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Models for prediction of hydrodynamic characteristics of gas–solid tapered and mini-tapered fluidized beds
Powder Technology 205 (2011) 224–230
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Models for prediction of hydrodynamic characteristics of gas–solid tapered and mini-tapered fluidized beds M.H. Khani ⁎ Nuclear Fuel Cycle Research School, Nuclear Science and Technology Research Institute, P.O. Box 14395, 836, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 18 May 2010 Received in revised form 18 August 2010 Accepted 15 September 2010 Available online 24 September 2010 Keywords: Tapered fluidized beds Mini-tapered fluidized beds Minimum fluidization velocity Maximum pressure drop Fluidization
a b s t r a c t Prediction of hydrodynamic characteristics is a prerequisite in the design and operation of tapered and minitapered fluidized beds. This paper has been focused on the development of generalized models for prediction of minimum fluidization velocity and maximum pressure drop in gas–solid tapered and mini-tapered fluidized beds. The empirical correlations were developed based on dimensionless analysis of empirical data. These correlations have the ability to predict the minimum fluidization velocity and maximum pressure drop in both tapered and cylindrical beds (the beds with tapered angle of zero). The empirical data were collected from tapered beds with different cone angles for various particles. The predicting capability of correlations has been discussed. Predicted values of minimum fluidization velocity and maximum pressure drop by the proposed models compared well with the empirical data. The effects of tapered angle are also discussed. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Much research has been focused on the hydrodynamic characteristics of cylindrical fluidized beds because of their extensive application in industry. Tapered fluidized beds, however are favored for some important processes, such as biological treatment of waste water [1,2], incineration of waste materials, granulation, drying [3], coating, crystallization, in nuclear fuel production cycle [4,5], metabolic gas production, [6] etc. In these fluidization processes, there may be reduction in particle size due to chemical reaction or attrition, which may adversely affect the efficiency and quality of fluidization due to channeling, slugging, entrainment, etc. Tapered beds are useful for fluidization of materials with wide particle size distribution. Because, gradual reduction of superficial velocity due to the increase in cross-sectional area with height, leads to unique hydrodynamic characteristics. Due to this the problems associated with fluidization in cylindrical beds like entrainment of particles, limitation of operating velocity, and increasing filter resistance, can be overcome by adopting tapered beds. In some industrial processes due to some technical reasons, very small tapered angle may be selected for fluidized bed reactor. Such reactors called mini-tapered fluidized bed [5]. In this paper, we will also consider mini-tapered beds and we'll try to obtain a comprehensive model for the calculation of some hydrodynamic characteristics of tapered and mini-tapered beds. Hydrodynamic characteristics of tapered beds differ from that of cylindrical beds. Much less is known about the characteristics of tapered beds than that of cylindrical ones, especially minimum fluidization velocity and corresponding maximum bed pressure drop. Prediction of ⁎ Tel.: + 98 2188221117; fax: + 98 88221116. E-mail address: [email protected]. 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.09.018
minimum fluidization velocity is a prerequisite in the design and operation of fluidized beds. Several studies have been made during the last few years. Peng and Fan [7] developed theoretical models for prediction of minimum fluidization velocity and maximum pressure drop for liquid–solid tapered fluidized beds of spherical particles, based on the dynamic forces exerted on the particles. Jing et al. [8] and Shan et al. [9], neglecting the kinetic energy change in the bed, developed models based on Ergun's equation for pressure drop prediction in gas–solid tapered fluidized beds of spherical particles. But all these works have addressed for spherical type and uniform size particles in tapered fluidized beds. Recently Sau et al. [10] have developed empirical correlations, based on dimensionless analysis for minimum fluidization velocity and maximum pressure drop in gas–solid tapered fluidized beds. The correlation reported by them for minimum fluidization velocity and maximum pressure drop are given in Eqs. (1) and (2) respectively: 0:3197
Fr = 0:2714ðArÞ
0:6092
ðsin αÞ
ε0 ts
0:6108
0:038 0:222 0:642 0:723 dp D Hs ρs ΔPmax = 7:457 1 : D0 D0 D0 ρg
ð1Þ
ð2Þ
Although, Sau et al. models have been developed for predicting the minimum fluidization velocity and maximum pressure drop for regular and irregular particles of gas–solid systems taking into account all the properties, i.e., particle diameter and density, tapered angle, porosity and sphericity, but their proposed model for predicting the minimum fluidization velocity cannot be true or accurate for minitapered beds (tapered beds with small tapered angle). Because the term of sin(α) in Eq. (1) will approach zero for mini-tapered beds and
M.H. Khani / Powder Technology 205 (2011) 224–230 Table 1 Properties of particles used in the study.
225
Table 3 Analysis of variance table for regression models of Umf prediction.
Materials
dp (mm)
ρp (kg/m3)
ε0
Øs
Source
DF
Sum of squares
Mean square
F ratio
Prob(F)
Quartz Glass beads Ceramic sphere FCC particles Limestone Sand UF4 particles
0.4,0.6,0.8,1 1 1.7 0.08 0.5,0.7,1 0.45,0.67,0.85,1,1.3 0.8,1.6
2650 2500 1650 1396 2782 2612 6700
0.44–0.52 0.3 0.45 0.375 0.24–0.32 0.38–0.54 0.47,0.51
0.86 1 1 1 0.82 0.7 0.91
Eq. (5) Regression Error Total
4 33 37
134120.6 1518.773 135639.3
33530.14 46.02342
728.5451
0
Eq. (6) Regression Error Total
4 17 21
90967.64 886.348 91853.99
22741.91 52.13812
436.1859
0
Eq. (7) Regression Error Total
4 27 31
278239.3 5032.083 283271.4
69559.83 186.3734
373.2282
0
ceramic sphere, FCC (fluid catalytic cracking) particles, limestone, sand and UF4 (uranium tetrafluoride) particles. Details of the particles are given in Table 1. The tapered prototypes, made of Plexiglas, are shown schematically in Fig. 1. The diameters of the bottom base of all these prototypes were 0.05 m. Three static bed heights (11, 15 and 20 cm) were tested. Air as fluidizing gas was supplied at ambient temperature into the tapered prototype through a gas distributor mounted at the lower base. Gas volume flow rate was controlled by a flow meter. To measure the bed pressure drop, two pressure taps were mounted on the wall along the bed and a tube monometer was used. The lower pressure tap was allocated close to the gas distributor. To prevent the particles from entering the probes, the opening of tabs was covered with a screen. 3. Development of correlations The prediction of hydrodynamic characteristics is essential for the operation of fluidized beds either through modeling or empirical correlations. At present study, correlations have been developed for the prediction of minimum fluidization velocity and maximum pressure drop in tapered and mini-tapered fluidized beds by dimensional analysis of empirical data. General equation forms, considered for dimensionless correlations for calculation of minimum fluidization velocity and maximum pressure drop, respectively are as follows:
Fig. 1. Schematic diagram of tapered prototypes.
in extreme point (extreme point correspond to cylindrical beds) will be exactly zero. The zero value of sin(α) will result zero Froude number consequently zero Umf, which is not consistent with physics of the process. According to the physics of the process, the minimum value of Umf should be positive and non-zero. Therefore, it is necessary to develop generalized correlations for calculation of minimum fluidization velocity and maximum pressure drop in tapered beds with applicability also for mini-tapered beds.
b
Remf = αðArÞ ΔPmax ρs gHs
2. Methodology
c d dp D0
ε0 ts
e
ðcos αÞ
ð3Þ
b c d dp e ρ Hs =α s ðcos αÞ : ρg
D0
ð4Þ
D0
The constant coefficients of the correlation were obtained by nonlinear regression analysis. Regression variable results and variance analysis were shown in Tables 2 and 3 for dimensionless correlations of minimum fluidization velocity and in Tables 4 and 5 for
The empirical data were collected from tapered beds with different cone angles (0–30°) for various particles such as quartz, glass beads, Table 2 Regression variable results of proposed models for prediction of Umf. Variable
a b c d e
Eq. (5)I
Eq. (7)II
Eq. (8)III
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
0.0302 0.6467 −0.8330 0.2378 −10.5444
0.026 0.036 0.128 0.118 0.354
1.15 18.16 −6.49 2.02 −29.79
0.25 0 0 0.05 0
7.1600 0.3927 −0.5783 0.9870 −275.486
0.619 0.038 0.038 0.116 18.26
11.55 10.28 −15.08 9.1 −15.08
0.09 0.03 0.01 0.03 0
10.3956 0.3673 −0.7312 0.8886 −10.4369
4.365 0.193 0.156 0.504 0.491
2.38 1.90 -4.67 1.76 −21.24
0.08 0.07 0 0.09 0
SE: standard error. I R2 = 0.9788, and adjusted R2 = 0.9774. II R2 = 0.9822, and adjusted R2 = 0.9796. III R2 = 0.9903, and adjusted R2 = 0.9881.
