Development and Validation of a Constitutive Model for Size-Segregation during Percolation

Segregation is a widely occurring undesirable phenomenon in industries that store, handle and process particulate materials. Size-segregation induced by the percolation mechanism is observed in several important processes that negatively impacts the product quality and mixing. To quantify sizesegregation, a constitutive model based on simultaneous convective and dif fusive demixing was developed and validated. The primary segregation shear cell (PSSC) was used to measure the fundamental parameters and validate the convective-dif fusive segregation model. Glass beads of size ratios of 10.9:1 (1250:115 μm), 8.7:1 (1000:115 μm), and 5.1:1 (1000:196 μm) were used for model parameter determination; whereas, size ratio of 6.4:1 (1250:196 μm) was used for model validation. As shown in a previous study, an effective segregation direction could be measured and used to validate the convective-dif fusive segregation model for percolation. This justified the use of an effective segregation direction to model the percolation of fines. When compared to the normalized measured data for size ratios larger than 8.7:1, the convective-dif fusive model resulted in standard deviations of 0.035. However, for size ratios smaller than 6.4:1, dif fusive demixing was occurring during shear with the absence of a rapid initial discharge phase, i.e., minimal contribution due to convective component. Estimating the percolation for the 6.4:1 size ratio was accomplished by using the mean data of the 5.1:1 size ratio, which resulted in standard deviation of 0.055. The initial rapid discharge present in 10.9:1 and 8.7:1 size ratios suggests that a critical size ratio exists that differentiates free-fall discharge segregation from random mixing segregation. This can be critical to powder manufacturers that could use this critical limit to define size distribution recommendations during manufacturing. * 229 AG Engineering Building University Park, PA 16802 1 Corresponding author † Accepted: September 9, 2003 place; it would provide a powerful tool in understanding and minimizing segregation. Percolation and sieving are two of the commonly observed mechanisms for segregation in industry. Percolation is similar to the sieving mechanism (Figure 1) that occurs during shear; however, a moving layer is absent. In the literature, percolation has been studied as a function of gravity and vibration (Tang et al., 2001). Vibration can cause a small individual particle to travel downward through the powder mass. Also, a smaller particle could travel through a larger granular mass due to gravity with diameter ratios less than 0.15. Due to the importance of percolation in industrial processing, a size-segregation mathematical model based on the convective-diffusive formulation for percolation is presented and validated in the paper. All continuum theory-based constitutive models have material parameters that need to be measured for a specific powder under prescribed loading conditions. For this purpose, a primary segregation shear cell (PSSC) was developed. The details of the PSSC are given in Duffy and Puri (2002), which is a vertically oriented segregation shear cell of 101.6 mm (high) 101.6 mm (wide) 50.8 mm (deep). The overall schematic of the PSSC is shown in Figure 2. This shear cell can be used to test the time-dependent segregation response of binary mixtures over a large range of size ratios ( 4:1). In addition, the tester can be operated at strains ranging from 5% to 25% to provide different energy inputs. Also, the rate of loading, i.e., cycle speed or strain rate, can be varied from 0.75 Hz to 1.7 Hz to test the material over a range of energy input rates. The bed depth of coarse particles can be preset ( 90 mm) to determine accurately the speed of movement of fines under different input energies, i.e., operating conditions. Additionally, the test material can be subjected to a constant confining pressure between 0 kPa (no confining pressure) to 10 kPa. In this study, the PSSC was used to measure the discharge of fine particles through a bed of coarse particles with the percolation parallel to the shear deformation and gravity directions. 2. LITERATURE REVIEW Segregation, while important in every aspect of powder technology, remains to this date elusive in terms of the fundamental understanding and primary test devices for constitutive formulations (for example, see Rosato and Blackmore, 2000). Most researchers define a segregation coefficient and explore segregation for a particular process (Duffy and Puri, 2002). However, a shear apparatus has been developed by Duffy and Puri (2002) to measure the movement of fine particles through a bed of coarse particles. This apparatus was used to collect data for constitutive model development and validation. Mixing and f low of particulate materials are important processes in powder industries such as agricultural and food, ceramic, chemical, mining, pharmaceutical, and powder metallurgy. Extensive work has been done in the f low of particulate materials especially out of hoppers. Most research on mixing has focused on determining a mixing efficiency. Sommer (1996) defined mixing as the blending of at least one solids component with another, where at least one property (such as size, shape, and density) of the two components is different. In this definition, segregation would be defined as demixing in which particles 152 KONA No.21 (2003) Fig. 1 Segregation by sieving Dection of low Fig. 2 Overall schematic of primary segregation shear cell (PSSC) Motor


