develop a new phenomenological slip velocity model. Levy and Kalman (2001) in their handbook provided an investigation done by Kamugasha and Mahgerefteh in which some model have been compared with each other. In this investigation, the models proposed by Richardson and Zaki (1954), Happel (1958), Batchelor (1972), Barnea and Mizrahi (1973), Garside and Al-Dibouni (1977) and Reed and Anderson (1980) have been compared. Their results emphasize that Richardson and Zaki' model (1954) has more acceptable predictions than others to predict particle slip velocity, because its results follow the experimental data closely. These results have been depicted in figure 4. Figure 4: A comparsion of experimentally detemined settling vlocities to those predicted from various theoritical models Levy and Kalman (2001). Majumder (2007) in his critical review compared the models provided by Richardson and (1954), Lee (1989) and Brauer et al. (1973). He concluded that Richardson and Zaki' model (1954) may be better to be used for predicting hindered-settling velocities in as assemblage of a mono-size and mono-density particles suspension. He also claimed that there is no well-accepted model which can be applicable in assemblage of poly-disperse and poly-dens suspension. Sarkar et al. (2010) compared accuracy of some models proposed by Galvin et al. (1999b), Van Der Wielen et al. (1996), Pathwardan et al., (1985) and Masliyah (1979) to model performance of hydraulic separators. They used the same situation when they applied each model to compute slip velocity. According to their conclusion, there are differences between accuracy of the models despite the fact that they all have been suggested to predict particle slip velocity. They concluded that the Galvin's model provides more accurate results than the other ones to predict the particle slip velocity. According to above explanation, it can be concluded that Richardson and Zaki' model (1954) has had more appropriate performance to determine particle slip velocity for hindered settling condition. 19

8 Besides, Galvin's model (1999b) which has been developed modifying Richardson and Zaki' model (1954) presents better results than other ones to describe separation process in hydraulic separator. Galvin's model has developed respecting the mentioned phenomena as well as its structure is simpler than other models. Hence it seems reasonable to assume that Galvin's model can be considered as a strong foundation to develop phenomenological model due to its accuracy, simple structure and used effective physical phenomena. Figure 5 demonstrates the main parameters to build up Galvin's model. Figure 5: Structure of the Galvin's model. Accordingly, the phenomenological slip velocity model has to take into account voidage, the terminal settling velocity and the Richardson and Zaki index. In order to develop a steady-state phenomenological slip velocity model, three aforementioned phenomena should be exactly studied because a comprehensive model must deal with most of matters which are involved in the process. Since Galvin and other researchers did not pay enough attention to particles interactions, teeter water, wall effect and voidage profile within the unit to develop their models, these crucial parameters should be factor in. Furthermore, effective parameters consisting of physical and operating parameters should be involved in the phenomenological model. Otherwise the proposed model will not be able to attain the project aims and objectives to generate a flexible simulator which will be applicable in a wide range of operating conditions. With respect to these conclusions, next section will be organized concerning model development and generating simulator. 3- Model development One of the main objectives of this project is to find a phenomenological model for the particle slip velocity. The phenomenological model has to be applicable for a wide variety of operating conditions, because previous empirical models don't have this essential feature. Hence it is better to develop a more comprehensive and simpler model than the preceding ones in order to predict the particle slip velocity. As it was mentioned, Galvin's model can be considered as initial idea to develop the slip velocity model for hydraulic separators. This model consists of three main parts named the terminal settling velocity, voidage and the Richardson and Zaki index. Galvin et al. (1999b) did not notice all aspects in separation process and they only modified the Richardson and Zaki's empirical model. Therefore, it is supposed in order to create a comprehensive phenomenological model, most of governed physical principals related to the process must be taken into consideration properly. In this section, the endeavor is to discuss the issues and new ideas to generate a new phenomenological model for the slip velocity respecting the Galvin's model and physical governed phenomena The Richardson and Zaki index Richardson and Zaki (1954) plotted log V against log