Powder Technology 235 (2013) Contents lists available at SciVerse ScienceDirect. Powder Technology

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1 Powder Technology 35 (013) Contents lists available at SciVerse ScienceDirect Powder Technology journal homepage: Scale-up strategy for continuous powder blending process Yijie Gao, Fernando J. Muzzio, Marianthi G. Ierapetritou Department of Chemical and Biochemical Engineering, Rutgers, The State University of New Jersey, 98 Brett Road, Piscataway, NJ 08854, USA article info abstract Article history: Received 30 March 01 Received in revised form 9 June 01 Accepted 15 September 01 Available online 3 September 01 Keywords: Powder blending Scale-up Sample size RTD DEM Periodic section modeling Continuous powder mixing has attracted a lot of interest within the pharmaceutical industry. Much work has been done recently that targets the characterization of continuous powder mixing. In this paper, a quantitative scaling up strategy is introduced that allows the transition from lab to industrial scale. The proposed methodology is based on the variance spectrum analysis, and the residence time distribution, which are key indexes in capturing scale-up of the batch-like mixing, and scale-up of the axial mixing and motion, respectively. Our simulation results are used as preliminary guidance for scaling up different powder mixing cases. 01 Elsevier B.V. All rights reserved. 1. Introduction Powder mixing is a significant unit operation in food, cement, and many other industrial fields. In the pharmaceutical industry, the concentration variability of active pharmaceutical ingredient (API) in each tablet directly depends on the quality of powder mixing in upstream unit operations. Batch processing provides reliable quality and process control when small volumes of powder are processed. While a traditional powder mixing process is typically performed in small batches, the recent development of continuous powder mixing processes has attracted a lot of interest due to its capacity in handling high flux manufacturing. Reviews on both batch [1 3] and continuous powder mixing [4] reflect our current understanding on these topics. Previous work in the area of powder mixing scale-up is based on the principle of similarity proposed by Johnstone and Thring [5]. According to this principle, the material flow patterns in two scales of mixing apparatuses are expected to be similar when the dimensions, velocity fields, and force distributions of corresponding points are proportional to each other. Generally, dimensional analysis is applied to identify these similarity criteria if the process governing equations are unknown, while dimensionless forms are derived if the governing equations are available. Applying this principle, Wang and Fan [6] investigated the scaling up of tumbling mixers. Their results indicate that the ratio of centrifugal force to the gravitational force, or the Froude number, can be used as the main similarity criterion for the scale-up of a tumbling mixer. Based on hydrodynamic similarity, Horio et al. [7] developed a scale-up rule for fluidized beds when the similarities in Corresponding author. Tel.: ; fax: address: marianth@so .rutgers.edu (M.G. Ierapetritou). bubble coalescence, bubble splitting, and interstitial flow pattern are satisfied. Surface velocity profiles in rotating cylinders were found to follow simple scaling rules through the normalization of a one-dimensional motion equation with constant acceleration [8,9]. Of the few studies concerning the scaling up of continuous powder mixing processes, most efforts are spent on the characterization of the axial mixing component, namely, scaling up of the RTD (residence time distribution). Based on dimensional analysis similar to the one used in batch mode studies, Abouzeid et al. [10] applied the characteristic time (the ratio of dimensionless hold up and feed rate) as the main scale-up criterion for the RTD of materials flowing in rotating drums. The dimensionless variance of the RTD, which is defined as the second moment of the RTD function [11], was approximately calculated as a function of Peclet number using the Fokker Planck dispersion model [1]. Similar analysis was performed in the polymer processing of a continuous twin-screw extruder [13], on which the scale-up issues of geometry, heat transfer, and power consumption were discussed. A more complex scaling relationship for rotating drums was developed for particles without considering cohesion and only for cases where the effect of interstitial fluid is negligible [14]. In this study, parameters and variables were normalized using governing equations based on the Eulerian approach, which assumes that the properties of the granular materials are a continuous function of position [15,16]. Despite the achievements described above, there are still two unresolved issues for developing a quantitative scale-up strategy for continuous powder mixing processes. Firstly, although it is generally believed that similar mixing performances can be achieved in geometries of different scales when the principle of similarity is satisfied [5,6], theoretical validation is not available yet. There is no doubt that we can get similar segregated structure in the blending process with a /$ see front matter 01 Elsevier B.V. All rights reserved.

