1 The Derivation o a Drag Coeicient Formula rom Velocity-Voidage Correlations By M. Syamlal EG&G, T.S.W.V, Inc. P.O. Box 880 Morgantown, West Virginia 6507-0880 T.J. O Brien U.S. Department o Energy Morgantown Energy Technology Center P.O. Box 880 Morgantown, WV 6507-0880 Abstract A ormula or the luid-solids drag coeicient or a multiparticle system is derived rom a Richardson-Zaki type velocity-voidage correlation. The ormula compares avorably with the Ergun equation in the void raction range o 0.5-0.6 and correctly reduces to a ormula or the single-particle drag coeicient, when the void raction becomes 1.0. The minimum luidization velocity calculated rom the ormula compares well with experimental data or Reynolds numbers greater than 10. keywords: multiphase low, luid-solids drag, minimum luidization, Richardson-Zaki equation April 1987
The Derivation o a Drag Coeicient Formula rom Velocity-Voidage Correlations Abstract A ormula or the luid-solids drag coeicient or a multiparticle system is derived rom a Richardson-Zaki type velocity-voidage correlation. The ormula compares avorably with the Ergun equation in the void raction range o 0.5-0.6 and correctly reduces to a ormula or the single-particle drag coeicient, when the void raction becomes 1.0. The minimum luidization velocity calculated rom the ormula compares well with experimental data or Reynolds numbers greater than 10. Introduction An important constitutive relation in any multiphase low model is the ormula or the luid-particle drag orce, which is oten expressed in ollowing orm (eq..9 in [1]): F = β (v vs) (1) The actor β can be expressed in terms o a drag coeicient as 3 ε (1 ε) ρ β = v vs () 4 dp The drag coeicient C is only a unction o the particle Reynolds number and the void raction D and must be determined rom experimental data.
3 One method is to derive a ormula or rom empirical correlations or the pressure drop in packed beds. For example, Gidaspow [1] uses the Ergun equation [], which is based on 00 (1 ε) 7 D = + (3) ε Re 3 ε C pressure-drop data or packed beds with void ractions in the range o 0.4-0.6: For values o the void raction greater than 0.6, the error in the value o C calculated rom the above equation increases with increasing void raction. To correct this problem, Gidaspow [1] uses a Wen and Yu [3] correlation or void ractions greater than 0.8: D = 4 ε Re 0.44 ε 0.687 ( 1+ 0.15( ε Re) ).65 ε.65 ( ε Re) < 1000 ( ε Re) 1000 (4) But such an approach makes C discontinuous at the switching void raction o 0.8, with the D magnitude o the discontinuity increasing with the Reynolds number. An alternative method is to derive a ormula or C rom the Richardson-Zaki equation [4], which expresses the ratio o the terminal settling velocity o a multiparticle system to that o an isolated particle as a unction o the void raction: D V V t V r = = ε n 1 (5) ts
4 The exponent is n 1, rather than n as usually written, because here we express the terminal velocity o the multiparticle system as the interstitial, rather than the supericial, velocity. The Richardson-Zaki exponent is given by n = 4.4 4.65 Re 4.4 Re.4 0.03 ts 0.1 ts Re ts 0. > < 0. Re < 1 1 > Rets < 500 Re > 500 ts ts (6) Sinclair and Jackson [5], or example, uses the ollowing ormula based on the Richardson-Zaki equation ρs g (1 ε) β = (7) n Vts ε The diiculty with the above ormula is that it depends upon the actor ρ s g. The presence o such a actor is not justiied because the drag orce experienced by a particle placed in a low ield with a given Reynolds number and void raction would not depend upon the particle density or the gravitational acceleration. The V in the denominator o the ormula, however, ts is proportional to ( ρ ρ ) g s g at Reynolds numbers less than 0.4 [6]. Thereore, the actor ρ s g gets cancelled at low Reynolds numbers (and or negligible gas density), making the ormula acceptable or low Reynolds numbers. At higher Reynolds numbers, however, a complete cancellation does not occur. For Reynolds numbers greater than 500, the ormula retains an
5 undesirable dependence on a actor o ρ g. s Another example o the use o the Richardson-Zaki equation is the ollowing ormula derived by Gibilaro et al. [7]: (4.8 n) ε n v vs 3.8 C D = s (Rets) ε (8) Vts To derive the above expression, they assumed that C D has a voidage dependency o ε 3.8. There is no need or such an assumption, as will be shown in this paper. Also the above ormula incorrectly depends upon constant. Vts and, hence, upon the particle density and the gravitational The objective o this paper is to derive a ormula or the multiparticle drag coeicient rom a Richardson-Zaki type velocity-voidage correlation and a ormula or the singleparticle drag coeicient. The ormula will be based on two parameters only, the Reynolds number and the void raction. Multiparticle drag coeicient The single-particle drag coeicient is deined as F s = C Ds π d 4 p ρ (v vs ) (9) From a dimensional analysis it can be shown that s is only a unction o the Reynolds
