e - coefficient of restitution [-] Subscript fs = Fanning friction factor for the solid particle [-] 0 = before collision Gs = solid mass velocity [kg/sec] 1 = after collision Ga = air mass velocity [kg/sec] H = impulsive force for Y direction [Kg -sec] I = momentof inertia of particle [kg-m] Literature Cited L =pipe length [m] Ms = particle mass [kg] 1) Belden, D.H. &L.S. Kassel: Ind. Eng. Chem., 41, 1174 m =mass flow ratio =GS/Ga [-] (1949) ni = local numberof particles in unit volume [1/cc] 2) CMOJiUBipeB, A. E.: Paipu Yuso (translated in Japanese P = impulsive force for Z direction [Kg -sec] by Z. Sotoo & M. Nagai) Gijutsu Shoin (1963) APa = pressure drop due to fluid friction [Kg/m2] 3) Doig, I.D. & G.H. Roper: Ind. Eng. Chem., Fundamen- APh - solid static head [Kg/m2] tals, 6, 247 (1967) APS = additional friction loss [Kg/m2] 4) Hariu, O.H. & M.C. Molstad: Ind. Eng. Chem., 41, APt - total pressure drop [Kg/m2] 1150 (1949) R =pipe radius [m] 5) Hinkle, B. L. : Ph. D. thesis. Georgia Inst. of Tech.(1953) r = particle radius [m] 6) Iinoya, K. & K. Goto: Kagaku Kogaku (Chem. Eng., u =local air velocity [m/sec] Japan), 27, 12 (1963) ua - meanair velocity [m/sec] 7) Jotaki, T. &H. Mori: Trans. J. S. M. E., 33, 1617(1967) Mmax= maximumair velocity [m/sec] 8) Julian, F.M. & A.E. Dukler: A 7. Ch. E. Journal, ll, Us =solid velocity 853 (1965) [m/sec][ m/sec] ut = free falling velocity of particle in air 9) Konno, H. S. Saito & S. Maeda: Kagaku Kogaku (Chem. vx - velocity componentof X direction [m/sec] Eng.y Japan), 31, 243 (1967) vy = velocity component of Y direction [m/sec] [m/sec] 10) Mori, Y., A. Suganuma & H. Sadotomo: Preprint of 31 vz - velocity componentof Z direction st annual meeting of Soc. Chem. Engrs. Japan y - distance from pipe axis [m] ll) Muschelknautz, E. : V. D. I. Forschungsheft, 476 (1959) X =direction at right angles to Y and Z Y = opposite gravitational direction 12) Ranz, W. K, G.R. Talandis &B. Gutterman: A. I. Ch. Z = longitudinal direction of the pipe E. Journal, 6, 124 (1960) p - coefficient of kinetic friction [-] 13) Tabor, D. & F.P. Bowden: Kotai no Masatsu to Jun- fis - coefficient of static friction [-] katsu (translated in Japanese by M. Soda) Maruzen (1964) pa -air density [kg/m3] 14) Uematsu, T. &T. Kanno: Trans. J S. M. E., 27, 1748 pds = dispersed density [kg/m3] (1961) ps -solid density [kg/m3] 15) Vogt, E.G. & R.R. White: Ind. Eng. Chem., 40, 1731 a) = angular velocity of the particle [ rad/sec] (1948) POWER REQUIREMENTS IN THE AGITATION OF NON-NEWTONIAN FLUIDS" NOBUO MITSUISH! AND NOBUYUKI HIRAI Department of Chemical Engineering, Kyushu University, Fukuoka, Japan A modified Reynolds number in the agitation, based on the Ellis and Sutterby models of three rheological constants for non-Newtonian fluids, is defined in the way that the relation between friction factor and Reynolds number in the case of non-Newtonian pipe flow maybe the same as that in the case of Newtonian pipe flow. Introducing this modified Reynolds number, the same correlation of the power requirements can be applied to both experimental results of Newtonian and non-Newtonian fluids with various types of impellers. I. Introduction Newtonianfluids is nowpossible except for somespe- cial cases. On the contrary, studies of power require- Agitation, which is one of the most important unit ments in the agitation of non-Newtonian fluids are operations, is not easy to approach theoretically and making slow progress. Such works have been carried quantitatively. The measurementof power require- out, by Metzner et al.8>9), Calderbank and Moo- ments in the agitation is, however, comparatively Young1'^, and others3l4'6>12). In many of these studies, easy, and the estimation of power consumption for a power-law model with two constants has been used * Received on May 29, 1968 to describe the rheological properties of non-Newtonian The first annual meeting in autumn, Soc. Chem. Eng. fluids. However, the power-law model cannot express Japan, Octorber 14, 1967 the properties of non-Newtonianfluids in a wide range VOL.2 NO.2 1969 217 Fig. I Correlation of T and JT of the Ellis model of the shear stress. The authors will correlate the power r_^._4Q_ (q) requirements in the agitation of Newtonian and non- Newtonian fluids by applying three constants-models T= - (10) for non-Newtonian pseudoplastic fluids based on the ri/2 works of Ellis7) and Sutterby13). The Ellis model is one obtains considered in this paper, and the Sutterby model is treated in the Appendix. r= t(i + ^3-r-) (id Eq.