ε resulted from many experiments. The concluded curves can be represented by an equation of the form: log V= n i log ε + log V i (18) Where n i is the slope of the curve and log Vi is the intercept on the log V axis corresponding to a value of ε equal to unity. n i has been named as the Richardson and Zaki index in many models. There are very close relationship between the Richardson and Zaki index and Reynolds number. Various equations have been introduced by some researchers to calculate this index. Most important of them have been published by Richardson and Zaki (1954) (RZ), Garside and Al- Dibouni (1977) (GA), Chien and Wan (1983) (CW) and Cheng (1997) (CH). Galvin et al. (1999b) used the model proposed by Garside and Al-Dibouni (1977) to generate their model, because they believed that this model has more accuracy as well as this model is structured simpler. This equation has been presented as follows: n i = Re p Re p 0.9 (19) 20

9 Where, Rep is Reynolds number related to particles. Pal et al. (2013) proposed new model to compute the index and endeavored to compare the published models with their new equation. More variables and n i = (1 C) 4/3 ln (1 C f C ) + 3lnf 7/8ln [ max ln (1 C) considerations have been involved in their new model compared with the previous models. Hence applying this model to create the slip velocity model may be more influential. This function can be seen as follows (Pal et al., 2013): Re p Re p 4 7 Where f can be obtained based on following function (Pal et al., 2013): ] 7/8 f = D m = [ (1 + p)(1 C) p 1 (1 C 2 ) (1 C) 1 ] 1/3 (21) D p C max Where, C is volumetric concentration, D m is modified non-dimensional particle diameter, D is non-dimensional particle diameter, p is (ρ p ρ f )/ρ f and C max is maximum volumetric concentration of suspended particle. To confirm accuracy of new equation compared with the previous published equations, some data Table 1: Results of the Richardson and Zaki index models (pal et al., 2013). Grain properties Sediment particle dp Rep ρ p ρ f Observed data (20) from a few articles were analyzed. They have been gathered in table 1: For assessment of results, equation (22) has been applied in order to compare the Richardson and Zaki index equations with each other. Modified Model RZ ni GA (Galvin's Model) Beach sand Filter sand Crushed sand CW CH 21

10 Gravel Glass (sphere) Crushed flint Crushed rock Error = Predicted value Observed Value Real Value 100 (22) The results of the comparison have been given in table 2: Table 2: Comparsion various equations for calculating the Richardson and Zaki index. PM RZ GA CW CH Average of Errors (%) Standard deviation of Errors (%) Interval of errors (%) When average of errors will be considered to assess the equations, it may be concluded that the Pal's equation will have better agreement with observed data. On the other hand, when interval of errors will be studied, it can be found that interval errors of all equations have overlap with each other. It causes to decline accuracy of this analysis. Although there is obvious overlap between intervals of errors, Pal's model has lower error. Thus the Pal's equation can be utilized to develop initial model at this stage, but for the final approval of this conclusion all equations have to be compared based on more data The terminal settling velocity The terminal settling velocity is another term in the slip velocity model which should be studied. This concept is defined as velocity of each particle in clear water. In Galvin's model, influential factors named fluid regime, teeter water flow rate and suspension condition have not been completely considered to generate the model very well. Thus, in order to build up the phenomenological model, these influential parameters should be taken into consideration. Relating to fluid regime, the terminal settling velocity can be changed in each fluid regime. Richardson and Zaki (1954) used the Stocks' law to find the particle velocity. Therefore, the terminal settling velocity of particle can be: U= d p 2 (ρ p ρ f )g 18 μ (23) Where, ρ p is density of particle, ρ f is density of fluid, dp is particle diameter and g is acceleration due to gravity. It should be mentioned that the equation (23) can be used when Reynolds number is under 1. Galvin et al. (1999b) used particle Reynolds number equation to compute the terminal settling velocity. By arranging the equation respecting the terminal settling velocity, we will have: U= μre p ρ d p (24) Where, ρ is particle density, U is the terminal settling velocity of particle, dp is particle diameter, Rep is particle Reynolds number and μ is fluid viscosity. Besides, they suggested that the terminal settling velocity can be obtained by considering three forces provided in equations (3), (4) and (5) in a balance. In this situation, the terminal settling velocity will be (Gupta et al., 1999): 22

