materials Article
Discrete Element Method Modeling of the Rheological Properties of Coke/Pitch Mixtures Behzad Majidi 1,2 , Seyed Mohammad Taghavi 2 , Mario Fafard 2 , Donald P. Ziegler 3 and Houshang Alamdari 1,2, * 1 2
3
*
Department of Mining, Metallurgical and Materials Engineering, Laval University, Quebec, QC G1V0A6, Canada;
[email protected] NSERC/Alcoa Industrial Research Chair MACE and Aluminum Research Center, Laval University, Quebec, QC G1V0A6, Canada;
[email protected] (S.M.T.);
[email protected] (M.F.) Alcoa Primary Metals, Alcoa Technical Center, 100 Technical Drive, Alcoa Center, New Kensington, PA 15069-0001, USA;
[email protected] Correspondence:
[email protected]; Tel.: +1-418-656-7666
Academic Editor: Martin O. Steinhauser Received: 2 February 2016; Accepted: 28 April 2016; Published: 4 May 2016
Abstract: Rheological properties of pitch and pitch/coke mixtures at temperatures around 150 ˝ C are of great interest for the carbon anode manufacturing process in the aluminum industry. In the present work, a cohesive viscoelastic contact model based on Burger’s model is developed using the discrete element method (DEM) on the YADE, the open-source DEM software. A dynamic shear rheometer (DSR) is used to measure the viscoelastic properties of pitch at 150 ˝ C. The experimental data obtained is then used to estimate the Burger’s model parameters and calibrate the DEM model. The DSR tests were then simulated by a three-dimensional model. Very good agreement was observed between the experimental data and simulation results. Coke aggregates were modeled by overlapping spheres in the DEM model. Coke/pitch mixtures were numerically created by adding 5, 10, 20, and 30 percent of coke aggregates of the size range of 0.297–0.595 mm (´30 + 50 mesh) to pitch. Adding up to 30% of coke aggregates to pitch can increase its complex shear modulus at 60 Hz from 273 Pa to 1557 Pa. Results also showed that adding coke particles increases both storage and loss moduli, while it does not have a meaningful effect on the phase angle of pitch. Keywords: discrete element method; simulation; pitch
viscoelastic;
particles;
dynamic shear rheometer;
1. Introduction Carbon anodes for the aluminum smelting process are made using a well-designed recipe of pitch binder and calcined coke particles. Anodes are consumed during the electrolysis process in Hall-Héroult cells, and it is estimated that around 17% of the cost of the aluminum smelting process comes from carbon anodes [1]. Thus, the properties and quality of the anodes have a direct effect on the performance and economy of the Hall-Héroult process. Anodes are made by mixing granulated coke aggregates with coal tar or petroleum pitch at 150 ˝ C and the mixture, known as anode paste, is then compacted and baked. It is essential to have a good understanding of the viscosity and rheological properties of pitch and pitch/coke mixtures to have better control of final compacted anode properties. Micromechanical models can provide considerable information about the flow and compaction properties of anode paste. Discrete element method (DEM) is used to simulate the behavior of granular materials in industrial applications especially where the dynamics and flow of a particulate material are of interest.
Materials 2016, 9, 334; doi:10.3390/ma9050334
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This method was introduced for the first time by Cundall and Strack [2] in 1979. In DEM simulations, rigid discrete elements, which are spheres in 3D and discs in 2D models, are used to model the granular material. The contact law between the elements defines the mechanical behavior of the bulk material. The discrete element approach has attracted the interest of researchers in mining, civil engineering, pharmaceutical industries, and materials engineering to simulate the flow [3,4], compaction [5,6], and mechanical properties [7,8] of single- and multi-phase materials. Recent advancements in the performance and power of computers have provided new insights to simulate the mechanical and rheological properties of asphalt concretes and mastics using two- and three-dimensional DEM models. For instance, Dondi et al. [9] studied the effects of aggregate size and shape on the performance of asphalt mixes. Khattak et al. [10] used an imaging technique coupled with DEM to create a micro-mechanical model of hot mix asphalts (HMA). They reported that the DEM model provided a good prediction of the dynamic modulus and strength of HMAs. Since materials properties in DEM are modeled by assigning appropriate contact models to the elements, different elastic, elasto-plastic, and viscoelastic models are developed. For asphalt mixes, usually Burger’s viscoelastic model is used for the mastic, and a simple elastic model is used for the aggregates [8,11]. Ma et al. [11] used Burger’s model in a three-dimensional DEM model to investigate the effects of air voids on creep behavior of asphalt mixtures. Burger’s model can be also embedded in traditional linear elastic contact models in which normal and shear stiffness of the contact changes by time to include viscous deformations. Vignali et al. [12] and Dondi et al. [13] adapted this approach of viscoelastic modeling to predict the rheological behavior of bituminous binders. In the present work, three-dimensional imaging is used to capture the irregular shapes of coke aggregates. Then, a 3D DEM model of pitch is developed to predict the rheological properties of pitch and coke/pitch mixtures at 150 ˝ C. 2. Theory A three-dimensional DEM model is composed of a combination of discrete spheres and walls. At the beginning, the position of all elements and walls are known so that the active contacts are easily determined. Then, according to the mechanical behavior of the material, an appropriate force-displacement law is applied to each contact, and the contact forces are calculated. Newton’s second law of motion is then used to update the position and velocity of each element. One common contact model widely used in DEM simulations is the linear contact model. This model is simply defined by assigning normal and shear stiffness values to the contacting elements. Contact forces can be calculated according to the extent of the overlap (for normal contact force) and tangent movement (for shear contact force): Fn “ Kn ˆ U n
(1)
F s “ ´K s ˆ δU s
(2)
Irregularly shaped particles can be generated as a clump composed of several touching or overlapping balls. Contact force calculations for balls within a clump is skipped during calculation cycle and only the contacts of a clump with neighboring clump/balls or walls are considered. Burger’s four-element model is the most common model used to simulate the viscoelastic properties of mastics and binders [14]. Burger’s model, as shown in Figure 1, is composed of the Maxwell model in series with the Kelvin model; thus, it is capable of representing the material behavior under both creep and relaxation [14].
