Prediction of void fraction and minimum fluidization velocity of a binary mixture of particles: Bed material and fuel particles

Article history: Received 14 August 2018 Received in revised form 8 January 2019 Accepted 16 March 2019 Available online 19 March 2019 For operational control and design of a fluidized bed reactor containing different types of solid particles, the bed void fraction and minimum fluidization velocity are vital parameters. This paper demonstrates a method for predicting the void fraction and minimum fluidization velocity of different binary mixtures of particles with improved accuracy. A newmodel for predicting the void fraction is presented. This model is non-linear and continuous, and it is developed by introducing a packing factor and establishing a mass balance between the solid phases in the packing environment. The results show that the model can accurately predict the void fraction of a binary mixture where the particles are well mixed, partially mixed or segregated. Using this void fraction model and the Ergun equation of pressure drop, the minimum fluidization velocity can be predicted with mean errors of 15.2% for a mixture of two inert materials and 7.0% for a mixture of biomass and inert particles. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).


Introduction
Some fluidized bed applications involve different types of solid particles. The difference in properties between these particle types may influence the bed behaviour. For example, in a bubbling fluidized bed biomass gasifier with sand as the bed material, the difference in density, size and shape between biomass and sand particles often leads to particle segregation [1]. Particle segregation in a biomass gasification reactor can also be influenced by devolatilization of the fuel particles and formation of bubbles around the particles [2,3]. For a bed of coarse particles characterized with large exploding bubbles, the quality of the fluidization can be improved by adding some amount of fine particles of the same material [4]. Due to the size difference between the fine and coarse particles, the void fraction of the mixture is lower than that of the coarse particles, resulting in flow of smaller bubbles in the fluidized bed. In addition to changes in bubbling behaviour, the difference in properties between different solid types in a bed also influences the minimum fluidization velocity of the bed. For operational control and design of a reactor containing two or more solid phases, the bed minimum fluidization velocity is a key parameter.
The minimum fluidization velocity of a bed of particles of the same size and density can be predicted using different correlations. Most of these correlations [5][6][7] were derived from the Ergun [8] equation but are independent of the bed void fraction. For a binary mixture of particles, similar correlations for predicting minimum fluidization velocity are also available [9][10][11]. Moreover, there are other models, which involve interpolations between the minimum fluidization velocities of the pure components [12,13]. Li et al. [14] and Asif [15] however, showed that the minimum fluidization velocity of a binary mixture can be predicted with a better accuracy by using a model that considers the void fraction. One major challenge in this approach is prediction of the bed void fraction at minimum fluidization condition. For a completely mixed binary system, the Westman [16] model can be used to predict the bed voidage with a good accuracy when the density difference between the solids in the mixture is very small [14]. In a bed where segregation occurs due to density difference, the Westman [16] model is inappropriate [15]. The void fraction of a completely mixed bed of two solid phases can also be predicted using other models classified as 2parameter [17,18], compressible [19] and 3-parameter [20] models. These models are linear, and according to Chan and Kwan [21], their accuracies depend on the size ratio between the two size classes in the mixture. Moreover, each of these models comprises a set of two equations, which are solved separately to determine the mixture void fraction based on the maximum value in the solution set. The models are thus discontinuous over the entire range of mixture composition.
This study therefore presents a new model for predicting the void fraction of a binary mixture and how it can be used to improve the prediction of minimum fluidization velocity of the mixture independent of particle types. The proposed model is analytically developed based on the mass balance between two solid phases in a packing environment. In developing the model, it is assumed that the smaller particles first fill the available void without changing the volume occupied by the larger particles in static conditions. The excess of these particles then occupies the space above the larger particles. On the basis that there is a limit to which solids can be packed in a given space, a packing factor is introduced. The packing factor compares the actual mass of smaller particles contained in the void of larger particles with the mass that would have occupied the maximum available void space. An expression for the packing factor is derived as a function of mass fraction of the smaller particles, the particle size ratio, and an interaction parameter between the two particle types in the mixture. The interaction parameter also depends on the size ratio as well as the density ratio between the particles. With the correlation proposed for the interaction parameter, which is obtained using some data in the literature, prediction of the mixture void fraction from the proposed model contains no adjustable parameter. For different binary mixtures, the results of the model are compared against experimental data in the literature.

