International Journal of ChemTech search CODEN( USA): IJCRGG ISSN : 0974-4290 Vol.4, No.4, 1314-1321, Oct-Dec 2012 Simulation Of Vertical Gas-Solid Flow: Comarison Of Correlations For article-wall Friction And Drag Coefficient K.S. Rajan School of Chemical & Biotechnology,SASTRA University,Tirumalaisamudram Thanjavur, India 613401. Corres.author: ksrajan@chem.sastra.edu h: 914362 304270 Abstract: neumatic conveying is a widely used oeration in ower lants, mineral industries for transortation of dust, ash, mineral etc. It forms an easy and eco-friendly method for the dry transortation of articulate solids. ressure dro lays an imortant role in the economic design of the neumatic conveying system. A review of correlations available for estimation of drag coefficient and the articlewall friction has been made. The ability of correlations to redict the ressure dro for the data available in the literature has been studied using a one-dimensional, two-fluid model. The simulation, along with statistical analysis indicates the suitability of few correlations for drag coefficient estimation for a wide range of article size and solid mass flow rates. For the estimation of article-wall friction, a correlation suitable for large articles at large solid feed rate needs to be develoed from exerimental results. Key words: neumatic conveying, drag coefficient, article-wall friction, ressure dro. INTRODUCTION The exerimental and comutational study of two hase gas-solid flow in various configurations is of interest to researchers owing to its wide-range of alications [1-6]. The conventional aroach to the simulation of two hase gas-solid flows is the use of one dimensional mass, momentum and energy conservation differential Equations for the gas hase and the solid hase using the cross-section averaged velocities, densities and volume fraction of the hases. These set of differential Equations are converted to system of algebraic (difference) Equations, the solution of which would rovide the cross sectional averaged information about velocity, volume fraction and density of the hases [5-6, 7]. This aroach is simle, and rovides the macroscoic information about the system. While the use of cross-section averaged one dimensional equations offer the advantage of comutationally less rigorous and hence easier and faster, the simulations carried out using multi-dimensional models offer higher insights into the system. However the success of the one dimensional and multi-dimensional model deends on the accuracy of the emirical correlations used and their suitability for the simulation roblem under question. The resent study is aimed at comaring the existing correlations for the simulation and suggests the more suitable of the available correlations for the articular case. roosal of a new correlation requires the rigorous analysis of more exerimental data under widely different oerating conditions and is not attemted here.
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1315 MODEL A one dimensional model for dilute hase neumatic conveying has been develoed using the momentum, mass and energy conservation Equations for both the hases and solved using the method reorted earlier [4, 5]. The conservation Equations contain source terms that utilize mass, momentum and energy interaction between the gas and articles. Since the model was develoed for dilute hase conveying, all tyes of article-article interactions are ignored. The model does not account for mass transfer between the hases. Hence only the sources due to the following were considered: article-wall friction, gas-wall friction, gas-article heat transfer, drag due to sli between the hases & gravitational force for both the hases. The detailed model, its assumtions, constitutive Equations, solution methodology and its erformance can be obtained from our earlier work by neglecting the terms that contain radial derivative [4]. Evaluation of source terms requires a number of constitutive Equations for various arameters and is available in [4]. Gaswall friction was determined by the well known Blasius Equation, which has been widely acceted to reresent the gas-wall friction accurately. In the resent work, the alicability of various correlations for drag coefficient and article-wall friction has been tested by comaring the simulation results with the ublished exerimental data [8, 9]. ESTIMATION OF DRAG COEFFICIENT A list of correlations that has been used widely for determining the single-article drag coefficient in the simulation of fast fluidized beds, circulating fluidized beds, risers & neumatic conveying by various researchers [10-17] is given in Table 1. These have been used in various one and two-dimensional simulation of gas-solid flow roblems. The ranges of ynolds number for which the correlations are alicable are also resented in Table 1. While considering the effects of resence of multile articles, the drag force er unit length that exists between gas and solid hases is given as [18] F d 3ACdAs 4A ( u g g 2.65 g u ) 2 ESTIMATION OF ARTICLE-WALL FRICTION Among several equations available for rediction of article-wall friction and initial solids velocity, choice of an aroriate equation is difficult [19]. From the mass flow rates of gas and solids, ressure dro and orosity data from neumatic conveying exeriments, the articlewall friction have been calculated and rovided in the literature using an Equation of the form, s 2D f ; 2 L u (2) where f reresents friction factor for solids. The detailed exeriment and the calculation of solids friction factor from a tyical neumatic conveying exeriment are available in the literature [20]. A list of correlations that have been develoed from such neumatic conveying studies and ublished in the literature [20-24] is given in Table 2 which covers the wide range of articles sizes, mass flow rate of gas and articles, void fraction etc. RESULTS & DISCUSSION ressure dro is an imortant arameter in the design of the neumatic conveying systems for disosal of dust, fly ash etc. The requirements of the blower/vacuum equiment are determined from the ressure dro along the conveying line. ressure dro in a neumatic conveying is due to the cumulative effects of article acceleration, gas-wall friction, gas-solid friction, drag and gravitational forces [9]. During the article acceleration, the articles gain momentum from gas by virtue of which their velocity increases to a large value from its initial velocity (a very low value). High ressure dros are exerienced in the acceleration region, due to high relative velocity in that region. After the acceleration region, the gas and solid hase velocities ractically remain constant. The length of acceleration region deends on the relative amounts of gas and solid flow, article size etc. (1) where Cd reresents the single article drag coefficient.
