AP06_6.vp 1 Introduction In a recent paper [1], the procedure for determining the rheological parameters from measurements on a viscometer with coaxial cylinders (see Fig. 1) was proposed on the basis of flow analysis. The relations for calculating of consistency coefficient K for power-law fluids and yield stress �0 for Bingham plastics were reported. These relations were derived for three reference radii – inner, mean and radius presented by Klein [2]. However, it is possible to find radius Rr at which Newto- nian and non-Newtonian shear rates are the same. If the experimental data are related to this radius, K and �0 can be obtained directly as the �-intercept of the measured data �� �� f ( � ) )straight line. 2 Solution A) Power-law fluids The following equation was derived for shear rate (eq.(9) in [1]) � ( ) � � � � � � � � � � � 2 1 2 1 2 n R rn n . (1) Inserting n � 1, the following equation can be obtained for the shear rate in Newtonian fluids � ( ) � � � � � � � � � � � 2 1 2 1 2 n R r . (2) The two values are the same at r = Rr and using eqs.(1) and (2) we get R R n r n n n 1 2 2 2 2 1 1 � � � � � � � � � � ( ) ( ) � � . (3) From this equation it can be seen that the R1/Rr ratio depends on n and �. The dependence of R1/Rr on n for se- lected values of ratio � is shown in Fig. 2. Comparison of R1/Rr with the ratio of R1 to the mean radius presented by Klein [2] R R R R R K � � 1 2 1 2 2 2 2 (4) for � � 0.8 is shown in Fig. 3. This figure shows that RK repre- sents the mean value of Rr in the presented n interval, and for this reason ratio K/KK is relatively small, as was shown in [1] (see Figs.12 and 13 in [1]). B) Bingham plastics Combining Eqs.(3), (11) and (13) presented in [1], the fol- lowing equation for shear rate can be obtained � ( ) ln� � � � � � � � � � � � � � � � � � � � � � � � � 0 0 2 2 1 22 1 2 1p p R r . (5) Again we can find radius Rr at which Newtonian and Bingham shear rates are the same by comparing equation (5) with the corresponding relation for a Newtonian fluid (2), and we get R Rr 1 21 2 1 � � � �ln( ) . (6) From this equation it can be seen that ratio R1/Rr depends on �. The graphical form of this dependence is shown in Fig. 4. 22 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 6/2006 Determination of Rheological Parameters from Measurements on a Viscometer with Coaxial Cylinders – Choice of the Reference Radius F. Rieger Knowledge about rheological behavior is necessary in engineering calculations for equipment used for processing concentrated suspensions and polymers. Power-law and Bingham models are often used for evaluating the experimental data. This paper proposes the reference radius to which experimental results obtained by measurements on a rotational viscometer with coaxial cylinders should be related. Keywords: viscometer with coaxial cylinders, power-law fluids, Bingham plastics Fig. 1: Viscometer with coaxial cylinders © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 23 Acta Polytechnica Vol. 46 No. 6/2006 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n R 1 /R r � � 0 . 8 � � 0 . 5 � � 0 . 2 Fig. 2: Dependence of R1/Rr on n for selected values of ratio � 0 9. 0 902. 0 904. 0 906. 0 908. 0 91. 0 912. 0 914. 0 916. 0.918 0 2. 0 3. 0 4. 0 5. 0 6. 0 7. 0 8. 0 9. 1 n R R1 K/ R R1/ r Fig. 3: Dependence of R1/RK resp. R1/Rr on n 0 0 2. 0 4. 0 6. 0 8. 1 1.2 0 1. 0 2. 0 3. 0 4. 0 5. 0 6. 0 7. 0 8. 0 9. 1 � R 1 /R r Fig. 4: Dependence of R1/Rr on � 3 Conclusion On the basis of the above paragraph, the following procedure for evaluating the experimental data can be recommended: 1) If the logarithmic plot of shear stress �1 on Newtonian shear rate ��1N is linear (the slope is equal to flow index n) the power-law model can be used and we can calculate R1/Rr from eq.(3). If the plot of shear stress �1 on the Newtonian shear rate ��1N is linear, the Bingham model can be used and we can calculate R1/Rr from eq.(6). 2) The shear stresses and shear rates related to radius Rr can be calculated from experimental values �1 and ��1N using the following relations � �r r R R � � � � � � � 1 1 2 , (7) � � �� � �r r r R RN N � � � � � � � � 1 1 2 (8) 3) If the logarithmic plot of shear stress �r on shear rate �� r is linear, the consistency coefficient K is �-intercept and flow index n is the slope of a straight line. If plot of shear stress �r on shear rate �� r is linear, the yield stress �0 is �-intercept and plastic viscosity �p is the slope of a straight line. 4 Nomenclature K coefficient of consistency L length of cylinder n flow index r radial coordinate R1 inner rotating cylinder radius R2 outer stationary cylinder radius RK radius presented by Klein Rr radius at which Newtonian and non-Newtonian shear rates are the same �� shear rate � R1/R2 ratio �p plastic viscosity � angular velocity � shear stress �0 yield stress subscripts 1 at radius R1 r at radius Rr N Newtonian Reference [1] Rieger, F.: Determination of Rheological Parameters from Measurements on a Viscometer with Coaxial Cylin- ders. Acta Polytechnica, Vol. 46 (2006), p. 42–51. [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 e-mail: frantisek.rieger@fs.cvut.cz. Czech Technical University in Prague Faculty of Mechanical Engineering Technická 4 166 07 Praha 6, Czech Republic 24 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 46 No. 6/2006