Isothermal Axial Laminar Flow of Non-Newtonian Fluids in Concentric Annuli (Power-Law and Sisko Fluids)

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1 IraJme Polymer lour al / Volume 5 Number 4 (1996) 1O26-I2096 Isothermal Axial Laminar Flow of Non-Newtonian Fluids in Concentric Annuli (Power-Law and Sisko Fluids) All Bahraini, Siroos Azizmohhmmadi, Mohammad-Reza Golkar Narenji and Vahid Taghikhani Department of Chemical Engineering, Amir Kabir University of Technology, Tehran, I.R.Iran Received : 5 February t995; accepted 19 October 1996 ABSTRACT In attention to industrial significance of non-newtonian fluids, determination of flow behaviour under various operational conditions is desirable. In this paper the isothermal and axial laminar flow of non-newtonian flulds(powerlaw and Sisk)) In annuli have been analyzed. In the first step the governing equations (equation of motion and rheological model) are solved with using a numerical method for power-law fluid, and the volumetric flow rate values are obtained. Then by comparing with a semi-analytical method and the methods proposed by the other workers, the accuracy of results is confirmed. Finally this numerical method is applied to Sisko fluids and the results are presented. Key Words: non-newtonlan flow, Sisko fluids, power-low fluids, numerical methods, concentric annuli INTRODUCTION The flow of non-newtonian fluids in annuli is of special industrial significance. More attention in this context has been paid to the charts that simplify calculations and design- So far some works have been done in this context. These works include simple non-newtonian fluids such as the Bingham plastic (Olphen 1950; Mori and Ototake 1930; Laird 1957 ; Fredrickson and Bird 1958; Slibar and Paslay 1957), Power-law (Fredrickson and Bird 1958 ; Tiu and Bhattacharyya 1973; Larsen and Hanks 1979), and some more complex models such as the Ellis model (Mc,Eachem 1966), Powell-Eyring model (Christiansen and Russell 1974), yield-pscudoplastic model (Gooier and Aziz 1972; Hanks 1979; Giicuyener and Mehmetoglii 1992), generalized Bingham (Cheng 1971, 1975) and yield power-law (Torrance 1963 ; Hanks and Ricks 1974; Hanks 1978), and Herschel-Bulkley model (Herschel and Bulkley 1926; Skelland 1967). Greases are non-newtonian fluids and the Sisko model is the best rheological model to predict their behaviour. Greases have high viscosities at low shear rates and low viscosities at high shear rates. Thus they do not flow away from the outside of gears and bearings, where shear rates are low, and they lubricate effectively at the bearing or gear surface where shear rates are high. The purpose of this paper is to present the theory of axial laminar flow of power-law and Sisko fluids, such as greases, in annuli together with appropriate design charts. This model (Sisko), due to complexities which govern its rheological equation, 271

2 lsolhenual Axial laminar Flow of Nou-Newtonian Fluids r* -r R Substitution of these parameters into Equation 1, the dimensionless form of rheological equation may be obtained KR ~1R R Figure 1. Velocity and shear stress distribution in annuli. has rarely been considered up to now (Figure 1). THEORETICAL DEVELOPMENT Power-low Model Power-low rheological model [1] may be expressed as : m du g, dr With dimensional analysis the following dimensionless parameters are suggested: ml tin u* 2,rrR g capr u r } ml l l ~n R LgcAPR r* rl APR (I) : l *= I ddu* (2) r* 't (. dr*j On the other hand the equation of motion in the x-direction may be expressed as: d (Tr) = OP (3) r dr L Substitution of dimensionless parameters into Equation 3, we obtain the following result: r**.dr* (l.'s') =1 (4) Integration of Equation 4 with the r'=o gives: r* 2 Lr* r* ~ (5) Combination of Equations 2 and 5 give the following equation: a du* I r du* r* l 2 r* 1 dr*! n ' L dr*j (6) Starting with an estimated value of A, Equation 6 is solved using direct or Newton-Raphson methods [2] and velocity gradients du*/dr* in the annulus are obtained. By the application of parabolic interpolation and the r* =K : u` =O, the velocity distribution in the annulus is determined. Finally with the r*=1: u*=o, accuracy of the estimated value of d is checked. The above operations are repeated until the latter condition is satisfied. In continuation, dimensionless volumetric flow rate is obtained applying the following two procedures (see Appendix). 272 Iranian Polymer Journal l Volume 5 Number 4 (IM)