Figure 1. Burger’s four-element model composed of the Maxwell (m) model in series with the Kelvin 3 of 13 (k) model.
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Deformation of Burger’s model is the sum of the deformations of Kelvin’s elements and Maxwell’s elements. This means that the total deformation can be written as: =
+
(3)
Deformation of the Maxwell element, in turn, comes from the dashpot and the spring: Figure 1. Burger’s four-element model composed of the Maxwell (m) model in series with the Kelvin
Figure 1. Burger’s(k)four-element model composed of the Maxwell (m) model in series with the Kelvin model. (4) = + + (k) model. Deformation of Burger’s model is the sum of the deformations of Kelvin’s elements and
The first andMaxwell’s secondelements. derivatives ofthat Equation (4) cancan bebewritten This means the total deformation written as: as:
(3) =the+deformations of Kelvin’s elements Deformation of Burger’s model is the sum and Maxwell’s (5) = of + + Deformation of the Maxwell element, in turn, comes from the dashpot and the spring: elements. This means that the total deformation can be written as: (6) (4) = = ++ + + The first and second derivatives of Equation written as: u “ u(4) `canube m can be written as: kmodel The force-displacement equation of Burger’s =
+
+
(3) (5)
1 in1 =turn, (6) + comes + Deformation of the Maxwell from + + element, + = the dashpot and the spring: + The force-displacement equation of Burger’s model can be written as:
(7)
1 u1 ` u “ u=mc (4) + + u + by + mk ` (7) k some Although Burger’s model is widely used researchers, mostly to simulate the mechanical behavior of asphalt mastic, it hasmodel notisbeen publicly YADE [15]. Therefore, authors Although Burger’s widely used by some implemented researchers, mostly toin simulate the mechanical The first and behavior secondof derivatives (4)implemented can be written asphalt mastic, it of has Equation not been publicly in YADE as: [15]. Therefore, authors use an in-house implementation of this model in YADE, creating a cohesive viscoelastic model to use an in-house implementation of this model in YADE, creating a cohesive viscoelastic model to . . . . simulate pitch. A new type of material, the so-called CohBurgersMat is defined in YADE. This simulate pitch. A new type of material, theu so-called is defined in YADE. This u“ umkdirection ` uCohBurgersMat (5) mcand four of Burger’s parameters in k` material takes four of Burger's parameters in normal material takes four of Burger's parameters in normal direction and four of Burger’s parameters in shear direction. Interactions of elements having CohBurgersMat are governed by Equation (7). .. .. .. .. Viscoelastic of properties of a material presented by two parameters of a complex shear direction. Interactions elements CohBurgersMat are governed by Equation (7). (6) uhaving “ are uk normally ` umk ` umc shear modulus, G*, and phase angle, δ [13]. These parameters can be measured using a parallel-plate Viscoelastic properties ofasaschematically materialshown are innormally by two(DSR) parameters of a complex measuring system Figure 2. Thepresented dynamic shear rheometer shown force-displacement equation of Burger’s can becan written as: properties in Figure is the laboratory widelymodel used to characterize thebe rheological of shearThe modulus, G*, and 3a phase angle, δequipment [13]. These parameters measured using a parallel-plate different types of binders and mastics [16]. As shown in Figures 2 and 3b, a disc of the tested material ˙ „ between twoˆplates measuring systemis sandwiched as schematically shown Figure 2. C TheIn this dynamic shear (DSR) shown desired test, a sinusoidal (stress) .temperature. . 1 at the1in C Ckforce Crheometer Ck m .. k m is applied to the sample, and the deformation (strain) is recorded. The induced strain has a sinusoidal ` C ` f ` f “ ˘C u u ˘ f ` m m in Figure 3a is the laboratory used to characterize theK rheological properties(7) of Kk lagequipment Kkfrom widely K Kk K form with a time that comes themmaterial’s viscose deformation. This time lag isk called the m different types of phase binders and mastics [16]. As shown in Figures 2 and 3b, a disc of the tested material angle (δ). The frequency sweep configuration is adopted to obtain the response of fluid-like materials model to different by measuring the complex shear modulus and angle Although Burger’s isloading widely by some researchers, mostly toaphase simulate theforce mechanical is sandwiched between two plates atfrequencies theused desired temperature. In this test, sinusoidal (stress) at different frequencies. behavior mastic, it has not been publicly implemented YADE [15]. Therefore, use is appliedoftoasphalt the sample, and the deformation (strain) is recorded.inThe induced strain has aauthors sinusoidal form with aimplementation time lag that comes from thein material’s viscoseadeformation. This time lag istocalled the an in-house of this model YADE, creating cohesive viscoelastic model simulate phase A angle frequencythe sweep configuration is adopted to obtain the response of fluid-like pitch. new(δ). typeThe of material, so-called CohBurgersMat is defined in YADE. This material takes materials to different loadinginfrequencies by measuring theofcomplex modulus phase angle four of Burger's parameters normal direction and four Burger’sshear parameters inand shear direction. at different frequencies. Interactions of elements having CohBurgersMat are governed by Equation (7).