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1316 Table 1: Correlations for estimation of single article drag coefficient. Correlation ference No. Equation No. 64 64 Cd 1 ; < 0.01 2 Cd 64 10 x 1 ; 0.01< < 1.5 x = -0.883+0.906 log( )-0.025[log( )] 2 64 0.792 Cd 1 0.138 ; 1.5< < 133 log(cd) = 2.0351-1.66(L)+0.3958(L 2 )-0.0306(L 3 ); [10] (3) g ( ug us) D Cd 24 ; < 1 g 40< <1000 [11] (4) Cd = 0.44 ; > 1000 4.8 Cd 0.63 0.5 Cd 0.336 2 17.3 Ag g ( ug us) D A Cd 0.4 Cd 0.28 24 g 21 4 0.5 6 0.5 0.72 1 0.125 ; < 1000; Cd 18 Cd = 0.445 ; > 1000 0.75 Cd 181 0.1 [12] (5) [13] (6) [14] (7) [15] (8) [16] (9) [17] (10) Table 2: Correlations for estimation of article-wall friction Correlation ference No. Equation No. 0.115 0.339 [21] W 0.0054 g u gd f ; W = m s /m g u D (11) 1.5 [20] 0.0017 ut ut f 4 u gu g g u (12) [20] f 12.2 gu (13) 1.1 55.5D [20] f 0.65 0.26 0.91 u D (14)
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1317 B u [22] t f A ; g ug u A = 0.0126 & B= -0.979 ; u g u t A= 0.041 & B = -1.021 ; u g < u t 1. f 0.048 u ; D = 20m, u = 3.5-22.9m/s [20] 22 0.75 f 0.074u ; u = 3.0-6.7m/s [20] f 1 0. 5 0.0285u gd ; D=26.5 46.8 mm [23] u = 1-10 m/s z = 0.082-0.3 Fr -0.86 Fr 0.25 s (D/ d ) 0.1 ;d 500m [24] z = 2.1-0.3 Fr -1 Fr 0.25 s (D/ d ) 0.1 ;d 500m (15) (16) (17) (18) (19) = z g U g 2 LW/(2D) ; W = m s /m g ; Fr = u 2 /gd Frs = u t 2 /gd COMARISON OF CORRELATIONS FOR DRAG COEFFICIENT Correlations in the Table 1 were tested for their alicability to redict the ressure dro. Some literature studies show the comarison of drag coefficients redicted from different correlations for a wide range of article ynolds number ( ). Since, ressure dro is more imortant from the economic design asects, these correlations were tested for their alicability to redict ressure dro. This may yield a wide choice of exressions suitable to design/simulate dilute hase neumatic conveying. The exerimental data were so chosen from the literature such that the simulations were carried out with fine and coarse articles, with a wide range of solid to gas mass ratios. oikoainen et al. [9] resented the exerimental data for the ressure dro in the neumatic conveying of ceramic articles of 64 m diameter through a duct of 19.2 cm diameter, at different solids feed rates and gas velocities. The ressure dro data were measured after 4 meters from the bottom of the conveying duct, thereby eliminating the ressure dro in the acceleration zone. A lot of solids friction factor and solids velocity, from their exerimental data was also resented. Their data was utilized for comarison of different exressions for drag coefficient. Simulations were done using the correlations listed in Table 1 for a articular value of gas and solid flow rates, thus for a fixed value of solid-gas mass ratio to redict the ressure dro for 5 m length after the acceleration zone. The simulations were reeated for different solid-gas mass ratios. The numerical value of wall-article friction factors were taken from their work. It was observed that almost all the correlations in the Table 1 redicted the ressure dro very closer to the exerimental value, with a very little absolute error of less than 1%. This behavior can be exlained as follows: Since the simulated and reorted ressure dros do not include the acceleration region, the velocities of both the hases in the ortion of the conveying duct under simulation (after acceleration region) are fairly constant. Also with finer articles, the sli velocity (difference between the gas and solid velocity) is less, further reducing the ressure dro due to drag. To comare the true erformance of correlations, the simulations were erformed for 1m length of the acceleration region from the bottom to redict the ressure dro for the solid-gas mass ratios of 0.415 & 3.178. The results are shown in the Figure 1.
