Powder Technology 306 (2017) 34–44
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An experimental approach to powder-binder separation of feedstock Dong Yong Park a,c, Youngmin Oh b, Hyung Ju Hwang b, Seong Jin Park c,⁎ a b c
Department of Solar Thermal, Korea Institute of Energy Research (KIER), 152, Gajeong-ro, Yuseong-gu, Daejeon 34129, Republic of Korea Department of Mathematics, Pohang University of Science and Engineering (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang, Kyeongbuk 37673, Republic of Korea Department of Mechanical Engineering, Pohang University of Science and Engineering (POSTECH), 77 Cheongam-ro, Nam-gu, Pohang, Kyeongbuk 37673, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 July 2016 Received in revised form 23 October 2016 Accepted 9 November 2016 Available online 10 November 2016 Keywords: Powder-binder separation Rheology Viscosity Powder injection molding
a b s t r a c t Powder-binder separation for a feedstock was investigated. A non-Newtonian viscosity model considering shear rate, temperature, and solids loading was employed to describe the rheological characteristics for a feedstock. The rheological parameters for the non-Newtonian viscosity model were obtained by torque and capillary rheometry experiments. The extracted feedstock in the designed jig, after injection through a capillary tube, was encapsulated in a mounting resin without pressure at room temperature to avoid destruction of the original powder distribution profile. The surface of the cross-section of the injected feedstock was polished with acetone to distinguish the powder from the binder. After the solids loading was measured in a radial direction by SEM, it was quantitatively calculated through image processing based on the intensity difference between the powder and binder. The ratio of the phenomenological coefficients, Kc/Kη, for the steady-state solution was obtained by the mean residual method. The values of Kc and Kη were determined using numerically and experimentally calculated convergence times. Based on the obtained values of the ratio of phenomenological coefficients, the powder-binder separation behavior in respect of the flux induced by shear rate, solids loading, and viscosity was numerically examined. © 2016 Elsevier B.V. All rights reserved.
1. Introduction In rapidly evolving powder injection molding (PIM) technology, the wide prevalence of complex shapes requires appropriate material or processing conditions to reduce the production of defective products. Furthermore, the material and processing conditions should be simultaneously considered, because the material conditions, such as the powder and binder characteristics, affect the processing conditions for the PIM process. For instance, variation of powder size influences the filling and densification [1–10]. Therefore, understanding the relationship between material and processing conditions in the PIM process is critical not only to determine the optimal processing conditions but also to produce defect-free parts. An inappropriate processing condition can cause unexpected problems in a PIMed part. One of the representative problems in the PIM process is powderbinder separation (P-B separation) which arises from various phenomena, including the spatially different shear rates and viscosities of feedstock [2–4,11,12]. Changes in flow direction during injection molding also lead to P-B separation, because the powder and binder with high and low densities are separately accumulated due to the different centrifugal forces induced by the difference in density. P-B separation results in various defects, such as distortion or cracking, ⁎ Corresponding author. E-mail address:
[email protected] (S.J. Park).
http://dx.doi.org/10.1016/j.powtec.2016.11.008 0032-5910/© 2016 Elsevier B.V. All rights reserved.
during debinding or sintering, because an inhomogeneously distributed powder in a green part, namely, an injection molded part, leads to non-uniform densification caused by the anisotropic shrinkage during sintering [9,13–15]. As a consequence of this problem, final products produced by PIM ultimately have poor physical and mechanical properties [16,17]. Several studies have been carried out regarding P-B separation in a non-Newtonian suspension, especially a feedstock in powder injection molding [2–10]. Phillips et al. [18] experimentally measured powder distribution by the non-invasive method of nuclear magnetic resonance (NMR) imaging through Couette flow for a concentrically rotating cylinder and cylindrical tube apparatus. The laser Doppler anemometry (LDA) method was used to measure particle distribution during injection molding [19]. SEM combined with energy dispersive X-ray (EDX) analysis was also employed to quantitatively evaluate the P-B separation phenomenon. Differential scanning calorimetry (DSC) was used to measure changes in the enthalpy of a feedstock ingredient (binder component) [20]. Mannschatz et al. [4] carried out a study to observe P-B separation phenomena on the cross-section in injection molded samples. They tried to use a special ceramographic method to overcome the destruction of the powder distribution profile. The broad ion beam technique that they employed enabled simultaneous polishing of the soft binder matrix and hard ceramic particles without destruction of the original powder distribution profile with the achievement of a relief at the surface.
