AP06_3.vp 1 Introduction A rotational viscometer with coaxial cylinders is widely used in rheological measurements. Its common configuration consists of an inner rotating cylinder with radius R1 and length L and an outer stationary cylinder with radius R2 – see Fig. 1. The dependence of shear stress � on the Newtonian shear rate �� N at the specified radius is usually obtained from measurements. The power law and Bingham model are the simplest rheological models. The aim of this paper is to show a way to calculate the parameters of these models from exper- imental values � and �� N . For this purpose, the flow in the viscometer must be analysed. 2 Theory If the influence of the bottom and the interface is ne- glected, the only non-zero velocity component u� depends in cylindrical coordinates on the radius r only. The component of Cauchy’s equation of motion for this case takes the form (see e.g. [1]) � �d dr r r 2 0� � � . (1) Integrating this equation, we obtain the following relation for shear stress � �r C r � 12 . (2) The shear rate for this type of flow is given by the relation (see e.g. [1]) �� �� � � � � r r u r d d . (3) 2.1 Power law fluids The power law is the simplest model that is widely used for describing the rheological behavior of non-Newtonian fluids. Using this model, the dependence of shear stress on shear rate can be expressed by the following relation � � �� �K n� �1 , (4) where K is the coefficient of consistency and n stands for flow behavior index. Inserting (2) and (3) into Eq. (4) and taking into consider- ation that the shear rate in the gap is negative, we obtain K r r u r C r n � � � � � � � � � � � � d d � 1 2 . (5) Integrating the above equation we obtain u r C K n r Cn n� � �� � � � � � 1 1 2 22 . (6) For determining integration constants, the following boundary conditions are necessary r R� 1, u r � �� , (7a) r R� 2, u r � � 0. (7b) 42 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 3/2006 Czech Technical University in Prague Determination of Rheological Parameters from Measurements on a Viscometer with Coaxial Cylinders F. Rieger The paper deals with measurements of non-Newtonian fluids on a viscometer with coaxial cylinders. The procedure for determining the rheological model parameters is recommended for power-law fluids and Bingham plastics. Keywords: viscometer with coaxial cylinders, power-law fluids, Bingham plastics. Fig. 1: Viscometer with coaxial cylinders Using them, the following expression for integration con- stant C1 is obtained � � C K n R Rn n n 1 1 2 2 2 2 � � � � � � � � � �� � � . (8) Inserting (8) into (5) the following equation for shear rate can be obtained after rearrangement � � �� � � � � � � � � � 2 1 2 1 2 n R rn n , (9) where � � R R1 2. The dependence of the dimensionless shear rate � � *� � �� � on the dimensionless coordinate defined by the relation y y R R* ( )� �2 1 (where y is the radial distance from the rotating cylinder) for ratio � � 0.5 and several n values is shown in Fig. 2. The same dependence for ratio � � 0.9 is shown in Fig. 3. From Figs. 2 and 3 it is obvious that the maxi- mum shear rate is at the inner cylinder (y* � 0) and the mini- mum is at the outer cylinder (y* � 1). From the above men- tioned figures for both � values it can also be seen that the © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 43 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 3/2006 = 0.5 0.001 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 y* * n = 1 n = 0.5 n = 0.2 Fig. 2: Dependence of dimensionless shear rate values ��* on dimensionless distance y* for � � 0.5 and selected flow behavior index values n = 0.9 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 y* n � 1 n � 0.5 n � 0.2 * Fig. 3: Dependence of dimensionless shear rate values ��* on dimensionless distance y* for � � 0.9 and selected flow behavior index values n effect of flow behavior index n on the shape of the shear rate profiles is more pronounced at smaller � value. At � � 1 (par- allel plate asymptote) the flow behavior index n has no effect on the shear rate profiles – the shear rate is constant. 