- Email: [email protected]
defluidization of metal powders based on solid surface energy
Powder Technology 301 (2016) 1144–1147
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Short communication
A thermodynamic model for agglomeration/defluidization of metal powders based on solid surface energy Yiwei Zhong a,⁎, Zhi Wang b,⁎, Jintao Gao a, Zhancheng Guo a a b
State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing 100083, PR China National Engineering Laboratory for Hydrometallurgical Cleaner Production Technology, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, PR China
a r t i c l e
i n f o
Article history: Received 1 February 2016 Received in revised form 26 April 2016 Accepted 29 July 2016 Available online 30 July 2016 Keywords: Metal powder Fluidized bed Defluidization Solid surface energy Thermodynamic model
a b s t r a c t A thermodynamic model was developed to associate the agglomeration tendency with the inherent properties of bed materials. Solid surface energy of metallic iron was calculated by the elastic modulus and lattice constant. Based on the relationship between solid surface energy and temperature, the thermodynamic spontaneity for particle agglomeration was explained, and the limiting temperature to defluidize was determined in a good agreement with the experimental data. Consequently, a criterion for the defluidization was proposed as a reference to select the bed materials. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Due to limiting the continuous operation and high productivity of process, the high-temperature agglomeration/defluidization in fluidized beds has received extensive research attention in many industry fields, such as metallurgical and mineral processes [1,2], coal/biomass/ waste conversion [3–5], and chemical looping process [6]. Defluidization phenomenon was attributed to an increase of particle stickiness at elevated temperature, and the agglomeration tendency depended strongly on the physical and chemical characteristics of the bed materials at high temperature range [7–10]. In theory, a given bed material had a corresponding limiting temperature, above which defluidization occurred. However, it is difficult to accurately predict this critical temperature and thus prevent the appearance of defluidization. Therefore, a model or criterion is needed to reliably predict the thermodynamic spontaneity of particle agglomeration based on the inherent properties of different bed materials. Many mathematical models have been developed to simulate the defluidization behavior at high temperatures, especially to predict the operating conditions where defluidization occurred in fluidized beds [10–16]. Most widely used approach in modeling was based on force balance acting on agglomerates and particles. By comparing the
⁎ Corresponding authors. E-mail addresses: [email protected] (Y. Zhong), [email protected] (Z. Wang), [email protected] (J. Gao), [email protected] (Z. Guo).
http://dx.doi.org/10.1016/j.powtec.2016.07.068 0032-5910/© 2016 Elsevier B.V. All rights reserved.
breaking forces, the local shear force and the cohesive force, the limiting gas fluidizing velocity [10–12], the defluidization temperature [13], the defluidization time [4,14,15] and the agglomerate size [16] were predicted as a function of the operating conditions. The agglomeration and defluidization behavior of metal powders were determined by particle adhesion. In the absence of liquid phases, the two aspects of physical interactions and chemical interactions can be considered for particle adhesion. Physical interactions involved the cohesive forces arising from the softening of bed particles (apparent surface viscosity) [9,13], whereas chemical interactions involved a surface bonding as a consequence of sintering between the bed particles [2,3,7–10]. Regardless of the type of interactions, the particle adhesion was essentially the surface diffusion/migration of atoms, and was dependent on crystal structure and atomic bonding of materials. Consequently, the thermodynamic spontaneity and limitation should be determined by the inherent properties of materials. However, studies were rarely conducted to associate the particle adhesion with the measured physical parameters at high temperature, which was highly relevant to understand the mechanisms of defluidization. Therefore, this work attempts to establish a relationship between the agglomeration tendency and the inherent properties of bed materials. The solid surface energy as a function of temperature is calculated by the elastic modulus and lattice constant. Based on this, a model is developed to predict the thermodynamic spontaneity for particle agglomeration in a fluidized bed, by which a criterion for the defluidization is proposed as a reference to select the bed materials.