Minimum fluidization velocity and mixture properties
Similar to pure solid components, the minimum fluidization velocity of a binary mixture of solids is generally obtained from curves of pressure drop against the superficial gas velocity. The measured minimum fluidization velocity depends on the procedure employed (i.e. whether the pressure drop is measured at increasing or decreasing gas velocity) and on the distribution of solids at the fixed bed condition [22]. The pressure drop curve at increasing gas velocity usually reviews the transition between the initial and full fluidization condition of the bed due to segregation effects. For this reason, several authors [23,24] reported the onset of full fluidization as the true minimum fluidization considering that the whole bed is capable of being fluidized beyond this gas velocity. However, for a well-mixed system, the difference between the initial and full fluidization velocities is insignificant [14]. To avoid the initial bed effect associated with increasing gas velocity procedure, the minimum fluidization velocity can be measured based on decreasing gas velocity procedure. Despite the measurement procedure, the reported minimum fluidization velocity for a given binary mixture often lies between those of the pure components of the mixture. Marzocchella et al. [23] concluded that neither of the initial and full fluidization velocities is related to the minimum fluidization velocities of the individual solids in the mixture. This means that the mixture minimum fluidization velocity is a weighted average of the pure component values [22]. This section presents the models for predicting the average minimum fluidization velocity of a binary mixture of particles and the corresponding bed void fraction. The average minimum fluidization velocity lies between the initial and full fluidization velocity, and it can be obtained from the pressure drop curve at the intersection of two extrapolation lines drawn through the fixed bed and fluidized bed conditions as noted in the literature.

Models for minimum fluidization velocity
At minimum fluidization, the required superficial gas velocity can be obtained from the force balance between the bed weight and the upward force exerted by the fluid on the particles. Using the Ergun [8] equation, the minimum fluidization velocity of a bed of mono-sized particles can be computed from here, U mf and ε mf are the superficial fluid velocity and bed void fraction at minimum fluidization condition, respectively. While d s is the particle diameter, φ s is the particle sphericity, and ρ f and ρ s are the fluid and particle densities, respectively. For a binary mixture, different correlations are derived from Eq. (1) for predicting the mixture minimum fluidization velocity, where the particle diameter, density and sphericity are replaced with their equivalent average properties. Some of these correlations are given in Table 1.
As shown in Table 1, there are different expressions for the average bed properties. For the methods based on the Ergun [8] equation, it can be shown briefly that the volume-average particle density and the surface-to-volume mean diameter are appropriate.
Considering a bed containing different types of particles with d si , φ si and ρ si the particle diameter, sphericity and density of each particle type, respectively, the total specific surface area, a of the bed can be expressed as giving the hydraulic diameter of the bed as where α i is the solid volume fraction of the individual particle type and ε m is the mean void fraction averaged over the bed height.
where x i is the mass fraction of each type of particles and ρ sm is the mean density of the solid mixture. Using Eqs. (4) and (5), a modified Ergun [8] equation can be expressed as Comparing Eq. (7) with the Ergun [8] equation derived for a bed of mono-sized particles, the equivalent mean particle size (Sauter mean diameter) d smeq for a bed of different types of particles is given by From the definition of particle sphericity, φ s , as the ratio of surface area of a sphere to surface area of a particle of the same volume as the spherical particle [4], it means that d si is the volume-equivalent spherical particle diameter of the individual solid in the mixture. Hence, the average volume-equivalent spherical particle diameter, d sm of the mixture can be obtained as and the average mixture particle sphericity φ sm as For a spherical particle, φ s = 1, and if all the particles are spherical, φ sm = 1. The particle sphericity can be found experimentally or computed from the particle geometry if well defined [4]. The mixture density and particle diameter given in Eq. (6) and Eq. (8) are described as the volume-average particle density and the surface-to-volume mean diameter, respectively. Hence, to obtain the minimum fluidization velocity of a bed of different particle types, ρ s , d s and φ s in Eq. (1) are replaced with the corresponding values for the mixture.
In addition to the correlations given in Table 1, U mf of a binary mixture of particles can also be obtained directly from Eq. (1) when ε mf of the mixture is known. For a completely mixed binary system, the bed void fraction can be obtained from the pure component values using the Westman [16] here, y S and y L are the volumetric fraction of the smaller and larger particles, respectively, and v S and v L are the respective specific volume, where The parameter G can be obtained from the correlation proposed by Yu et al. [25] or Finkers and Hoffmann [26].