Figure 2. Schematic illustration of a dynamic shear rheometer.
Figure 2. 2. Schematic Schematic illustration illustration of of aa dynamic dynamic shear shear rheometer. rheometer. Figure
Viscoelastic properties of a material are normally presented by two parameters of a complex shear modulus, G*, and phase angle, δ [13]. These parameters can be measured using a parallel-plate measuring system as schematically shown in Figure 2. The dynamic shear rheometer (DSR) shown in Figure 3a is the laboratory equipment widely used to characterize the rheological properties of different types of binders and mastics [16]. As shown in Figures 2 and 3b, a disc of the tested material is sandwiched between two plates at the desired temperature. In this test, a sinusoidal force (stress) is
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applied to the sample, and the deformation (strain) is recorded. The induced strain has a sinusoidal form with a time lag that comes from the material’s viscose deformation. This time lag is called the phase angle (δ). The frequency sweep configuration is adopted to obtain the response of fluid-like materials to different loading frequencies by measuring the complex shear modulus and phase angle at different frequencies. Materials 2016, 9, 334 4 of 12
(a)
(b)
Figure 3. 3. Dynamic Dynamic shear shear rheometer rheometer (DSR) (DSR) (a); (a); and and aa pitch pitch sample sample being being tested tested (b). (b). The The black black part part in in Figure the DSR instrument is the sample bath which keeps the temperature constant during the test. the DSR instrument is the sample bath which keeps the temperature constant during the test.
The following equations are used to calculate G*: The following equations are used to calculate G*: 2 (8) = τ 2T π τmax “ 3 (8) πr θ (9) γ = ℎ θr γ “ (9) is the maximum shear stress, maxis theh maximum applied torque, is the radius of in which τ thewhich specimen, is the rotationshear angle,stress, and ℎTisis the the specimen. the complex in τmax isθ the maximum the thickness maximumofapplied torque, Finally, r is the radius of the shear modulus is obtained specimen, θ is the rotation from: angle, and h is the thickness of the specimen. Finally, the complex shear modulus is obtained from: τ ∗ = τmax (10) (10) G˚ “ γ γmax The phase phase angle angle (δ), (δ), in in turn, turn, was was determined determined at at different different frequencies frequencies by by measuring measuring the the delay delay in in The seconds between the peaks in the stress and strain functions. seconds between the peaks in the stress and strain functions. Having obtained obtained G˚∗ and frequency are are Having and δ, δ, the the storage storage and and loss loss moduli moduli at at 150 150 ˝°C C for for each each frequency calculated by: calculated by: ˚ G1 “ ˆ cos cosδδ = G∗ ×
(11) (11)
˚ G2" “ ˆsin sinδ δ = G∗ ×
(12) (12)
3. Experimental 3. Experimental Dynamic shear tests were conducted in strain-controlled mode using an ARES rheometer (see Dynamic shear sweep tests were conducted in adopted strain-controlled using ARES rheometershear (see Figure 3). Frequency configuration was at 150 ˝ C mode from 0.06 Hz an to 60 Hz. Complex Figure 3). Frequency sweep configuration was adopted at 150 °C from 0.06 Hz to 60 Hz. Complex modulus and phase angle were measured at each frequency. Then, the obtained data were used to shear modulus and phase angle were measured estimate the four parameters of Burger’s model at of each pitchfrequency. at 150 ˝ C. Then, the obtained data were used to estimate the four parameters of Burger’s model of pitch at 150 °C. Numerical Method Numerical Method The open-source discrete element code, YADE [15], was used in this work to simulate the DSR test. The open-source discrete model element YADE wasisused in is this work4.toPitch simulate the DSR The geometry of the numerical ofcode, the DSR test [15], of pitch shown Figure is modeled by test. The geometry of the numerical model of the DSR test of pitch is shown is Figure 4. Pitch is an assembly of spheres with the radius of 0.08 mm. Spheres are generated in hexagonal closed pack modeled by antoassembly with of 0.08 Spheres are generated in hexagonal configuration make upofa spheres disc of 12 mmthe in radius diameter andmm. 2 mm in thickness. Size of the elements closed pack configuration to make up a disc of 12 mm in diameter and 2 mm in thickness. Size of the elements has been chosen considering the balance between the resolution of the model (size of the elements compared to the size of the sample) and the computation time.
