AP08_5.vp 1 Introduction This paper presents viscous heating in a rotational visco- meter with coaxial cylinders (see Fig. 1), which is often used for measuring rheological behaviour. Power-law and Bingham models are often used to describe rheological behaviour [1]. The power-law model is the simplest model widely used for describing the rheological behaviour of non-Newtonian fluids. Using this model, the dependence of shear stress � on shear rate can be expressed by the relation: � �� K n� , (1) where K is the coefficient of consistency and n stands for the flow behaviour index. The Bingham model is the simplest model used for de- scribing the rheological behaviour of viscoplastic materials. Using this model, the relation of shear stress � and shear rate �� can be expressed by the following relation: � � � �� �p� 0 for � �� 0, (2) where �0 is the yield stress and �p stands for plastic viscosity. If the inner to outer cylinder diameter ratio does not dif- fer significantly from 1, the curvature can be neglected and the flow reduces to the flow between the moving and station- ary plates. The temperature distribution can be obtained by solving the Fourier-Kirchhoff equation [1] � �� d d 2 2 T y � � � , (3) where y is the distance from the stationary plate. The equa- tion will be solved with the following boundary conditions y H T y y T y T T � � � � � , , ( ) d d d d f 0 0 � � (4) i.e., we assume an insulated moving plate (rotating cylinder) and a stationary plate (cylinder) tempered to temperature Tf. 2 Solution 2.1 Power-law fluids Inserting (1) for � into (3), we get � � d d 2 2 1T y K n� � �� (5) and after integration we obtain T T K H y H y H Bi n � � � � � � � � � � � � � � � f 2 1 21 2 1�� � , (6) where Bi H� � �. The solution is shown in graphical form in Fig. 2, where T T T K H y y Hn * ( ) � , *� � � � f � �2 1 . (7) 2.2 Bingham plastics Inserting (2) for � into (3), we get � � � � � �� d d 2 2 2 0 1T y p� � �( � � (8) after integration and rearrangement we obtain © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 37 Acta Polytechnica Vol. 48 No. 5/2008 Dissipative Heating in a Rotational Viscometer with Coaxial Cylinders F. Rieger Rotational viscometers with coaxial cylinders are often used for measuring rheological behaviour. If the inner to outer cylinder diameter ratio does not differ significantly from 1, the curvature can be neglected and the flow reduces to the flow between moving and stationary plates. The power-law and Bingham models are often used for describing rheological behaviour. This paper deals with the temperature distribution obtained by solving the Fourier-Kirchhoff equation and in the case of negligible inner heat resistance it also covers temperature time dependence. The solution is illustrated by a numerical example. Keywords: viscous heating, viscometer with coaxial cylinders, power-law fluids, Bingham plastics. Fig. 1: Rotational viscometer with coaxial cylinders T T H y H y H Bi � � � � � � � � � � � � � � �f p� � � � 2 2 0 2 1 1 2 1� ( )* , (9) where � � � � 0 0* � � p . (10) The solution of (9) for Bi � is shown in graphical form in Fig. 3, where T T T H * ( ) � � � f p � � �2 2 . (11) Fig. 3 shows that increasing plasticity (�0 * ) increases the temperature rise. 38 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 5/2008 0.01 0.1 1 10 100 0 0.2 0.4 0.6 0.8 1 y* T * Bi=1 Bi=10 Bi=100 Fig. 2: Dimensionless temperature profiles 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 y* T * ����� ������� ����� Fig. 3: Dimensionless temperature profiles From