6 number Res. Correlations or C Ds have been developed rom experimental data and theoretical analysis and are well-established, or example see [8]. Here we use the ollowing simple ormula given by Dalla Valle [9]: 4.8 s = 0.63 + (10) Res Under terminal settling conditions, the drag orce on a particle is equal to its buoyant weight, and the momentum balance is given by C Ds π d 4 p ρ V ts π d = 6 3 p ( ρ s ρ ) g (11) which can be written in a dimensionless orm as 3 s Rets = Ar (1) 4 The multiparticle drag coeicient is deined in a similar manner, as shown by eq. (). is a unction o the void raction in addition to the Reynolds number. Under terminal settling conditions, the momentum balance is given by 3 Ret = Ar (13) 4 which, or example, is a dimensionless orm o eq..17 in [1] with the riction and the solids
7 pressure terms ignored. From eqs. (1) and (13) we get C D Rets (Ret, ε ) = s (Rets) (14) Ret Although eqs. (1) and (13) were written or a particular value o the magnitude o the drag orce -- the buoyant weight o a particle -- the magnitude o the drag orce does not explicitly appear in eq. (14). Thereore, we claim that eq. (14) can be used or calculating any magnitude o the drag orce, or equivalently, by dropping the subscript t or the terminal settling condition. This amounts to changing the question rom "What is the Ret o a multiparticle system o void raction ε, consisting o particles o known Rets?" to "What is the Rets o certain (ictitious) particles that will be under terminal settling conditions or the given ε and Re?" The validity o the method, thereore, hinges only on the uniqueness o the inversion o the velocity voidage equation V r (Re ts Ret, ε ) =, ε = (Ret, ε) (15) which is demonstrated or the Richardson-Zaki [4] and the Garside and Al-Dibouni [10] equations in this study. Thus, replacing by and by and substituting Ret Re Rets Res Res = Re / (16)
8 in eq. (14), we get s (Re/ V ) D (Re, ε ) = (17) V C r r which is a ormula or calculating rom the velocity-voidage correlation and the singleparticle drag coeicient s and, as desired, Re and ε are the only parameters needed. To determine C rom the Richardson-Zaki equation [4] with this method, a numerical D procedure, as shown in Table I, is required. First, is calculated iteratively, as shown by steps Table 1 Calculation o C D rom Richardson-Zaki Equation 1. Guess a value or V r, say 1.. Calculate Re s rom eq. (16). 3. Calculate n rom eq. (6). 4. Calculate V r rom eq. (5). 5. Check or convergence. I not converged, update V r and go to step. 6. Calculate C D rom eq. (17) and eq. (10). through 5 in the table. A successive substitution method converges to a unique solution or 5 within a tolerance o 10 usually under 10 iterations. Ater obtaining a converged value or, can be calculated rom eq. (17) and a suitable ormula or, e.g., eq. (10). s
9 An analytical ormula or and, hence, or can be derived, rom the ollowing velocity-voidage correlation proposed by Garside and Al-Dibouni [10]: where V B r A = 0.06Re s (18) A = ε 4.14 (19) and 0.8 B = ε.65 ε 1.8 ε 0.85 ε > 0.85 (0) Substituting Res = Re/ in eq. (18) and solving or we get [ 0.06Re + 0.0036 Re + 0.1Re(B A) + A ] = 0.5 A (1) Eqs (10), (17), and (1) give the desired ormula or. Figure 1 shows a plot o as a unction o Re or three dierent values o ε. C calculated rom the Garside and Al-Dibouni equation, the Richardson and Zaki equation, and the Ergun equation are shown. The Garside and Al-Dibouni equation is always in reasonable agreement with the Richardson and Zaki equation. At a void raction o 0.6, all three o the correlations are in good agreement. However, as mentioned, the Ergun equation deviates signiicantly rom the other two equations at a void raction o 0.9. D
10 Minimum luidization velocity From the Garside and Al-Dibouni ormula or, an explicit ormula or the minimum luidization velocity is derived as ollows. Substituting eq. (10) in eq. (1) and solving or the Reynolds number we get 4Ar 4.8 +.5 4.8 3 Rets = () 1.6 which is the Reynolds number based on the terminal settling velocity o a single-particle. Since the right-hand side o eq. () is only a unction o Ar, we will call it Ar *. Substituting eq. () in eq. (18) and solving or we get * A + 0.06 B Ar r = (3) 1+ 0.06 Ar V * Now using the identity Re t = Re ts V r and eq. (), we get the ollowing ormula or the Reynolds number at minimum luidization condition:
11 * * A + 0.06 B Ar Re t = Ar (4) * 1+ 0.06 Ar The Reynolds number calculated rom eq. (4) is compared with experimental data in Fig.. The data are or spherical particles or sand, covering a wide range o conditions usually encountered in luidized beds: void raction, 0.36-0.48; temperature, 98-113 K; pressure, 100-3500 kpa; particle diameter, 15-6350 µm; particle density, 1100-7840 kg/m3. Four data points are or a water luidized bed; all others are or air or nitrogen luidized beds. The agreement between the theory and the experiment is very good or Reynolds numbers larger than 10. For smaller Reynolds numbers, however, the theory systematically over predicts the Reynolds number. Summary Based on a correlation proposed by Garside and Al-Dibouni [10], an analytical ormula or the multiparticle drag coeicient is where is given by 0.63 4.8 C D (Re, ε) = + (5) Re [ 0.06Re + 0.0036 Re + 0.1Re(B A) + A ] = 0.5 A (6)
1 A = ε 4.14 (7) 0.8 ε B =.65 ε 1.8 ε 0.85 ε > 0.85 (8) The above ormula compares avorably with the Ergun equation [] in the void raction range o 0.5-0.6 and correctly reduces to a ormula or the single-particle drag coeicient, when the void raction becomes 1.0. The derivative Re is a continuous unction o Re. C and its D derivative with respect to ε are continuous, except at ε = 0.85 where C is continuous (rounded D o to three signiicant igures), but its derivative is discontinuous. The minimum luidization velocities calculated rom the ormula compares well with experimental data, especially or Reynolds numbers greater than 10. LIST OF SYMBOLS A A unction o void raction deined by eq. (19) 3 Ar Archimedes number, dp ρ ( ρ ρ ) g / µ Ar * A unction o Ar deined by the right hand side o eq. () B A unction o void raction deined by eq. (0) s Multiparticle drag coeicient s Single-particle drag coeicient dp Particle diameter, m F The drag orce per unit volume in a two-phase system, N/m 3
13 Fs The drag orce on an isolated particle, N g gravitational acceleration, m/s Re Reynolds number or a multiparticle system, d p ρ v vs / µ Res Reynolds number or a single-particle, d p ρ v vs / µ Ret Reynolds number or a multiparticle system under terminal settling conditions, dp ρ Vt / µ Rets Reynolds number or a single-particle under terminal settling conditions, d ρ Vts µ p / v Fluid velocity (interstitial), m/s vs Solids velocity, m/s The ratio o the terminal settling velocity o a multiparticle system to that o an isolated single particle Vt v vs or a multiparticle system under terminal settling conditions, m/s Vts v vs or an isolated, single particle under terminal settling conditions, m/s Greek symbols β ε A coeicient deined by eq. (1), kg/(m 3 s) Void raction µ Fluid viscosity, Pa s ρ Fluid density, kg/m 3 ρ s Solids density, kg/m 3
14 REFERENCES 1 D. Gidaspow, Multiphase Flow and Fluidization, Academic Press, New York, 1993, pp.35-38. S. Ergun, Chem. Eng. Progr., 48 (195) 89. 3 C.Y. Wen and Y.H. Yu, Chem. Eng. Progress Symp. Ser., 6 (1966) 100. 4 J.F. Richardson and W.N. Zaki, Trans. Instn. Chem. Engrs., 3 (1954) 35. 5 J.L. Sinclair and R. Jackson, AIChE J., 35, (1989) 1473. 6 D. Kunii and O. Levenspiel, Fluidization Engineering, Robert E. Krieger Publishing Company, New York, (1977) p. 76. 7 L.G. Gibilaro, R. Di Felice, S.P. Waldram, and P.U. Foscolo, Chemical Engineering Science, 40, (1985) 1817. 8 A.R. Khan and J.F. Richardson, Chem. Eng. Comm., 6 (1987) 135. 9 J.M. Dalla Valle, Micromeritics, Pitman, London, (1948) p. 3. 10 J. Garside and M.R. Al-Dibouni, Ind. Eng. Chem., Process Des. Dev., 16, (1977) 06. 11 L.E.L. Sobreiro and J.L.F. Monteiro, Powder Technol., 33 (198) 95. 1 N.S. Grewal and S.C. Saxena, Powder Technol., 6 (1980) 9. 13 S.E. George and J.R. Grace, AIChE Symp. Ser. No. 176, 74 (1978) 67. 14 J.S.M. Botterill and Y. Teoman, in J.R. Grace and J.M. Matsen (eds.), Fluidization, Plenum Press, New York, 1980, p. 93. 15 J.M. Rockey, J.S. Mei, C.V. Nakaishi, and E.H. Robey, in Proc. o the 10th International Fluidized Bed Combustion Conerence, San Francisco, Caliornia, May 1989.
16 R.R. Pattipati and C.Y. Wen, Ind. Eng. Chem. Proc. Des. Dev., 0 (1981) 705. 15
16 List o Figures Figure 1. Comparison o multiparticle drag coeicients Figure. Experimental and predicted Reynolds numbers at minimum luidization conditions
17 Void raction = 0.4 10000 Garside-Al-Dibouni Richardson-Zaki 1000 Ergun 100 10 1 0.1 1 10 100 1000 10000 Re Figure 1a Figure 1. Comparison o multiparticle drag coeicients
18 Void raction = 0.6 10000 Garside-Al-Dibouni Richardson-Zaki 1000 Ergun 100 10 1 0.1 1 10 100 1000 10000 Re Figure 1b Figure 1. Comparison o multiparticle drag coeicients
19 Void raction = 0.9 1000 Garside-Al-Dibouni Richardson-Zaki 100 Ergun 10 1 0.1 0.1 1 10 100 1000 10000 Re Figure 1c Figure 1. Comparison o multiparticle drag coeicients
Figure Experimental and predicted Reynolds numbers at minimum luidization conditions. 0