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147
2. Experimental High purity iron powders (N99.99%) provided by Sinopharm Chemical Reagent Co., Ltd. were used in the present study. In order to obtain the lattice parameter, the diffraction patterns at different temperatures were determined by in situ X-ray diffractometer (X'Pert PRO MPD, Netherland). The X-ray diffraction data were collected in a broad 2θ range using Cu-Kα1 radiation (45 kV, 40 mA) with a step size of 0.02° and a scanning speed of 6°/min. The in situ hot-stage unit was a stainless steel chamber. The sample was placed in the chamber and heated at a rate of 10 °C/min from 25 to 800 °C. When the designated temperature was achieved, the temperature was maintained for 10 min before data collecting. To avoid oxidation at high temperature, all XRD measurements were carried out under an inert atmosphere of Ar (N99.99%). 3. Results and discussion 3.1. Calculation of solid surface energy Surface energy is one of the most fundamental parameters directly related to the binding forces of the solid material [17,18]. Thermodynamically, the surface with a higher potential energy had a tendency to shrink the surface of particles. This trend can be considered as a decrease of surface viscosity and thus responsible for particle agglomeration [2,9]. Nevertheless, very few experimental data of solid surface energy was reported, especially its temperature dependence. Therefore, based on the material mechanics theory, in this study the surface energy of solid metal was theoretically calculated at different temperatures by combined with the experimental data of physical properties of materials. According to the definition, the surface tension was equal to the energy to rebuild two new surfaces after separating a solid into two parts. If considering surface separation as a continuous process, the surface tension of solids was represented as: Z
RF 0
Fy dy A
ð1Þ
where γ is the surface tension of solids; ΔE is the energy to generate two new surfaces; A is the surface area; RF is the distance between the two new surfaces; Fy is the force to break the surface. If the distance between atoms was much larger than 1 × 10−10 m, the interaction force was nearly zero, Hooke's law was not applicable under this condition. And the strain related to Young's modulus which was only available for small displacement. Therefore, it was assumed that RF was 1 × 10−10 m (approximately the magnitude of the distance between atoms). The Hooke's law can be expressed as:
Z
ð6Þ
where K is the elasticity modulus, N/m2; K0 is a regressive constant, N/ m2; β is the correction coefficient of the elasticity modulus; α is the linear expansion coefficient, 11.7 × 105/K; Ω is the orientation factor, 0.5; ΔF is the activation of dislocation motion, 0.5 eV; vc is the sonic velocity in metal, 5.05 × 105 cm/s; b is the Burgers vector of dislocation, 2.90 × 10−8 cm; ρm is the density of dislocation, 100/cm2; C is the strain rate, 0.0092/s; k is Boltzmann constant. All the values were from Ref. [19]. The calculation results were shown in Fig. 1. The elasticity modulus of metal irons significantly decreased as temperature increased. The theoretical values were in a good agreement with the experimental data [19], suggesting a good accuracy of this model. To obtain the lattice constants as a function of temperature, in situ Xray diffraction measurements were carried out. As shown in Fig. 2, there were apparent diffraction peaks at position with diffraction angles of 44°–45°, 64°–66° and 81°–83°, referring to [100], [200] and [211] crystal plane of Fe, respectively. As temperature increased, the position of diffraction peak for each crystal plane gradually shifted to the left, indicating the inter-atomic distance of metal irons was increasing and the lattice was expanding. The diffraction angle for each crystal plane at different temperatures was listed in Table 1. Since Fe was cubic crystalline, its lattice constant was calculated by Bragg equation: a¼
λ 2 sinθ
RF
ð7Þ
where a is the lattice constant; λ is the wavelength of the radiation (Cu-
24
Experimental Calculated
22
ð3Þ
20 18
dγ dT
where Es is solid surface energy; T is Kelvin temperature.
ð4Þ
16
×
K
0
y R F 2 K ðT Þ dy ¼ 4 aðT Þ a
where K is Young's modulus of solid; a is the lattice constant. Unlike liquids, the surface energy of solids was not be equated with its surface tension, because solid separation was divided to the formation of new surface and the rearrangement of surface atoms. Thus, the surface energy of solids was represented as its interfacial free energy [17,18]: Es ¼ γ−T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 h þk þl
-10
ed: K 2
Ωρm bvc expð−ΔF=kT Þ β ¼ exp − C
ð2Þ
Taking Eq. (2) into Eq. (1), the surface tension of solids was calculat-
γ ðT Þ ¼
ð5Þ
2
Fy ΔL y ¼K ¼K A L a
K ðT Þ ¼ K 0 ð1−25αT Þβ
(N/m )
ΔE 1 ≈ 2A 2
According to this calculation, the solid surface energy of materials was related to its bulk properties. As shown in Eq. (4), since the elasticity modulus (K) and lattice constant (a) were dependent on temperature (Eq. (3)), the solid surface energy was also a function of temperature. Consequently, the solid surface energy was calculated by the elasticity modulus and lattice constant of materials. For most metallic materials, the elasticity modulus depended on the inter-atomic bonding force and decreased with increasing temperature. Besides of the elastic strain, the dislocation motion also occurred inside metals due to the thermal activation within the elastic range at high temperature, leading to local small plastic strain. Thus, the empirical correlation of elasticity modulus and temperature for Fe, Al and lowcarbon steel was expressed as [19]:
10
γ¼
1145
14
10
K0 = 24.25 × 10 N/m
2
12 10 200
400
600
800
1000
1200
Temperature (K) Fig. 1. Calculated and experimental elasticity modulus of metallic iron as a function of temperature.