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has been chosen considering the balance between the resolution of the model (size of the elements compared to9,the Materials 2016, 334 size of the sample) and the computation time. 5 of 12
Figure 4. 4. Discrete element method method (DEM) (DEM) simulation simulation of of the the DSR DSR test test of of pitch. pitch. Figure Discrete element
A small small compressive compressive load load was was initially initially applied applied to to ensure ensure the the contacts contacts between between the the plates plates and and the the A testing material. material. Frequency sweep dynamic dynamic shear shear tests tests were were run run at at frequencies frequencies ranging ranging from from 0.06 0.06 Hz Hz testing Frequency sweep to 60 Hz, according to the DSR experiments. to 60 Hz, according to the DSR experiments. Equations (8)–(12) (8)–(12) were were then then used used to to calculate calculate the the rheological rheological parameters. parameters. A A cohesive cohesive contact contact Equations model based on Burger’s model (Equation (7)) is used to simulate the viscoelastic behavior of pitch model based on Burger’s model (Equation (7)) is used to simulate the viscoelastic behavior of pitch ˝ at 150 °C. The spherical elements representing pitch have Burger’s model in both shear and normal at 150 C. The spherical elements representing pitch have Burger’s model in both shear and normal direction in in all all contacts. contacts. In In this this work, work, the the same same values values were were used used for for Burger’s Burger’s model model of of pitch pitch in in both both direction shear and normal directions. shear and normal directions. In 2011, 2011, Liu Liu et In et al. al. [17] [17] proposed proposed aa new new method method in in using using the the frequency-temperature frequency-temperature superposition superposition (FTS). Traditionally, Traditionally,after afterconducting conductingdynamic dynamicmodulus modulustests testsatat different temperatures, FTS is used (FTS). different temperatures, FTS is used in in calculating shift factors to build up the master curve at the reference temperature. However, in the calculating shift factors to build up the master curve at the reference temperature. However, in the ∗ approach proposed proposedbyby δ are measured atfrequencies different frequencies at desired approach LiuLiu et al.,etG˚al.,and δ and are measured at different at desired temperatures temperatures in the laboratory and the Burger’s model parameters are obtained. Then, Burger’s in the laboratory and the Burger’s model parameters are obtained. Then, Burger’s dashpot viscosities ∗ ˚ dashpot viscosities are modified to predict and δ at amplified frequencies. This method of are modified to predict G and δ at amplified frequencies. This method is of considerable interestisfor considerable interest for discrete element by simulations as for example by using amplifying discrete element simulations as for example using the amplifying coefficient of ξ “ the 1000, dynamic coefficient of ξ = 1000, dynamic modulus test at the frequency of 0.06 Hz can be simulated at f = 60 modulus test at the frequency of 0.06 Hz can be simulated at f = 60 Hz, resulting in significantly Hz, resulting in significantly computation theused present work, = 1000 was used in reduced computation time. Inreduced the present work, ξ “time. 1000In was in the DSR ξtests simulations. the DSR tests simulations. 4. Results and Discussion 4. Results and Discussion 4.1. DSR of Pitch and Model Verification 4.1. DSR of Pitch anddata Model Verification Experimental of the DSR tests of pitch at 150 ˝ C are given in Figure 5. The obtained data is then used to fit the Burger’s model parameters. The°C fitting procedure is performed by the data solver Experimental data of the DSR tests of pitch at 150 are given in Figure 5. The obtained is option in Microsoft Excel by minimizing the following function: then used to fit the Burger’s model parameters. The fitting procedure is performed by the solver option in Microsoft Excel by minimizing «the following function: ff2 ff2 « m ÿ G1 j G2 j f “ p ´ 1 ` " ´1 q 10 G2 0j− 1 ) = j“1 (G j − 1 + ′ " 0
(13) (13)
1 20 where where G ′j and and G "j are are respectively respectively the the storage storage and and loss loss moduli moduli experimentally experimentally measured measured at at jth jth 1 2 frequency; G and G are respectively the predicted storage and loss moduli at jth frequency; and m j j frequency; and " are respectively the predicted storage and loss moduli at jth frequency; and is the number of data points, which is 16. m is the number of data points, which is 16.