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M.H. Khani / Powder Technology 205 (2011) 224–230
Table 4 Regression variable results of proposed models for prediction of ΔPmax. Variable
a b c d e
Eq. (6)I
Eq. (9)II
Eq. (10)III
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
49.182 −0.397 0.216 −0.936 −3.700
9.884 0.026 0.015 0.066 0.174
4.98 −15.43 14.45 −14.21 −21.25
0 0 0 0 0
106.729 −0.522 0.309 −0.379 −10.858
12.221 0.014 0.010 0.041 4.389
8.73 −37.05 30.83 −9.19 −2.47
0 0 0 0 0.01
163.419 −0.524 0.269 −0.976 −3.277
58.886 0.045 0.022 0.117 0.201
2.77 −11.55 12.25 −8.34 −16.27
0.008 0 0 0 0
SE: standard error. I R2 = 0.9468, and adjusted R2 = 0.9436. II R2 = 0.9714, and adjusted R2 = 0.9702 III R2 = 0.9628, and adjusted R2 = 0.9679.
dimensionless correlations of maximum pressure drop respectively. The “t” statistic is computed by dividing the estimated value of the parameter by its standard error. This statistic is a measure of the likelihood that the actual value of the parameter is not zero. The larger the absolute value of t, the less likely that the actual value of the parameter could be zero. The “Prob(t)” value is the probability of obtaining the estimated value of the parameter if the actual parameter value is zero. The smaller the value of Prob(t), the more significant the parameter and the less likely that the actual parameter value is zero. As shown in Tables 2 and 4, the value of Prob(t) is small for all of the parameters. Also the R2 values (coefficients of multiple determination), shown in Tables 2 and 4, are acceptable for all models. This indicates that the functions can predict the dependent variable correctly. Also, the analysis of variance tables (shown in Tables 4 and 5) show that the regression models, being fitted, are significant. The low value of Prob(F) implies that the regression equations have validity in fitting the data (i.e., the independent variables are not purely random with respect to the dependent variable). As a result, the dimensionless correlations obtained for minimum fluidization velocity and maximum pressure drop in total range of tapered angle are given in Eqs. (5) and (6) respectively. −2
Remf = 3:02 × 10 ΔPmax ρs gHs
0:6467
ðArÞ
0:238 −0:833 ε0 ts
dp D0
−10:544
ðcos αÞ
−0:397 0:216 −0:936 dp −3:700 ρ Hs = 49:182 s ðcos αÞ ρg
D0
0:393
Remf = 7:16ðArÞ
0:987 −0:833 ε0 ts
dp D0
−275:486
ðcos αÞ
; For 0≤α≤4:5∘ ð7Þ
0:367
Remf = 10:396ðArÞ
0:889 −0:731 dp D0
ε0 ts
−10:437
ðcos αÞ
; For α N 4:5∘: ð8Þ
The set of equations for maximum pressure drop calculation: ΔPmax ρs gHs
−0:522 0:309 −0:379 dp −10:858 ρ Hs = 106:729 s ðcos αÞ ; For 0≤α≤4:5 ρg
D0
D0
ð9Þ
ð5Þ
ð6Þ
D0
fluidized beds. But, by adopting cos(α) as tapered angle parameter true calculation values are obtainable for mini-tapered beds. Also, the results of empirical data have been divided in two ranges of tapered angle one includes mini-tapered angles (0 b α ≤ 4.5°) and the other includes larger tapered angles (α N 4.5°) and equations have been obtained that minimizes the error in the calculation of Umf and ΔPmax. The set of equations for minimum fluidization calculation:
ΔPmax ρs gHs
−0:524 0:269 −0:976 dp −3:277 ρ Hs = 163:419 s ðcos αÞ ; For α N 4:5∘: ρg
D0
D0
ð10Þ
In these correlations, the term of cos(α) was used to incorporate the effect of tapered angle, which is an important parameter in tapered fluidized beds hydrodynamics. This term is much appropriate than sin(α), which has been used in Sau et al. model (Eq. (1)). As discussed in section of introduction, if α → 0, then sin(α) and consequently Umf → 0. This cannot be a true result. For this reason, the model of Sau et al. cannot give accurate results for mini-tapered
By considering two different correlations in two separate ranges of tapered angles, the accuracy of equations will be improved in the calculation of minimum fluidization velocity and maximum pressure drop (as has been shown in results and discussion section).
Table 5 Analysis of variance table for regression models of ΔPmax prediction. Source
DF
Sum of squares
Mean square
F ratio
Prob(F)
Eq. (6) Regression Error Total
4 106 110
3.130527 0.284173 3.414701
0.782632 0.00268
291.9308
0
Eq. (9) Regression Error Total
4 91 95
2.190303 6.44E-02 2.254669
0.547576 0.000707
774.1494
0
Eq. (10) Regression Error Total
3 48 51
1.789429 0.149737 1.939166
0.596476 0.00312
191.2076
0 Fig. 2. Total pressure drop as a function of superficial gas velocity for a bed of glass beads with tapered angle of 20.4°.
M.H. Khani / Powder Technology 205 (2011) 224–230
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Fig. 3. Total pressure drop as a function of superficial gas velocity for a bed of glass beads with tapered angle of 3.2°.
Fig. 4. Comparison of calculated and empirical values of minimum fluidization velocity.