INTRODUCTION
In many particulate materials industries, the handling, storage, f low, and mixing represent important processing steps.During these and similar processing steps, product quality may be lowered by a phenomenon known as segregation.Accordingly, quantitative analysis of segregation plays a significant role in evaluating powder-related processing, manufacturing, storing, or conveying systems.Segregation is defined as a demixing process in which components of a mixture separate as long as one component of the mixture is different than another.Of all the particle attributes, size is considered to be the most dominant variable (Williams, 1976, Duffy andPuri, 2002).Segregation has been measured using a coefficient, mechanism, and model (Rosato and Blackmore, 2000).The coefficient technique is by far the most common.However, it only describes the degree of segregation taking place for a particular set of operating conditions.The mechanism technique provides insight for processes that exhibit a dominating segregation mechanism.The most encompassing method is to model a particular process.It is neither possible nor practical to model every mechanism for a given process.However, if a specific mechanism is identified that explains the majority of segregation taking place; it would provide a powerful tool in understanding and minimizing segregation.
Percolation and sieving are two of the commonly observed mechanisms for segregation in industry.Percolation is similar to the sieving mechanism (Figure 1) that occurs during shear; however, a moving layer is absent.In the literature, percolation has been studied as a function of gravity and vibration (Tang et al., 2001).Vibration can cause a small individual particle to travel downward through the powder mass.Also, a smaller particle could travel through a larger granular mass due to gravity with diameter ratios less than 0.15.Due to the importance of percolation in industrial processing, a size-segregation mathematical model based on the convective-diffusive formulation for percolation is presented and validated in the paper.
All continuum theory-based constitutive models have material parameters that need to be measured for a specific powder under prescribed loading conditions.For this purpose, a primary segregation shear cell (PSSC) was developed.The details of the PSSC are given in Duffy and Puri (2002), which is a vertically oriented segregation shear cell of 101.6 mm (high)҂101.6 mm (wide)҂50.8mm (deep).The overall schematic of the PSSC is shown in Figure 2.This shear cell can be used to test the time-dependent segregation response of binary mixtures over a large range of size ratios (4:1).In addition, the tester can be operated at strains ranging from 5% to 25% to provide different energy inputs.Also, the rate of loading, i.e., cycle speed or strain rate, can be varied from 0.75 Hz to 1.7 Hz to test the material over a range of energy input rates.The bed depth of coarse particles can be preset (90 mm) to determine accurately the speed of movement of fines under different input energies, i.e., operating conditions.Additionally, the test material can be subjected to a constant confining pressure between 0 kPa (no confining pressure) to 10 kPa.In this study, the PSSC was used to measure the discharge of fine particles through a bed of coarse particles with the percolation parallel to the shear deformation and gravity directions.