the dependencies shown in Fig. 2 it can be seen that at small Biot number values the outer temperature resistance prevails and the temperature of the liquid is practically con- stant, and in a unsteady-state it depends on time. For this case the enthalpy balance can be written in the form � �� �cH T t H T T d d f � � �� ( ) (12) and after integration it transforms to T T T T T K K t* * * ( *) exp( *)� � � � � � �f f0 1 , (13) where K t H T T t t cH * � ( ) , *� � � � � � �0 f . (14) The dependence is shown in Fig. 4, where the line for K * � 1designates the equilibrium state at which all dissipative heat is removed by convection. The dependence of the dimensionless time t* after which the dimensionless temperature attains 99 % of the steady- -state value on K* is shown in Fig. 5. In the case when initial temperature T0 is equal to temper- ature Tf it is suitable to define the dimensionless temperature as T T T t H � � �( ) � f � � (15) after integration (11) we can obtain the relation T t� � � �1 exp( *) . (16) The application of the above relations will be illustrated in the following example. 3 Example A rotational viscometer with inner cylinder diameter 48 mm and outer cylinder diameter 50 mm contains a New- tonian liquid with density � � 1000 kgm�3, heat capacity c � 4200 Jkg�1K�1 and heat conductivity � �0.5 Wm�1K�1. Calculate the inner and outer cylinder temperature 1) when the tempering temperature is 20 °C and the heat transfer coefficient � � 100 Wm�2K�1 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 39 Acta Polytechnica Vol. 48 No. 5/2008 0 0.5 1 1.5 2 2.5 3 3.5 0 5 10 15 t* T * * K����� K����� K��� K��� K��� Fig. 4: Dependence of dimensionless temperature on dimensionless time 0 1 2 3 4 5 6 7 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 K* t * Fig. 5: Dependence of the dimensionless time t* after which the dimensionless temperature attains 99 % of the steady-state value on K* 2) at no tempering, outer temperature 20 °C and � � 5 Wm�2K�1. The temperature dependence of viscos- ity is described by the relation � [ ] . exp( . [ ])Pa s C� � � �0 82 0 025 T . Measurement is carried out at shear rate a) 100 s�1, b) 1000 s�1. Calculate also the temperature after 5 minutes, when the ini- tial temperature is �) 20 °C, �) 15 °C. 4 Solution 4.1 Calculations of final temperatures First the Biot number will be calculated 1) Bi H � � � � � � 100 0 001 0 5 0 2 . . . . Inserting n � 1 and K � � into Eq. (6) or inserting �0 0 * � and � �p � into Eq. (9), the inner cylinder temperature Ti will be calculated for y H� from a) T H Bi Ti f� � � � � � � � � � � � � � � � 2 2 6 4 1 2 1 0 5 10 10 0 5 5 5 20 20 � . . . .055 �C and the outer cylinder temperature Te will be calcu- lated for y � 0 from T H Bi Te � � � � � � � � � � � � � � � � 2 2 6 4 0 1 0 5 10 10 0 5 5 20 20 05 � . . . f �C. b) T H Bi Ti f� � � � � � � � � � � � � � � � 2 2 6 6 1 2 1 0 44 10 10 0 5 5 5 20 2 � . . . 4 9. �C, T H Bi Te � � � � � � � � � � � � 2 2 6 6 1 0 44 10 10 0 5 5 20 24 4 � . . . f C. The viscosities at the mean liquid temperature were in- serted into the above equations. 2) Bi H � � � � � � 5 0 001 0 5 0 01 . . . and the two temperatures will be calculated from the same equations a) T H Bi Ti f� � � � � � � � � � � � � � � 2 2 6 4 1 2 1 0 485 10 10 0 5 100 5 2 � . . . 