4. Results and discussion
materials, using ΔP-U0 diagram. For instance, Figs. 2 and 3 show the total pressure drop as a function of superficial gas velocity (ΔP-U0 diagram) for a bed of glass beads with tapered angle of 20.4 and 3.2° respectively. As seen in the diagrams, the pressure drop in region I is in quasi-linear relationship with the superficial gas velocity. This region corresponds to fixed bed regime and indicates substantial
4.1. The ΔP-U0 diagram A conventional method was used for determination of minimum fluidization velocity and maximum pressure drop for various
Table 6 Comparison of Umf values calculated by proposed models and Sau et al. model with the empirical values. Particle
Quartz
Glass beads FCC Quartz
UF4 powder Ceramic sphere Limestone
Glass beads FCC Sand
Glass beads Ceramic sphere Limestone
Sand Quartz
Glass beads Ceramic sphere sand
Glass beads FCC
dp (mm)
α
0.4 0.6 0.8 1.0 1.0 0.08 0.4 0.6 0.8 1.0 0.8 1.6 1.7 0.5 0.7 1.0 1.2 1.0 0.08 0.45 0.67 0.85 1.0 1.3 1.0 1.7 0.5 0.7 1.0 0.45 0.67 0.4 0.6 0.8 1.0 1.7 0.45 0.67 1.0 1.3 1.0 0.08
0° 0° 0° 0° 0° 0° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 3.2° 3.2° 3.2° 3.2° 3.2° 3.2° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 10.2° 10.2° 10.2° 10.2° 10.2° 20.4° 20.4° 20.4° 20.4° 20.4° 30° 30° 30° 30° 30° 30°
Umf,exp (m/s) 0.13 0.24 0.36 0.44 0.45 0.009 0.14 0.25 0.38 0.35 0.47 1.18 0.68 0.26 0.37 0.52 0.70 0.48 0.016 0.19 0.35 0.49 0.58 0.78 0.53 0.96 0.31 0.47 0.64 0.23 0.41 0.28 0.52 0.82 1.12 1.69 0.62 1.1 2.2 2.6 2.6 0.12
Umf,cal (m/s) by
Absolute error (%) of models based on empirical data
Two-range model
Single-range model
Sau et al. model
Two-range model
Single-range model
Sau et al. model
0.10 0.16 0.24 0.31 0.32 0.01 0.10 0.17 0.26 0.34 0.38 0.87 0.55 0.23 0.34 0.54 0.69 0.48 0.02 0.22 0.37 0.5 0.61 0.86 0.73 1.17 0.34 0.49 0.74 0.22 0.34 0.37 0.58 0.84 1.10 1.63 0.83 1.29 2.01 2.69 2.52 0.18
0.16 0.26 0.41 0.56 0.55 0.02 0.16 0.26 0.41 0.56 0.76 1.83 0.81 0.26 0.40 0.66 0.86 0.56 0.02 0.16 0.26 0.36 0.45 0.63 0.57 0.83 0.30 0.47 0.77 0.18 0.30 0.31 0.52 0.80 1.10 1.60 0.69 1.16 1.97 2.79 2.53 0.10
0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.07 0.12 0.18 0.17 0.48 0.34 0.11 0.18 0.32 0.43 0.29 0.01 0.09 0.16 0.23 0.30 0.45 0.36 0.69 0.22 0.37 0.65 0.14 0.26 0.20 0.38 0.61 0.89 1.71 0.26 0.49 0.92 1.38 1.11 0.03
23.08 33.33 33.33 29.55 28.89 11.11 28.57 32.00 31.58 2.86 19.15 26.27 19.12 11.54 8.11 3.85 1.43 0.00 25.00 15.79 5.71 2.04 5.17 10.26 37.74 21.88 9.68 4.26 15.63 4.35 17.07 32.14 11.54 2.44 1.79 3.55 33.87 17.27 8.64 3.46 3.08 50.00
23.08 8.33 13.89 27.27 22.22 122.22 14.29 4.00 7.89 60.00 61.70 55.08 19.12 0.00 8.11 26.92 22.86 16.67 25.00 15.79 25.71 26.53 22.41 19.23 7.55 13.54 3.23 0.00 20.31 21.74 26.83 10.71 0.00 2.44 1.79 5.33 11.29 5.45 10.45 7.31 2.69 16.67
100.00 100.00 100.00 100.00 100.00 100.00 71.43 72.00 68.42 48.57 63.83 59.32 50.00 57.69 51.35 38.46 38.57 39.58 37.50 52.63 54.29 53.06 48.28 42.31 32.08 28.13 29.03 21.28 1.56 39.13 36.59 28.57 26.92 25.61 20.54 1.18 58.06 55.45 58.18 46.92 57.31 75.00
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M.H. Khani / Powder Technology 205 (2011) 224–230
to the different models results, which prove that it could also be applicable to calculate the minimum fluidization velocity in cylindrical beds. The maximum pressure drop values were obtained for different materials with various stagnant bed heights. The calculated values of maximum pressure drop by proposed models (Eq. (6), and set of Eqs. (9) and (10)) and Sau et al. model (Eq. (2)) with empirical data for a stagnant bed height of 0.11 m, for different particles are given in Table 9 (data for other stagnant bed heights are not shown). Comparison of calculated values and empirical data are shown in Fig. 5 and standard and average deviations of the models are given in Table 10. The comparison shows that the lowest deviation belongs to the two-range model which includes Eqs. (9) and (10), which prove that it is in the best accordance with empirical data. According to Tables 7 and 10, the standard and average deviations of Sau et al. model in the case of maximum pressure drop are smaller than that in the case of minimum fluidization velocity. The larger deviation arises due to the extent of difference of calculated values with empirical data for small tapered angles especially for zerotapered angles.
Table 7 Deviation of the proposed models for prediction of Umf. Model
Average deviation, ∑
Umf ;exp −Umf ;cal Umf ;exp N
Two-range model Single-range model Sau et al. model
× 100
±0.886 ±9.32 ±51.96
Standard deviation, 11=2 0 Umf ;exp −Umf B∑ C Umf ;exp B C × 100 @ A N−1 21.71 29.04 58.76
viscous effects in gas–solid interaction. In this region the bed voidage remains unchanged. In region II, the pressure drop across the bed lowered with the superficial gas velocity. Thus, at this region the total pressure drop should be a function of superficial gas velocity. This results from restructuring of the bed followed by an increase in the bed voidage in lower bed layers. At point A the transition from fixed bed to partially fluidized bed occurs. The region III corresponds to the condition of fully fluidized bed. In this region the total pressure drop remains constant. Fig. 3 shows that for small tapered angle, the region II disappeared and the transition from fixed bed to fully fluidized bed occurred directly. In fact, the area of region II tends to be zero and can be neglected.
4.3. Effect of tapered angle According to calculated and empirical results shown in Tables 6 and 9, the minimum fluidization velocity and maximum pressure drop increase with increasing tapered angles. This result has also been reported in literature (Peng–Fan, Jing et al., and Sau et al.). Also, comparison of Fig. 2 with Fig. 3 shows that for larger tapered angles, the partially fluidized bed regime (region II) occurred in a wider range of superficial gas velocities and for mini-tapered beds the area of region II becomes smaller.
4.2. Model comparison The values of minimum fluidization velocity calculated by proposed models (Eqs. (5), (7) and (8)) and also Sau et al. model (Eq. (1)) have been represented against the empirical values in Table 6. It can be seen that there is a good agreement between calculated and empirical values for the tapered angles of higher than 4.5°. But for the lower tapered angles the results of Eq. (1) cannot be accurate. Also, Table 6 shows that the calculated results of Eq. (7) have the best agreement with the empirical data. A comparative study has been carried out between the results obtained from Eqs. (1), (5) and set of Eqs. (7) and (8) and those obtained empirically is shown in Fig. 4. Table 7 shows the performance of the proposed equations in term of standard and average deviations. As seen from Table 7, the deviations in case of two-range model (set of Eqs. (7) and (8)) are smaller in comparison with singular-range models (Eq. (5)) and Sau et al. model (Eq. (1)). That's why a separate correlation (Eq. (7)) was obtained for the range of tapered angles lower than 4.5° (mini-tapered beds). Eq. (7) can also predict the minimum fluidization velocity of cylindrical beds (the beds with tapered angle of zero). Table 8 shows the comparison of proposed model (substitution of unity to cos(α) for cylindrical beds in Eq. (7)) with the most well-known published models in literature [11–16] to predict the minimum fluidization velocity in cylindrical beds. As it can be observed, the proposed model predictions are close
5. Conclusion The hydrodynamic characteristics of tapered and mini-tapered fluidized gas–solid systems with different cone angles for different materials have been studied. The results show that the hydrodynamic characteristics especially minimum fluidization velocity and maximum bed pressure drop are dependent on tapered angle of bed. Three different flow regimes: (I) fixed bed, (II) partially fluidized bed and (III) fully fluidized bed take place depending on the superficial gas velocity. The range of partially fluidized bed is enlarged by increasing the tapered angle. Correlations have been developed for the prediction of minimum fluidization velocity and maximum bed pressure drop for different particles in gas–solid system of tapered beds and mini-tapered beds. Average and standard deviations of models from empirical data were obtained. Comparison shows that the calculated results are in good agreement with empirical data. The least deviation (average and