2 56 Yijie Gao et al. / Powder Technology 35 (013) satisfied principle of similarity. However, the key difficulty is that in most cases the sampling size applied for mixing of different geometry scales is constant, as the size of samples is predetermined by the size of the tablets, which is the final product of pharmaceutical manufacturing process. This constant-sample effect on measuring the mixing performance is rarely considered in previous scale-up studies. Secondly, few investigations focus on continuous powder mixing compared to the well-studied batch mixing process. While it is well-known that the continuous powder mixing process is a combination of axial and cross-sectional mixing [17], quantitative modeling for both mixing components, especially for the cross-sectional mixing, has not been well established. Thus, development of quantitative scale-up strategies is difficult for the same process. In this work, the theoretical basis of scaling up continuous powder mixing is established by solving the two issues described above. For the first issue, variance spectrum analysis is introduced. Since it clearly illustrates the influence of sampling size on the relationship between flow pattern similarity and mixing performance similarity, it provides a theoretical means to analyze the scale-up of batch mixing processes and the scale-up of the cross-sectional mixing in continuous mixing processes. For the second issue, the periodic sectional modeling proposed in our previous work [18] is applied to characterize the cross-sectional mixing component, which is considered as the combination of a batch-like mixing process and the axial motion of the powder along the continuous mixer. The rest of the paper is organized as follows. In the next section, modeling of the continuous powder mixing process, the variance spectrum, and the simulation designs are introduced. Two methodologies to determine the variance spectrum, the Fourier series and the Lomb Scargle Periodogram, are described. Conceptual simulations with typical mixing cases of non-segregating and cohesive segregating particles are then performed in Section 3 to show the applicability of the developed scale-up strategy, followed by the conclusion in Section 4.. Proposed scaling-up methodology.1. Modeling of continuous powder mixing The continuous powder mixing process consists of axial and cross-sectional mixing. Since the axial mixing component can be analyzed through the attenuation of feeding fluctuations by the axial dispersive effect of residence time distribution [19 1], scaling up of this mixing component involves the quantitative interaction of feeding fluctuations and RTD when feed rate increases. However, as the focus of this work is on scaling up of the mixing process, the effects of feeding are not discussed. As a result, only the characteristics of residence time distribution are considered in the scaling up study of the axial mixing component. For the scaling up of the cross-sectional mixing component, which is the mixing of initially segregated materials fed continuously into the system, the periodic section modeling proposed in our previous work [18] is applied. The model considers cross-sectional mixing in the continuous mixing process in a cylindrical vessel as a batch-like mixing process, superimposed by the convective dispersive axial motion of particles. A periodic section that shares the same geometry, operation, fill level, flow rate and thus the same mixing efficiency with the whole mixer is used to estimate the batch-like mixing process. The following empirical relationship is proposed to predict the variance decay along the mixing axis of the whole continuous mixer: σ c ðþ¼ x 0 σ bðþet; t ð xþdt ð1þ where E(t, x) is the mass-based RTD measured between the inlet of the continuous mixer and the transverse plane at location x; σ c (x) and σ b (t) are the location dependent variance decay curve along the axis of the continuous mixing, and the time dependent variance decay curve of the batch-like mixing process in the periodic section, respectively, both measured using the same sampling size. The cross-sectional mixing component of a continuous mixing system is equivalent to a batch mixing system, which shares the same mixing efficiency with the continuous system and is operated for a time equal to the residence time distribution of the system. Notice that since the mixing process is affected by the powder fill level in the mixer[ 4],the fill level in the periodic section should be consistent with that of the whole mixer, which leads to thesamemixingefficiency. In sum, the variance decay in the batch-like mixing of the periodic section, as well as the axial motion reflected in the residence time distribution, is the major factor in characterizing the continuous mixing process. Successful scale-up of both factors is a prerequisite to developing a reliable overall scaling up strategy. On the other hand, it is also important to understand the effect of sampling size on scale-up of mixing processes, as mentioned in the Introduction. Therefore, the variance spectrum is introduced next... Variance spectrum analysis..1. Fourier analysis Variance spectrum is defined as the distribution of variance components in different frequency domains, which can be obtained through the application of Fourier transform. Shin and Fan [5] used the application of discrete Fourier transform (DFT) to investigate the characterization of random and ordered powder mixtures. While the amplitude of the spectrum indicates the intensity of segregation at different frequency domains, the location of the maximum spectrum component illustrates the scale of segregation intuitively [,6]. For simplicity, let us consider a one-dimensional discrete mixture structure x n, where n is the index of evenly sampled points ranging from 1 to N. The interval between adjacent points is Δx. Notice that the series of x n is centered on zero. The discrete Fourier transform X k of the kth frequency and the corresponding variance spectrum s(f) can be expressed as [7]: X k ¼ a k þ ib k ¼ N XN 1 n¼0 sf ðþ ¼ skδf ð Þ j ¼ X kj Δf x n e πi N kn ; k ¼ 1; ; N 1 where Δf=1/NΔx is the minimum detectable frequency interval in the discrete Fourier transform X k (NΔx is the overall scale of the mixture), and i represents the imaginary unit. To calculate the variance of the mixture, the sampling size cannot be smaller than one single point. If more than one point is collected in a sample, we consider that this mixture is sampled at frequency f s, where the value of 1/f s indicates the number of adjacent points collected in one sample. The corresponding variance value σ (f s ) is calculated as the integration of the spectrum from zero to half of the sampling frequency: σ ðf s Þ ¼ 1 f s = 0 sf ðþ df : Eqs. () (4) indicate the relationship between variance spectrum, sampling size and the measurement of variance. The variance spectrum is only determined by the structure of the mixture, while the measured value of the variance depends on the sampling frequency. Therefore, it can be expected that: (i) mixtures with similar structures but different scales will share similar variance spectra; (ii) mixing processes of different scales will share similar amounts of decay within the variance spectra when the principle of similarity is satisfied; and (iii) before the interaction of sampling frequency and variance spectrum decay is clarified, we cannot claim that similar flow patterns and segregated blend structure always lead to similar variance decay when a constant ðþ ð3þ ð4þ