3 Rabrami A. el I , K K rigure 2. Dimensionless flow rate of pseudoplastics in Figure 2. Dimensionless flow rate of pseudoplastics in annuli. annuli, Completely Numerical Method Dimensionless volumetric flow rate is expressed by the following relation : Completely Numerical Method Dimensionless volumetric flow rate is expressed by the following relation: Q* Jr*u*dr a (7) r By application of numerical integration, the value of Q* is obtained. Results for Q* and). are shown against different values of K and n in Figures 2 and 3, and Table 1, respectively. # Z 2 1+1/n j l 2) I + 1 /n r*dr*+ (Aar* x Semi-analytical Method We can obtain from Equation 6 [3,4] : A (9) du* " =±1 L dr 2 (2 2 r * (8) On the other hand an expression for Q* may r* be obtained by integrating Equation 7 by parts, as: Where (+) is used for K- r*5d and ( ) for d5r*5]. According to Equation 7 : Q* 2 r*adu* dr* (10) 2 dr* K 1 ~. 1 Q*= j r*u*dr*= J r*u*dr*+ r*u*dr* By application of Equation 8 and integrating by K K parts the Equation 10 may be written as: Ganurn Polymer Journal 1 Volume 5 Number 4 (1996) 273

4 l Isothermal Axial Laminar Flow of Non-Newtonian Fluids Table 1. A for power-law fluids in annuli V5.K and n. n , , , , Q*= 1 X 22+11n [1+ n L 1] r 142] 1+1/n _ K1_11n ( n -1 J ~ 2 P, [ A 2_ K2 ] 1+1/n 2l l+1/n r*_11n(it + r* it (r* 2 l+1/n r*_ 11ndr * A2) + (11) Combining Equations 9 and 11, and rearranging the resulting equation, the following expression is obtained for the dimensionless volumetric flow rate: Q* n (3+ ] n (142) 1+1/n K 1 lln (12K2) 1+1/n] (12) With determination of A from procedure mentioned above, one is able to compute Q. The results of two methods are in good agreement. The results of the numerical method is compared to Fredrickson and Bird [1] in Table 2. Thus the numerical method has the required accuracy to be applied to complex rheological models that are difficult to be analyzed analytically. Sisk') Model The Sisko model [5] is a model that appropriately predicts the rheological behaviour of greases. Greases have high viscosities at low shear rates and low viscosities at high shear rates. Thus they do not 274 Iranian Polymer Journal 1 Volume 5 Number 4 (1996)

5 aarami A. et al Table 2. Comparing values of 2.. from reference [1] and the experimental data. n K Ref. [1] Exp. data Ref. [1] Exp. data Ref. [1] Exp. data Ref. [1] Exp. data Ref. [1] Exp. data , flow away from the outside of gears and bearings where shear rates are low, and they lubricate effectively the bearings or gear surface where shear rates are high_ The Sisko rheological model can be expressed as: r= k+b I dr I t J l 1 J (13) By application of dimensional analysis the following dimensionless parameters are obtained: Q*-. Q 27rR 3 [A 1 C1 B AP* = u*= &APR c c.i 2L LB to R[B C1 Substitution of these parameters into Equation 13, results the dimensionless form of rheological equation as: r * - [ i + I r t r * I c t ) L dr*j (14) The dimensionless form of equation of motion (Equation 3) with new parameters may be written as: I -a r* " dr * (r*r*) =2AP* Integration of this equation together r* =A: r'=o gives the following result: s [r*-] r* (15) (16) Iranian Polymer Joam& I Volume 5 Number 4 {1996) 275