1146
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147
800 °C
(110)
(200)
(211)
700 °C
Counts
600 °C 500 °C 400 °C 300 °C 200 °C 100 °C 25 °C 30 35 40 45 50 55 60 65 70 75 80 85 90
2θ θ (°) Fig. 3. The lattice constant of crystal iron as a function of temperature.
Fig. 2. X-ray diffraction patterns of metallic iron at different temperature.
Kα1, 0.154051 nm); θ is the diffraction angle; h, k, l are Miller indices. The measured values of lattice constant of metal irons at different temperatures were shown in Fig. 3. Obviously, the lattice constant increased with increasing temperature. The expression of a with temperature was obtained by a statistical regression, and the regression results supported the linear relation between K and temperature as followed:
For the whole fluidized bed system, the change of Gibbs free energy was expressed as:
a ¼ 2:8586−3:124 10−5 T
where G is the total Gibbs free energy in system; A is the total surface area. In thermodynamic equilibrium, dG = 0, which leads to
ð8Þ
3.2. A thermodynamic criterion for particle agglomeration
dG ¼ AdEs þ Es dA
ð9Þ
Es dA ¼ −AdEs
Table 1 Crystalline structure parameters of pure iron at different temperatures. Temperature (K)
298 373 473 573 673 773 873 973 1073
Diffraction angle θ110 (°)
θ200 (°)
θ211 (°)
44.7218 44.6976 44.6684 44.6188 44.5619 44.5009 44.4460 44.3898 44.3236
65.0750 64.9894 64.9564 64.9309 64.8128 64.7277 64.7051 64.5718 64.4768
82.3745 82.3231 82.2646 82.1485 82.0328 81.9151 81.8072 81.6822 81.5296
ð10Þ
Differentiating with respect to T gives Es
dA dEs ¼ −A dT dT
ð11Þ
Thermodynamically, a single particle or granular group tended to a state in which the total energy of the system was minimization. When the state of granular group in a fluidized bed transformed from normal fluidization to defluidization, agglomeration and sintering between particles occurred. In this process the total superficial area of the system was decreased. According to Fig. 4, at temperature below 600 K, the solid surface energy did not change with temperature. It was assumed
3.6
Surface energy (J/m2)
The linear trend of the bcc lattice parameter as a function of temperature was related to the intrinsic nature of thermal expansion as a result of increasing atomic thermal vibrations from their equilibrium positions [20]. Consequently, by combining Eqs. (3)–(8), the solid surface energy of metal irons was calculated as a function of temperature. As shown in Fig. 4, when the temperature was lower than 600 K, the solid surface energy was almost a constant value of about 2.1 J/m2. However, when the temperature was higher than 600 K, the solid surface energy significantly increased as the temperature increased in the range of 673 K to 873 K. This range was just the Huttig temperature of metal irons (500–600 ° C) [9,21]. In a view of solid lattice dynamics [21], the amplitude of atoms/ions around the equilibrium position was increasing when the temperature increased. Atoms/ions began to overcome the lattice binding force, and exchanged places inside or on the surface of lattices, then further diffused into adjacent lattices. Simultaneously, atoms and defects inside lattices started to activate. The diffusion was intensified, and the sintering rate accelerated significantly. Therefore, the metal materials presented a higher solid surface energy.
3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8
300 400 500 600 700 800 900 1000 1100
Temperature (K) Fig. 4. The surface energy of solid iron as a function of temperature.