3 Yijie Gao et al. / Powder Technology 35 (013) sampling size is used. Thus, the constant-sample effect of scaling up needs further investigation. By using the one-dimensional discrete Fourier transform (1D-DFT), Shin and Fan [5] theoretically confirmed the linear relationship between the decay of the spectrum and the variance for mixing processes obeying the Fickian diffusion equation. However, this 1D-DFT is not applicable in the study of practical powder mixing processes due to the following reasons. First, almost all industrial processes of powder mixing are three-dimensional, which cannot be characterized by one-dimensional Fourier analysis. Secondly, the definition of Fourier analysis requires evenly distributed samples with equal sampling sizes in a cube-shape space. This is usually not satisfied in practice, since most powder mixing spaces are not cube-shaped, and the rotating motions of blades (convective mixers) or mixing apparatuses (tumbling mixers) hinder the formation of evenly distributed samples through mixing processes. To overcome the first problem, the three-dimensional discrete Fourier transform (3D-DFT) is introduced [8]: X h;j;k ¼ a h;j;k þ ib h;j;k ¼ N 3 1 XN 1 XN 1 l¼0 m¼0 n¼0 X sðþ f ¼ sf ð 1 ; f ; f 3 Þ ¼ shδf ð ; jδf ; kδf Þ h;j;k ¼ Δf 3 x l;m;n e πi N ðhlþjmþknþ ð5þ where l, m, n are the indexes of evenly sampled points within the three dimensions of the space domain, and h, j, k are the indexes of frequency components in the corresponding dimensions of the spatial-frequency domain. The overall variance measured at the sampling frequency f s 3 (the corresponding sampling size is 1/f s 3 in unit of volume) can be expressed as σ f 3 s ¼ 1 f 1 þf þf 3 3 f s sðþ f df 1 df df 3 where f 1,f,f 3 [0,NΔf], and the limit 3f s / is applied to eliminate the effect of aliasing [18]. As indicated in Eqs. (6) and (7), the 3D variance spectrum s(f) itself is a three-dimensional function. This makes it difficult to visualize the four-dimensional profile s(f,t) in the mixing process when we try to capture the spectrum decay as a function of time. Therefore, it is necessary to convert the 3D variance spectrum to a 1D spectrum by averaging the spectrum components with the same magnitude of frequency. In this work, the variance spectrum values s(f) following Eq. (8) are considered to share the same magnitude of frequency f: f 1 þ f þ f 3 ¼ 3f : The conversion of the variance spectrum is then expressed as follows: sf ðþ ¼ σ f 3 s f 1 þf þf 3 ¼3f f 1 þf þf 3 ¼3f sðþ f ds ¼ 1 ds wf ðþ ¼ 1 f s = 0 sf ðþ wf ðþdf f 1 þf þf 3 ¼3f sðþ f ds ð6þ ð7þ ð8þ ð9þ ð10þ where w(f) is the weight of the averaged 1D variance spectrum s(f) ; ds indicates the infinitesimal area on the plane indicated by Eq. (8) in the averaging process. By using Eq. (9) instead of Eq. (6), the variance spectrum decay s(f,t) can be visualized in a contour plot. The second issue regarding the application of a Fourier transform is that the particles are not evenly distributed. To overcome this problem, the Lomb Scargle Periodogram is introduced in the following subsection.... Lomb Scargle Periodogram (LSP) To characterize stellar cycles where the observation points are usually unevenly spaced, Lomb Scargle Periodogram was first proposed in the analysis of astronomical data in order to detect periodic signals hidden in noise [9]. The Lomb Scargle Periodogram is equivalent to a linear least-square fit of sine and cosine functions to the observed signal series, which is an alternative to the Fourier series in estimating the profile of variance spectra [30]. For one-dimensional unevenly sampled data (t n x n ), where t n indicates the location of sample x n, its linear least-square fit of sine and cosine functions can be expressed as follows: a x f ðþ¼a t cos ωt þ b sin ωt ¼ ½ cos ωt sin ωt Š ð11þ b EðωÞ ¼ XN h i : x n x f ðt n Þ ð1þ Notice that x n is also with mean of zero. To determine the value of E(ω) min for the best fitting result, the following relationship should hold: E a ¼ E b ¼ 0 ð13þ and the difference of the least square fitting is calculated as follows: ΔEðωÞ ¼ XN ½Xt ð n ÞŠ E min ðωþ: ð14þ Note that ΔE(ω)/N represents the variance component measured at frequency ω. Based on Lomb and Scargle's derivation [31,3], an estimation of the variance spectrum s(f) can be expressed as follows: 3 sf ðþ ¼ ΩΔf 6 4 τ ¼ 1 ω tan x n cos ωðt n τþ cos ωðt n τþ 3 sinωt n 7 5 cosωt n! þ x n sin ωðt n τþ sin ωðt n τþ! 7 5 ð15þ ð16þ where ω=πf is the angular frequency, and Ω represents the dimension of the variance spectrum analysis [33]. τ is a redundant parameter to simplify the formula of the solution. The value of τ is chosen so that cos ωðt n τþsin ωðt n τþ ¼ 0: ð17þ Eqs. (11) (17) represent the generation of original formula of the Lomb Scargle Periodogram which is widely used in astronomical research. To apply this concept for the analysis of powder mixing, two aspects of the mathematical formula should be modified to account for a three-dimensional problem structure and an uneven sampling size distribution. To achieve the first modification, all the scalar product ωt n in previous equations are replaced by a dot product of two vectors ω t n,where ω=πf (ω 1, ω, ω 3 ) is the angular form of the three-dimensional spatial frequency, and t n (t n1, t n, t n3 ) is the three-dimensional location of the nth sample unevenly taken in the three-dimensional mixture. The modified

4 58 Yijie Gao et al. / Powder Technology 35 (013) Eqs. (15) and (16) for three-dimensional analysis are then expressed as follows: sðþ f ¼ Ω 3 Δf θ ¼ 1 tan x n cos ðω t n θþ cos ðω t n θþ! þ x n sin ðω t n θþ sin ðω t n θþ! ð18þ 3 sinω t n 7 5 : ð19þ cosω t n To achieve the second modification, the generalized Lomb Scargle Periodogram (LSP) [34] is applied in this work, which introduces the weight of samples in the calculation of variance spectrum. Eq. (1) is thus modified as follows: EðωÞ ¼ XN h i w n x n x f ðt n Þ ð0þ where w n represents the weight of the nth sample of the threedimensional mixture. When uneven sampling size distribution is present, which is the case in powder mixing, samples with different sampling sizes are assigned different weights. For the samples with the desired sampling size, we set w n =1; for the samples with sampling size far away from the desired sampling size, we set w n =0. A simple linear weight distribution is used in this study: the sample weight increases in a manner proportional to the sampling size until the desired sampling size is reached, and then decreases in a similar manner. By introducing the sample weight, the 3D formula of LSP in Eqs. (18) and (19) is now expressed as follows:!! 