D.Y. Park et al. / Powder Technology 306 (2017) 34–44 Table 1 Powder characteristics of SUS17-4PH. Powder size (μm) D10
D50
D90
Distribution slope parameter (Sw)
1.72
3.17
5.26
5.25
Apparent density (g/cm3)
Tap density (g/cm3)
True density (g/cm3)
2.14
3.82
7.77
Table 2 Chemical compositions of SUS17-4PH. Chemical compositions [wt.%] C
Si
Mn
P
S
Ni
Cr
Mo
Cu
Co
Nb
Fe
0.04
0.38
0.14
0.02
0.01
4.06
16.36
0.01
4.02
0.03
0.30
Bal.
Density (g/cm ) Melting temperature (°C) Decomposition temperature (°C)
objective was to determine the phenomenological coefficients of Phillip's model combined with a non-Newtonian viscosity model through experimental and numerical approaches. The material parameters for the viscosity model were empirically obtained by capillary rheometry experiments. After extraction of the injected feedstock in the designed jig, the surface of the cross-section was polished with acetone to expose the powder in the binder. Then, the solids loading in the radial direction of the injected feedstock was measured and quantified by SEM and image processing based on the intensity difference, respectively. The powder distribution profile in the radial direction was finally obtained. As compared with experimental and numerical results, i.e. the mean residual method, the ratio of the phenomenological coefficients was determined. The individual phenomenological coefficients of Kc and Kη were derived by considering the convergence time to steady-state time. Finally, a parametric study based on the obtained phenomenological coefficients was carried out to understand each term of the governing equation. 2. Preliminary backgrounds for P-B separation
Table 3 Characteristics of binder components.
3
35
Wax
PP
PE
SA
0.90 42–62 180–320
0.90 110–150 350–470
0.92 60–130 420–480
0.94 74–83 263–306
Hausnerova et al. [2] evaluated separation behavior using a specially developed testing mold to observe the geometric effects. They employed SEM combining EDX (for the observation of chemical compositions including iron) to quantify the separation. Even though this approach was suitable to evaluate the P-B separation induced by inertia effects, it was difficult to theoretically identify the contributing factors, such as a shear rate, viscosity, and solids loading, that resulted in P-B separation from a theoretical point of view. In this study, new approaches to determine the phenomenological coefficients for a governing equation representing separation behavior were investigated, and then each term of the governing equation combining the obtained phenomenological coefficients was analyzed. The research
2.1. Non-Newtonian viscosity model The rheological property of a feedstock depends on the material and processing conditions, such as the solids loading, shear rate, and injection temperature. Many studies have characterized viscosity as one of the representative rheological characteristics. Several viscosity models for a Newtonian and non-Newtonian suspension or feedstock have been proposed [21,22] since the development of the viscosity model of Newtonian dilute suspension was introduced by Einstein [23]. In this study, a viscosity model combining power law and Krieger and Dougherty [24,25] was employed to describe the viscosity behavior in terms of the shear rate, temperature and solids loading. In addition, the maximum critical solids loading was taken into account. The viscosity model as a function of the shear rate, temperature, and solids loading is expressed as : Q : n−1 ϕ −Bϕm 1− ; γ ηðγ ; T; ϕÞ ¼ m exp RT ϕm
Fig. 1. (a) Experimental setup for capillary rheometer, and (b) design of jig [mm].