Figure 5. Dynamic shear tests results of pitch at 150 °C.
′
(13)
"
where ′ and " are respectively the storage and loss moduli experimentally measured at jth frequency; and " are respectively the predicted storage and loss moduli at jth frequency; and Materials 2016, 9, 334 6 of 13 m is the number of data points, which is 16.
Figure 5. Dynamic results pitch at ˝150 Figure 5. Dynamicshear shear tests tests results of of pitch at 150 C. °C. Calculated values for Burger’s model of pitch have been presented in Table 1. These parameters are called macroscale parameters, and they determine the global material behavior. The Burger’s model parameter values for each contact in the DEM model of pitch, however, depend on the size of two contacting spheres. In our implementation of Burger’s model on YADE, these parameters are obtained by the following equation for each contact: Pm “ L ˆ PM
(14)
in which PM is the macroscale parameter of Burger’s model as given in Table 1, L is the diameter of the element, and Pm is the microscale parameters of Burger’s model of pitch. For a contact formed by two overlapping elements, however, the contact model parameters are obtained as: Pcontact “ 2 ˆ
P1 P2 P1 ` P2
(15)
Table 1. Calculated Burger’s model parameters of pitch at 150 ˝ C. Km (Pa)
Cm (Pa.s)
Kk (Pa)
Ck (Pa.s)
3867.136
37.111
10.047
7.375
In the DEM code here, as a contact is formed, its rheological parameters are obtained from Equation (15) and then the force-displacement law, Equation (7), is solved at each cycle as long as the contact is active. Using the values obtained for pitch and the configuration previously explained in Section 3, DSR tests were simulated in YADE. The DEM model of the DSR test was calibrated by changing the stiffness of the loading plate. Stiffness of the plate was modified to obtain the complex shear modulus of pitch at f = 60 Hz. Then, this value was kept constant in DSR tests at all other frequencies. Using this method, stiffness of the loading plate was determined as 40,000 N/m. Figure 6 presents the simulation results of a DSR test of pitch in terms of stress and strain curves by which, using previously mentioned equations, G˚ and δ are calculated. Experimentally measured rheological data (G*, G', and G") of pitch at 150 ˝ C in Figures 7–9 are compared with those obtained by the DEM simulation with the viscoelastic parameters given in Table 1.
of pitch at f = 60 Hz. Then, this value was kept constant in DSR tests at all other frequencies. Using of pitch at f = 60 Hz. Then, this value was kept constant in DSR tests at all other frequencies. Using this method, stiffness of the loading plate was determined as 40,000 N/m. this method, stiffness of the loading plate was determined as 40,000 N/m. Figure 6 presents the simulation results of a DSR test of pitch in terms of stress and strain curves Figure 6 presents the simulation results of a DSR test of pitch in terms of stress and strain curves by which, using previously mentioned equations, ∗∗ and δ are calculated. Experimentally measured by which, using previously mentioned equations, and δ are calculated. Experimentally measured *, G', and G") of pitch at 150 °C in Figures 7–9 are compared with those obtained rheological Materials 2016, data 9, 334(G 7 of 13 rheological data (G*, G', and G") of pitch at 150 °C in Figures 7–9 are compared with those obtained by the DEM simulation with the viscoelastic parameters given in Table 1. by the DEM simulation with the viscoelastic parameters given in Table 1.
Figure 6. 6. DEM results results as strain strain and stress stress curves of of pitch at at f = 60 Hz. Figure Figure 6. DEM DEM results as as strain and and stresscurves curves ofpitch pitch atff == 60 60 Hz. Hz.
Figure 7. Experimental and DEM simulation results for complex modulus of pitch at 150 °C.
Figure 7. 7. Experimental and DEM simulation results for complex modulus of pitch at 150 ˝°C. Figure Materials 2016, 9, 334 Experimental and DEM simulation results for complex modulus of pitch at 150 C. Materials 2016, 9, 334
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Figure 8. 8. Experimental and and DEM DEM simulation simulation results results for for storage storage modulus modulus ofpitch pitch at150 150˝°C. Figure C. Experimental modulus of of pitch at at 150 °C.
Figure 9. Experimental and DEM simulation results for loss modulus of pitch at 150˝°C. Figure 9. 9. Experimental and DEM DEM simulation simulation results results for for loss loss modulus modulus of of pitch pitch at at 150 150 °C. Figure Experimental and C.