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1318 80 655 ressure dro (a/m) 75 Eqn. (3) Eqn. (5) Eqn. (6) Eqn. (7) Eqn. (8) Eqn. (9) Eqn. (10) Eqn. (3) Eqn. (5) Eqn. (6) Eqn. (7) Eqn. (8) Eqn. (9) Eqn. (10) 650 645 640 635 630 625 70 620 0 0.5 1 1.5 2 2.5 3 3.5 Solids-gas mass ratio (-) Figure 1: Comarison of ressure dro redicted for acceleration region using the Equations (3) to (10) for the dilute hase conveying of 64 m ceramic articles through a duct of 0.192 m. 19 17 ressure dro (mbar/m) 15 13 11 9 7 5 Ex. Eqn. (3) Eqn. (4) Eqn. (5) Eqn. (6) Eqn. (7) Eqn. (8) Eqn. (9) Eqn. (10) 0 2 4 6 8 10 Solids-gas mass ratio (-) Figure 2: Comarison of ressure dro redicted using the Equations (3) to (10) and the exerimental data of [8] for the dilute hase conveying of 5 mm olythene articles through a tube of 5.34 cm. It can be seen from the Figure 1 that the ressure dro in the acceleration region increases with the increase in solids-gas flow ratio. This is attributed to the fact that the resence of more solids results in their decreased acceleration, thereby increasing the sli velocity and hence the high ressure dro in the acceleration region. The RMS errors in the ressure dro rediction among the different correlations were 0.9 and 9.92 for the solid-gas mass ratio of 0.4175 and 3.178 resectively. Equations (3), (4), (6), (7) & (8) were found to redict ressure dro closer to each other with much lesser RMS errors. Drag force increases with the article size, due to slower acceleration or larger sli velocities. Hence, the comarison of these correlations for a larger article size was necessary to comlete the analysis. Woodhead et al. [8] resented the ressure dro data for the neumatic conveying of 5 mm olythene articles through a duct of 5.34 cm at different gas velocities and susension densities. The ressure dro included the acceleration region also. Caes and Nakamura Equation [20] was used to estimate the article-wall friction. Simulations were erformed using the correlations for various solids-gas mass ratios from 1.7 to 7.8 and the results are lotted in Figure 2. Figure 2 shows the measured ressure dro and the ressure dro redicted from correlations as a function of the solids-gas mass ratio. The deviations are large at the larger solids-gas ratio, due to higher sli and higher drag. A statistical analysis was erformed and the RMS errors for these
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1319 correlations were estimated. The Equations (3), (4), (5), (7) & (9) had RMS errors lesser than 0.35, while the other correlations showed RMS errors in the range of 0.65-1.55 for the rediction of ressure dro. Hence, it can be concluded that the Equations (3), (4), (5), (7) & (9) can be satisfactorily utilized for the design and simulation of neumatic conveying of large articles (5 mm) and for wide range of solids-gas mass ratios (1.7 7.8). The best correlation under review for the simulation of data in [8] is the correlation of Kaskas [14] with a RMS error of 0.22 for ressure dro rediction. From the simulations carried out in the resent study and the comarison with exerimental data in [8] and [9], it can be concluded that the Equations (3), (4) & (7) can be used for the simulation of ressure dro in neumatic conveying of articles in the size range 64 m to 5 mm and solids-gas mass ratio in the range of 0.4 7.8. COMARISON OF CORRELATIONS FOR WALL-ARTICLE FRICTION Correlations in Table 2 were tested for their alicability to redict the ressure dro through estimation of wall-article friction. The simulations were erformed for the conveying of ceramic sherical articles of 64 m through the duct of 19.2 cm for various solids-gas mass ratios. The correlations in the Table 2 were used to redict the article-wall friction in the model for each solids-gas mass ratio. For the estimation of drag coefficient, the Equation 5 was used, as roosed in the revious section of this article. The results were comared with the exerimental data in [9]. The range of solids-gas mass ratio for the simulation is 0.4-4. The results are lotted in Figure 3, as ressure dro-solids-gas mass ratio relationshi redicted using different correlations for wall-article friction. It can be seen from the Figure 3 that most of the correlations redict the ressure dro with large