ð1Þ
36
D.Y. Park et al. / Powder Technology 306 (2017) 34–44
Fig. 2. Encapsulated injection feedstock and measurement position.
where m is the material constant, Q is the activation energy, R is the universal gas constant (8.314 J·mol−1 K−1), T is the temperature,γ_ is the shear rate, ϕ is the solids loading, η0 is the binder viscosity, n is the power law index, ϕm is the critical solids loading, and B is the intrinsic viscosity. The activation energy physically means the sensitivity between viscosity and temperature. For instance, a material with a higher activation energy is more sensitive to temperature.
condition R b 1, and P = P0z. The steady-state solution combining Eqs. (5) and (6) is derived as K η 1 ϕs ðr Þ ðϕm −ϕs ðr ÞÞ−l K c −n r−1=n K η ¼ −1=n : ϕs ðRÞ −l K c −1n R ðϕm −ϕs ðRÞÞ
ð7Þ
2.2. Conservation equation for P-B separation Assuming the flow of a feedstock in a cylindrical tube with a low Reynolds number (Re b b1) due to polymeric fluid, Stokes flow approximation is applicable. The continuity and momentum equations are given as ∇ u ¼ 0; ∇ τ þ ∇p ¼ 0;
ð2Þ ð3Þ
whereτ is the shear stress, u is the velocity vector, and p is the hydrostatic pressure. The z-component of momentum Eq. (3) is expressed as −
∂P ∂τ rz τrz ∂τzz þ þ þ ¼ 0: r ∂z ∂r ∂z
ð4Þ
The assumptions for Stokes flow in a pressure-driven tube flow are the following: i) steady-state, ii) ∂/∂z b b ∂/∂r , ur b b uz , uz = uz(r, z) and p = p(z), and iii) isothermal and incompressible flow. Under the assumptions of Stokes flow, Eq. (4) can be simplified as ∂P 1 ∂ ∂uz ¼ 0: − rη ∂z r ∂r ∂r
ð5Þ
The conservation equation proposed by Phillip et al. [18] for the fluxes induced by the spatially varying viscosity, shear rate, and solids loading was employed to exhibit the powder migration behavior in a wax-based polymeric binder system. The conservation equation for powder migration in a cylindrical tube under the axisymmetric condition is expressed as : : 1 ∂η ∂ϕ : ∂ϕ ∂ϕ a2 ∂ ∂γ þ K η γ ϕ2 ¼ 0; ¼ r K c ϕ2 þ ϕγ r ∂r η ∂ϕ ∂r ∂t ∂r ∂r
ð6Þ
where Kc is the phenomenological constant for the flux induced by interaction frequency, a is the particle radius, ϕ is the solids loading, and γ_ is the shear rate. Here, Kη is the phenomenological constant for the flux induced by viscosity, and η is the viscosity. The steady-state solution for Eq. (6) with assumptions of Poiseuille flow as a steady unidirectional pressure-driven flow and axisymmetric
Fig. 3. Mixing behaviors with different volumetric solids loadings; (a) torque, and (b) error bar.