11 U = 4 3 (ρ p ρ f )gd p ρ f C D (25) Furthermore, Concha and Almendra (1979) proposed an empirical equation based on experimental data to find this term in Stokes and Newton's regions. Kim (2002) used their equation to model particle separation process in hydraulic separator. This equation can be found as follows: U= 20.52μ {[ ( d p (ρ s ρ f )ρ f g ) ] 1} 2 (26) d p ρ f 0.75μ 2 With regard to above explanation, it seems logical to use equations (23) and (25) in Reynolds number of under 1 and above 1 respectively. To approve this assumption, equations (23), (24), (25) and (26) have been applied to compute the terminal settling velocity. Results have been provided in table 3: The results have been compared with each other using equation (22). It can be clearly concluded that new assumption for computing the terminal settling velocity is more accurate. The results have been provided in table 4: Table 3: Results of the terminal seetling velocity models (Tomikns et al, 2005). Materials dp (mm) Rep U (mm/s) ρ p ρ f the terminal failing velocity based on new assumption (mm/s) the terminal settling velocity based on Galvin's model (mm/s) the terminal settling velocity based on Concha and Almendra's equation (mm/s) Beach sand Beach sand Filter sand Filter Sand crushed sand Table 4: Comparsion various equation for calculating the terminal settling velocity. Average of Errors (%) New assumption Galvin Concha and Almendra Relating to the terminal settling velocity, it should be noticed that there is teeter water flow rate in hydraulic separator as well. Due to this fluid flow, all previous calculations can be significantly affected. Hence it is vital to consider velocity of teeter water in computing the terminal settling velocity. For this purpose, the concept of the relative velocity can be utilized. For a particle of volume Vp and density ρp that is moving in a flowing fluid of velocity UT (here it is velocity of teeter water), the relative velocity (Ur) is (Gupta et al., 1999): U r= U U T (27) 23

12 Thus Ur can be applied to develop the slip velocity model, because it is computed regarding teeter water as a significant flow within the unit. Based on another alternative idea to compute the relative terminal settling velocity, it is better to employ water bias instead of teeter water. Water bias has so crucial role in column flotation which its direction and value can have significant effect on performance of flotation (Finch and Dobby, 1990). Since there is net-fluid within hydraulic separator the same as column flotation, it may seem reasonable to utilize water bias concept. This net is downward water stream in column flotation which is called the bias rate (Finch and Dobby, 1990). This net has to take place in hydraulic separator as well. To compute the bias rate in hydraulic separator, water mass balance should be taken into account for whole of unit and for each zone separately. In this situation, there will be several mass balance equations which are directly dependent on each other. By writing all water mass balance equations and solving them properly the bias rate will be: j b= j u-j T j b= j O-j f (28-a) (28-b) Where, jb is the bias water rate, ju is water rate in underflow stream, jt is teeter water rate, jf is water rate in feed stream and jo is water rate in overflow stream. jb can be substituted with UT in equation (27) to find the relative settling velocity. Perhaps it will be more accurate to use the bias rate, but both approaches should be studied experimentally to draw a valid conclusion Voidage Voidage is defined as distances between particles in suspension environments. Voidage has very important impact on the particle slip velocity, because it has a direct sharp effect on slip velocity of particles. Indeed, increasing voidage causes that the particles slip velocity enhance and vice versa. This enchantment and reduction are due to available space in suspension environment for particles. Voidage can be generally defined based on the following equations (Yang, 2003): ε= 1- C (29-a) ε= ρ p ρ sus ρ p ρ f ε= V f V f +V p (29-b) (29-c) Here, ρ sus is density of suspension, C is volume concentration, Vf is volume of fluid, Vp is volume of all particles and ε is voidage. At issue here is to introduce two approaches based on the profile concentration and the pressure drop to compute voidage inside the unit. Relating to the profile concentration, there is a very close relationship between voidage and particle volume concentration in hindered settling. Moreover, volume concentration changes point to point owing to settling velocity of particle. In other word, particles which have higher density will be settled down sooner than lighter particle through particle settling. In this situation, heavier particles will accumulate in the lower part and lighter particles will go to upper part of the unit. Hence there are particle density and size distribution and consequently changes in particle concentration throughout the unit vertically. According to this explanation, the profile concentration has to be taken into consideration. Seibert et al. (1996) utilized profile concentration