Model development for bed voidage
For direct application of Eq. (1) in a binary mixture of particles, this section introduces a new model for void fraction of the bed mixture.
In a given mixture of two solid phases, we define the packing factor θ as follows: where m * is the mass of the smaller particles occupying the interstices between the larger particles, and V 0 is the initial total volume occupied by the larger particles. The subscripts 1 and 2 denote the larger and smaller particles, respectively. The modulus |θ| indicates that θ can be negative or positive. When θ b 0, the bed is contracting and when θ N 0, the bed is expanding. A binary mixture of particles contracts if the bulk volume of the mixture is lower than the sum of the bulk volumes of the two particle types in the mixture. Bed expansion occurs when the volume of an initially well-mixed system increases due to particle segregation. The packing factor is a measure of packing density of a binary system. The larger the value of |θ|, the lower the void fraction of the mixture. Assuming that N 1 and N * are the respective number of particles in the packed bed, Eq. (14) can be simplified to where Substituting Eq. (16) into Eq. (15) and using the relation, α i + ε i = 1, where ε i is the pure component void fraction, the packing factor can be expressed as when d s2 /d s1 = 1, m * = 0. For Eq. (17) to satisfy this condition, the term (ε 1 − α * )/ε 1 must be a function of d s2 /d s1 in addition to the amount of smaller particles present in the mixture. Thus, here, β can be described as the interaction parameter between the two solid phases. When a bed contracts during solid mixing, the value of m * is high. On the contrary, m * is lower when the particles segregate. To account for these effects, β b 0 for a well-mixed system and β N 0 for a segregated mixture. Table 1 Correlations for predicting the minimum fluidization velocity in binary mixtures.

Correlation
Application Reference All binary mixtures Cheung et al. [12] U Biomass -inert mixture Si and Guo [30] Re m f ¼ f30: Biomass -inert mixture Paudel and Feng [31] Arm Furthermore, the total mass of the bed is expressed as where ΔV is the total volume occupied by the smaller particles above the larger particles, and it can be obtained from here, m 2 is the total mass of the smaller particles in the bed. Substituting Eq. (20) into Eq. (19) and noting that Substituting Eq. (18) into Eq. (21) and replacing the subscripts 1 and 2 with the corresponding letters, yields Eq. (22) can be used to predict the void fraction in a binary mixture of different particle types. As can be seen, the equation requires the solids/void fraction of the pure components and contains only one fitting parameter, β. The value of β depends on the relative difference between the properties of the different particle types in the mixture and on whether the bed is well mixed, partially mixed or segregated as shown in section 4. It should be noted that the value of ε m predicted from Eq. (22) is the bed voidage averaged over the bed height which may differ from the local void fractions in the bed. Depending on the particle size ratio, d sS /d sL , the local void fraction can vary along the bed axis due to segregation effect [27]. For a mixture containing biomass particles, the higher the value of d sS /d sL , the wider the deviation of ε m from the local void fraction at the segregated layers. The accuracy of Eq. (22) with a correctly assigned value of β is demonstrated in section 4.