Table 8 Comparison of Umf values in cylindrical beds calculated by proposed model with the values calculated by the published models in literature and absolute error based on empirical values. Model
Wen-Yu [11] Sexana-Vogel [12] Babu et al. [13] Richardson-Jeronimo [14] Bourgeois-Grenier [15] Lucas et al. [16] Proposed model Eq. (5)
Umf,cal, [m/s] (Error [%]) for materials (dp [mm]) Quartz (0.4)
Quartz (0.6)
Quartz (0.8)
Quartz (1)
Glass beads (1)
FCC (0.08)
0.13 0.24 0.27 0.15 0.16 0.13 0.10
0.27 0.44 0.49 0.30 0.32 0.27 0.16
0.42 0.63 0.70 0.45 0.47 0.41 0.24
0.57 0.80 0.87 0.59 0.61 0.55 0.31
0.54 0.77 0.84 0.57 0.59 0.52 0.32
0.003 0.01 0.01 0.004 0.004 0.003 0.01
(00.00) (84.61) (107.7) (15.38) (23.08) (00.00) (23.08)
(12.50) (83.33) (104.2) (25.00) (33.33) (12.50) (33.33)
(16.67) (75.00) (94.44) (25.00) (30.56) (13.89) (33.33)
(29.54) (81.82) (97.72) (34.09) (38.63) (25.00) (29.55)
(20.00) (71.11) (86.67) (26.67) (31.11) (15.56) (28.89)
(66.67) (11.11) (11.11) (55.56) (55.56) (66.67) (11.11)
M.H. Khani / Powder Technology 205 (2011) 224–230
229
Table 9 Comparison of ΔPmax values calculated by proposed models and Sau et al. model with the empirical values. Particle
Quartz
Glass beads FCC Quartz
UF4 powder Ceramic sphere Limestone
Glass beads FCC Sand
Glass beads Ceramic sphere Limestone
Sand
Quartz
Glass beads Ceramic sphere sand
Glass beads FCC
dp (mm)
α
0.4 0.6 0.8 1.0 0.08 0.4 0.6 0.8 0.8 1.6 1.7 0.5 0.7 1.0 1.0 0.08 0.45 0.67 1.0 1.3 1.0 1.7 0.5 0.7 1.0 1.2 0.45 0.67 0.85 0.4 0.6 0.8 1.0 1.0 1.7 0.45 0.67 0.85 1.0 1.3 1.0 0.08
0° 0° 0° 0° 0° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 3.2° 3.2° 3.2° 3.2° 3.2° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 10.2° 10.2° 10.2° 10.2° 10.2° 10.2° 10.2° 20.4° 20.4° 20.4° 20.4° 20.4° 20.4° 30° 30° 30° 30° 30° 30° 30°
ΔPexp (Pa)
967 1107 1192 1248 452 975 1112 1224 1995 2451 1124 1137 1174 1302 1265 476 1005 1202 1302 1392 1306 1234 1445 1562 1723 1547 1385 1567 1370 1322 1489 1596 1679 1685 1638 1514 1693 1989 1791 1915 1845 786
ΔPcal (Pa) by
Absolute error (%) of models based on empirical data
Two-range model
Single-range model
Sau et al. model
Two-range model
Single-range model
Sau et al. model
931 1055 1153 1201 417 934 1058 1156 1801 2231 1164 1037 1151 1285 1221 424 989 1119 1266 1373 1240 1198 1220 1336 1470 1544 1151 1281 1366 1318 1470 1588 1686 1640 1552 1751 1948 2077 2170 2329 2125 816
1137 1242 1321 1339 546 1139 1243 1323 2315 2689 1170 1237 1330 1437 1347 549 1170 1276 1391 1472 1355 1182 1304 1403 1515 1576 1227 1337 1408 1446 1578 1679 1762 1702 1485 1970 2147 2261 2341 2478 2280 930
1105 1209 1289 1298 486 1121 1227 1308 2558 2983 1098 1237 1333 1443 1336 500 1163 1271 1389 1473 1346 1121 1274 1373 1486 1548 1189 1299 1370 1198 1311 1398 1468 1408 1173 1236 1350 1423 1475 1564 1429 535
3.72 4.70 3.27 3.77 7.74 4.21 4.86 5.56 9.72 8.98 3.56 8.80 1.96 1.31 3.48 10.92 1.59 6.91 2.76 1.36 5.05 2.92 15.57 14.47 14.68 0.19 16.90 18.25 0.29 0.30 1.28 0.50 0.42 2.67 5.25 15.65 15.06 4.42 21.16 21.62 15.18 3.82
17.58 12.20 10.82 7.29 20.80 16.82 11.78 8.09 16.04 9.71 4.09 8.80 13.29 10.37 6.48 15.34 16.42 6.16 6.84 5.75 3.75 4.21 9.76 10.18 12.07 1.87 11.41 14.68 2.77 9.38 5.98 5.20 4.94 1.01 9.34 30.12 26.82 13.68 30.71 29.40 23.58 18.32
14.27 9.21 8.14 4.01 7.52 14.97 10.34 6.86 28.22 21.71 2.31 8.80 13.54 10.83 5.61 5.04 15.72 5.74 6.68 5.82 3.06 9.16 11.83 12.10 13.76 0.06 14.15 17.10 0.00 9.38 11.95 12.41 12.57 16.44 28.39 18.36 20.26 28.46 17.64 18.33 22.55 31.93
standard deviations within respectively ±0.886% and 21.71% for minimum fluidization velocity and respectively ±2.89% and 9.65% for maximum bed pressure drop) belongs to two-range model. The results also were compared with Sau et al. model results. The comparison shows that the Sau et al. model is not highly accurate for the calculation of minimum fluidization velocity in mini-tapered beds.
The capability of the proposed model for the prediction of minimum fluidization velocity and maximum bed pressure drop in mini-tapered beds was also tested for cylindrical beds (zero-tapered beds) and a comparison was performed with published well-known equations for cylindrical beds. The proposed correlations could find practical utility in designing and operation of tapered and mini-tapered fluidized beds for various gas–solid systems.
Table 10 Deviation of the proposed models for prediction of ΔPmax. Model
Average deviation,
Umf ;exp −Umf ;cal ∑ Umf ;exp N
Fig. 5. Comparison of calculated and empirical values of maximum bed pressure drop.
Two-range model Single-range model Sau et al. model
± 2.89 ± 8.79 ± 0.93
× 100
Standard deviation, 0 2 11=2 Umf ;exp −Umf ;cal C B∑ Umf ;exp C B × 100 C B A @ N−1 9.65 14.39 14.77
230
M.H. Khani / Powder Technology 205 (2011) 224–230
Nomenclature Ar Archimedes number (= gd3p (ρs − ρg) ρg / μ2g ) dp particle diameter, m D0 bottom diameter of the tapered bed, m D1 top diameter of the tapered bed, m Fr Froude number (= Umf / (gdp)0.5) g gravity, m s−2 Hs stagnant height of the bed, m ΔPmax maximum bed pressure drop, Pa Remf Reynolds number at Umf (= dpUmfρg / μg) Umf minimum fluidization velocity, m s−1
Greek letters α tapered angle (°) ε0 voidage of the stagnant bed (–) Øs sphericity of solid particle (–) μg gas viscosity, Pa s ρg gas density, kg m3 ρs solid particles density, kg m3
Subscripts cal calculated value exp experimental value mf minimum fluidization Acknowledgements The authors gratefully acknowledge Nuclear Science and Technology Research Institute of Iran for the financial support provided for this work.
References [1] R. Parthiban, P.V.R. Iyer, G. Sekaran, Anaerobic digestion in tapered fluidized bed reactor and modeling using radial basis function neural network, Biotechnology 6 (2007) 534–542. [2] C.D. Scott, C.W. Hancher, Use of a tapered fluidized bed as a continues bioreactor, Biotechnology and Bioengineering 18 (1976) 1393–1403. [3] H. Tanfara, T. Pugsley, C. Winters, Effect of particle size distribution on local voidage in a bench-scale conical fluidized bed dryer, Drying Technology 20 (2002) 1273. [4] R.L. Levery, A.D.L. Garga, S.C. Jacobs, H.M. Heidt, D.E. Trent, Fluid bed conversion of UO3 to UF4, Chemical Engineering Progress 56 (1960) 43. [5] M.H. Khani, M. Ghannadi, H. Pahlavanzadeh, Fluorination of UF4 in a minitapered fluidized bed and mathematical modeling, Annals of Nuclear Energy 37 (2010) 1241–1247. [6] C.S. Wu, J.S. Huang, R. Ohara, Hydrodynamics of tapered anaerobic fluidized beds for metabolic gas production, Chemical Engineering Journal 148 (2009) 179–289. [7] Y. Peng, L.T. Fan, Hydrodynamic characteristics of fluidization in liquid–solid tapered beds, Chemical Engineering Science 52 (14) (1997) 2277–2290. [8] S. Jing, Q. Hu, J. Wang, Y. Jin, Fluidization of coarse particles in gas–solid conical beds, Chemical Engineering and Processing 39 (2000) 379–387. [9] J. Shan, C. Guobin, M. Fan, B. Yu, W. Jingfu, J. Yong, Fluidization of fine particles in conical beds, Powder Technology 118 (2001) 271–274. [10] D.C. Sau, S. Mohanty, K.C. Biswal, Minimum fluidization velocities and maximum bed pressure drops for gas–solid tapered fluidized beds, Chemical Engineering Journal 132 (1–3) (2007) 151–157. [11] C.Y. Wen, Y.H. Yu, A generalized method for predicting the minimum fluidization velocity, American Institute of Chemical Engineering Journal 12 (1966) 610–612. [12] S.C. Saxena, G.J. Vogel, The measurment of incipient fluidization velocities in a bed of coarse dolomite at high temperature and pressure, Transactions of the Institute of Chemical Engineers 55 (1977) 184. [13] S.P. Babu, B. Shah, A. Talwalker, Fluidization correlation for coal gasification materials-minimum fluidization velocity and fluidized bed expansion, AIChE Symposium Series 74 (1978) 176. [14] J.F. Richardson, M.A.S. Jeromino, Velocity voidage relations for sedimentation and fluidization, Chemical Engineering Science 39 (1979) 1419. [15] P. Bourgeois, P. Grenier, The ratio of terminal velocity to minimum fluidization velocity for spherical particles, Canadian Journal of Chemical Engineering 46 (1968) 325. [16] A. Lucas, J. Arnaldos, J. Casal, L. Puigjaner, Improved equation for the calculation of minimum fluidization velocity, Industrial Engineering Chemical Process Design and Development 25 (1986) 426–429.