3 sðþ f ¼ x n w n cosðω t n θþ x n w n sinðω t n θþ þ Ω 3 Δf 3 6 w n cos 7 4 ðω t n θþ w n sin ðω t n θþ 5 θ ¼ 1 tan w n sinω t n 7 5 w n cosω t n ð1þ ðþ which are the final formulas of LSP that are ready for characterizing the variance spectrum decay in powder mixing processes. Ω 3 represents the dimension of the 3D variance spectrum, and Δf the minimum detectable frequency under the current uneven sampling condition. As mentioned before, the method of Lomb Scargle Periodogram is an estimation of the variance spectrum for the case where the standard Fourier analysis cannot be applied. When fewer samples are retrieved, less information is available in the investigation of the mixture structure, which leads to large deviation between the LSP and the real spectrum. Previous studies on astronomical fields have investigated the statistical significance of the spectrum obtained through the LSP method [35 37]. However, these works are mainly focused on the ability of the method in detecting solar cycles, namely, how to distinguish single spectrum peaks from the original intensity measurement with a noisy background. As the variance components in most of the frequency domains are considered as white noise and are useless information, their role as the spectrum distribution of variance is seldom referred. As a result, we only summarize a few qualitative conclusions from these works that are useful for our study: (i) When the mixture is ideally random, the variance spectrum of the mixture is the same everywhere for both the DFT and the LSP method. In this case, the corresponding variance measurement is proportional to the integration interval in Eq. (4) from zero to half of the sampling frequency, thus being anti-proportional to the sampling size. (ii) When the concentration in the mixture has a sinusoidal distribution, the variance spectrum derived through DFT shows a sharp peak at the frequency of the sinusoid curve, while its LSP estimation is a smear of the peak at the same frequency. The fewer the samples that are used in the LSP method, the more smears the peak looks like. This smear results from the lack of mixture information in the cube-shape characteristic space. In other words, in practical systems, a large fraction of the segregated structure within the blend should be sampled experimentally to perform variance spectrum analysis. This can be achieved by freezing and slicing the particle bed when the blender itself can be sacrificed, or we can pour melted paraffin wax onto a metal plate support in the studies of large vessels [38]. Examples of these characteristics of the LSP method are discussed within the results in later sections..3. Simulation experiments.3.1. Simulation environment As mentioned in Section.1, periodic section modeling is used in the study of scaling up continuous mixing processes. Due to the difficulty in realizing the periodic section in practice [18], the Discrete Element Method (DEM) is applied as the simulation environment for examining the scaling up issue, which refers to a family of numerical methods for computing the motion of large numbers of particles by solving Newton's equation of motion for each particle. The approach has been used in our previous studies [39,40], as well as in a number of previous efforts [4,41,4] and was found to be able to accurately capture the dynamics of mixing processes. In this work, the software EDEM by DEM Solutions was used to develop the desired geometries and mixing operations. Sphere particles are simulated, where the contact forces are calculated using the default Hertz Mindlin no slip contact model [43].The physical and numerical parameters used in this simulation are summarized in Table 1. The values are chosen so that simulations can reproduce experimental trends in previous studies [18,,44 48]. The computation time is about 1 h of CPU time with a Dell Precision Table 1 Physical and numerical parameters used in the DEM model. Parameter Cohesiveness Value Particle 1 Particle Particle 3 No-cohesive k c1 =0 J m 3 Particle diameter m Particle density kg m 3 Poisson's ratio 0.5 Shear modulus Pa Coefficient of 0.5 restitution Coefficient of static 0.5 friction Coefficient of rolling 0.01 friction Simulating time step 10 6 s Recorded time step 0.01 s Sampling time 0.1 s interval Self-cohesive with k c = J m 3 Self-cohesive with k c3 = J m 3

5 Yijie Gao et al. / Powder Technology 35 (013) Workstation T7400 computer with a.50 GHz Quad Core Intel Xeon Processor E540 for 1 s of simulation time..3.. Simulation design In order to investigate the effects of different mixing behaviors on the scaling up of continuous mixing processes, three kinds of particles with the same diameters and densities were used in the simulations (Table 1). In the present study, we investigate two case studies: in the first one a mixture consisting of only particle type 1 having different colors is considered whereas in the second case a mixture of particle types and 3 is considered. The first case is representative of a non-segregating mixture, while in the second case the two types of cohesive particles (types and 3) will tend to agglomerate with only the same type of particles, which results in segregation. A linear cohesion model is used here based on the Hertz Mindlin no slip contact model, where the cohesive force can be represented by the following equation: F c ¼ k c A ð3þ where F c represents the cohesive force between two particles, A is the contact area, and k c represents the coefficient of cohesion. The values of k c for the three types of particles are listed in Table 1. Thevalueof k c is selected so that both the cohesive segregation and the blending process can be easily observed in the simulations, and the corresponding variance spectrum including both features can be generated. The simulation design is shown in Fig. 1. Two periodic sections with geometries that are similar in shape and scale are constructed within the DEM environment, while the 10 mm-scale section has a size that is 1.5 fold larger than the 80 mm-scale one in all dimensions. Details of the geometry of the 80 mm-scale periodic section are plotted in Fig.. Twofill levels (5%, 50%) were investigated for both geometries. Inordertounderstandtheinfluence of the initial scale of segregation on the scaling up issue, two loading patterns, the half loading and the quarter loading were tested. An overall 50:50 volumetric ratio of particles Fig.. Geometry of the 80 mm scale periodic section. Blades rotate counter-clockwise and push particles forward along the x axis. (a) Isometric view. (b) Side view with displayed dimensions. was loaded for all the simulations (Table ). The empirical principle of similarity was also considered, which requires a constant Froude number for the mixing of different scales. Based on the expression of Froude number Fr=ω r/g, whereω is the angular velocity of blade rotation, r the radius of blade and g the gravity, 0 RPM is set as the blade speed for all the 80 mm scale mixing cases, while 180 RPM is set for all the 10 mm scale simulations. This leads to a constant Froude number value of 4.35 in both mixing scales..4. Mixing characteristics.4.1. Variance spectrum The variance spectrum of mixing in the studied periodic sections was calculated as follows. Cubes were divided with a side length of 8 mm or 1 mm for the mixing cases of 80 mm and 10 mm scales respectively, which leads to a sampling matrix of for the mixing on both scales. For each time point, all the cubes with at least one particle are counted as samples, and the corresponding location t n, zero-centered concentration x n are recorded and calculated in sequence. The reciprocal of the mixing scale Δf=1/scale, the maximum detectable frequency ΩΔf, and the weight of sample w n, which is the ratio of the current sampling size and the maximum number of particles randomly packed in the same cube space, are also calculated. By using Eqs. (1) and (), the four-dimensional variance Table The number of initial loading particles for both mixing cases. Fig. 1. Illustration of simulation designs. Fill level Particle number 80 mm scale 10 mm scale Blue (type 1 or ) Red (type 1 or 3) Blue (type 1 or ) 5% % Red (type 1 or 3)