2.2 Bingham plastics The simplest model for viscoplastic behavior is the Bing- ham model �� � 0 for � �� 0 (10 a) � � � � �� � � � � � � p 0 � � for � �� 0, (10 b) where �p is plastic viscosity and �0 stands for yield stress. Inserting (2) and (3) into equation (10b) and taking into consideration that the shear rate in the gap is negative, we obtain � � � p d d r r u r C r � � � � � �0 1 2 (11) 44 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 3/2006 Czech Technical University in Prague 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 y* * � � � 0 � � � 0.1 � � � 1 Fig. 4: Dependence ��* on y* for � � 0.5 and selected values �* (�c* � 1.24) 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 y* � � � 0 * � � � 5 � � � 50 Fig. 5: Dependence � *� on y* for � � 0.9 and selected values �* (�c* � 83.9) and after integration u r r C r C� � � � � � �0 12 2 1 2p p ln . (12) Using boundary conditions (7), we get, after some manipulation C R R R R R R R R R R1 2 2 1 2 1 2 2 2 0 1 2 2 2 1 2 2 2 1 2 2 2� � � � � � � p ln . (13) Combining Eqs. (3), (11) and (13) the equation for shear rate distribution can be obtained and the shear rate profiles shown in Figs. 4 and 5 can be depicted. From both figures it can be seen that ��* distribution depends on � and the dimen- sionless parameter � � � � * � 0 p . (14) The distribution is more pronounced for greater �* and lower � values. However �* must be lower than critical at which shear stress value at the outer cylinder is equal to the yield stress � �2 0� . At greater values of �*, equation (10b) does not hold in the whole gap between the cylinders. The critical value �c* can be calculated from the equation � � � � � c* ln( ) � � � 2 1 2 1 2 2 2 (15) obtained from the condition � ��r � � 0 at r R� 2. The curves in Figs. 4 and 5 for �* � 0 are related to Newtonian fluids. 3 Evaluation of rheological measurements The evaluation procedure is influenced by the radius to which the primary measured data (shear stress � and the New- tonian shear rate �� N ) are related. 3.1 Power law fluids a) Data are related to inner radius R1 Equation (4) at the inner cylinder surface takes the form � �1 1� K n � , (16) where �1 and ��1 are positive values of shear stress and shear rate at the inner cylinder surface. Using (9) we can obtain � � �� � � 1 2 2 1 � �n n . (17) However, the dependence of shear stress �1 on the Newto- nian shear rate ��1N at the surface of inner the cylinder is ob- tained from measurements. For this reason, equation (16) can be rewritten to the form � � � � � � � � �1 1 1 1 2 2 1 1 1 � � � �� � � � � � � � � � � � K K nN n N n n n N � � � � n N nK� 1 1�� (18) where ��1N was expressed from (17) for n � 1 �� � � 1 2 2 1 N � � (19) The dependences of the ratio � �� �1 1N on flow behavior index n for cylinder ratio values � � 0.5 and 0.9 are depicted in Fig. 6. This figure shows that the values of this ratio (ne- cessary for Newtonian shear rate correction) increase with decreasing flow behavior index n and are significantly greater at � � 0.5 than at � � 0.9. From (18) it can be seen that dependence �1 on ��1N is a straight line with slope n in logarithmic coordinates and the coefficient of consistency K can be calculated by the equation � � K K n n n � � � � � � � � � � 1 2 2 1 1 � � . (20) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 45 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 3/2006 1 2 3 4 5 6 7 8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n � � 0.5 � � 0.9 Fig. 6: Dependence of ratio � �� �1 1N on flow behavior index n for selected � values The dependence of ratio K/K1 on n for several values � is shown in Fig. 7. From this figure it can be seen that it exhibits the minimum that is dependent on � (for � � 0, minimum K/K1 = 0.692 at n = 1/e = 0.368). b) Data related to mean radius R R Rm � �( )1 2 2 Equation (4) written for the mean radius takes the form � �m m� K n � , (21) where �m and �� m are positive values of shear stress and shear rate at the mean radius. Using (9) we can obtain � � � � �� � � � � � � m m � � � � �� � � � � � � � � 