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147 s that dE ¼ 0, and thus dA ¼ 0. Consequently, the total superficial area was dT dT constant, suggesting that particle agglomeration did not appear. However, at temperatures above 600 K, the surface energy increased with in-
dEs dT
creasing temperature, N0 , thus indicating that the total superficial area of granular group decreased as temperature increased. Since the surface of single solid particles hardly changed, the only way to decrease the total superficial area of the system was through sintering and agglomeration. As a result, the bed defluidization occurred at the macro-level. These results were consistent with the experimental observations in previous works [2,9,13]. Based on this model, solid surface energy was associated with the inherent properties of bed materials. The thermodynamic conditions of defluidization in a fluidized bed were theoretically determined by the relationship between solid surface energy and temperature. Thus, a criterion for the defluidization was proposed as followed: 8 dEs > > N0 b0 Dispersion > > > dT dA < dEs ¼ 0 Stabilization ¼0 dT > dT > > > dE > s : b0 N0 Agglomeration dT
dA b0 , dT
ð12Þ
For a given bed material, the granular group tended to stable fluidization if
dEs dT
≤0, whereas the agglomeration/defluidization occurred
spontaneously if
dEs dT
N0 . The limiting temperature of defluidization
s N0, which was considered (about 600 K for Fe) was the point where dE dT as a reference to select the bed materials. When operated above this limiting temperature, the risk of agglomeration was increased. It was s also inferred that if dE of the bed material was adjusted to be ≤0 by surdT face coating or modification, defluidization can be avoided.
4. Conclusions In this study, the solid surface energy of metallic iron was calculated by the elastic modulus and lattice constant, which increased with increasing temperature. Based on thermodynamic analysis, a model was developed to predict the spontaneity for particle agglomeration. Thus, the relationship between the agglomeration tendency and the inherent properties of bed materials was established. For a given bed material, s the fluidization state tended to be stable if dE dT ≤0, whereas agglomeras N0. Consequently, tion and defluidization occurred spontaneously if dE dT the limiting temperature of defluidization was determined as a point s N0, which was in accordance with the experimental results. where dE dT
1147
Acknowledgement This work was financially supported by the National Natural Science Foundation of China (Grant 91534121), Project Funded by China Postdoctoral Science Foundation (Grant 2015M570931) and Fundamental Research Funds for the Central Universities (Grant FRF-TP-15-013A1).
References [1] L.S. Johannes, Recent status of fluidized bed technologies for producing iron input materials for steelmaking, Particuology 9 (2011) 14–23. [2] Y. Zhong, Z. Wang, X. Gong, Z. Guo, Sticking behaviour caused by sintering in gas fluidisation reduction of haematite, Ironmak. Steelmak. 39 (2012) 38–44. [3] B.J. Skrifvars, M. Hupa, M. Hiltunen, Sintering of ash during fluidized bed combustion, Ind. Eng. Chem. Res. 31 (1992) 1026–1030. [4] W. Lin, K. Dam-Johansen, F. Frandsen, Agglomeration in bio-fuel fired fluidized bed combustors, Chem. Eng. J. 96 (2003) 171–185. [5] C.L. Lin, T.H. Peng, W.J. Wang, Effect of particle size distribution on agglomeration/ defluidization during fluidized bed combustion, Powder Technol. 207 (2011) 290–295. [6] P. Cho, T. Mattisson, A. Lyngfelt, Defluidization conditions for a fluidized bed of iron oxide-, nickel oxide-, and manganese oxide-containing oxygen carriers for chemical-looping combustion, Ind. Eng. Chem. Res. 45 (2006) 968–977. [7] J.P.K. Seville, High-temperature defluidization, Powder Technol. 38 (1984) 13–22. [8] P. Compo, R. Pfeffer, G.I. Tardos, Minimum sintering temperatures and defluidization characteristics of fluidizable particles, Powder Technol. 51 (1987) 85–101. [9] Y. Zhong, Z. Wang, Z. Guo, Q. Tang, Defluidization behavior of iron powders at elevated temperature: influence of fluidizing gas and particle adhesion, Powder Technol. 230 (2012) 225–231. [10] T. Mikami, H. Kamiya, M. Horio, The mechanism of defluidization of iron particles in a fluidized bed, Powder Technol. 89 (1996) 231–238. [11] J.P.K. Seville, H. Silomon-Pflug, P.C. Knight, Modelling of sintering in high temperature gas fluidisation, Powder Technol. 