LITERATURE REVIEW
Segregation, while important in every aspect of powder technology, remains to this date elusive in terms of the fundamental understanding and primary test devices for constitutive formulations (for example, see Rosato and Blackmore, 2000).Most researchers define a segregation coefficient and explore segregation for a particular process (Duffy and Puri, 2002).However, a shear apparatus has been developed by Duffy and Puri (2002) to measure the movement of fine particles through a bed of coarse particles.This apparatus was used to collect data for constitutive model development and validation.
Mixing and flow of particulate materials are important processes in powder industries such as agricultural and food, ceramic, chemical, mining, pharmaceutical, and powder metallurgy.Extensive work has been done in the flow of particulate materials especially out of hoppers.Most research on mixing has focused on determining a mixing efficiency.Sommer (1996) defined mixing as the blending of at least one solids component with another, where at least one property (such as size, shape, and density) of the two components is different.In this definition, segregation would be defined as demixing in which particles with one similar property, usually size, accumulate together.Sommer (1996) outlined four major models of powder mixing.A limited number of researchers have used stochastic formulation to describe segregation.Law and Kelton (1991) defined deterministic models as simulations that do not contain any probabilistic (random) components.A stochastic model contains at least one time-varying random component.
Model 1 Ҁ The Fokker-Planck Equation (1), similar to Fick's second law of diffusion, is used to describe mixing that arises from convective and random motions of particles.
This equation is used to describe the concentration (c) at a given position (x) with respect to time (t) in the mixer.Equation ( 1) contains two material parameters which are u, the transport coefficient, and D, the dispersion coefficient.The transport coefficient u (units of L/T) describes the convective flow present in mixing, and the dispersion coefficient D (units of L 2 /T) is a measure of the random motion available in the mixer.This is referred to as the T is a characteristic time (҃L 2 /D) formed from a characteristic mixer length L and dispersion coefficient D. If rigid walls, i.e., there is no material flow at x҃0 and x҃L, and a highly concentrated side ini- This is simply the Fokker-Planck Equation (1) for each stream where c 1 , c 2 , and u 1 , u 2 and D 1 , D 2 are concentrations, convective and diffusion parameters of components 1 and 2, respectively.Some work has been accomplished using Equation ( 5) to demonstrate that mixing efficiency was strongly dependent on the dispersion coefficient.However, simulations that varied the parameters showed the most inf luential parameter was the residence time (a parameter related to the characteristic fluctuation time T of the entrance streams).Model 4 Ҁ The final mixing model described by Sommer was silo mixing.Silo (i.e., bin) mixing is described as various components mixed in a bin via external or internal blending and recirculation.This means different components introduced into the bin at the same time can leave the bin at different times.A cell/layer model was used to describe the time-dependent concentration distribution, c k , given in Equation (6).
where, M is the mixer matrix and contains the residence time spectrum data.Williams (1986) outlined statistical calculations of random mixtures.A random mixture is defined as a mixture with the probability of finding a particular component of the mixture throughout the sample independent of sampling location and equal to the percentage of the component in the entire mixture.A random mixture composed of two sets of identical particles was studied.Quantitative relationships for locating a component were developed.The models described herein and their variations have been applied to chute-f low (Vallance andSavage, 2000, Hwang, 1978), drum-f low (Khakhar et al., 2001), heap-f low (Shinohara and Golman, 2002), and constitutive model (Bridgwater, 1994).
The overall goal of this research was to develop and validate a percolation-induced segregation constitutive model to predict the movement of fines during shear of a binary, i.e., coarse-fine, mixture.In order to fulfill this goal, the specific objectives were: 1. To test several binary mixtures comprising varying size ratios at different boundary and loading conditions, i.e., strain, cycle speed, coarse particle bed depth, in the PSSC.2. To quantify the fundamental parameters of the percolation constitutive model using experimental data from the PSSC.3. To validate the constitutive model by performing tests with the PSSC under conditions different from the mechanism parameter determination and comparison with the results from the constitutive model.

EXPERIMENTAL METHODOLOGY
Four binary particulate material mixtures of glass beads were selected.Glass beads were used based on the availability of narrow cut sizes, sphericity, and non-hygroscopic properties under controlled ambient test conditions.All tests were conducted in an environment controlled laboratory with average temperature of 21°CȀ3°C and relative humidity less than 40% to minimize the effects of moisture on the test results.The glass spheres were considered dry (i.e., moisture content was equal to zero).A dehumidifier, placed near the shear apparatus, was used to reduce the ambient moisture in the environment.
Additional variables considered during data collection were strain, cycle speed and coarse particle bed depth.For determination of material parameters of the convective-diffusive constitutive model given in Equation ( 1), three strain values, two cycle speeds, and three beds depth shown in Table 2 were used.Each combination was repeated four times for calculation of a representative mean response.This series of experiments is the same as those reported in Duffy and Puri (2002).They provide further information on the rationale for these variables and methodology of tests.To validate the segregation constitutive model, experiments given in Table 3 were performed.The purpose of validation experiments was to provide a data set at an intermediate operating condition so that the effectiveness of the candidate segregation constitutive model could be evaluated.