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Table 4 Material parameters for viscosity model and activation energy. Solid loading (vol.%) 40 50 60
Temperature (°C)
Material parameter
130
150
170
Activation energy (kJ/mol)
n ln A1 n ln A1 n ln A1
0.43 6.65 0.39 7.78 0.33 8.92
0.43 6.27 0.36 7.64 0.32 8.78
0.42 12.00 6.34 0.38 12.40 7.45 0.32 12.95 8.57
ln A2
B
ln m
3.00 0.99 2.67 4.09 5.07
3. Experimental procedures 3.1. Material preparation Water-atomized 17-4PH stainless steel powder was prepared by Epson Atmix Corporation (Japan). The mean powder size was 3.17 μm. The distribution slope parameter (Sw) for the size distribution defined by Eq. (8) was 5.25. A larger value of Sw indicates a narrower powdersize distribution: Sw ¼
2:56 ; D90 log10 D10
ð8Þ
where Sw is the distribution slope parameter, and D10 and D90 are the 10% and 90% points of the cumulative powder size distribution, respectively. The powder characteristics and chemical compositions (wt.%) of SUS17-4PH are summarized in Tables 1 and 2. A wax-based polymeric binder system composed of wax (57%), polypropylene (PP, 25%), polyethylene (PE, 15%) and stearic acid (SA, 3%) was used as a temporary vehicle to move the powder. The binder characteristics are summarized in Table 3. The critical solids loading (vol.%), i.e. the maximum packing fraction, is regarded as an adjustable parameter of the viscosity model in Eq. (1). The critical solids loading was determined through a torque rheometry experiment. Based on the results of the critical solids loading, mixing was carried out to attain a uniform distribution of the powder in the binder system by using a twin-extrude type of mixer (made by CetaTech Inc.). The viscosity of the feedstock was measured by the capillary rheometer (Rosand RH7, Malvern). A twin-bore-type rheometer and Rabinowitch correction [26] were employed to compensate the pressure drop at the entrance and exit as well as to consider the non-
Fig. 4. Viscosity for three different temperatures, solids loading; (a) 40, (b) 50, and (c) 60 vol.%.
Fig. 5. Activation energies for different initial solids loadingϕ0.
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Fig. 6. Intrinsic viscosity.
Newtonian characteristics. The diameter and L/D of the die for the capillary rheometry experiments were 1 mm and 16, respectively. The viscosities for the three temperatures of 130, 150, and 170 °C and solids
loadings of 40, 50 and 60 vol.% were measured within the shear rate range from 100 to 5000 rad/s. The jig to encapsulate the feedstock injected from a barrel was inserted into the barrel. As soon as the jig contained the feedstock after injection, it was separated from the barrel, and was rapidly quenched in alcohol. Fig. 1 shows the experimental setup for injection and the design of the jig. The solids loading (powder volume fraction) on the surface of extracted feedstock was measured by SEM and then calculated by image processing. Before the powder volume fraction was measured, cold mounting was carried out to avoid destruction of the original powder distribution profile because the injected feedstock was sensitive to heat and the pressure of compression mounting. The injected feedstock in the jig was encapsulated in the cold mounting resin (Quickcure acrylic, ALLIED). QuickCure acrylic powder and liquid were mixed in the volume ratio of 2 to 1. The curing time and temperature were 20 min and 25 °C, respectively. The encapsulated feedstock was polished to expose the powder on the surface of the injected feedstock composed of hard SUS powder and a soft binder. To maintain the powder distribution in the injected feedstock, polishing was carried out by polishing cloths (MD/DP-Nap, Struers) with acetone because the mounting resin and binder were soluble in acetone. The solids loading was measured along the radial direction under same measurement conditions, including the brightness, contrast, and magnitude. Fig. 2 shows the
Fig. 7. Polished surfaces on the cross-section of injection molded feedstock, (a) SiC paper with grit of 1200 (b) SiC paper with grit of 1500, (c) debinding, (d) electro etching, (e) smooth cloth with water polishing, and (f) smooth cloth with acetone.
D.Y. Park et al. / Powder Technology 306 (2017) 34–44
measurement positions. To ensure the reliability of the data acquisition, the solids loading measurement was repeated five times for various locations at the same r, and an average value was presented except the maximum and minimum values.
39
versus ϕm ln (1 − ϕ/ϕm) shows the empirical value of B as shown in Fig. 6. The obtained values are summarized in Table 4.