97 (1998) 160–169. [12] P.C. Knight, J.P.K. Seville, H. Kamiya, M. Horio, Modelling of sintering of iron particles in high-temperature gas fluidization, Chem. Eng. Sci. 55 (2000) 4783–4787. [13] Y. Zhong, Z. Wang, Z. Guo, Q. Tang, Prediction of defluidization behavior of iron powder in a fluidized bed at elevated temperatures: theoretical model and experimental verification, Powder Technol. 249 (2013) 175–180. [14] C.L. Lin, M.Y. Wey, C.Y. Lu, Prediction of defluidization time of alkali composition at various operating conditions during incineration, Powder Technol. 161 (2006) 150–157. [15] J.H. Kuo, K.M. Shih, C.L. Lin, M.Y. Wey, Simulation of agglomeration defluidization inhibition process in aluminum-sodium system by experimental and thermodynamic approaches, Powder Technol. 224 (2012) 395–403. [16] J.M. Valverde, A. Castellanos, Fluidization of nanoparticles: a simple equation for estimating the size of agglomerates, Chem. Eng. J. 140 (2008) 296–304. [17] D. Myers, Surfaces, Interfaces, and Colloids: Principles and Applications, second ed. John Wiley & Sons, Inc., New York, 1999. [18] Z. Wu, B. Wang, Thin Film Growth, Science Press, Beijing, 2001. [19] D. Feng, Metal Physics, Science Press, Beijing, 1975. [20] B.D. Cullity, Elements of X-ray Diffraction, second ed. Adison-Wesley Publishing Company, London, 1978. [21] C.N. Satterfield, Heterogeneous Catalysis in Industrial Practice, second ed. McGrawHill, New York, 1991.
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Short communication
A thermodynamic model for agglomeration/defluidization of metal powders based on solid surface energy Yiwei Zhong a,⁎, Zhi Wang b,⁎, Jintao Gao a, Zhancheng Guo a a b
State Key Laboratory of Advanced Metallurgy, University of Science and Technology Beijing, Beijing 100083, PR China National Engineering Laboratory for Hydrometallurgical Cleaner Production Technology, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, PR China
a r t i c l e
i n f o
Article history: Received 1 February 2016 Received in revised form 26 April 2016 Accepted 29 July 2016 Available online 30 July 2016 Keywords: Metal powder Fluidized bed Defluidization Solid surface energy Thermodynamic model
a b s t r a c t A thermodynamic model was developed to associate the agglomeration tendency with the inherent properties of bed materials. Solid surface energy of metallic iron was calculated by the elastic modulus and lattice constant. Based on the relationship between solid surface energy and temperature, the thermodynamic spontaneity for particle agglomeration was explained, and the limiting temperature to defluidize was determined in a good agreement with the experimental data. Consequently, a criterion for the defluidization was proposed as a reference to select the bed materials. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Due to limiting the continuous operation and high productivity of process, the high-temperature agglomeration/defluidization in fluidized beds has received extensive research attention in many industry fields, such as metallurgical and mineral processes [1,2], coal/biomass/ waste conversion [3–5], and chemical looping process [6]. Defluidization phenomenon was attributed to an increase of particle stickiness at elevated temperature, and the agglomeration tendency depended strongly on the physical and chemical characteristics of the bed materials at high temperature range [7–10]. In theory, a given bed material had a corresponding limiting temperature, above which defluidization occurred. However, it is difficult to accurately predict this critical temperature and thus prevent the appearance of defluidization. Therefore, a model or criterion is needed to reliably predict the thermodynamic spontaneity of particle agglomeration based on the inherent properties of different bed materials. Many mathematical models have been developed to simulate the defluidization behavior at high temperatures, especially to predict the operating conditions where defluidization occurred in fluidized beds [10–16]. Most widely used approach in modeling was based on force balance acting on agglomerates and particles. By comparing the
⁎ Corresponding authors. E-mail addresses: [email protected] (Y. Zhong), [email protected] (Z. Wang), [email protected] (J. Gao), [email protected] (Z. Guo).
http://dx.doi.org/10.1016/j.powtec.2016.07.068 0032-5910/© 2016 Elsevier B.V. All rights reserved.