SEGREGATION CONSTITUTIVE MODEL
Based on literature review and its rational basis, the convective-diffusive segregation model (Equation 1) was selected.The convective-diffusive constitutive model can be used to describe one-, two-, or three- dimensional analyses.A one-dimensional analysis was selected for the test conditions used in this study.This is an appropriate assumption based on the mechanism of shear.As discussed by Duffy and Puri (2002), the percolation of binary mixtures of glass beads is nearly spatially uniform.Furthermore, as shown by Duffy and Puri, an effective direction exists wherein any resultant anisotropies due to the loading conditions and test material can be fully addressed.This is shown as the θ-direction in Figure 3.An effective percolation direction (one-dimensional) was identified by combining the mass versus time relationship from the six load measurement locations (Figure 3).The effective direction was a resultant mass discharge vector of the six load cell measurements (Duffy, 2001).
The convective-diffusive segregation model for percolation mechanism was rewritten in the θ-direction as shown in Equation (7).
Equation ( 7) quantifies the mass of fines (m) at any given time (t) and location along the θ-direction of the bed depth (h) using the material parameters D Ҁ θ and u Ҁ θ , which are the fundamental material parameters and represent the diffusive and convective components, respectively.The units of D Ҁ θ and u Ҁ θ are L 2 /T and L/T, respectively.The data collected from the six load cells (Cell #4 BR-LM, #6 FR-LM, #8 MR-LC, #11 MR-RC, #13 BR-RM, and #15 FR-RM) in Figure 4 were used to get the mass versus time relationship in the θ-direction by calculating a resultant percolation vector as a function of time.Geometrical distances from origin to the center of load cell grid in the mesh The finite difference representation for Equation (7) and the associated boundary conditions are summarized in Equations ( 8) and (9).All values at the current time t are known.In addition, the subscript n denotes the n-th layer along the coarse bed depth.

µ҃ δ҃
The last term in Equation ( 8) represents a diffusive f lux that is opposite to the direction of gravity.Based on physical observations at the top of the cell and the amount of time for the fines to reach the collection pan, δm(θ nѿ∆n , t)҃0.The process of fine particles dropping out of the test cell and into the collection pan gives minimal resistance in the direction of gravity.It should be noted that if the bottom of the test cell was blocked, this term could not be deleted.Therefore, the finite difference representation for the convectivediffusive model simplifies to Equation (10).
One of the two purposes of the finite difference formulation was to estimate the material specific parameters µ and δ in the convective-diffusive segregation constitutive model.The second purpose was to use the estimated µ and δ values to validate the segregation constitutive model.The parameters, µ and δ, were varied to minimize the cumulative squared error between the numerically solved and measured effective mass value m(θ n , t).In order to determine µ and δ, ∆t҃1 s and ∆θ * cos θ҃5.08 mm, 10.16 mm, and 15.24 mm for bed depths of 25.4 mm, 50.8 mm, and 76.2 mm, respectively, were used.Considerations θ ∆t ∆θ such as convergence and number of cases were factored in the determination of ∆t and ∆θ values.The Solver utility in Microsoft Excel ® was used to solve the finite difference equations in Equation (10) with the boundary and initial conditions shown in Equation (9).The precision of µ and δ was held to three decimal places (i.e., 0.000).Equation (11) defines the standard deviation between the actual and predicted data.The term, df, represents the degrees of freedom for the data.The values of µ and δ that produced the smallest error were taken as the material parameter for the given set of operating conditions.In Equation (11), M(t i ) represents the experimentally measured mass value at the exit, i.e., corresponding to θ r .