6 60 Yijie Gao et al. / Powder Technology 35 (013) spectra decay s(f, t) as a function of time can be calculated on the interval [0, ΩΔf] for all three dimensions, where Δf is set as the resolution in the frequency domain for this LSP calculation. The 1D formula s(f, t) of the 3D-LSP profile is then derived using Eq. (9), which is shown in the results section..4.. Variance and RSD To determine the time-dependent batch-like mixing in the periodic section, samples are retrieved inside the mixing space. The mixing space is divided into cubes with the side length of 10 mm, and all the samples in a time interval of 0.1 s of the simulation process were considered as samples of the same time point. In order to gather enough samples (more than 80) for each time point, samples with 1 particles were selected corresponding to the number of particles that can be randomly packed within the sampling size of 0.45 cm 3. Notice that this sampling size is comparatively smaller than in practice. Based on the expression of variance σ, the relative standard deviation (RSD) of blue particles of each time point can be calculated as follows: RSD ¼ σ Standard deviation C ¼ Average concentration σ ¼ i¼1 C i C N 1 ð4þ where C is the average mass concentration of N samples, and C i is the concentration of each sample. In order to characterize the batch-like mixing process with fewer parameters, the variance profile σ b (t) is modeled by the exponential decrease relationship applied in many of our previous works [,44,45,49], σ bðþ¼σ t SS þ σ 0 σ SS expð k b tþ ð5þ where σ SS and σ 0 are the steady state variance of the investigated sampling size, and the initial variance when particles are totally segregated, respectively; and k b is the variance decay rate of the batch mixing process. For 50:50 volumetric mixing ( C ¼ 0:5), σ 0 can be considered as equal to 0.5 using the following equation: σ 0 ¼ C 1 C ð6þ which leads to the initial RSD value of RSD 0 =1 by applying Eq. (4) Residence time distribution The method of determining the residence time distribution (RTD) can be found in our previous work [7]. Instead of using the pulse test method by Danckwerts [19], RTD was measured by considering the trajectory of particles inside the flow of the periodic section. When a studied particle reaches the right side of the section, it reappears at the other side. In this study, the time interval between seven transverses at one end of the periodic space is used as one sample, which is applied to calculate the RTD equivalent to that of a continuous mixer with six periodic sections linearly connected. The RTD can be derived using the following equation: n i Et ð i Þ ¼ n i Δt i¼1 ð7þ where Δt is the sampling time interval, and n i the number of samples in which the residence time of each sample lies in (t i, t i +Δt). To eliminate noise from the measurements of RTD, around 6000 samples of particle trajectory were retrieved for each RTD measurement. The Taylor dispersion model is introduced to clarify the physical meaning of the RTD [3,50,51]: Eðξ; θþ ¼ Pe1= Pe exp ð ξ θ 1= ð4πθþ 4θ Þ! ð8þ Fig. 3. Variance spectrum decay contours of non-segregating mixing cases. The left four plots (a, c, e, g) show the 80 mm scale mixing cases. The right four plots (b, d, f, h) show the 10 mm scale mixing cases.

7 Yijie Gao et al. / Powder Technology 35 (013) where θ=t/τ, and ξ=x/l represent the dimensionless time, and location in RTD measurement, respectively; τ is the mean residence time and l the length of the studied flowing system; Pe=v x l/e x represents the Peclet number; v x and E x are the axial velocity and dispersion coefficient of the system. Based on the Taylor dispersion model, the statistical RTD data is fitted to get a smoothed curve, which will be applied in the scaling up analysis of axial mixing and motion in the next section. 3. Simulation results 3.1. Cross-sectional mixing Non-segregating particle mixing case Fig. 3 refers to the overall variance spectrum decay contours s(f,t) of non-segregating mixing cases. The left four graphs show the spectrum decay of the 80 mm scale cases, while the right four show the 10 mm scale cases. The maximum detectable frequencies ΩΔf of the 80 mm and 10 mm scales are 0.15 mm 1 and mm 1, respectively. The limits of the vertical axes are then selected as [0, 0.09]mm 1 and [0, 0.06]mm 1 in order to cover the main decay characteristics of the two scales. Notice, that a constant Froude number is used, which indicates different blade speeds between the two mixing scales. For comparison purposes, the mixing processes are measured using revolution of blade rotation instead of time. An accurate ratio of spectrum amplitude can be expected between the two scales based on the qualitative conclusion (ii) in Section.. Since mixing on both scales shares the same initial variance and similar initial variance spectra, we can claim the following Eq. (10): s 1 ðf 1 Þ s ðf Þ ¼ scale3 1 scale 3 ð9þ where f 1 /f =scale /scale 1. Based on Eq. (9), the color maps of the contours in Fig. 3 are set with a ratio of (10/80) 3/ =1.84 for the two scales, namely, the color map interval [0, 60] for the 80 mm scale and the interval [0, 110] for the 10 mm scale. A level step of 6 or 11 is applied for the corresponding color map so that the spectra with different amplitudes can be visually compared. In order to identify mixing similarity of the two scales, the left four plots in Fig. 3 are compared with the right four plots. Although the initial amplitudes are different, the variance spectrum decay processes of the two scales are similar for all fill levels and initial loadings. Therefore, we can state that similar flow patterns due to a satisfied principle of similarity lead to similar decay processes of variance spectra for the non-segregating mixing cases. To further illustrate this observation, we plot the initial spectrum profile (at the beginning of the first revolution) and the steady state spectrum (at the end of 18 revolutions) in Fig. 4. The vertical and horizontal axis ratios are the same as in Fig. 4. As shown in the plots,the initial variance spectra are very similar for mixing on different scales. When a steady state is reached, the same horizontal spectra, representing ideal random mixtures, are obtained for all plots (dot lines, Fig. 4). Figs. 3 and 4 also show the effect of loading pattern and fill level on the initial variance spectrum. A lower peak for the initial spectrum is found in the quarter loading than the half loading patterns, while the frequency of the peak in the quarter loading is twice that of the half loading. This can also be explained using Eq. (10), since the weight w(f) of the 1D converted variance spectrum s(f) increases with the magnitude of frequency f. While a constant overall variance is kept for different loadings, a smaller peak is expected when it is in a higher frequency domain. When comparing different fill levels, fewer smears occur in the spectra with higher fill levels, due to more samples being obtained in that condition. This observation corresponds with qualitative conclusion (iii) in the previous section. Fig. 4. The initial and steady state profiles of variance spectrum of the non-segregating mixing cases. (a) 80 mm scale at 5% fill. (b) 10 mm scale at 5% fill. (c) 80 mm scale at 50% fill. (b) 10 mm scale at 50% fill. Half: half loading. Quarter: quarter loading. rev: revolution.