breaking forces, the local shear force and the cohesive force, the limiting gas fluidizing velocity [10–12], the defluidization temperature [13], the defluidization time [4,14,15] and the agglomerate size [16] were predicted as a function of the operating conditions. The agglomeration and defluidization behavior of metal powders were determined by particle adhesion. In the absence of liquid phases, the two aspects of physical interactions and chemical interactions can be considered for particle adhesion. Physical interactions involved the cohesive forces arising from the softening of bed particles (apparent surface viscosity) [9,13], whereas chemical interactions involved a surface bonding as a consequence of sintering between the bed particles [2,3,7–10]. Regardless of the type of interactions, the particle adhesion was essentially the surface diffusion/migration of atoms, and was dependent on crystal structure and atomic bonding of materials. Consequently, the thermodynamic spontaneity and limitation should be determined by the inherent properties of materials. However, studies were rarely conducted to associate the particle adhesion with the measured physical parameters at high temperature, which was highly relevant to understand the mechanisms of defluidization. Therefore, this work attempts to establish a relationship between the agglomeration tendency and the inherent properties of bed materials. The solid surface energy as a function of temperature is calculated by the elastic modulus and lattice constant. Based on this, a model is developed to predict the thermodynamic spontaneity for particle agglomeration in a fluidized bed, by which a criterion for the defluidization is proposed as a reference to select the bed materials.
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147
2. Experimental High purity iron powders (N99.99%) provided by Sinopharm Chemical Reagent Co., Ltd. were used in the present study. In order to obtain the lattice parameter, the diffraction patterns at different temperatures were determined by in situ X-ray diffractometer (X'Pert PRO MPD, Netherland). The X-ray diffraction data were collected in a broad 2θ range using Cu-Kα1 radiation (45 kV, 40 mA) with a step size of 0.02° and a scanning speed of 6°/min. The in situ hot-stage unit was a stainless steel chamber. The sample was placed in the chamber and heated at a rate of 10 °C/min from 25 to 800 °C. When the designated temperature was achieved, the temperature was maintained for 10 min before data collecting. To avoid oxidation at high temperature, all XRD measurements were carried out under an inert atmosphere of Ar (N99.99%). 3. Results and discussion 3.1. Calculation of solid surface energy Surface energy is one of the most fundamental parameters directly related to the binding forces of the solid material [17,18]. Thermodynamically, the surface with a higher potential energy had a tendency to shrink the surface of particles. This trend can be considered as a decrease of surface viscosity and thus responsible for particle agglomeration [2,9]. Nevertheless, very few experimental data of solid surface energy was reported, especially its temperature dependence. Therefore, based on the material mechanics theory, in this study the surface energy of solid metal was theoretically calculated at different temperatures by combined with the experimental data of physical properties of materials. According to the definition, the surface tension was equal to the energy to rebuild two new surfaces after separating a solid into two parts. If considering surface separation as a continuous process, the surface tension of solids was represented as: Z
RF 0
Fy dy A
ð1Þ
where γ is the surface tension of solids; ΔE is the energy to generate two new surfaces; A is the surface area; RF is the distance between the two new surfaces; Fy is the force to break the surface. If the distance between atoms was much larger than 1 × 10−10 m, the interaction force was nearly zero, Hooke's law was not applicable under this condition. And the strain related to Young's modulus which was only available for small displacement. Therefore, it was assumed that RF was 1 × 10−10 m (approximately the magnitude of the distance between atoms). The Hooke's law can be expressed as:
Z
ð6Þ
where K is the elasticity modulus, N/m2; K0 is a regressive constant, N/ m2; β is the correction coefficient of the elasticity modulus; α is the linear expansion coefficient, 11.7 × 105/K; Ω is the orientation factor, 0.5; ΔF is the activation of dislocation motion, 0.5 eV; vc is the sonic velocity in metal, 5.05 × 105 cm/s; b is the Burgers vector of dislocation, 2.90 × 10−8 cm; ρm is the density of dislocation, 100/cm2; C is the strain rate, 0.0092/s; k is Boltzmann constant. All the values were from Ref. [19]. The calculation results were shown in Fig. 1. The elasticity modulus of metal irons significantly decreased as temperature increased. The theoretical values were in a good agreement with the experimental data [19], suggesting a good accuracy of this model. To obtain the lattice constants as a function of temperature, in situ Xray diffraction measurements were carried out. As shown in Fig. 2, there were apparent diffraction peaks at position with diffraction angles of 44°–45°, 64°–66° and 81°–83°, referring to [100], [200] and [211] crystal plane of Fe, respectively. As temperature increased, the position of diffraction peak for each crystal plane gradually shifted to the left, indicating the inter-atomic distance of metal irons was increasing and the lattice was expanding. The diffraction angle for each crystal plane at different temperatures was listed in Table 1. Since Fe was cubic crystalline, its lattice constant was calculated by Bragg equation: a¼
λ 2 sinθ
RF
ð7Þ
where a is the lattice constant; λ is the wavelength of the radiation (Cu-
24
Experimental Calculated
22
ð3Þ
20 18
dγ dT
where Es is solid surface energy; T is Kelvin temperature.