As can can be be seen seen from from Figures Figures 7–9, 7–9, the the DEM DEM model model of of pitch pitch is is capable capable of of predicting predicting the the complex complex As modulus at at aa wide wide range range of of frequencies frequencies with with aa very very good good precision. precision. However, However, storage storage modulus modulus at at modulus high frequencies is over-estimated. This comes from the under-estimation of phase angle at higher high frequencies is over-estimated. This comes from the under-estimation of phase angle at higher frequencies. Similar Similar observation observation has has been been reported reported by by Liu Liu and and You You [17]. [17]. They They reported reported that that the the errors errors frequencies. in predictions predictions of of the the complex complex modulus modulus and and phase phase angle angle for for the the amplified amplified frequencies frequencies of of less less than than in
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As can be seen from Figures 7–9 the DEM model of pitch is capable of predicting the complex modulus at a wide range of frequencies with a very good precision. However, storage modulus at high frequencies is over-estimated. This comes from the under-estimation of phase angle at higher frequencies. Similar observation has been reported by Liu and You [17]. They reported that the errors in predictions of the complex modulus and phase angle for the amplified frequencies of less than 25 kHz is around 4%. However, it increases up to 15% for higher frequencies. Using a smaller time-step and not applying the FTS to avoid its probable inertial effects can improve the predictions of the phase angle. 4.2. DEM Simulation of Coke/Pitch Pastes The verified discrete element model of pitch is then used to predict the viscoelastic properties of pitch/coke mixtures at 150 ˝ C. It should be noted that DSR test measurements on coke/pitch mixtures are not always possible or precise due to the small thickness of the sample disc in the DSR tests and limitations of the setup. Thus, a well-designed DEM model is of considerable practical interest. Calcined coke has particles with irregular shapes and, as used in anodes, a wide size distribution. For discrete element models in the present work, the method previously developed by the authors [18] was applied to model the coke aggregates by overlapping spheres. Coke particles of the size range of ´30 + 50 mesh (0.297–0.595 mm) were modeled as clumps in YADE. An example of a modeled coke particle is shown in Figure 10. Materials 2016, 9, 334 8 of 12
Figure 10. 3D image of a coke particle and its twin DEM model. Figure 10. 3D image of a coke particle and its twin DEM model.
Four numerical mixes of coke and pitch with 5, 10, 20, and 30 wt % of coke aggregates were
Four numerical of was cokesimulated and pitch with 5, 10,with 20,the and 30configuration wt % of coke aggregates generated. Then, themixes DSR test on the samples same of the onlywerepitch generated. Then, the DSR test was simulated on the samples with the same configuration sample. The DEM model of coke/pitch mixtures is composed of two phases: spheres in hexagonal closed packedThe (hcp) configuration and is clumps representing of the only-pitch sample. DEM model ofrepresenting coke/pitch pitch mixtures composed of twocoke phases: aggregates. spheres in hexagonal closed packed (hcp) configuration representing pitch and clumps representing To create the coke/pitch sample models, first, clumps of coke aggregates with the intended size coke aggregates. distribution andcoke/pitch numbers are randomly placed in aclumps disc withofa coke radius of 6 mm and a height of 2 mm. size To create the sample models, first, aggregates with the intended The model is cycled to remove the possible overlaps between aggregates. At this step, zero gravity is distribution and numbers are randomly placed in a disc with a radius of 6 mm and a height of 2 mm. used in the model to prevent the particles from sinking to the bottom of the cylindrical container. The model is cycled to remove the possible overlaps between aggregates. At this step, zero gravity Then, using spheres with radii of 0.08 mm, the sample volume is tessellated in an hcp configuration. is used in the model to prevent the particles from sinking to the bottom of the cylindrical container. Since there are already clumps in the space, there will obviously be some overlaps between the Then, using spheres with radiimust of 0.08 mm, the to sample is tessellated in an hcp configuration. clumps and spheres, which be managed have avolume stable sample. Since there are already in the space, thereare will obviously some overlaps between the Using a Python script,clumps big clump-sphere overlaps removed. Thebe code detects the interactions clumps and spheres, which must be managed to have a stable sample. in which the penetration depth is larger than 0.02 mm and deletes the standalone sphere (pitch Using a Python script, clump-sphere overlaps are removed. Theofcode detects thesphere interactions element). However, if thebig overlap value is less than 0.02 mm, the size the standalone is in which theto penetration depth is larger it. than mm and spherein (pitch element). reduced 0.04 mm without deleting An0.02 example of a deletes result ofthe thisstandalone method is shown Figure 11. However, if the overlap value is less than 0.02 mm, the size of the standalone sphere is reduced to 0.04 mm without deleting it. An example of a result of this method is shown in Figure 11.
Since there are already clumps in the space, there will obviously be some overlaps between the clumps and spheres, which must be managed to have a stable sample. Using a Python script, big clump-sphere overlaps are removed. The code detects the interactions in which the penetration depth is larger than 0.02 mm and deletes the standalone sphere (pitch element). However, if the overlap value is less than 0.02 mm, the size of the standalone sphere9 of is 13 Materials 2016, 9, 334 reduced to 0.04 mm without deleting it. An example of a result of this method is shown in Figure 11.