4.3. Measurement of powder distribution 4. Results and discussion 4.1. Determination of critical solids loading Torque rheometer experiments were carried out to determine the critical solids loading (vol.%). Fig. 3 shows the mixing behavior of feedstock with solids loadings. The torque was measured by adding 1% of solids loading for each step at the temperature of 150 °C. The torque of the feedstock was increased by additional solids loading, and it fluctuated until the powder and binder were uniformly distributed. The homogeneity of the feedstock was evaluated by consideration of the steady-state value of torque. The indicative features of critical solids loading are regarded as a rapid rise in the torque and an emergence of an erratic value [1]. However, the result was unclear to determine the critical solids loading because there was no distinguishable feature representing the critical solids loading as shown in Fig. 3(a). As an alternative method, the average value and error bar for torque were considered to find the critical solids loading [27,28]. Fig. 3(b) shows the slope of the curve and error bar with the torque. Even though a sharp increase of torque was not observed, the critical solids loading was determined by the onset point, namely, the intersection point of linear fitting curves between the low and high ranges of solids loading. The critical solids loading was determined to be approximately 61 vol.%. 4.2. Viscosity parameters As mentioned in sub-section 2.1, the power law model combining the Krieger and Dougherty model for viscosity was employed. Viscosity measured by a capillary rheometer showed shear-thinning behavior regardless of processing conditions. From the view point of temperature, an increase in temperature led to decreased viscosity. On the other hand, an increase in solids loading resulted in increased viscosity: ln η ¼ ln m þ
: Q ϕ ; þ ðn−1Þ ln γ −Bϕm ln 1− RT ϕm
ln A1 ¼ lnm þ
Q ϕ ; −Bϕm ln 1− RT ϕm
ϕ ; ln A2 ¼ lnm−Bϕm ln 1− ϕm
Various polishing methods have been tried to measure the solids loading on the surface of an extracted feedstock because the original powder distribution profile can be destroyed by polishing. Fig. 7 shows the resulting polished surfaces. Fig. 7(a, b) surfaces that were mechanically polished by SiC papers with grits of 1200 and 1500. After mechanical polishing, the surfaces were rough. Furthermore, pores left by removed powder are observed. For these reasons, another method not only to form a smooth surface but also to solve the powder removal problem was considered. To realize a smooth surface without mechanical force, the binder on the surface was decomposed by debinding as shown in Fig. 7(c). Even though no pore were formed by the removal of powder, the surface was still rough because the binder was nonuniformly removed by a solvent. Electro-etching was also carried out. An encapsulated feedstock was dipped into ethyl alcohol with perchloric acid (volumetric ratio 4:1) for 10 min. As shown in Fig. 7(d), the binder was inhomogeneously etched due to the previously mentioned debinding. Fig. 7(e) shows a surface polished by a smooth cloth (MD-Dac, Struers) with water. Although the surface was smooth, the binder was deformed by polishing. Finally, a surface was polished by a smooth cloth with acetone to avoid deformation of the binder. The results were sufficient to measure the solids loading as shown in Fig. 7(f). The solids loading was quantitatively calculated by image processing based on the difference of intensity between the powder and binder. Fig. 8 shows the results of the image analysis. In the original image, the binder was recognized as a dark color, while the powder was brightly colored. Based on the intensity difference, the powder was labeled by different colors, as shown in Fig. 8(b), and then the boundaries between the powder and binder were marked as shown in Fig. 8(c). Fig. 8(d) shows the final image calculated by
ð9Þ
ð10Þ
ð11Þ
where A1 is the constant when ln γ_ is 0 in Eq. (9), and A2 is the constant when Q/RT is 0 in Eq. (10). From Eq. (9), by taking the natural logarithm of Eq. (1), the values of the power law index (n) were determined by the slope of the curves as shown in Fig. 4. The power law index of the feedstock was less than the unit value, which indicated that all of the feedstock had pseudo-plastic behavior. The obtained values are summarized in Table 4. The activation energy represents the dependency of the temperature on viscosity. When the activation energy is high, a small change in temperature gives rise to a large change in viscosity, which can result in defects, such as a warping or distortion. From the slopes of these curves in Fig. 5, the activation energies (Q) were calculated by linear fitting of the relationship between the y-intersection point of ln A1 and 1/RT. For the results for three different solids loadings of ϕ, the intrinsic viscosity of B was calculated. Here, B physically means the sensitivity of solids loading to viscosity. For instance, if B is a high value, the viscosity might be more sensitive to changes in ϕ. The slope of the linear fitted curve representing ln A2
Fig. 8. Image analysis, (a) original image measured by SEM (× 700), (b) individually labeled powder, (c) boundary measurement, and (d) powder measurement.