deviation from the exerimental value. It can also be seen that the Equation (19) redicts the ressure dro cl osely with the exerimental data, when comared with the other correlations. The RMS error for the data redicted by Equation (19) was 5 a/m, while the other correlations had RMS errors in the range of 16-90 a/m for the rediction of ressure dro. Since the rediction of Equations (11), (12), (16) & (18) are fairly closer to each other and have the same behavior as that of the exerimental data; they can be imroved by altering the emirical constants in these Equations. This also indicates the strong effect of article size and mass flow ratios on the wallarticle friction. Hence Equation (19) and the Equations (11), (12), (16) & (18) (with different constants) may be used for design and simulation for articles in the micron size range. ressure dro (a/m) 300 250 200 150 100 50 Eqn. (11) Eqn. (12) Eqn. (13) Eqn. (14) Eqn. (15) Eqn. (16) Eqn. (17) Eqn. (18) Eqn. (19) ex ressure dro (mbar/m) 35 30 25 20 15 10 5 Ex Eqn. (11) Eqn. (12) Eqn. (14) Eqn. (15) Eqn. (16) Eqn. (17) Eqn. (18) Eqn. (19) 0 0 1 2 3 4 5 Solids-gas mass ratio (-) Figure 3: Comarison of ressure dro redicted using the Equations (11) to (19) and the exerimental data of [9] for the dilute hase conveying of 64 m ceramic articles through a duct of 0.192m. 0 0 2 4 6 8 10 12 14 Solids-gas mass ratio (-) Figure 4: Comarison of ressure dro redicted using the Equations (11) to (19) and the exerimental data of [8] for the dilute hase conveying of 5 mm olythene articles through a tube of 5.34 cm. To study the erformance of these correlations for large sized articles, simulations were
K.S. Rajan/Int.J.ChemTech s.2012,4(4) 1320 erformed for the conveying of ceramic sherical articles of 5 mm diameter through the tube of 5.34 cm for various solids-gas mass ratios in the range of 1.7 12.4. Figure 4 shows the comarison of ressure dros redicted by different correlations and the exerimental data. Again, the redictions from different correlations are different to a large extent. Statistical analysis shows that the correlation of Caes and Nakamura [20] redicts the ressure dro with a RMS error of 0.67 mbar/m. Equations (11), (12), (13), (17) & (18) had RMS errors between 1.3 and 2.9 mbar/m for ressure dro rediction. Since the ressure dro values are less, higher deviations will result in too an erroneous rediction. The deviations are more ronounced at higher solids-gas mass ratios, indicating that a correlation suitable for the simulation of large article flow at high solids-gas mass ratios needs to be develoed. CONCLUSIONS The rediction of ressure dro is critical for the design and simulation of neumatic conveying equiments. The economy of a neumatic conveying oeration for dust disosal deends on the ressure dro in the conveying line. Since the ressure dro estimation involves determination of number of forces acting on the solids and the gas, various correlations roosed in the literature for the rediction of drag coefficient and wall-article friction were tested. Some of the existing correlations for the rediction of drag coefficient have been found to be satisfactory for articles of small and large size (64 m & 5 mm) for a wide range of mass flow ratios (0.4 7.8). The existing correlations for the rediction of wall-article friction estimate the ressure dro with a large error, for the large articles at high solids-gas mass flow ratios, indicating the necessity for more exerimental investigations in these oerating conditions. Acknowledgements: The author acknowledges Dr. Visa oikoainen for roviding the exerimental data for the simulation and for his valuable suggestions through electronic mail. NOMENCLATURE ALHABETS A Cross-sectional Area (m 2 ) C d Drag Coefficient (-) D Diameter of duct (m) D Diameter of article (m) f article-wall friction factor (-) m Mass flow rate (kg/s) ressure dro (a) article ynolds number (-) ynolds number (-) u Actual velocity (m/s) W Solids-gas mass flow ratio (-) L Length (m) F d Drag force er unit length (N/m) Fr Froude number (-) GREEK SYMBOLS Density (kg/m 3 ) Viscosity (kg/m.s) Volume fraction of hase (-) Friction factor in Equation 19 (-) SUBSCRITS & SUERSCRITS g Gas hase article s Solid hase t Terminal
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