Convective-Diffusive Segregation Model Parameters
The convective-diffusive segregation model parameters, µ and δ, for strains of 15%, 10%, and 5% are given in Tables 4 through 6, respectively.In Tables 4 through 6, the convective term (µ) is reported first then the diffusive term (δ) separated by a slash.used in the model development and validation studies.The normalized mass values were calculated by dividing the mass at any given location, i.e., cell locations as shown in Figures 3 and 4, with the measured mass of fines at that location at the end of data collection.
The average standard deviations in Table 4 for 10.9:1, 8.7:1, and 5.1:1 are 0.075, 0.065, and 0.013, respectively.In addition, the convective model, without the diffusive terms was used.The average standard deviations for 10.9:1, 8.7:1 and 5.1:1 were 0.118, 0.146 and 0.071, respectively.At all strain values, the convective segregation model produced larger deviations between measured and predicted values as compared with the convective-diffusive segregation model.This suggests that random mixing is occurring which is characterized by the δ parameter.The convective parameter characterizes the initial rapid discharge or free fall percolation occurring in the larger size ratios.As shown in Tables 4 through 6, the convective parameter of the 5.1:1 treatments is near zero.This suggests that the percolation of fines at the 5.1:1 treatments is dominated by the diffusive mixing mechanism.Average standard deviations in Tables 5 and 6 are below 0.050 for all treatments with the standard deviations for 5.1:1 less than 0.030 for all strain levels.
Measured and modeled normalized mass versus time relationships for three different treatments are given in Figures 5 through 7. The first set of results, Figure 5, is for strain of 15%, cycle speed of 0.75 Hz, and bed depth of 7.62 cm.The three responses are for different size ratios.The measured data are represented by symbols and the predicted values are shown as solid lines.The convective-diffusive segregation constitutive model represents the measured data very well (standard deviations for 10.9:1, 8.7:1, and 5.1:1 are 0.060, 0.059, and 0.010, respectively).As shown in Figure 5, the majority of error in the standard deviation is within the first 5 seconds of the percolation profile.The model has a lag followed by a step which is not present in the measured data.This can be explained physically by assuming that some of the fine particles did migrate through the bed during deposition but did not fall out of the screen.The same trend is apparent in Figure 6, which is for the size ratio of 10.9:1 at a strain rate of 1.33 Hz and bed depth of 7.62 cm (standard deviations are 0.057, 0.038, and 0.009 for strains of 15%, 10%, and 5%, respectively).Figure 7 is for the size ratio of 5.1:1 at cycle speed of 1.33 Hz and bed depth of 7.62 cm.For the size ratio of 5.1:1, the initial lag of fines in the collection pan dur-ing data collection is present.Therefore, the model also predicts the initial lag very well (standard deviations are 0.015, 0.008, and 0.010 for strains of 15%, 10%, and 5%, respectively).The coefficient of variation for the convective term was large (50%).This was inf luenced by the initial lag time.The coefficient of variation for the diffusive term ranged from 10 to 20%.
From this discussion it was concluded that the convective-diffusive segregation constitutive model describes the percolation of fines through a bed of coarse particles better than simply the convective segregation constitutive model.Generally, the standard deviations declined by a factor of three, and the calculated values follow the measured mass eff lux trends.The validation of the convective-diffusive segregation model is presented in the next section.