40
D.Y. Park et al. / Powder Technology 306 (2017) 34–44
Fig. 9. Powder distribution for various values of Kc/Kη and experimental results by image analysis: (a–c) solids loading of 40, 50, and 60 vol.% for 130 °C; (d–f) solids loading of 40, 50, and 60 vol.% for 150 °C; and (g–i) solids loading of 40, 50, and 60 vol.% for 170 °C.
image processing. The solids loadings at r = 0, 0.2, 0.4, 0.6, 0.8, and 1 mm were calculated as Z
rþΔr
2π ϕr ¼
Z
r
ϕrdr
rþΔr
rdr
2π
¼
Aϕ ; A
ð12Þ
r
where r = 0, 0.2, 0.4, 0.6, 0.8, and 1.0 mm; ϕr is the solids loading at r; Aϕ is the area calculated by the image analysis; and A is the total area. 4.4. Determination of Kc/Kη The solids loading obtained from the image analysis should be corrected because some numerical error might be involved. As shown
in Fig. 8(b), some of the binder near the powder was mistakenly identified as powder. In consideration of the this numerical error, the correction factor based on the volume conservation was defined as Z rϕs dr ; ð13Þ correction factor ¼ Z rϕe dr where ϕs is the solids loadings calculated by the numerical analysis, and ϕe is the solids loadings calculated by experiment (calculated by image processing). Given the value of Kc/Kη, the value of the mean residual was determined. It was natural that the best estimator of Kc/Kη would minimize the deviation, which was represented by the mean residual. The value of Kc/Kη that minimized the residual between ϕs and ϕe was determined by the mean residual method as
Table 5 Kc/Kη with different conditions.
Residual ¼
Temperature (°C)
1 X
2
ϕs;r −ϕe;r ;
ð14Þ
r¼0
Solid loading (vol.%)
130
150
170
40 50 60
0.10 0.10 0.16
0.10 0.10 0.21
0.10 0.15 0.20
where r = 0, 0.2, 0.4, 0.6, 0.8, and 1 mm; ϕs,r is the numerically calculated solids loading at r; and ϕe,r is the experimentally measured solids loading at r.
D.Y. Park et al. / Powder Technology 306 (2017) 34–44
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Fig. 10. (a) Experimentally measured convergence time to steady-state under a fixed shear rate condition for various solids loadings and temperatures, and (b) measured viscosity and shear rate for solids loading of 50% and temperature of 150 °C.
As shown in Fig. 9, the lower Kc/Kη, the more uniformly distributed the powder. A high value of Kc/Kη means that the net force induced by the shear rate difference is larger than the net force induced by the viscosity difference. As Kc/Kη increases, the powder dramatically moves to the center of the cylinder. These results indicate that the interaction frequency induced by powder collision had a greater influence at the region of the wall. The obtained values of Kc/Kη are summarized in Table 5. All values of Kc/Kη at the 40 vol.% of solids loading were 0.1 regardless of the injection temperature. This result means that a relatively small amount of P-B separation occurs. For the solids loading of 50 vol.%, Kc/ Kη at the injection temperature of 170 °C increased. The higher temperature resulted in more P-B separation. On the other hand, all Kc/Kη increased at the solids loading of 60 vol.%. Even though the Kc/Kη increased at the ϕ0 of 60 vol.%, P-B separation in consideration of the scale of the y-axis with ϕ0 decreased. Therefore, Kc/Kη is not an absolute criterion for P-B separation; rather, it is a relative criteria for P-B separation under given material and processing conditions (Table 6). 4.5. Determination of Kc and Kη Individual values of Kc and Kη were obtained by comparison with experimental and numerical convergence times until a steady-state
powder distribution profile was obtained. Even though the ratio of phenomenological coefficients, Kc/Kη, was obtained as described in section 4.5, an additional variable is required to obtain each coefficient. In this regard, the convergence time to steady state was calculated. Fig. 10(a) shows the measured convergence time to steady-state under a fixed shear rate condition. As the shear rate had a constant value, the starting time was determined as shown in Fig. 10(b). The end of the convergence time in Fig. 10(b) was determined by Eq. (15). When the volume difference defined in Eq. (15) was b 5%, the numerical convergence time was determined as shown in Fig. 11. When the experimental and numerical convergence times were identical, the values of Kc and Kη were determined as summarized in Table 6. Z 2 2π r ϕs;r −ϕt;r Volume difference ¼ Z 2 100; 2π r ϕs;r −ϕ0;r
ð15Þ
where r = 0, 0.2, 0.4, 0.6, 0.8, and 1 mm; ϕs,r is the steady-state solids loading at r; ϕt,r is the solids loading at time t; and r, ϕ0,r is the initial solids loading at r.