Results and discussion
In this section, the validation of the proposed model, Eq. (22) and its application to predicting the minimum fluidization velocity of a binary mixture are demonstrated using different experimental data from the literature. Since it is often difficult to measure void fractions at minimum fluidization condition, a systematic procedure in using Eq. (22) to predict the mixture U mf is also highlighted. Fig. 1 compares the void fraction at static condition predicted using Eq. (22) against the experimental data given in Marzocchella et al. [23] for a mixture of glass particles with mean diameter 500 μm and sand particles with mean diameter 125 μm at different mixture compositions. The data obtained from Tharpa et al. [28] at fixed bed condition are also shown for a mixture of 3500 μm plastic and 709 μm zirconium oxide particles. At minimum fluidization condition, the model results are compared against the experimental data obtained from Li et al. [14] and Formisani et al. [22] for different binary mixtures: two glass powders with mean sizes (385 and 163) μm and two glass powders with mean sizes (612 and 154) μm, respectively. The particle properties in these mixtures are shown in Table 2. As can be seen, the results from Eq. (22) strongly agree with the experimental data shown in both figures. With the correlation of Yu et al. [25], the Westman [16] equation also agrees well with the experimental data at the minimum fluidization condition. For the mixtures given in Li et al. [14], the Westman [16] equation and Eq. (22) predict the same results for all values of x 1 (mass fraction of the larger particles). However, for the data obtained at fixed bed condition as shown in Fig. 1(a), the Westman [16] equation does not give good predictions. Fig. 2 compares the accuracy of Eq. (22) with that of the Westman [16] equation against the experimental data. The experimental data include those shown in Fig. 1 and those obtained from Formisani et al. [22] for a binary mixture of two different glass particles with mean diameters 499 and 271 μm. The figure shows that Eq. (22) predicts the experimental data with a very good accuracy. The mean prediction error associated with Eq. (22) is 1.5%. The prediction error using the Westman [16] equation can be as high as ±15% due to poor prediction of the bed voidage reported in Marzocchella et al. [23] and Tharpa et al. [28] at static conditions. However, the mean errors using the Westman [13] model are 4.0% based on the Yu et al. [25] correlation and 4.1% based on the Finkers and Hoffman [26] correlation.

Correlation for β
As can be seen in Fig. 1, β varies from one system to another. The individual value of β used in the results is obtained by fitting the experimental data to the model, Eq. (22). To successfully apply Eq. (22) without experimental data, a correlation for β is required. Analysis of some literature data obtained at the minimum fluidization condition shows that the absolute value of β decreases with the ratio d sS ρ sS / (d sL ρ sL ) as represented in Eq. (23).
In the subsequent sections, β computed from Eq. (23) is used in Eq. (22) for prediction of the mixture void fraction.

Minimum fluidization velocity of binary mixtures
The results in Fig. 1 show that the voidage of a binary mixture can be predicted with a good accuracy from the void fractions of the pure components. Since accurate prediction of void fraction of a pure component at minimum fluidization condition is a challenge, we present a method where U mf of the solid phases in a binary mixture are inputs to Eq. (22). As illustrated in Fig. 3, ε mf of the pure components are computed from the respective U mf values using Eq. (1). For a given mixture composition (mass fraction or volumetric fraction of the solid phases), the average particle properties and void fraction of the mixture are calculated from the relevant equations. From the values of ε m , average density, sphericity and particle diameter of the mixture, the mixture U mf is computed using Eq. (1). Due to the cohesiveness of biomass particles, the minimum fluidization velocity of a pure biomass is much higher than that predicted by Eq. (1) even when the volume equivalent spherical diameter of the particle is used. Since the sphericity of most practical biomass can be as low as 0.2, using the actual sphericity of biomass in Eq. (1) will result in a much lower value of U mf for the particles. Hence, for a mixture involving biomass and inert particles, φ si = 1 should be used in the proposed algorithm. Fig. 4 shows the predicted values of U mf based on four different models at different mass fraction of the larger particles. For each of the models, U mf of the different particle types are used as inputs. As shown in Fig. 4(a), the predicted results from the different models are in good agreement with the experimental data. However, the results in Fig. 4(b) shows that a combination of Eq. (1) with the Westman  H. char = hollow char, ZrO 2 = zirconium oxide.  a better prediction than those given by Cheung et al. [12] and Rincon et al. [13]. Furthermore, Fig. 5 compares the calculated values of U mf from these four models against the experimental data obtained from different literature [14,22,23]. The result is based on the binary mixtures (I, III, IV, V and VI) given in Table 2. By using any of the four models, Fig. 5 shows that the minimum fluidization velocity of the beds can be predicted with an error within ±35%. On average, the predictions based on the present study give the best results with mean absolute error of 15.2%, whereas those based on the Westman [16] equation with Yu et al. [25] correlation have a mean error of 15.5%. The models given by Cheung et al. [12] and Rincon et al. [13] show very high prediction errors with mean values 27.6% and 30.5%, respectively.