2 1 2 1 2 12 1 2 2 2 n R R nn n n n . (22) However, the dependence of shear stress �m on the Newto- nian shear rate �� mN at the mean radius is obtained from measurements. For this reason, equation (21) can be rewritten to the form � � � � �m m m m m m� � � �� � �K K N n N n N n� � � � , (23) where �� mN can be obtained from (22) for n � 1 �� � � � � mN � � � � � � � 2 1 2 12 2 . (24) Dependencies of � �� �m mN ratio on flow behavior index n for cylinder ratio values � � 0.5 and 0.9 are depicted in Fig. 8. This figure shows that the values of this ratio are less than 1 46 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 3/2006 Czech Technical University in Prague 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 0 0.2 0.4 0.6 0.8 1 n K /K 1 � � 0 � � 0.5 � � 0.9 Fig. 7: Dependence of the ratio K/K1 on flow behavior index n for selected � values 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.3 0.5 0.7 0.9 n � � 0.5 � � 0.9 Fig. 8: Dependence of ratio � �� �m mN on flow behavior index n for selected � values and decrease with decreasing flow behavior index n and are significantly lower at � � 0.5 than at � � 0.9. From (23) it can be seen that dependence �m on �� mN is a straight line with slope n in logarithmic coordinates and coef- ficient of consistency K can be calculated by the equation K K N n � � � �� � m m m � � � � (25) the dependence of ratio K/Km on n for � � 0.5 and 0.9 is shown in Fig. 9. This figure shows that the ratio K/Km is greater than 1 and it increases with decreasing n. c) Data related to the mean radius presented by Klein [2] R R R R R K � � 1 2 1 2 2 2 2 Equation (4) written for this mean radius takes the form � �K K� K n � , (26) where �K and �� K are positive values of shear stress and shear rate at mean radius RK. Using (9) we can obtain � � � � �� � � � � � K K � � � � �� � � � �� � � � � 2 1 2 1 1 22 1 2 2 2 1 n R R nn n n n . (27) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 47 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 3/2006 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 0.1 0.3 0.5 0.7 0.9 n K /K m � � 0.5 � � 0.9 Fig. 9: Dependence of the ratio K/Km on flow behavior index n for selected � values 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.1 0.3 0.5 0.7 0.9 n � � 0.5 � � 0.9 Fig. 10: Dependence of ratio � �� �K KN on flow behavior index n for selected � values However, the dependence of shear stress �K on the New- tonian shear rate �� KN at mean radius RK is obtained from measurements. For this reason equation (26) can be rewritten to the form � � � � �K K K K K K� � � �� � �K K N n N n N n� � � � , (28) where �� KN can be obtained from (27) for n � 1 �� � � �KN � � � 1 1 2 2 . (29) The dependencies of � �� �K KN ratio on flow behavior in- dex n for cylinder ratio values � � 0.5 and 0.9 are depicted in Fig. 10. This figure shows that the values of this ratio are ap- proximately 1 for average values of n and rapidly decrease with decreasing flow behavior index n at low n values espe- cially at � � 0.5 . From (28) it can be seen that dependence �K on �� KN is a straight line with slope n in logarithmic coordinates, and coef- ficient of consistency K can be calculated by the equation K K N n � � � �� � K K K � � � � (30) the dependence of ratio K/KK on n for � � 0.5 and 0.9 is shown in Fig. 11. This figure shows that ratio K/KK is approximately 1 for usual values of n and rapidly increases with decreasing flow behavior index n at low n values, especially at � � 0.5. 48 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 3/2006 Czech Technical University in Prague 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n K /K k � � 0.5 � � 0.9 Fig. 11: Dependence of the ratio K/KK on flow behavior index n for selected � values � = 0.9 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 0.05 0.25 0.45 0.65 0.85 n K K/ m K K/ K K K1/ Fig. 12: Dependence of coefficient consistency ratios on flow index n for � � 0.9 The ratios of the real and apparent consistency coeffi- cients related to one of the radii mentioned above are com- pared in Figs. 12 and 13 for � � 0.9 and 0.5. Figs. 12 and 13 show that for � � 0.9 and for � � 0.5 and n � 0.36 no correc- tion of K is practically necessary (error is smaller than 3 %) when the data are related to the radius defined according to Klein. 