expression provided by Heimenz (1986) to simulate particle settling in suspension with tiny particles. They considered matching the sedimentation and diffusion fluxes for modeling the concentration profile. The expression has been given as follows (Heimenz, 1986): dc dx = m p KT (1 ρ f ρ p ) gc (30) Where mp is the mass of a particle, k is Boltzmann s constant, T is the system operating temperature, ρ p is the density of the colloidal particle, ρ f is the density of the surrounding fluid, and C is the concentration at any point, x, in the container. This idea to use profile concentration may be applicable for hydraulic separator because there should be the same situation inside of the unit as well. Since voidage is directly dependent on particle concentration, thus voidage can have the same behavior with particle concentration. By substituting equation (28) in equation (29), the voidage profile can be obtained based on the following expression: dε dx = m p KT (ρ f ρ p 1) g(1 ε) (31) To obtain an equation to present voidage dispersion in each place of unit, equation (30) has to be solved algebraically. Thus, equation for computing voidage will be: 24

13 mp KT ε=1-e (ρ f ρp 1)g x (32) Here, x is related to height of unit. In the previous models, the voidage have been considered as a constant value, but in fact this parameter is changed in the unit. By considering equation (32) to compute voidage, voidage will be computed in each point of unit. It can improve calculation of voidage and consequently calculation of the slip velocity. Based on the second approach, the pressure drop can be utilized to compute voidage in the unit. According to this idea, height, diameter and weight of particle have vital role to find voidage, because there is relationship between the pressure drop and geometry of unit and also weight of materials. Equation (33) presents relationship between voidage and the pressure drop: P H = (1 ε)(ρ p ρ f )g (33) Where P is the pressure drop which can be computed between two point of equipment and H is height of equipment. To compute voidage, the equation (33) can be rearranged respecting voidage based on following equation: P ε = 1 Hg(ρ p ρ f ) (34) It should be explained that equation (30) is frequently utilized for suspension environment with very tiny particles. Accordingly, equation (32) which has been obtained from equation (30) can have the same situation as well, while the drop pressure can be applied for suspension with bigger particle. On the other hand, measuring pressure drop in hydraulic separator is feasible and achievable. These two facts can make equation (34) more sufficient than equation (32) to compute voidage. It seems true to conclude that equation (34) may be more applicable for hydraulic separator, but it demands to be validated practically not theoretically. As the second idea which is related to the pressure drop may be more applicable, it is better to pay more attention to relationship between voidage and pressure drop. Regarding relationship between pressure drop and voidage, many researchers proposed various equations (Yang, 2003). The most widely used equation is suggested by Ergun (1952). Ergun (1952) found his equation empirically, hence using this equation to develop the phenomenological slip velocity model may be inappropriate. Gibilaro et al (1985) modified the Ergun equation and proposed an alternative pressure drop equation on the basis of theoretical considerations. Their model was significantly improved Ergun equation so that its results represent high accuracy. Their model comprises two equations for laminar flow regime and turbulent flow regime which are respectively: P 2 H = ( ) ρ fu f Cε 4.8 Re p d p (35) P = ρ fu f 2 H d p Cε 4.8 (36) By rearranging equations (34) and (35) respecting voidage, following equations are concluded: ε = 4.8 ( ) ρ fu f Re p 4.8 ε = P d p ρ f U 2 f H C d p P 2 H Where, Uf is superficial fluid velocity. C (37) (38) Nonetheless, equations (37) and (38) are more complicated than equation (34), it seems reasonable to consider them as good approach for voidage calculation. These equations take into account more effective variables, hence they may provide more accurate results. For example, particle Reynolds number and superficial fluid velocity have been involved in model. Particle Reynolds number is dependent on fluid and particles properties and the superficial fluid velocity is directly related to teeter water. Thus, more influential parameters have been utilized to compute voidage. Heretofore three approaches related to computing voidage were discussed. It seems reasonable to consider the third one (equations (37) and (38) for the model development, but to lead to one accurate conclusion more valid reasons are needed. Hence the proper equation must be chosen and validated between equations (32), (34), (37) and (38) through experimental data. 4- Conclusion In this investigation, the endeavor was to study hydraulic separator in order to understand essential 25