ð4Þ
16
×
K
0
y R F 2 K ðT Þ dy ¼ 4 aðT Þ a
where K is Young's modulus of solid; a is the lattice constant. Unlike liquids, the surface energy of solids was not be equated with its surface tension, because solid separation was divided to the formation of new surface and the rearrangement of surface atoms. Thus, the surface energy of solids was represented as its interfacial free energy [17,18]: Es ¼ γ−T
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 h þk þl
-10
ed: K 2
Ωρm bvc expð−ΔF=kT Þ β ¼ exp − C
ð2Þ
Taking Eq. (2) into Eq. (1), the surface tension of solids was calculat-
γ ðT Þ ¼
ð5Þ
2
Fy ΔL y ¼K ¼K A L a
K ðT Þ ¼ K 0 ð1−25αT Þβ
(N/m )
ΔE 1 ≈ 2A 2
According to this calculation, the solid surface energy of materials was related to its bulk properties. As shown in Eq. (4), since the elasticity modulus (K) and lattice constant (a) were dependent on temperature (Eq. (3)), the solid surface energy was also a function of temperature. Consequently, the solid surface energy was calculated by the elasticity modulus and lattice constant of materials. For most metallic materials, the elasticity modulus depended on the inter-atomic bonding force and decreased with increasing temperature. Besides of the elastic strain, the dislocation motion also occurred inside metals due to the thermal activation within the elastic range at high temperature, leading to local small plastic strain. Thus, the empirical correlation of elasticity modulus and temperature for Fe, Al and lowcarbon steel was expressed as [19]:
10
γ¼
1145
14
10
K0 = 24.25 × 10 N/m
2
12 10 200
400
600
800
1000
1200
Temperature (K) Fig. 1. Calculated and experimental elasticity modulus of metallic iron as a function of temperature.
1146
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147
800 °C
(110)
(200)
(211)
700 °C
Counts
600 °C 500 °C 400 °C 300 °C 200 °C 100 °C 25 °C 30 35 40 45 50 55 60 65 70 75 80 85 90
2θ θ (°) Fig. 3. The lattice constant of crystal iron as a function of temperature.
Fig. 2. X-ray diffraction patterns of metallic iron at different temperature.