Mixtures of biomass and inert materials
Unlike the mixture of two inert materials with more or less the same particle density, a mixture of biomass and inert particles can show some degree of segregation. Hence, application of the Westman [16] equation in Eq. (1) will not be appropriate. However, this section shows that the proposed model, Eq. (22) can also be applied for prediction of minimum fluidization velocity of a mixture of biomass and inert particles. To be able to predict the volume expansion in the binary mixture, a positive value of the parameter β, which can be computed from Eq. (23), is required. Fig. 6(a) shows the void fraction computed using Eq. (22) at the minimum fluidization condition for a mixture of plastic particles with effective particle diameter 2550 μm and sand particles with particle diameter 550 μm. The plastic particles have a density of 1761 kg/m 3 and sphericity of 0.87 while the corresponding properties for the sand particles are 2664 kg/m 3 and 1.0. The experimental data are obtained from Asif [15] where water is used as the fluidizing fluid at 20°C. With β N 0, the result shows that Eq. (22) predicts the bed voidage with a good accuracy when the mass of the plastic particles is considerably high, i.e. x 1 N 0.4. At a lower mass fraction, the bed is partly mixed and partly segregated. Thus, Eq. (22) with β = 1.35 (computed from Eq. (23)) over predicts the mixture voidage. However, when the value of β is reduced to -0.38, Eq. (22) predicts the voidage with a better accuracy when x 1 b 0.4. This result and those presented above therefore show that with β N 0, Eq. (22) gives the voidage for a well-segregated bed. With β b 0, the model provides results where there is some degree of mixing. When β b 0 and the magnitude of β is computed from Eq. (23), Eq. (22) predicts the voidage for a well-mixed bed. For prediction of β in a bed exhibiting partial mixing behaviour, a different correlation than Eq. (23) is required. In addition, a model for predicting the mixture composition at which the bed begins to segregate is also required. In spite of the error in predicting the void fraction where the bed exhibits partial segregation, Fig. 6(b) shows that the minimum fluidization velocity of the bed can be well predicted using the combination of Eq. (1) and Eq. (22) for all values of x 1 . For the result where β = 1.35 is used over the entire values of x 1 , the prediction error of the proposed model is 11.3% as against 27.5% and 27.6% errors obtained from the Cheung et al. [12] and Rincon et al. [13] models, respectively. If the value β = − 0.38 is used for the compositions x 1 b 0.4, the proposed model predicts the minimum fluidization velocity shown in Fig. 6(b) with a better accuracy and the mean prediction error is reduced to 7.5%.
As the main aim of this study is to predict with improved accuracy the minimum fluidization velocity of a biomass-inert mixture, which often exhibits segregation behaviour, the results in Fig. 6 show that this can be achieved. The properties of different mixtures of biomass and inert particles used for this demonstration are given in Table 3 and the beds as described subsequently are fluidized with air at the ambient condition. For all computations in this section, Eq. (23) is used to predict the absolute value of β.
For the mixture of 856 μm walnut shell and 241 μm sand particles, Fig. 7(a) shows the predicted values of U mf compared with the   Table 2. experimental data. The results obtained for a mixture of 1560 μm rice husk and 350 μm sand particles are shown in Fig. 7(b). As can be seen in Fig. 7(a), the computed values of U mf using the Paudel and Feng [31] model are closer to the experimental values although the model does not capture the expansion behaviour of the bed at increasing mass of biomass particles. The Kumoro et al. [32] model under predicts the bed expansion at higher values of x 1 , giving a lower value of U mf for the biomass mixture. The Si and Guo [30] model gives the best prediction when x 1 ≤ 0.4 but shows the greatest prediction error at higher mass of biomass particles. However, in Fig. 7(b), the Si and Guo [30] model gives the least prediction error for biomass mass fraction within 0.3 b x 1 b 0.8. The Kumoro et al. [32] model over predicts the U mf value at a higher mass fraction of the rice husk particles even though the experimental data were used in the model development. Unlike these two latter models, which also predict the expansion and contraction behaviour of the bed, the Paudel and Feng [31] model