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Powder Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Models for prediction of hydrodynamic characteristics of gas–solid tapered and mini-tapered fluidized beds M.H. Khani ⁎ Nuclear Fuel Cycle Research School, Nuclear Science and Technology Research Institute, P.O. Box 14395, 836, Tehran, Iran
a r t i c l e
i n f o
Article history: Received 18 May 2010 Received in revised form 18 August 2010 Accepted 15 September 2010 Available online 24 September 2010 Keywords: Tapered fluidized beds Mini-tapered fluidized beds Minimum fluidization velocity Maximum pressure drop Fluidization
a b s t r a c t Prediction of hydrodynamic characteristics is a prerequisite in the design and operation of tapered and minitapered fluidized beds. This paper has been focused on the development of generalized models for prediction of minimum fluidization velocity and maximum pressure drop in gas–solid tapered and mini-tapered fluidized beds. The empirical correlations were developed based on dimensionless analysis of empirical data. These correlations have the ability to predict the minimum fluidization velocity and maximum pressure drop in both tapered and cylindrical beds (the beds with tapered angle of zero). The empirical data were collected from tapered beds with different cone angles for various particles. The predicting capability of correlations has been discussed. Predicted values of minimum fluidization velocity and maximum pressure drop by the proposed models compared well with the empirical data. The effects of tapered angle are also discussed. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Much research has been focused on the hydrodynamic characteristics of cylindrical fluidized beds because of their extensive application in industry. Tapered fluidized beds, however are favored for some important processes, such as biological treatment of waste water [1,2], incineration of waste materials, granulation, drying [3], coating, crystallization, in nuclear fuel production cycle [4,5], metabolic gas production, [6] etc. In these fluidization processes, there may be reduction in particle size due to chemical reaction or attrition, which may adversely affect the efficiency and quality of fluidization due to channeling, slugging, entrainment, etc. Tapered beds are useful for fluidization of materials with wide particle size distribution. Because, gradual reduction of superficial velocity due to the increase in cross-sectional area with height, leads to unique hydrodynamic characteristics. Due to this the problems associated with fluidization in cylindrical beds like entrainment of particles, limitation of operating velocity, and increasing filter resistance, can be overcome by adopting tapered beds. In some industrial processes due to some technical reasons, very small tapered angle may be selected for fluidized bed reactor. Such reactors called mini-tapered fluidized bed [5]. In this paper, we will also consider mini-tapered beds and we'll try to obtain a comprehensive model for the calculation of some hydrodynamic characteristics of tapered and mini-tapered beds. Hydrodynamic characteristics of tapered beds differ from that of cylindrical beds. Much less is known about the characteristics of tapered beds than that of cylindrical ones, especially minimum fluidization velocity and corresponding maximum bed pressure drop. Prediction of ⁎ Tel.: + 98 2188221117; fax: + 98 88221116. E-mail address: [email protected]. 0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.powtec.2010.09.018
minimum fluidization velocity is a prerequisite in the design and operation of fluidized beds. Several studies have been made during the last few years. Peng and Fan [7] developed theoretical models for prediction of minimum fluidization velocity and maximum pressure drop for liquid–solid tapered fluidized beds of spherical particles, based on the dynamic forces exerted on the particles. Jing et al. [8] and Shan et al. [9], neglecting the kinetic energy change in the bed, developed models based on Ergun's equation for pressure drop prediction in gas–solid tapered fluidized beds of spherical particles. But all these works have addressed for spherical type and uniform size particles in tapered fluidized beds. Recently Sau et al. [10] have developed empirical correlations, based on dimensionless analysis for minimum fluidization velocity and maximum pressure drop in gas–solid tapered fluidized beds. The correlation reported by them for minimum fluidization velocity and maximum pressure drop are given in Eqs. (1) and (2) respectively: 0:3197
Fr = 0:2714ðArÞ
0:6092
ðsin αÞ
ε0 ts
0:6108
0:038 0:222 0:642 0:723 dp D Hs ρs ΔPmax = 7:457 1 : D0 D0 D0 ρg
ð1Þ
ð2Þ
Although, Sau et al. models have been developed for predicting the minimum fluidization velocity and maximum pressure drop for regular and irregular particles of gas–solid systems taking into account all the properties, i.e., particle diameter and density, tapered angle, porosity and sphericity, but their proposed model for predicting the minimum fluidization velocity cannot be true or accurate for minitapered beds (tapered beds with small tapered angle). Because the term of sin(α) in Eq. (1) will approach zero for mini-tapered beds and
M.H. Khani / Powder Technology 205 (2011) 224–230 Table 1 Properties of particles used in the study.
225
Table 3 Analysis of variance table for regression models of Umf prediction.
Materials
dp (mm)
ρp (kg/m3)
ε0
Øs
Source
DF
Sum of squares
Mean square
F ratio
Prob(F)
Quartz Glass beads Ceramic sphere FCC particles Limestone Sand UF4 particles
0.4,0.6,0.8,1 1 1.7 0.08 0.5,0.7,1 0.45,0.67,0.85,1,1.3 0.8,1.6
2650 2500 1650 1396 2782 2612 6700
0.44–0.52 0.3 0.45 0.375 0.24–0.32 0.38–0.54 0.47,0.51
0.86 1 1 1 0.82 0.7 0.91
Eq. (5) Regression Error Total
4 33 37
134120.6 1518.773 135639.3
33530.14 46.02342
728.5451
0
Eq. (6) Regression Error Total
4 17 21
90967.64 886.348 91853.99
22741.91 52.13812
436.1859
0
Eq. (7) Regression Error Total
4 27 31
278239.3 5032.083 283271.4
69559.83 186.3734
373.2282
0
ceramic sphere, FCC (fluid catalytic cracking) particles, limestone, sand and UF4 (uranium tetrafluoride) particles. Details of the particles are given in Table 1. The tapered prototypes, made of Plexiglas, are shown schematically in Fig. 1. The diameters of the bottom base of all these prototypes were 0.05 m. Three static bed heights (11, 15 and 20 cm) were tested. Air as fluidizing gas was supplied at ambient temperature into the tapered prototype through a gas distributor mounted at the lower base. Gas volume flow rate was controlled by a flow meter. To measure the bed pressure drop, two pressure taps were mounted on the wall along the bed and a tube monometer was used. The lower pressure tap was allocated close to the gas distributor. To prevent the particles from entering the probes, the opening of tabs was covered with a screen. 3. Development of correlations The prediction of hydrodynamic characteristics is essential for the operation of fluidized beds either through modeling or empirical correlations. At present study, correlations have been developed for the prediction of minimum fluidization velocity and maximum pressure drop in tapered and mini-tapered fluidized beds by dimensional analysis of empirical data. General equation forms, considered for dimensionless correlations for calculation of minimum fluidization velocity and maximum pressure drop, respectively are as follows:
Fig. 1. Schematic diagram of tapered prototypes.