8 6 Yijie Gao et al. / Powder Technology 35 (013) An interesting phenomenon is observed in Fig. 3 when comparing the spectrum decays for different loading patterns. For the half loading cases, the decay of the spectrum peak is always at the lowest detectable frequency f=δf, which indicates a decreasing intensity of segregation with the scale of segregation unchanged. However, when the loading is the quarter pattern, the spectrum peak decays as well as shifts from the second lowest frequency domain f=δf to the lowest one. Therefore, it can be expected that no matter how the initial variance spectrum appears, the scale of segregation always tends to approach the scale of the overall mixing space in the process of mixing. Due to a similar mechanism of mixing, the same phenomenon is also expected in tumbling and many other mixing processes. Although similar contours are observed in Fig. 3, quantitative criteria are required to estimate the similarity of the corresponding variance spectrum decays of the two scales. The main decay processes of the spectrum occur at the lowest and the second lowest frequency domains in the current simulations. Thus, the decay curves on both frequency domains are applied to quantify the similarity of spectra decay. To eliminate the disturbance of amplitude discrepancy of the two scales, the spectrum decay should be normalized first. The standard Z score is used here for this normalization: Z ¼ X μ σ ð30þ where X is the original data, μ and σ the mean and standard deviation of X. Once the normalized spectrums are obtained, a linear regression between points of corresponding spectra is performed. The coefficient of determination R,andthepairedt-test are then calculated to indicate the similarity of the original curves. Based on this method, the comparison of normalized variance spectra is plotted in Fig. 5, with the criteria of similarity listed in Table 3. Since the normalized spectrum decay Table 3 Similarity criteria of the standard scores, non-segregating mixing cases. Fill level Loading Frequency h p-value C.I. (y 1 y ) R 5% Half f=δf f=δf Quarter f=δf f=δf % Half f=δf f=δf Quarter f=δf f=δf curves fit each other very well for both frequency domains, no significant difference is found between the curves for two scales using the paired t-test (h=0 for all cases), indicating similar decay processes in all non-segregating mixing cases. The RSD-t curves measured at the sampling size 0.45 cm 3 are plotted in Fig. 6. Based on the number of particles in each sample, the steady state RSD value of the mixing process can be calculated using the binominal theorem []. Specifically, the steady state RSD is approximately 0. for the sampling size of 0.45 cm 3 (1 particles per sample). As shown in the plots, mixing is significantly faster in the quarter loading pattern, while it is slightly slower when the fill level is increased. However, the RSD decay curves almost overlap with each other for the two scales of mixing for all conditions, which illustrates good scaling characteristics on mixing performance under current sampling size. To clarify the constant-sample effect we described in the Introduction, the effect of spectrum integration interval on the corresponding RSD decay curve is investigated. The half-loading mixing at 5% fill level (Fig. 3a and b) is applied for this illustration. The same limits of vertical axis of frequency are plotted for both scales ([0, 0.1]). The RSD decay curves measured at a larger sampling size (3.6 cm 3 )arealso Fig. 5. The standard score of spectrum decay curves at the lowest two frequency domains, non-segregating mixing cases. (a, c, e, g) f=δf. (b, d, f, h) f=δf. (a, b) 5% fill, half loading. (c, d) 5% fill, quarter loading. (e, f) 50% fill, half loading. (g, h) 50% fill, quarter loading.

9 Yijie Gao et al. / Powder Technology 35 (013) Fig. 6. RSD-t decay curves measured at sampling size of 0.45 cm 3, non-segregating mixing cases. (a) 5% fill, half loading. (b) 50% fill, quarter loading. (c) 50% fill, half loading. (d) 50% fill, quarter loading. involved besides the 0.45 cm 3 cases.theresultsareshowninfig. 7. Whenthesamplesizeof0.45cm 3 is applied, the integration interval of s(f,t) in the variance calculation is [0, 0.065]mm 1. This interval covers all the spectrum decay components for the mixing of both scales (Fig. 7a). As a result, well fitted RSD decay curves are observed. In contrast, the integration interval in the variance calculation is [0, Fig. 7. Illustration of the constant-sample effect on the scaling up of powder mixing. (a) Sampling size of 0.45 cm 3. (b) Sampling size of 3.6 cm 3. Different integration coverage on the main spectrum decay components produces different RSD-t curves in the case of 3.6 cm 3 sampling size.