0 21� �C, T H Bi Te � � � � � � � � � � � � � � � � 2 2 6 4 0 1 0 485 10 10 0 5 100 20 2 � . . f 1 �C. b) T H Bi Ti f� � � � � � � � � � � � � � � 2 2 6 6 1 2 1 019 10 10 0 5 100 5 20 � . . . � �58 3. C, T H Bi Te � � � � � � � � � � � � 2 2 6 6 1 019 10 10 0 5 100 20 58 2 � . . . f C. These results show that at a high shear rate the tempera- ture rise is unacceptable especially without tempering. It can also be seen that the difference between the inner and outer cylinder temperature is not high due to the low Biot number values, especially in case 2). 4.2 Calculations of temperatures after 5 minutes of measurement �) In the case when the initial temperature T0 is equal to temperature Tf, Eq.(15) will be usedin the calculations 1a) T H t T� � � � � � � � � � � � � [ exp( *)] . . exp 2 2 1 0 497 0 001 100 100 1 1 f 00 300 1000 4200 0 001 20 20 05 � � � � �� � �� � � � . . C, 1b) T H t T� � � � � � � � � � � � � [ exp( *)] . . exp 2 2 1 0 47 0 001 1000 100 1 1 f 00 300 1000 4200 0 001 20 24 7 � � � � �� � �� � � � . . C, 2a) T H t T� � � � � � � � � � � � � � [ exp( *)] . . exp 2 2 1 0 5 0 001 100 5 1 5 300 f 1000 4200 0 001 20 20 3 � � � �� � �� � � � . . C, 2b) T H t T� � � � � � � � � � � � � � [ exp( *)] . . exp 2 2 1 0 375 0 001 1000 5 1 5 f 300 1000 4200 0 001 20 42 5 � � � �� � �� � � � . . C, where viscosity was calculated at the mean temperature ( )T T� f 2. 40 © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ Acta Polytechnica Vol. 48 No. 5/2008 ) In cases when the initial temperature T0 is not equal to temperature Tf, Eq.(12) will be used in the calculations 1a) T T T T K K t� � � � � � � � � f f( )[ * ( *) exp( *)] [( . ) exp 0 1 20 5 1 0 0106 ( *) . ] .� � � �t 0 0106 20 05 C, 1b) T T T T K K t� � � � � � � � � f f( )[ * ( *) exp( *)] [( . ) exp( 0 1 20 5 1 0 995 � � � �t*) . ]0 995 25 C, 2a) T T T T K K t� � � � � � � � � � f f( )[ * ( *) exp( *)] [( . ) exp( 0 1 20 5 1 0 22 t*) . ] .� � �0 22 16 8 C, 2b) T T T T K K t� � � � � � � � � � f f( )[ * ( *) exp( *)] [( . ) exp( 0 1 20 5 1 16 3 t*) . ]� � �16 3 41 C, where viscosity was calculated at the mean temperature ( )T T0 2� . The results presented above show that the temperature rise and the experimental error is considerable especially at high shear rate in cases 1b) and 2b). 5 Conclusion It was shown that dissipative heating can play an impor- tant role in measurement of highly viscous fluids. The tem- perature of measured liquid can be significantly higher than tempering temperature, which can cause significant experi- mental error. The time necessary for temperature stabilisa- tion is often not negligible. Measurement without tempering can lead to a significant temperature rise and unacceptable error of measurement. List of symbols Bi Biot number, 1 c specific heat capacity, J�kg�1�K�1 H distance of planes (cylinder walls), m K consistency coefficient, Pa�sn n flow index, 1 R1 inner cylinder radius, m R2 outer cylinder radius, m t time, s T temperature, K y coordinate, m � heat transfer coefficient, W�m�2�K�1 �� shear rate, s�1 ratio R1/R2, 1 � heat conductivity, W�m�1�K�1 � viscosity, Pa�s �p plastic viscosity, Pa�s � density, kg�m�3 � shear stress, Pa �0 yield stress, Pa lower indexes f fluid 0 initial upper indices *+ dimensionless Acknowledgment This work was supported by research project of the Minis- try of Education of the Czech Republic MSM6840770035. Reference [1] Bird, R. B., et al.: Transport Phenomena, J. Wiley, N. York, 1960. Prof. Ing. František Rieger, DrSc. Phone: +420 224 352 548 e-mail: frantisek.rieger@fs.cvut.cz Department of Process Engineering Czech Technical University in Prague Faculty of Mechanical Engineering Technická 4 166 07 Prague 6, Czech Republic © Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 41 Acta Polytechnica Vol. 48 No. 5/2008