Validation of Convective-Diffusive
Segregation Constitutive Model It was initially the goal to predict the normalized mass versus time relationship for the validation size ratio of 6.4:1.However, as discussed in the preceding section, the prediction is hindered by the presence or absence of an initial discharge of fines at the beginning of the test.As shown hereafter, the normalized mass versus time relationship for 6.4:1 is similar to the 5.1:1 ratio, i.e., does not exhibit an initial free fall discharge.Therefore, prediction attempts produced very large standard deviations when comparing the predicted values to the collected/calculated values of the 6.4:1 ratio.The convective and diffusive parameters were estimated using a linear interpolation based on the size ratio.For a size ratio of 6.4:1, the convective parameter (µ) and diffusive parameter (δ) were estimated as 0.137 and 0.263 for strain of 15% and 0.031 and 0.218 for strain of 5%, respectively.Graphical representations between the measured data and modeled data using the linear interpolation based on size are given in Figures 8 and 9 for strains of 15% and 5%, respectively.The average standard deviations for 15% and 5% strain for the 6.4:1 size ratio using linear interpolation were 0.338 and 0.174, respectively.These deviations are much higher than the optimized models in the previous sections.
Since the 6.4:1 data were similar to the 5.1:1 data (i.e., no initial rapid discharge) and significantly different (p0.05) than the data for the 8.7:1 and 10.9:1 size ratios (Duffy and Puri, 2002), the average convective and diffusive parameters at strains of 15% and 5% (of 5.1:1) were used to estimate the percolation of 6.4:1.For a size ratio of 6.4:1, the convective parameter (µ) and diffusive parameter (δ) were estimated as 0.025 and 0.206 for a strain of 15% and 0.004 and 0.104 for a strain of 5%, respectively, using the average calculated parameters of 5.1:1.Figures 10 and 11 compare the measured data and the calculated normalized mass discharge values for the size ratio of 5.1:1.The parameters averaged from the 5.1:1 data reduced the standard deviations to 0.061 and 0.048 for the 15% and 5% strains, respectively.This is an average reduction in standard deviation of 77% when compared with the linear interpolation values.
In addition, the convective and diffusive parameters of the segregation constitutive model were calculated by minimizing the standard deviation in the same manner as the other size ratios.The optimal solutions are given in Tables 7 and 8.The standard deviations from optimal solutions are comparable for this size ratio (6.4:1) to the three other ratios (10.9:1, 8.7:1, 5.1:1).The standard deviation values using 5.1:1 data to predicted 6.4:1 percolation profiles were 0.061 and 0.048 for strain of 15% and 5%, respectively.The corre-sponding standard deviation values for optimal solutions are 0.009 and 0.007, respectively.While the magnitudes of standard deviation values when using 5.1:1  values is six-fold compared with optimal solutions, the absolute errors between measured and calculated values are approximately 5.5%.Such absolute errors are acceptable considering that no interpolation of parameter values was possible.Thus as a first approximation, 5.1:1 ratio parameters may be used to qualitatively assess the response of 6.4:1 size ratio mixtures.

CONCLUSIONS
A primary segregation shear cell (PSSC) was used to measure the material parameters of a convectivediffusive segregation constitutive model for percolation of fines through a bed of coarse particles.The effects of size ratio, strain, cycle speed, and bed depth on the percolation of fine particles in binary mixtures of glass spheres were modeled.Based on the tests conducted in this research, the following conclusions were made.
1.The convective-diffusive segregation constitutive model represented the normalized mass versus time relationships better than the convective model.On average, the range of standard deviation was 0.105 to 0.025, a 76% reduction, when comparing the PSSC convective model with the PSSC convective-diffusive model.This is due to the random mixing occurring during the diffusive behavior either during the entire duration of test or after the initial discharge.2. The convective-diffusive model was limited in predicting the percolation of the 6.4:1 ratio due to the absence of an initial discharge.Estimating the percolation for the 6.4:1 size ratio was accomplished by using the mean normalized mass vs.
time data of the 5.1:1 size ratio.The average data of the 5.1:1 size ratio reduced the standard deviation by 77% when comparing to size ratio linear interpolation using all three size ratios (10.9:1, 8.7:1, and 5.1:1).The initial rapid discharge present in 10.9:1 and 8.7:1 treatments increased the standard deviation of the prediction.3. A region of size ratios that defines an initial rapid discharge for binary mixtures of spherical glass beads lies between 6.4:1 and 8.7:1.

ACKNOWLEDGEMENT S
This study was partially funded by the U.  along the direction shown in Figure 3

Fig. 3 Fig. 4
Fig. 3 Schematic of global coordinates for the PSSC and the collection pan, underlined cell numbers are the measurement locations and h is the coarse material fill depth

Table 3
Validation schedule for binary mixtures of spherical glass beads* *Each combination was repeated four times

Table 4
treatments are given in Figures 5 through 7. Based on detailed analyses of experimental data by Duffy and Puri (2002), the normalized mass represents the preferred dependent variable rather than the absolute mass values.Therefore, normalized mass values are

Table 5
Convective-diffusive segregation constitutive model parameter values (µ/δ) at 10% strain with standard deviations between data and model in parentheses* * To convert µ to u θ , multiply the entries by 1.

Table 6
Convective-diffusive segregation constitutive model parameter values (µ/δ) at 5% strain with standard deviations between data and model in parentheses*

Table 7
Convective-diffusive segregation constitutive model parameter values (µ/δ) at 15% strain with standard deviations between model and data in parentheses* S. Department of Agriculture and Pennsylvania Agricultural Experiment Station.