in extreme point (extreme point correspond to cylindrical beds) will be exactly zero. The zero value of sin(α) will result zero Froude number consequently zero Umf, which is not consistent with physics of the process. According to the physics of the process, the minimum value of Umf should be positive and non-zero. Therefore, it is necessary to develop generalized correlations for calculation of minimum fluidization velocity and maximum pressure drop in tapered beds with applicability also for mini-tapered beds.
b
Remf = αðArÞ ΔPmax ρs gHs
2. Methodology
c d dp D0
ε0 ts
e
ðcos αÞ
ð3Þ
b c d dp e ρ Hs =α s ðcos αÞ : ρg
D0
ð4Þ
D0
The constant coefficients of the correlation were obtained by nonlinear regression analysis. Regression variable results and variance analysis were shown in Tables 2 and 3 for dimensionless correlations of minimum fluidization velocity and in Tables 4 and 5 for
The empirical data were collected from tapered beds with different cone angles (0–30°) for various particles such as quartz, glass beads, Table 2 Regression variable results of proposed models for prediction of Umf. Variable
a b c d e
Eq. (5)I
Eq. (7)II
Eq. (8)III
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
0.0302 0.6467 −0.8330 0.2378 −10.5444
0.026 0.036 0.128 0.118 0.354
1.15 18.16 −6.49 2.02 −29.79
0.25 0 0 0.05 0
7.1600 0.3927 −0.5783 0.9870 −275.486
0.619 0.038 0.038 0.116 18.26
11.55 10.28 −15.08 9.1 −15.08
0.09 0.03 0.01 0.03 0
10.3956 0.3673 −0.7312 0.8886 −10.4369
4.365 0.193 0.156 0.504 0.491
2.38 1.90 -4.67 1.76 −21.24
0.08 0.07 0 0.09 0
SE: standard error. I R2 = 0.9788, and adjusted R2 = 0.9774. II R2 = 0.9822, and adjusted R2 = 0.9796. III R2 = 0.9903, and adjusted R2 = 0.9881.
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M.H. Khani / Powder Technology 205 (2011) 224–230
Table 4 Regression variable results of proposed models for prediction of ΔPmax. Variable
a b c d e
Eq. (6)I
Eq. (9)II
Eq. (10)III
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
Value
SE
t-ratio
Prob(t)
49.182 −0.397 0.216 −0.936 −3.700
9.884 0.026 0.015 0.066 0.174
4.98 −15.43 14.45 −14.21 −21.25
0 0 0 0 0
106.729 −0.522 0.309 −0.379 −10.858
12.221 0.014 0.010 0.041 4.389
8.73 −37.05 30.83 −9.19 −2.47
0 0 0 0 0.01
163.419 −0.524 0.269 −0.976 −3.277
58.886 0.045 0.022 0.117 0.201
2.77 −11.55 12.25 −8.34 −16.27
0.008 0 0 0 0
SE: standard error. I R2 = 0.9468, and adjusted R2 = 0.9436. II R2 = 0.9714, and adjusted R2 = 0.9702 III R2 = 0.9628, and adjusted R2 = 0.9679.
dimensionless correlations of maximum pressure drop respectively. The “t” statistic is computed by dividing the estimated value of the parameter by its standard error. This statistic is a measure of the likelihood that the actual value of the parameter is not zero. The larger the absolute value of t, the less likely that the actual value of the parameter could be zero. The “Prob(t)” value is the probability of obtaining the estimated value of the parameter if the actual parameter value is zero. The smaller the value of Prob(t), the more significant the parameter and the less likely that the actual parameter value is zero. As shown in Tables 2 and 4, the value of Prob(t) is small for all of the parameters. Also the R2 values (coefficients of multiple determination), shown in Tables 2 and 4, are acceptable for all models. This indicates that the functions can predict the dependent variable correctly. Also, the analysis of variance tables (shown in Tables 4 and 5) show that the regression models, being fitted, are significant. The low value of Prob(F) implies that the regression equations have validity in fitting the data (i.e., the independent variables are not purely random with respect to the dependent variable). As a result, the dimensionless correlations obtained for minimum fluidization velocity and maximum pressure drop in total range of tapered angle are given in Eqs. (5) and (6) respectively. −2
Remf = 3:02 × 10 ΔPmax ρs gHs
0:6467
ðArÞ
0:238 −0:833 ε0 ts
dp D0
−10:544
ðcos αÞ
−0:397 0:216 −0:936 dp −3:700 ρ Hs = 49:182 s ðcos αÞ ρg
D0
0:393
Remf = 7:16ðArÞ
0:987 −0:833 ε0 ts
dp D0
−275:486
ðcos αÞ
; For 0≤α≤4:5∘ ð7Þ
0:367
Remf = 10:396ðArÞ
0:889 −0:731 dp D0
ε0 ts
−10:437
ðcos αÞ
; For α N 4:5∘: ð8Þ
The set of equations for maximum pressure drop calculation: ΔPmax ρs gHs
−0:522 0:309 −0:379 dp −10:858 ρ Hs = 106:729 s ðcos αÞ ; For 0≤α≤4:5 ρg
D0
D0
ð9Þ
ð5Þ
ð6Þ
D0
fluidized beds. But, by adopting cos(α) as tapered angle parameter true calculation values are obtainable for mini-tapered beds. Also, the results of empirical data have been divided in two ranges of tapered angle one includes mini-tapered angles (0 b α ≤ 4.5°) and the other includes larger tapered angles (α N 4.5°) and equations have been obtained that minimizes the error in the calculation of Umf and ΔPmax. The set of equations for minimum fluidization calculation:
ΔPmax ρs gHs
−0:524 0:269 −0:976 dp −3:277 ρ Hs = 163:419 s ðcos αÞ ; For α N 4:5∘: ρg
D0
D0
ð10Þ
In these correlations, the term of cos(α) was used to incorporate the effect of tapered angle, which is an important parameter in tapered fluidized beds hydrodynamics. This term is much appropriate than sin(α), which has been used in Sau et al. model (Eq. (1)). As discussed in section of introduction, if α → 0, then sin(α) and consequently Umf → 0. This cannot be a true result. For this reason, the model of Sau et al. cannot give accurate results for mini-tapered
By considering two different correlations in two separate ranges of tapered angles, the accuracy of equations will be improved in the calculation of minimum fluidization velocity and maximum pressure drop (as has been shown in results and discussion section).
Table 5 Analysis of variance table for regression models of ΔPmax prediction. Source
DF
Sum of squares
Mean square
F ratio
Prob(F)
Eq. (6) Regression Error Total
4 106 110
3.130527 0.284173 3.414701
0.782632 0.00268
291.9308
0
Eq. (9) Regression Error Total
4 91 95
2.190303 6.44E-02 2.254669
0.547576 0.000707
774.1494
0
Eq. (10) Regression Error Total
3 48 51
1.789429 0.149737 1.939166
0.596476 0.00312
191.2076
0 Fig. 2. Total pressure drop as a function of superficial gas velocity for a bed of glass beads with tapered angle of 20.4°.
M.H. Khani / Powder Technology 205 (2011) 224–230
227
Fig. 3. Total pressure drop as a function of superficial gas velocity for a bed of glass beads with tapered angle of 3.2°.
Fig. 4. Comparison of calculated and empirical values of minimum fluidization velocity.