(ll) is the equation of flow in a circular pipe for 2. Development of Correlation an Ellis model fluid. The relationship between Tand F is plotted with a as a parameter in Fig. 1. 2-1 Evaluation of rheological constants The three rheological constants can be evaluated by The rheological equations of the Ellis model are measuring the flow rate, Q, and the pressure drop, expressed by the following equations. AP, with a capillary viscometer and then plotting 4Q/ T= ~VJ (1) nR* against tr on a logarithmic scale. From a curve in Fig. 1, the gradient of which is as same as that of the experimental curve, one can determine the value V=Vt/[1+ ^2___ ) (2) of a, and then n/2 and Vo can be determined from where, A is the rate of deformation tensor with the deviations on the ordinate and abscissa7\ components 2-2 Modified Reynolds number From Eq.(7), one obtains jIJ = (p- + tojL) (3) -**<£>-=-^fi+-^(-^-H (12) T1/2D ri/2L a+3\n/2/ J and (r:r) = S i Sj tijTji (4) Therefore, tr/ti/2 is expressed as a function of « and When an Ellis model fluid which is described by Eqs. 8y<i<Vz>/Ti/2D as follows : / 8yo<^>\ , , (l) and (2) flows in a circular pipe, the velocity pro- = Fe[a, ---7s- (13) file is expressed by Ti/2 \ T1/2JLJ Since - (^)[1M£)1 -^- Tl/2= T (10) +iM£TV-GDI « and 8^o<t;2> = yo.4Q ^ ,14^ where, rR is the shear stress at the wall T1/2D T1/2 ?riv tB ~ ^r~ {6) the function Fe is obtained from Fig. 1. For Newtonian laminar flow in a circular pipe, there From Eq.(5) the volumetric flow rate Q is obtained as *R3 I"-._l 4 (try-1] ,n, is the following relation between friction factor, f, and ^ Reynolds number, Re. Q=-aw^Y1 +-^tjK^r) \ (7) Rewriting Eq. (7) , /" p<vz>2 " i^c U5; If the Reynolds number is denned so that Eq.(l5) may hold also for the Ellis model, Eqs.(l2) and (13) Substituting the following equations into Eq. (8), yield 218 JOURNAL OF CHEMICAL ENGINEERING OF JAPAN Fig. 2 Experimental apparatus for flow characteristics Fig. 4 Typical results of fluid flow characteristics measurements Table 1 Values of K for piano wires Diameter Length X[G.cm/rad] [mm] LmmJ No. 1 2.703 1560 2. 895X HT3 No.2 3.710 1540 10.240 " No. 3 4.509 1560 22.540 " Table 2 Range of Variables Studied Impeller type d/DT b/DT H/DT 2-Bladed-paddle 0. 5 0. 2 1. 0 0.8 0.2 1.0 6-Bladed-turbine 0. 56 0. 112 1. 0 0.67 0.134 1.0 4-Multiple-paddle 0. 5 0. 2 1. 5 Screw with draft tube 0.6 - 1.3 Tank diameter DT=l0, 30 [cm] Modified Reynolds number NRe =0. 13~520 Impeller speed n=0.617-9.70 [rev./sec] Fluids used Carboxy-methyl-cellulose solutions in water Hydroxy-ethyl-cellulose solutions in water Polyvinyl-alcohol solutions in water Clay suspensions in water Fig. 3 Experimental apparatus for measurement of agitation power requirements diameter of the pipe was determined from flow rate and pressure drop data for aqueous cane sugar solutions p Dp<vt>r 4 f / 87o<^>y-1! and "millet" jelly, which are Newtonian fluids of ^ = 1 H tvk4«' 7s-J known viscosity. As the manometer fluid, mercury (16) was used. In order to apply Eq.(l6) to an agitated tank, one 3-2 Power requirements of agitation conventionally replaces the pipe diameter D by the The experimental apparatus was similar to that of impeller diameter d and the meanvelocity <vz> by Honjo et al.5) and is shown in Fig.3. The agitated the impeller tip velocity dn. tank was hung by a torsion wire. The torque for agitation was evaluated from the angle of deflection of the turn table, which was measured by a lamp Fe in Eq.(l7) is then obtained from the ordinate cor- scale and a mirror. For the attenuation of vibrations, responding to the value of 8?zWri/2 on the abscissa in a damper was installed on the turn table. Piano wires, Fig.1. specifications of which are shown in Table 1, were used as the torsion wire. The above-mentioned measuring device was replaced 3. Experiment by a torque-meter only for a 30cm diameter tank. 3-1 Flow characteristics Agitation speed was obtained by measuring the pick-up The experimental apparatus, shown in Fig. 2, is the signal pulses with a photoelectric cell installed near the same one described in a previous report10}. The internal VOL.2 NO.2 1969 219 Fig. 5 Correlation of power requirements 2-Bladed-paddle (Ellis model). Each impeller was positioned 5.Ocm from the bottom of a lOcm diameter unbaffled tank Fig.