3.2 Bingham plastics The dependences of the ratio of the shear rate to the New- tonian shear rate (related to the reference radii) on �* for � � 0.5 and 0.9 are shown in Figs. 14 and 15. These figures shows that the values of shear rate related to radii R1 and Rm are greater than the Newtonian values and the shear rate values related to radius RK are smaller than the Newtonian values. Inserting (13) into (2) we derive that the dependence of shear stress on Newtonian shear rate values (related to refer- ence radii) can be expressed in the form � � � �� �0ref p � N . (31) The ratio � �0 0ref depends on � and the reference radius to which the measured data are related and the following for- mulas can be derived: a) Data related to inner radius R1 � � � � 01 0 2 2 1 1 � � ln( ) (32) © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 49 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 3/2006 � = 0.5 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 0.05 0.25 0.45 0.65 0.85 n K K/ m K K/ K K K1/ Fig. 13: Dependence of coefficient consistency ratios on flow index n for � � 0.5 � 0.5� 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 0.1 0.3 0.5 0.7 0.9 1.1 �� 1 m K Fig. 14: Dependences of shear rate to Newtonian shear rate ratio (related to reference radii) on �* for � � 0.5 b) Data related to mean radius R R Rm � �( )1 2 2 � � � � � � 0 0 2 2 8 1 1 1 m 2 ) ) � � � ln( ( )( (33) c) Data related to the mean radius presented by Klein [2] R R R R R K � � 1 2 1 2 2 2 2 � � � � � 0 0 2 2 1 1 1K � � � ln . (34) The dependences of ratio � �0 0ref on � for different ref- erence radii are depicted in Fig.16. This figure shows that practically no correction is necessary for � � 0.74 when the data are related to the radius defined according to Klein (the error is smaller than 3 %). 4 Conclusion The following procedure can be recommended for deter- mining the rheological model parameters: 1) Comparing the values of � and �� N reported by the manu- facturer of the viscometer, we determine reference radius to which measured data are related (Eq.(19), (24), (29)). 2) From the measured shear stress and Newtonian shear rate values the values of Kref and n are obtained for power-law 50 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 3/2006 Czech Technical University in Prague 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 � m 1 K Fig. 16: Dependences of ratio � �0 0ref on � for different reference radii � 0.9� 0.8 1 1.2 1.4 1.6 1.8 2 0 10 20 30 40 50 60 70 80 1 m K �� Fig. 15: Dependences of shear rate to Newtonian shear rate ratio (related to reference radii) on �* for � � 0.9 fluids (Eq.(18), (23), (28)) or �0ref and �p are obtained for Bingham plastics (Eq.(31)). 3) Values of the rheological model parameters, i.e., consis- tency coefficient K (Eq.(20), (25), (30)) or yield stress �0 (Eq.(32), (33), (34)) must be determined. List of symbols K coefficient of consistency L length of cylinder n flow index r radial coordinate R1 inner rotating cylinder radius R2 outer stationary cylinder radius u velocity �� shear rate � tangential coordinate � R1/R2 ratio �p plastic viscosity � angular velocity � R1/R2 � shear stress �0 yield stress References [1] Middleman, S.: The Flow of High Polymers. New York: Interscience Publishers, 1968. [2] Klein, G.: Basic Principles of Rheology and the Applica- tion of Rheological Measurement Methods for Evaluat- ing Ceramic Suspensions. In: Ceramic Forum International Yearbook 2005 (Edited by H. Reh). Baden-Baden: Göller Verlag, 2004, p. 31–42. Prof. Ing. František Rieger, DrSc. phone: +420 224 352 548 email: frantisek.rieger@fs.cvut.cz Czech Technical University in Prague Faculty of Mechanical Engineering Technická 4 166 07 Praha 6, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 51 Czech Technical University in Prague Acta Polytechnica Vol. 46 No. 3/2006