10 64 Yijie Gao et al. / Powder Technology 35 (013) Fig. 8. Variance spectrum decay contours of cohesive segregating mixing cases. The left four plots (a, c, e, g) show the 80 mm scale mixing cases. The right four plots (b, d, f, h) show the 10 mm scale mixing cases. The same level step of color map is used for all contours to show the spectrum profile of segregation when steady state is reached. Fig. 9. The steady state profiles of variance spectrum of the cohesive segregating mixing cases. (a) 80 mm scale at 5% fill. (b) 10 mm scale at 5% fill. (c) 80 mm scale at 50% fill. (b) 10 mm scale at 50% fill. Half: half loading. Quarter: quarter loading. rev: revolution.

11 Yijie Gao et al. / Powder Technology 35 (013) ]mm 1 for the case of 3.6 cm 3 sampling size. In this case, the integration interval covers almost all the main decay components for the 10 mm scale mixing, while some components in higher frequency domains are missed in the case of the 80 mm scale. This leads to different fitted RSD decay curves (Fig. 7b). Therefore, with the principle of similarity satisfied, similar variance decay process can be achieved only when the sampling size is much smaller than the scale of both mixing geometries in the scale-up process Cohesive particle mixing case Fig. 8 shows the overall variance spectrum decay contours of cohesive segregating mixing cases. In order to visualize spectrum profiles of cohesive segregation, the same level step value of the color map (6 mm 3/ ) is used for both scales. As previously, similar spectra decays are found when the contour regions at high amplitudes are compared. On the other hand, when steady state is reached, segregation can be observed, the intensity of which is mainly located in the frequency domain interval between 0.01 and 0.05 mm 1 for all mixing cases. The steady state profiles of variance spectra are plotted in Fig. 9. The black dot lines indicate the horizontal spectrum of an ideal random mixture. Due to the higher spectrum in the cohesive cases, variance values larger than the one calculated by binominal theorem will be observed. As Fig. 9 shows, the steady state spectra of cohesive segregation are scarcely influenced by fill level, loading pattern or mixing scale. The corresponding snapshots of steady state mixture can be seen in Fig. 10, where agglomerations with the same scale are found in all mixing cases. The standard scores of the two lowest spectra components are plotted and compared in Fig. 11, with the similarity criteria listed in Table 4. Although some slight differences can be visually observed in Fig. 11a, b, c and g between the decay of the two scales, no significant difference is claimed by the paired t-test, the same as previous results. Corresponding to the standard score results, similar decays of RSD curves are observed in Fig. 1 for all cases despite some insignificant difference between fitted curves. Notice the larger value of RSD around 0.40 compared to the binominal theorem calculation (0.) as has been discussed above. 3.. Axial mixing and motion The axial mixing and motion of the two mixing cases are shown in Figs. 13 and 14. Fig. 13 shows the original data and the fitted curves of the residence time distribution measured using the method described in Section.4. The plots in this figure are rescaled using the unit revolution instead of time for comparison purposes. Since the residence time distribution does not depend on loading pattern, only the cases of half loading are plotted. As illustrated in Fig. 13a and b, the RTD in non-segregating cases is similar for both fill levels, indicating good scaling up characteristics of non-segregating particles on axial mixing and motion. In contrast, Fig. 13c and d shows larger RTDs of cohesive mixing in the 80 mm scale mixing, while RTDs in the 10 mm scale mixing cases are similar to those in the non-segregating cases. Therefore, the effect of cohesion on the axial motion of particles is significant only when the scale of mixing space is small, under which condition the cohesive particles move much slower than the non-segregating particles. To further illustrate this, the mean residence time and mean axial velocity are calculated from the fitted RTD curves, which are plotted in Fig. 14. While the mean residence time is slightly shorter at 5% fill in Fig. 10. The snapshots of steady state mixture in the cohesive segregating mixing cases. (a) 80 mm scale at 5% fill. (b) 80 mm scale at 50% fill. (c) 10 mm scale at 5% fill. (b) 10 mm scale at 50% fill. Agglomerations of the same scale are observed for all four cases.

12 66 Yijie Gao et al. / Powder Technology 35 (013) Fig. 11. The standard score of spectrum decay curves at the lowest two frequency domains, cohesive segregating mixing cases. (a, c, e, g) f=δf. (b, d, f, h) f=δf. (a, b) 5% fill, half loading. (c, d) 5% fill, quarter loading. (e, f) 50% fill, half loading. (g, h) 50% fill, quarter loading. non-segregating mixing cases, particles stay much longer in 80 mm scale mixing space in the cohesive cases, especially when the lower fill level is used. It is worth to note that the axial velocity measured in 10 mm scale is around 1.5 fold of that in the 80 mm scale due to the scale ratio of mixing space as indicated in Fig. 14c andd Flow rate evaluation For similar mixing geometries with scale ratio k and constant Froude number the flow rates can be estimated as follows: r ¼ kr 1 ; l ¼ kl 1 ω ¼ k 0:5 ω 1 : ð31þ ð3þ Since the mean residence time distribution measured by revolution (ωl/v x ) is kept the same in the process of scaling up, it follows that: v x ¼ k 0:5 v x1 : ð33þ The scaling relationship of flow rate can then be derived using the formula F=πv x r and be expressed as follows: F ¼ k :5 F 1 : ð34þ Therefore, when a four-fold scaling up is applied on the mixing apparatus, the flow rate of the larger continuous mixer is 3 fold of the original lab scale apparatus, which indicates a tremendous increase in production rate for the continuous powder mixing system. This shows the merit of using the model for continuous powder mixing on the scaling up characteristics. Similar analysis can be performed on the batch mixing systems, where the production rate can be measured as F=l 3 k b where l 3 represents the batch volume and k b the mixing rate described in Eq. (5). Since similar mixing rates, measured by the reciprocal of revolution (k b /ω), are achieved, the scaling relationship of mixing rate is expressed as k b ¼ k 0:5 k b1 : ð35þ Table 4 Similarity criteria of the standard scores, cohesive segregating mixing cases. Fill level Loading Frequency h p-value C.I. (y 1 y ) R 5% Half f=δf f=δf Quarter f=δf f=δf % Half f=δf f=δf Quarter f=δf f=δf As a result, a scaling up expression of production rate the same as Eq. (34) is derived for the case of batch mixing, indicating the inherent coincidence of batch and continuous mixing processes. For the cohesive cases, Eq. (34) can be directly used for the scaling up of batch-like powder mixing process, due to the scaling relationship observed in Figs However, larger flow rate will be observed in the scaled-up continuous powder mixing process, due to a relatively fast axial velocity in that case. This effect should be considered in the design of scaling up strategy of cohesive mixing cases.