Figure 11. Coke/pitch mixture generation method. (a) Creating coke aggregates; (b) adding pitch Figure 11. Coke/pitch mixture generation method. (a) Creating coke aggregates; (b) adding pitch spheres and handling the overlaps. spheres and Materials 2016, 9, 334handling the overlaps. 9 of 12
Figure 11a shows the first step in which only clumps are generated. In Figure 11b, pitch spheres have been added, and the overlapping spheres have been deleted or reduced in diameter to avoid huge overlaps. The The resulting resulting paste paste then then undergoes undergoes aa triaxial triaxial compression compression to have an integrated compacted sample. In Figure 12, the numerical sample sample of of pitch pitch and and coke coke mixture mixture ready ready for for the the DSR DSR tests tests is is shown. shown. The dimensions and test procedure were the same as those for pitch. Four pastes of coke and pitch mixture were numerically prepared with 5, 10, 20, and 30 wt % of coke particles. Size distribution of coke aggregates was the same for all samples in the the range range of of ´30 −30 ++50 50mesh. mesh.
Figure 12. 12. Dynamic shear shear test test of of coke/pitch coke/pitch mixture. Figure mixture.
The effect of the content of coke particles on the complex shear modulus of coke/pitch pastes has The effect of the content of coke particles on the complex shear modulus of coke/pitch pastes has been shown in Figure 13. As the content of coke in the mixture increases, the complex shear modulus been shown in Figure 13. As the content of coke in the mixture increases, the complex shear modulus increases. This raise in ∗ ˚is more pronounced for higher frequencies. However, the addition of coke increases. This raise in G is more pronounced for higher frequencies. However, the addition of particles does not have a meaningful effect on the value phase angle and, as the content of the coke coke particles does not have a meaningful effect on the value phase angle and, as the content of the particles increases, both storage and loss moduli increase. Similar results have been reported by coke particles increases, both storage and loss moduli increase. Similar results have been reported by Pasquino [19] for glass bead suspensions in viscoelastic polymers. Pasquino [19] for glass bead suspensions in viscoelastic polymers.
The effect of the content of coke particles on the complex shear modulus of coke/pitch pastes has been shown in Figure 13. As the content of coke in the mixture increases, the complex shear modulus increases. This raise in ∗ is more pronounced for higher frequencies. However, the addition of coke particles does not have a meaningful effect on the value phase angle and, as the content of the coke particles both storage and loss moduli increase. Similar results have been reported10by Materials 2016,increases, 9, 334 of 13 Pasquino [19] for glass bead suspensions in viscoelastic polymers.
Figure 13. Effect of coke aggregate content on complex shear modulus of coke/pitch mixtures. Figure 13. Effect of coke aggregate content on complex shear modulus of coke/pitch mixtures.
Figures 13–15 show that there is a considerable rise in the dynamic moduli of pitch by adding coke Figures show that there is also a considerable rise in thetwo dynamic moduli of pitchfor by the adding particles. In13–15 Figure 14, results have been compared with equations developed case coke of particles. In Figure 14, results have also been compared with two equations developed for the spherical rigid particles in Newtonian fluids, the Hashin-Shtrikman [20] and Krieger-Doughertycase [21] of spherical rigid particles in Newtonian fluids, equations, which are respectively written as: the Hashin-Shtrikman [20] and Krieger-Dougherty [21] equations, which are respectively written as: ′(φ) 2 + 3φ (16) = ′ 22 − `2φ 3ϕ G1 pϕq “ (16) ′(φ)G1 0 2 ´ 2ϕ (17) = (1 − 1.5625φ) . ′ G1 pϕq “ p1 ´ 1.5625ϕq´1.6 (17) where φ is the volume content of particles. 1 G 0 ( ) ratio for pitch mixed coke particles are larger than those predicted by the In Figure 14, the where ϕ is the volume content of particles. equations. It shouldGbe that these equations were proposed for Newtonian fluids including 1pϕnoted q In Figure 14, the G10 ratio for pitch mixed coke particles are larger than those predicted by the Materials 2016, 9, 334 10 of 12 equations. It should be noted that these equations were proposed for Newtonian fluids including rigid spherical particles. However, pitch ispitch a non-Newtonian viscoelastic fluid fluid above its softening point. rigid spherical particles. However, is a non-Newtonian viscoelastic above its softening Added particles here are not spherical either, and they are relatively coarse. Therefore, the case seems point. Added particles here are not spherical either, and they are relatively coarse. Therefore, the case complex, and we have found analytical or empirical equation to describe the effects of adding seems complex, and we havenofound no analytical or empirical equation to describe the effects of rigid particles a viscoelastic fluid. Performing experimental measurement waswas notnot possible due adding rigid to particles to a viscoelastic fluid. Performing experimental measurement possible to the ofsize theofparticles (in the of 0.297–0.595 mm) ofthe thesample, sample, which due size to the the particles (in range the range of 0.297–0.595 mm)and andthe the stiffness stiffness of which exceeded thethe limits ofof our withfine fineparticles particlessmaller smallerthan than 150 micron exceeded limits oursetup. setup.Therefore, Therefore,aa mix mix of pitch with 150 micron made experimental measurements. The results DSR tests these samplescompared comparedtotothe waswas made forfor experimental measurements. The results ofof DSR tests onon these samples the above-mentioned equations are in given in Figure 16. Again, it can seen that the experimental above-mentioned equations are given Figure 16. Again, it can be seenbethat the experimental point of G1 of the point ratio for pitch/coke mixtures are larger than the upper limit predicted by the Kriegerthe G10 ratio for pitch/coke mixtures are larger than the upper limit predicted by the Krieger-Dougherty relation. Thus,relation. it can be concluded these models maymodels not bemay appropriate to study the rheological Dougherty Thus, it can bethat concluded that these not be appropriate to study the parameters of particles mixed with non-Newtonian viscoelastic fluids. rheological parameters of particles mixed with non-Newtonian viscoelastic fluids.