predicts a steady increase in U mf with an increase in the amount of rice husk particles. As the models given by Si and Guo [30] and Kumoro et al. [32] consider particle sphericity, these results show that particle shape plays a significant role in prediction of U mf . It should be noted that inclusion of particle sphericity in these two models also means that the models indirectly consider the bed voidage since these two properties are closely related. Moreover, the results in Fig. 7 show that by using the proposed model, U mf is predicted with a better accuracy in both different bed mixtures. The results given by the proposed model is based on β N 0 where β value is as given in Eq. (23). The results also show that the predicted U mf using Eq. (1) and Eq. (22) gets better at increasing amount of biomass particles due to higher degree of segregation effect. Where there is some degree of bed contraction as shown in Fig. 7(a), the proposed model slightly over predicts the U mf value due to the steady expansion behaviour predicted by Eq. (22) when β N 0 is used as demonstrated in Fig. 6(a).
In addition, Fig. 8 compares the prediction accuracy of the proposed model with those of the existing models for biomass-inert systems. The experimental data are based on different mixtures of biomass and inert particles given in the literature [31][32][33]; see Table 3. As shown in the figure, the Cheung et al. [12] model under predicts the mixture U mf with an error as high as 40%. The accuracy of the Cheung et al. [12] model increases with increasing size ratio d s1 /d s2 and with increasing amount of biomass in the mixture. The high prediction errors shown by the models of Si and Guo [30], Paudel and Feng [31] and Kumoro et al. [32] are associated with the size ratio and density difference. The higher the values of d s1 /d s2 and ρ s2 − ρ s1 , the better the model accuracies. For d s1 /d s2 b 3.5, these models over predict the mixture U mf with an error N40%. However, the method proposed in this study as described in Fig. 3 using Eq. (1) and Eq. (22) predicts the mixture U mf with a better accuracy for all values of d s1 /d s2 and ρ s2 − ρ s1 . The mean prediction error using the proposed model is 7.0%, whereas those using the models of Cheung et al. [12], Si and Guo [30], Paudel and Feng [31] and Kumoro et al. [32] are 23.4%, 24.4%, 27.0% and 27.7%, respectively.  In summary, the accuracy of Eq. (22) in predicting the void fraction of a binary mixture depends on the value of the interaction parameter, β used. As shown in Figs. 2 and 6, Eq. (22) can predict the experimental data with a very good accuracy if a correct value of β is assigned. For the results shown in Figs. 4-8, Eq. (23) was used to estimate the values of β. Although the figures show that the U mf values of the binary mixtures are predicted to a reasonable accuracy, the results can also be better with an improvement in the correlation for β. In its current form, Eq. (23) was derived from data of six binary pairs of solids. If a larger data set is analysed, the model for the interaction parameter can be improved.

Conclusion
In a binary mixture, the difference in properties between the two different particle types greatly influences the bed behaviour. For this reason, accurate prediction of minimum fluidization velocity of binary mixtures, especially those involving biomass particles, has been a challenge. This paper presents a new model for predicting the bed void fraction and its application to predicting the minimum fluidization velocity of a binary mixture.
For prediction of the bed void fraction, the proposed model requires the void fractions of the pure components in the mixture. However, with known values of minimum fluidization velocities of the different particles in the mixture, the approach presented in this paper avoids the challenge in determining the bed voidage.
For a completely mixed system involving two inert materials, the proposed model can predict the minimum fluidization velocity with a mean error of 15.2%. For a bed mixture of biomass and inert materials, the model can predict the minimum fluidization velocity with an error of 7.0%.
Finally, for accurate prediction of the voidage and minimum fluidization velocity in a partly mixed bed of two types of particles, further work is required to establish a correlation for the binary interaction parameter as well as the mixture composition at the transition to the segregation behaviour.    Table 3.

Funding sources
This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.