6 Isothermal Axial Laminar Bow of Non-s Ionian Fluids Table 3. A for Sisko fluids in annuli with Ic=0.5 vs. C and log AP*. IogAP C B Combination of Equations 14 and 16 may be ex- and 4 for A, against different values of qp* and C, pressed as : at a constant K. C r*-- I du* AP*= (I+ I I c tl. dt~* `(17) r* dr* 1J I l dr*j CONCLUSION The axial laminar flow of power-law fluids in annuli This equation is solved by application of the has been studied by Fredrickson and Bird [1]. method applied to power-law (in the previous Using the charts in the mentioned work to obtain a section), and velocity distribution and volumetric value of Q* is not as straightforward as it could be. flow rate are obtained (see Appendix). The results In the present work one can obtain Q* from are shown in Figures 4 and 5 for Q*, and Tables 3 charts directly. Furthermore Fredrickson and Bird D e a D AP.!1P* 5 6 Figure 4. Dimensionless flow rate of Sisko fluids in annuli. Figure 5. Dimensionless flow rate of Sisko fluids in annuli. K=0.5. K= Iranian PoOner 7oomal 1 Volume 5 Number 4 (1990

7 Bahraini A et a1. Table 4. x for Sisko fluids in annuli with K=0.9 vs. C and log AP*. IogAP C [1] expanded the integral of Q to a binomial expansion and computed Q* by using this procedure. In the present work this integral has been obtained by virtue of numerical methods directly. The numerical method applied to power-law model can be used for complex rheological models. Here this method has been used for Sisko fluids. REFERENCES Arnold G. Fredrickson and R. Byron Bird Ind. Eng. Chem., 50, 3, , Shacham M., Computers. Chem. Eng., 14, 6, 621-9, Richard W. Hanks, Ind Eng. Chem. Process Des. Develop., 18, 3, , Hakki I. Cllcdyener and Tanju Mehmetoglil, 4IChE7., 38, 7, , Sisko A. W., Ind. Eng Chem., 50, 12, , APPENDIX In this appendix, the numerical method used to determine the velocity distribution of the flow of Sisko fluids in annuli, is described. The following algorithm shows how one can proceed to determine the velocity distribution, for given K, AP*, and C. The same algorithm may he applied to power-law fluids, with minor modifications. 1. Input K, AP*, and C. 2. Divide r* domain to desired intervals. 3. Assume an initial value for 2 satisfying K <A C I. * 4. For every r*, evaluate dr* L,f. To do this, an initial value for du* dr* _ (e.g., for r; CA, for rr>x, and 0 for r*=a.). This value of du* fr= is used in Equation 17 in the form dr tu* dr* AP* re--] r 1+ du* C-1 dr' to iterate to final value. 5. To every two points r* and r;shaving obtained du* c du* 1 = and u*i,. a parabola is dr dr* +t fitted. u * l,_,i is the corresponding value of the fitted parabola for r;'+,. This procedure is started from i=1(r*-k) for which u*=0 and repeated for other r; up to r*=1 (outer wall). 6. If from the previous step u at r*=1 is evaluated to be zero, calculations are terminated. Iranian Pohmer k,umal I Volume 5 Number 4 {19% 277

8 lsolhermal Apxl Laminar Flow of Non-Newoonian Fluids Otherwise, a new value for A is guessed and the calculations are resumed from step 4. A method to update d is to use Newton iteration 1k+1_ 2 k u* Ak+1 4k k+1 k k r*=1 u I r =1 since u t lr=_1 is a function of A chosen. 7. From the velocity distribution, flow rat O* may be obtained by integrating from r* =K to r*=1. SYMBOLS k r m. 1 'r shear stress, lb fft- 2 m consistency coefficient, lb, nft- l s n-2 n flow behaviour index, 1 & Newton's law proportionality factor, hmfflbF I s 2 u velocity in axial direction, ft.s- 1 r radial distance, ft Q volumetric flow rate, ft 3 s- 1 R inner radius of outer cylinder, ft L annulus length, ft OP axial pressure gradient, lbfft A value of dimensionless radial coordinate r* for which shear stress is zero, 1 K ratio of radius of inner cylinder to that of outer cylinder, 1 A a constant of Sisko model, 1bmft- 1s-1 B a constant of Sisko model, lb mft- l ac-2 C a constant of Sisko model, 1 SUPERSCRIPT. the dimensionless form of a physical quantity. 278 Iranian Polymer loam' J Volume 5 Number 4 (1996)