13 Yijie Gao et al. / Powder Technology 35 (013) Fig. 1. RSD-t decay curves measured at sampling size of 0.45 cm 3, cohesive segregating mixing cases. (a) 5% fill, half loading. (b) 50% fill, quarter loading. (c) 50% fill, half loading. (d) 50% fill, quarter loading. 4. Conclusions A new methodology based on the concept of variance spectrum has been developed for studying the scale-up of powder mixing in continuous processes. The Lomb Scargle Periodogram technique widely applied in astronomical studies has been used to estimate the variance spectrum of unevenly distributed samples in powder mixing processes. The periodic section modeling, proposed in our previous work [18],permits us to consider the batch-like cross-sectional mixing and axial motion separately for the scale-up of continuous powder mixing process. Such quantitative investigations are used to clarify the effect of constant sampling size on mixing performance, and the effect of scaling up on both cross-sectional and axial mixing within continuous mixing. For the non-segregating mixing cases, similar scale-up characteristics are observed for both cross-sectional and axial mixing in our simulation results. While a satisfying principle of similarity leads to similar decay contours of variance spectra, a sampling size much smaller than the scales of both apparatuses should be used to achieve similar mixing performance Fig. 13. Residence time distribution of all mixing cases. (a) Non-segregating cases, 5% fill. (b) Non-segregating cases at 50% fill. (c) Cohesive segregating cases, 5% fill. (d) Cohesive segregating cases, 50% fill. rev: revolution.

14 68 Yijie Gao et al. / Powder Technology 35 (013) Fig. 14. The mean residence time and the mean axial velocity of all mixing cases. (a) The mean residence time, non-segregating cases. (b) The mean residence time, cohesive segregating cases. (c) The mean axial velocity, non-segregating cases. (b) The mean axial velocity, cohesive segregating cases. rev: revolution. measurements. When non-segregating materials are processed, a.5 fold increase in production rate is obtainable for the scale-up of continuous powder mixing. However, in processes where cohesive materials segregate due to a tendency to agglomerate, an even larger increase of flow rate is expected. This results from the effect of cohesion on axial motion especially when the scale of the mixing process is small. Our preliminary simulation results illustrate that the proposed analysis can be used to achieve a successful scale-up strategy for continuous powder blending of large sphere particles. Whether the proposed scaling relationships are applicable to practical systems consisting of small, cohesive, and irregular powder requires further investigation. Nomenclature C concentration ( ) E(t i ) discrete RTD measurement (s 1 ) E(t, x) location dependent RTD in continuous blending (s 1 ) E(ω) sum of square fit ( ) f one dimensional spatial frequency (mm 1 ) f three dimensional spatial frequency (mm 1 ) f s one dimensional sampling spatial frequency (mm 1 ) Δf one dimensional minimum detectable spatial frequency (mm 1 ) F volumetric flow rate (mm 3 /s) or cohesive force (N) Fr Froude number ( ) g gravity (9.81 m /s) k scale ratio of blending geometries k b exponential variance decay rate of batch-like blending (s 1 ) k c coefficient of cohesion (J m 3 ) l scale of geometry Pe Peclet number ( ) r blender radius (mm) RSD relative standard deviation ( ) s(f) one dimensional variance spectrum (mm 1/ ) s(f) three dimensional variance spectrum (mm 3/ ) s(f,t) one dimensional variance spectrum decay (mm 1/ ) s(f,t) three dimensional variance spectrum decay (mm 3/ ) t time of batch like periodic section blending (s) Δt time interval (s) t n t n one dimensional location in Lomb Scargle Periodogram (mm) three dimensional location in Lomb Scargle Periodogram (mm) v x axial velocity (mm/s) w(f) weight of averaged one dimensional variance spectrum ( ) w n weight of the nth sample in Lomb Scargle Periodogram ( ) x axial location in continuous blender (mm) x n sample as a function of location ( ) x f (t) one dimensional fit in Lomb Scargle Periodogram X sample in a population ( ) X k one dimensional discrete Fourier transform sample ( ) X h,j,k three dimensional discrete Fourier transform sample ( ) Z standard score θ dimensionless time ( )or phase lag in Lomb Scargle Periodogram (rad) μ mean value of a population ( ) ξ dimensionless location ( ) σ standard deviation ( ) σ (f s ) variance as a function of sample size ( ) σ 0 initial variance of totally segregated blend ( ) σ SS steady state variance of random blend ( ) σ b (t) time dependent variance decay curve in periodic section blending ( ) σ c (x) location dependent variance decay curve in continuous blending ( ) τ mean residence time (s) or location lag in Lomb Scargle Periodogram (mm) ω angular frequency (rad/mm) or angular velocity of blade (rad/s) ω three dimensional angular frequency (rad/mm) Ω one dimensional scale of the variance spectrum analysis (mm) Acknowledgments This work is supported by the National Science Foundation Engineering Research Center on Structured Organic Particulate Systems, through grant NSF-ECC , and by grant NSF