4. Results and discussion
materials, using ΔP-U0 diagram. For instance, Figs. 2 and 3 show the total pressure drop as a function of superficial gas velocity (ΔP-U0 diagram) for a bed of glass beads with tapered angle of 20.4 and 3.2° respectively. As seen in the diagrams, the pressure drop in region I is in quasi-linear relationship with the superficial gas velocity. This region corresponds to fixed bed regime and indicates substantial
4.1. The ΔP-U0 diagram A conventional method was used for determination of minimum fluidization velocity and maximum pressure drop for various
Table 6 Comparison of Umf values calculated by proposed models and Sau et al. model with the empirical values. Particle
Quartz
Glass beads FCC Quartz
UF4 powder Ceramic sphere Limestone
Glass beads FCC Sand
Glass beads Ceramic sphere Limestone
Sand Quartz
Glass beads Ceramic sphere sand
Glass beads FCC
dp (mm)
α
0.4 0.6 0.8 1.0 1.0 0.08 0.4 0.6 0.8 1.0 0.8 1.6 1.7 0.5 0.7 1.0 1.2 1.0 0.08 0.45 0.67 0.85 1.0 1.3 1.0 1.7 0.5 0.7 1.0 0.45 0.67 0.4 0.6 0.8 1.0 1.7 0.45 0.67 1.0 1.3 1.0 0.08
0° 0° 0° 0° 0° 0° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 3.2° 3.2° 3.2° 3.2° 3.2° 3.2° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 10.2° 10.2° 10.2° 10.2° 10.2° 20.4° 20.4° 20.4° 20.4° 20.4° 30° 30° 30° 30° 30° 30°
Umf,exp (m/s) 0.13 0.24 0.36 0.44 0.45 0.009 0.14 0.25 0.38 0.35 0.47 1.18 0.68 0.26 0.37 0.52 0.70 0.48 0.016 0.19 0.35 0.49 0.58 0.78 0.53 0.96 0.31 0.47 0.64 0.23 0.41 0.28 0.52 0.82 1.12 1.69 0.62 1.1 2.2 2.6 2.6 0.12
Umf,cal (m/s) by
Absolute error (%) of models based on empirical data
Two-range model
Single-range model
Sau et al. model
Two-range model
Single-range model
Sau et al. model
0.10 0.16 0.24 0.31 0.32 0.01 0.10 0.17 0.26 0.34 0.38 0.87 0.55 0.23 0.34 0.54 0.69 0.48 0.02 0.22 0.37 0.5 0.61 0.86 0.73 1.17 0.34 0.49 0.74 0.22 0.34 0.37 0.58 0.84 1.10 1.63 0.83 1.29 2.01 2.69 2.52 0.18
0.16 0.26 0.41 0.56 0.55 0.02 0.16 0.26 0.41 0.56 0.76 1.83 0.81 0.26 0.40 0.66 0.86 0.56 0.02 0.16 0.26 0.36 0.45 0.63 0.57 0.83 0.30 0.47 0.77 0.18 0.30 0.31 0.52 0.80 1.10 1.60 0.69 1.16 1.97 2.79 2.53 0.10
0.00 0.00 0.00 0.00 0.00 0.00 0.04 0.07 0.12 0.18 0.17 0.48 0.34 0.11 0.18 0.32 0.43 0.29 0.01 0.09 0.16 0.23 0.30 0.45 0.36 0.69 0.22 0.37 0.65 0.14 0.26 0.20 0.38 0.61 0.89 1.71 0.26 0.49 0.92 1.38 1.11 0.03
23.08 33.33 33.33 29.55 28.89 11.11 28.57 32.00 31.58 2.86 19.15 26.27 19.12 11.54 8.11 3.85 1.43 0.00 25.00 15.79 5.71 2.04 5.17 10.26 37.74 21.88 9.68 4.26 15.63 4.35 17.07 32.14 11.54 2.44 1.79 3.55 33.87 17.27 8.64 3.46 3.08 50.00
23.08 8.33 13.89 27.27 22.22 122.22 14.29 4.00 7.89 60.00 61.70 55.08 19.12 0.00 8.11 26.92 22.86 16.67 25.00 15.79 25.71 26.53 22.41 19.23 7.55 13.54 3.23 0.00 20.31 21.74 26.83 10.71 0.00 2.44 1.79 5.33 11.29 5.45 10.45 7.31 2.69 16.67
100.00 100.00 100.00 100.00 100.00 100.00 71.43 72.00 68.42 48.57 63.83 59.32 50.00 57.69 51.35 38.46 38.57 39.58 37.50 52.63 54.29 53.06 48.28 42.31 32.08 28.13 29.03 21.28 1.56 39.13 36.59 28.57 26.92 25.61 20.54 1.18 58.06 55.45 58.18 46.92 57.31 75.00
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M.H. Khani / Powder Technology 205 (2011) 224–230
to the different models results, which prove that it could also be applicable to calculate the minimum fluidization velocity in cylindrical beds. The maximum pressure drop values were obtained for different materials with various stagnant bed heights. The calculated values of maximum pressure drop by proposed models (Eq. (6), and set of Eqs. (9) and (10)) and Sau et al. model (Eq. (2)) with empirical data for a stagnant bed height of 0.11 m, for different particles are given in Table 9 (data for other stagnant bed heights are not shown). Comparison of calculated values and empirical data are shown in Fig. 5 and standard and average deviations of the models are given in Table 10. The comparison shows that the lowest deviation belongs to the two-range model which includes Eqs. (9) and (10), which prove that it is in the best accordance with empirical data. According to Tables 7 and 10, the standard and average deviations of Sau et al. model in the case of maximum pressure drop are smaller than that in the case of minimum fluidization velocity. The larger deviation arises due to the extent of difference of calculated values with empirical data for small tapered angles especially for zerotapered angles.
Table 7 Deviation of the proposed models for prediction of Umf. Model
Average deviation, ∑
Umf ;exp −Umf ;cal Umf ;exp N
Two-range model Single-range model Sau et al. model
× 100
±0.886 ±9.32 ±51.96
Standard deviation, 11=2 0 Umf ;exp −Umf B∑ C Umf ;exp B C × 100 @ A N−1 21.71 29.04 58.76
viscous effects in gas–solid interaction. In this region the bed voidage remains unchanged. In region II, the pressure drop across the bed lowered with the superficial gas velocity. Thus, at this region the total pressure drop should be a function of superficial gas velocity. This results from restructuring of the bed followed by an increase in the bed voidage in lower bed layers. At point A the transition from fixed bed to partially fluidized bed occurs. The region III corresponds to the condition of fully fluidized bed. In this region the total pressure drop remains constant. Fig. 3 shows that for small tapered angle, the region II disappeared and the transition from fixed bed to fully fluidized bed occurred directly. In fact, the area of region II tends to be zero and can be neglected.
4.3. Effect of tapered angle According to calculated and empirical results shown in Tables 6 and 9, the minimum fluidization velocity and maximum pressure drop increase with increasing tapered angles. This result has also been reported in literature (Peng–Fan, Jing et al., and Sau et al.). Also, comparison of Fig. 2 with Fig. 3 shows that for larger tapered angles, the partially fluidized bed regime (region II) occurred in a wider range of superficial gas velocities and for mini-tapered beds the area of region II becomes smaller.