Figure Effect cokeparticle particlecontent contenton onGGʹ/Gʹ ratio (Gʹ storage moduli 1 /G1 0 ratio Figure 14.14. Effect ofof coke (G1 and andGʹ G100are arerespectively respectivelythe the storage moduli 0 of the mixture and pitch). of the mixture and pitch).
Figure Effect of coke particle content on Gʹ/Gʹ0 ratio (Gʹ and Gʹ0 are respectively the storage moduli Materials 2016,14. 9, 334 11 of 13 of the mixture and pitch). Figure 14. Effect of coke particle content on Gʹ/Gʹ0 ratio (Gʹ and Gʹ0 are respectively the storage moduli of the mixture and pitch).
Figure 15. 15. Effect Effect of of coke coke particle particle content content on on G Gʺ/Gʺ ratio (G (Gʺ and G Gʺ are respectively respectively the the loss loss moduli moduli 2 G2 0 ratio 2 and 2 0 are Figure 0 0 of the mixture and pitch). of the mixture and Figure 15. Effect ofpitch). coke particle content on Gʺ/Gʺ0 ratio (Gʺ and Gʺ0 are respectively the loss moduli of the mixture and pitch).
Figure 16. Effect of fine coke particle content on Gʹ/Gʹ0 ratio. Experimental data are compared with the Hashin-Shtrikman Krieger-Dougherty equations. Figure 16. Effect Effectof offine fineand coke particlecontent content 0 ratio. Experimental data are compared with Figure 16. coke particle onon G1Gʹ/Gʹ /G1 0 ratio. Experimental data are compared with the the Hashin-Shtrikman and Krieger-Dougherty equations. Hashin-Shtrikman and Krieger-Dougherty equations.
5. Conclusions A cohesive viscoelastic contact model based on Burger’s model was implemented in YADE. The rheological behavior of pitch was measured at 150 ˝ C using a dynamic shear rheometer. The obtained data were then used to calibrate the DEM model parameters for pitch. DSR test of pitch was then simulated by a three-dimensional DEM model in which pitch is modeled by an assembly of spheres of radius of 0.08 mm in hcp configuration. Results confirm that Burger’s model is a superior choice to describe the complex rheological behavior of pitch. The simulation results are in a very good agreement with the experimental data. However, the storage modulus is over-estimated at high frequencies. It is believed that using a smaller time-step and not applying the frequency-temperature superposition (which of course results in very long computation times) can provide a better prediction of phase angle and thus a better prediction of storage modulus at higher frequencies. Calibrated DEM model of pitch was then used to predict the rheological properties of coke/pitch pastes with different content of coke aggregates. Since experimental measurements using a DSR machine was not possible, the proposed DEM numerical model is of great practical interest. Results showed that as the content of coke aggregates increases, both storage and loss moduli of the mixture increase, resulting in an near-constant phase angle. The rise in complex modulus value is more pronounced at higher frequencies. Comparison of the obtained result with the available literature shows that the empirical models developed for the case of rigid spheres dispersed in Newtonian fluids fail to accurately predict the effects of adding coke particles to pitch. More investigations are required
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to elucidate the hydrodynamic interactions between two particles in a viscoelastic suspension and the rheological behavior of the mixture. Acknowledgments: The authors gratefully acknowledge the financial support provided by Alcoa Inc., the Natural Sciences and Engineering Research Council of Canada, and the Centre Québécois de Recherche et de Développement de l'Aluminium. A part of the research presented in this article was financed by the Fonds de recherche du Québec-Nature et Technologies by the intermediary of the Aluminium Research Centre–REGAL. Author Contributions: Behzad Majidi and Houshang Alamdari conceived and designed the experiments; Behzad Majidi performed the experiments; Mario Fafard, Donald P. Ziegler and Seyed Mohammad Taghavi contributed in data analysis; Behzad Majidi wrote the paper and all co-authors commented/corrected it. Conflicts of Interest: The authors declare no conflict of interest.
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