Kα1, 0.154051 nm); θ is the diffraction angle; h, k, l are Miller indices. The measured values of lattice constant of metal irons at different temperatures were shown in Fig. 3. Obviously, the lattice constant increased with increasing temperature. The expression of a with temperature was obtained by a statistical regression, and the regression results supported the linear relation between K and temperature as followed:
For the whole fluidized bed system, the change of Gibbs free energy was expressed as:
a ¼ 2:8586−3:124 10−5 T
where G is the total Gibbs free energy in system; A is the total surface area. In thermodynamic equilibrium, dG = 0, which leads to
ð8Þ
3.2. A thermodynamic criterion for particle agglomeration
dG ¼ AdEs þ Es dA
ð9Þ
Es dA ¼ −AdEs
Table 1 Crystalline structure parameters of pure iron at different temperatures. Temperature (K)
298 373 473 573 673 773 873 973 1073
Diffraction angle θ110 (°)
θ200 (°)
θ211 (°)
44.7218 44.6976 44.6684 44.6188 44.5619 44.5009 44.4460 44.3898 44.3236
65.0750 64.9894 64.9564 64.9309 64.8128 64.7277 64.7051 64.5718 64.4768
82.3745 82.3231 82.2646 82.1485 82.0328 81.9151 81.8072 81.6822 81.5296
ð10Þ
Differentiating with respect to T gives Es
dA dEs ¼ −A dT dT
ð11Þ
Thermodynamically, a single particle or granular group tended to a state in which the total energy of the system was minimization. When the state of granular group in a fluidized bed transformed from normal fluidization to defluidization, agglomeration and sintering between particles occurred. In this process the total superficial area of the system was decreased. According to Fig. 4, at temperature below 600 K, the solid surface energy did not change with temperature. It was assumed
3.6
Surface energy (J/m2)
The linear trend of the bcc lattice parameter as a function of temperature was related to the intrinsic nature of thermal expansion as a result of increasing atomic thermal vibrations from their equilibrium positions [20]. Consequently, by combining Eqs. (3)–(8), the solid surface energy of metal irons was calculated as a function of temperature. As shown in Fig. 4, when the temperature was lower than 600 K, the solid surface energy was almost a constant value of about 2.1 J/m2. However, when the temperature was higher than 600 K, the solid surface energy significantly increased as the temperature increased in the range of 673 K to 873 K. This range was just the Huttig temperature of metal irons (500–600 ° C) [9,21]. In a view of solid lattice dynamics [21], the amplitude of atoms/ions around the equilibrium position was increasing when the temperature increased. Atoms/ions began to overcome the lattice binding force, and exchanged places inside or on the surface of lattices, then further diffused into adjacent lattices. Simultaneously, atoms and defects inside lattices started to activate. The diffusion was intensified, and the sintering rate accelerated significantly. Therefore, the metal materials presented a higher solid surface energy.
3.4 3.2 3.0 2.8 2.6 2.4 2.2 2.0 1.8
300 400 500 600 700 800 900 1000 1100
Temperature (K) Fig. 4. The surface energy of solid iron as a function of temperature.
Y. Zhong et al. / Powder Technology 301 (2016) 1144–1147 s that dE ¼ 0, and thus dA ¼ 0. Consequently, the total superficial area was dT dT constant, suggesting that particle agglomeration did not appear. However, at temperatures above 600 K, the surface energy increased with in-
dEs dT
creasing temperature, N0 , thus indicating that the total superficial area of granular group decreased as temperature increased. Since the surface of single solid particles hardly changed, the only way to decrease the total superficial area of the system was through sintering and agglomeration. As a result, the bed defluidization occurred at the macro-level. These results were consistent with the experimental observations in previous works [2,9,13]. Based on this model, solid surface energy was associated with the inherent properties of bed materials. The thermodynamic conditions of defluidization in a fluidized bed were theoretically determined by the relationship between solid surface energy and temperature. Thus, a criterion for the defluidization was proposed as followed: 8 dEs > > N0 b0 Dispersion > > > dT dA < dEs ¼ 0 Stabilization ¼0 dT > dT > > > dE > s : b0 N0 Agglomeration dT
dA b0 , dT
ð12Þ
For a given bed material, the granular group tended to stable fluidization if
dEs dT
≤0, whereas the agglomeration/defluidization occurred
spontaneously if
dEs dT
N0 . The limiting temperature of defluidization
s N0, which was considered (about 600 K for Fe) was the point where dE dT as a reference to select the bed materials. When operated above this limiting temperature, the risk of agglomeration was increased. It was s also inferred that if dE of the bed material was adjusted to be ≤0 by surdT face coating or modification, defluidization can be avoided.
4. Conclusions In this study, the solid surface energy of metallic iron was calculated by the elastic modulus and lattice constant, which increased with increasing temperature. Based on thermodynamic analysis, a model was developed to predict the spontaneity for particle agglomeration. Thus, the relationship between the agglomeration tendency and the inherent properties of bed materials was established. For a given bed material, s the fluidization state tended to be stable if dE dT ≤0, whereas agglomeras N0. Consequently, tion and defluidization occurred spontaneously if dE dT the limiting temperature of defluidization was determined as a point s N0, which was in accordance with the experimental results. where dE dT
1147
Acknowledgement This work was financially supported by the National Natural Science Foundation of China (Grant 91534121), Project Funded by China Postdoctoral Science Foundation (Grant 2015M570931) and Fundamental Research Funds for the Central Universities (Grant FRF-TP-15-013A1).
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