4.2. Model comparison The values of minimum fluidization velocity calculated by proposed models (Eqs. (5), (7) and (8)) and also Sau et al. model (Eq. (1)) have been represented against the empirical values in Table 6. It can be seen that there is a good agreement between calculated and empirical values for the tapered angles of higher than 4.5°. But for the lower tapered angles the results of Eq. (1) cannot be accurate. Also, Table 6 shows that the calculated results of Eq. (7) have the best agreement with the empirical data. A comparative study has been carried out between the results obtained from Eqs. (1), (5) and set of Eqs. (7) and (8) and those obtained empirically is shown in Fig. 4. Table 7 shows the performance of the proposed equations in term of standard and average deviations. As seen from Table 7, the deviations in case of two-range model (set of Eqs. (7) and (8)) are smaller in comparison with singular-range models (Eq. (5)) and Sau et al. model (Eq. (1)). That's why a separate correlation (Eq. (7)) was obtained for the range of tapered angles lower than 4.5° (mini-tapered beds). Eq. (7) can also predict the minimum fluidization velocity of cylindrical beds (the beds with tapered angle of zero). Table 8 shows the comparison of proposed model (substitution of unity to cos(α) for cylindrical beds in Eq. (7)) with the most well-known published models in literature [11–16] to predict the minimum fluidization velocity in cylindrical beds. As it can be observed, the proposed model predictions are close
5. Conclusion The hydrodynamic characteristics of tapered and mini-tapered fluidized gas–solid systems with different cone angles for different materials have been studied. The results show that the hydrodynamic characteristics especially minimum fluidization velocity and maximum bed pressure drop are dependent on tapered angle of bed. Three different flow regimes: (I) fixed bed, (II) partially fluidized bed and (III) fully fluidized bed take place depending on the superficial gas velocity. The range of partially fluidized bed is enlarged by increasing the tapered angle. Correlations have been developed for the prediction of minimum fluidization velocity and maximum bed pressure drop for different particles in gas–solid system of tapered beds and mini-tapered beds. Average and standard deviations of models from empirical data were obtained. Comparison shows that the calculated results are in good agreement with empirical data. The least deviation (average and
Table 8 Comparison of Umf values in cylindrical beds calculated by proposed model with the values calculated by the published models in literature and absolute error based on empirical values. Model
Wen-Yu [11] Sexana-Vogel [12] Babu et al. [13] Richardson-Jeronimo [14] Bourgeois-Grenier [15] Lucas et al. [16] Proposed model Eq. (5)
Umf,cal, [m/s] (Error [%]) for materials (dp [mm]) Quartz (0.4)
Quartz (0.6)
Quartz (0.8)
Quartz (1)
Glass beads (1)
FCC (0.08)
0.13 0.24 0.27 0.15 0.16 0.13 0.10
0.27 0.44 0.49 0.30 0.32 0.27 0.16
0.42 0.63 0.70 0.45 0.47 0.41 0.24
0.57 0.80 0.87 0.59 0.61 0.55 0.31
0.54 0.77 0.84 0.57 0.59 0.52 0.32
0.003 0.01 0.01 0.004 0.004 0.003 0.01
(00.00) (84.61) (107.7) (15.38) (23.08) (00.00) (23.08)
(12.50) (83.33) (104.2) (25.00) (33.33) (12.50) (33.33)
(16.67) (75.00) (94.44) (25.00) (30.56) (13.89) (33.33)
(29.54) (81.82) (97.72) (34.09) (38.63) (25.00) (29.55)
(20.00) (71.11) (86.67) (26.67) (31.11) (15.56) (28.89)
(66.67) (11.11) (11.11) (55.56) (55.56) (66.67) (11.11)
M.H. Khani / Powder Technology 205 (2011) 224–230
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Table 9 Comparison of ΔPmax values calculated by proposed models and Sau et al. model with the empirical values. Particle
Quartz
Glass beads FCC Quartz
UF4 powder Ceramic sphere Limestone
Glass beads FCC Sand
Glass beads Ceramic sphere Limestone
Sand
Quartz
Glass beads Ceramic sphere sand
Glass beads FCC
dp (mm)
α
0.4 0.6 0.8 1.0 0.08 0.4 0.6 0.8 0.8 1.6 1.7 0.5 0.7 1.0 1.0 0.08 0.45 0.67 1.0 1.3 1.0 1.7 0.5 0.7 1.0 1.2 0.45 0.67 0.85 0.4 0.6 0.8 1.0 1.0 1.7 0.45 0.67 0.85 1.0 1.3 1.0 0.08
0° 0° 0° 0° 0° 1.4° 1.4° 1.4° 1.4° 1.4° 1.4° 3.2° 3.2° 3.2° 3.2° 3.2° 4.5° 4.5° 4.5° 4.5° 4.5° 4.5° 10.2° 10.2° 10.2° 10.2° 10.2° 10.2° 10.2° 20.4° 20.4° 20.4° 20.4° 20.4° 20.4° 30° 30° 30° 30° 30° 30° 30°
ΔPexp (Pa)
967 1107 1192 1248 452 975 1112 1224 1995 2451 1124 1137 1174 1302 1265 476 1005 1202 1302 1392 1306 1234 1445 1562 1723 1547 1385 1567 1370 1322 1489 1596 1679 1685 1638 1514 1693 1989 1791 1915 1845 786
ΔPcal (Pa) by
Absolute error (%) of models based on empirical data
Two-range model
Single-range model
Sau et al. model
Two-range model
Single-range model
Sau et al. model
931 1055 1153 1201 417 934 1058 1156 1801 2231 1164 1037 1151 1285 1221 424 989 1119 1266 1373 1240 1198 1220 1336 1470 1544 1151 1281 1366 1318 1470 1588 1686 1640 1552 1751 1948 2077 2170 2329 2125 816
1137 1242 1321 1339 546 1139 1243 1323 2315 2689 1170 1237 1330 1437 1347 549 1170 1276 1391 1472 1355 1182 1304 1403 1515 1576 1227 1337 1408 1446 1578 1679 1762 1702 1485 1970 2147 2261 2341 2478 2280 930
1105 1209 1289 1298 486 1121 1227 1308 2558 2983 1098 1237 1333 1443 1336 500 1163 1271 1389 1473 1346 1121 1274 1373 1486 1548 1189 1299 1370 1198 1311 1398 1468 1408 1173 1236 1350 1423 1475 1564 1429 535
3.72 4.70 3.27 3.77 7.74 4.21 4.86 5.56 9.72 8.98 3.56 8.80 1.96 1.31 3.48 10.92 1.59 6.91 2.76 1.36 5.05 2.92 15.57 14.47 14.68 0.19 16.90 18.25 0.29 0.30 1.28 0.50 0.42 2.67 5.25 15.65 15.06 4.42 21.16 21.62 15.18 3.82
17.58 12.20 10.82 7.29 20.80 16.82 11.78 8.09 16.04 9.71 4.09 8.80 13.29 10.37 6.48 15.34 16.42 6.16 6.84 5.75 3.75 4.21 9.76 10.18 12.07 1.87 11.41 14.68 2.77 9.38 5.98 5.20 4.94 1.01 9.34 30.12 26.82 13.68 30.71 29.40 23.58 18.32
14.27 9.21 8.14 4.01 7.52 14.97 10.34 6.86 28.22 21.71 2.31 8.80 13.54 10.83 5.61 5.04 15.72 5.74 6.68 5.82 3.06 9.16 11.83 12.10 13.76 0.06 14.15 17.10 0.00 9.38 11.95 12.41 12.57 16.44 28.39 18.36 20.26 28.46 17.64 18.33 22.55 31.93
standard deviations within respectively ±0.886% and 21.71% for minimum fluidization velocity and respectively ±2.89% and 9.65% for maximum bed pressure drop) belongs to two-range model. The results also were compared with Sau et al. model results. The comparison shows that the Sau et al. model is not highly accurate for the calculation of minimum fluidization velocity in mini-tapered beds.
The capability of the proposed model for the prediction of minimum fluidization velocity and maximum bed pressure drop in mini-tapered beds was also tested for cylindrical beds (zero-tapered beds) and a comparison was performed with published well-known equations for cylindrical beds. The proposed correlations could find practical utility in designing and operation of tapered and mini-tapered fluidized beds for various gas–solid systems.
Table 10 Deviation of the proposed models for prediction of ΔPmax. Model
Average deviation,
Umf ;exp −Umf ;cal ∑ Umf ;exp N
Fig. 5. Comparison of calculated and empirical values of maximum bed pressure drop.
Two-range model Single-range model Sau et al. model
± 2.89 ± 8.79 ± 0.93
× 100
Standard deviation, 0 2 11=2 Umf ;exp −Umf ;cal C B∑ Umf ;exp C B × 100 C B A @ N−1 9.65 14.39 14.77
230
M.H. Khani / Powder Technology 205 (2011) 224–230
Nomenclature Ar Archimedes number (= gd3p (ρs − ρg) ρg / μ2g ) dp particle diameter, m D0 bottom diameter of the tapered bed, m D1 top diameter of the tapered bed, m Fr Froude number (= Umf / (gdp)0.5) g gravity, m s−2 Hs stagnant height of the bed, m ΔPmax maximum bed pressure drop, Pa Remf Reynolds number at Umf (= dpUmfρg / μg) Umf minimum fluidization velocity, m s−1
Greek letters α tapered angle (°) ε0 voidage of the stagnant bed (–) Øs sphericity of solid particle (–) μg gas viscosity, Pa s ρg gas density, kg m3 ρs solid particles density, kg m3
Subscripts cal calculated value exp experimental value mf minimum fluidization Acknowledgements The authors gratefully acknowledge Nuclear Science and Technology Research Institute of Iran for the financial support provided for this work.
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