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D.Y. Park et al. / Powder Technology 306 (2017) 34–44
Fig. 11. Numerically calculated convergence time.
4.6. Parameter study Based on the obtained values of material parameters for the viscosity model and Kc/Kη, the physical meanings of each term, including shear rate, solids loading, and viscosity, were examined. First of all, the directions of net forces induced by each term were investigated. Fig. 12 shows the quantified profile for each parameter calculated based on the experimental results of powder distribution. Here, ∇γ_ has a positive value with the overall range of r/R. A positive value means that the net force induced by ∇γ_ acts on the center of the cylinder. That is, the powder migrates from the wall to the center of the cylinder. On the other hand, ∇ϕ and ∇ η are negative values with the overall range of r/R. This implies that the net force induced by ∇ ϕ and ∇ η acts on the wall of the cylinder. Therefore, it is confirmed that the net force induced by ∇γ_ is applied in the opposite direction of the net force induced by ∇ϕ and ∇ η. As shown in Fig. 13, the fluxes of the conservation equation were calculated based on the : obtained Kc/Kη. The terms applying to opposite directions, K c ϕ2 ∂ γ =∂r : 2 and K η ðγ ϕ Þð1=ηÞð∂η=∂ϕÞð∂ϕ=∂rÞ, had a greater influence on P-B sepa: ration than the solids loading, K c ϕ γ ∂ϕ=∂r: Considering the scale of the : : y-axis in Fig. 13(d), K c ϕ2 ∂ γ =∂r and K η ðγ ϕ2 Þð1=ηÞð∂η=∂ϕÞð∂ϕ=∂rÞ com: pared to K c ϕ γ ∂ϕ=∂r have relatively high values. This result indicates that P-B separation is mainly dominated by fluxes induced by ∇γ_ and ∇η. 5. Conclusions Based on experimental and numerical approaches, the phenomenological coefficients of a governing equation combined with a non-Newtonian viscosity model were obtained. In the process of deriving the phenomenological coefficients, it was confirmed that the proposed polishing and image processing methods were suitable to quantify the solids loading. Separation behavior was evaluated by a parametric study using each term of the governing equation combining the Table 6 Kc and Kη under various conditions.
obtained coefficients. From the results, it was confirmed that the effect induced by shear rate was applied in the opposite direction induced by solids loading and viscosity.
Temperature (°C) 130
150
Fig. 12. Parameter study, effects of (a) shear rate, (b) solids loading, and (c) viscosity.
170
Solid loading (vol.%)
Kc
Kη
Kc
Kη
Kc
Kη
40 50 60
27.7 2.54 0.17
277 25.4 0.92
54 0.66 0.15
540 6.6 0.71
22.3 2.85 0.11
223 19 0.56
Acknowledgments This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (no. 2011-
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Fig. 13. Fluxes induced by (a) shear rate, (b) solids loading, (c) viscosity, and (d) total flux.
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