Dissipative Heating in a Rotational Viscometer with Coaxial Cylinders F

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 describing 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 differ significantly from 1, the curvature can be neglected and the flow reduces to the flow between the moving and stationary plates. The temperature distribution can be obtained by solving the Fourier-Kirchhoff equation [1] d d 2


Introduction
This paper presents viscous heating in a rotational viscometer 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 t on shear rate can be expressed by the relation: where K is the coefficient of consistency and n stands for the flow behaviour index.
The Bingham model is the simplest model used for describing the rheological behaviour of viscoplastic materials.Using this model, the relation of shear stress t and shear rate & g can be expressed by the following relation: where t 0 is the yield stress and m p stands for plastic viscosity.
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 the moving and stationary plates.
The temperature distribution can be obtained by solving the Fourier-Kirchhoff equation [1] where y is the distance from the stationary plate.The equation will be solved with the following boundary conditions i.e., we assume an insulated moving plate (rotating cylinder) and a stationary plate (cylinder) tempered to temperature T f .

Power-law fluids
Inserting (1) for t into (3), we get and after integration we obtain where Bi H = a l.The solution is shown in graphical form in Fig. 2, where .

Bingham plastics
Inserting (2) for t into (3), we get after integration and rearrangement we obtain where The solution of (9) for Bi ® ¥ is shown in graphical form in Fig. 3, where (11) From the dependencies shown in Fig. 2 it can be seen that at small number values the outer temperature resistance prevails and the temperature of the liquid is practically constant, and in a unsteady-state it depends on time.For this case the enthalpy balance can be written in the form and after integration it transforms to where 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 T 0 is equal to temperature T f it is suitable to define the dimensionless temperature as after integration (11) we can obtain the relation The application of the above relations will be illustrated in the following example.

Example
A rotational viscometer with inner cylinder diameter 48 mm and outer cylinder diameter 50 mm contains a Newtonian liquid with density r = 1000 kgm -3 , heat capacity c = 4200 Jkg -1 K -1 and heat conductivity l =0.5 Wm -1 K -1 .Calculate the inner and outer cylinder temperature 1) when the tempering temperature is 20 °C and the heat transfer coefficient a = 100 Wm -2 K -1

Calculations of final temperatures
First the Biot number will be calculated Inserting n =1 and K = m into Eq.( 6) or inserting t 0 0 * = and m m p = into Eq.( 9), the inner cylinder temperature T i will be calculated for y H = from a) The viscosities at the mean liquid temperature were inserted into the above equations.These results show that at a high shear rate the temperature 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).

Calculations of temperatures after 5 minutes of measurement a)
In the case when the initial temperature T 0 is equal to temperature T f , Eq.( 15) will be usedin the calculations 1a) b) In cases when the initial temperature T 0 is not equal to temperature T f , Eq.( 12) will be used in the calculations 1a) 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).

Conclusion
It was shown that dissipative heating can play an important role in measurement of highly viscous fluids.The temperature of measured liquid can be significantly higher than tempering temperature, which can cause significant experimental error.The time necessary for temperature stabilisation is often not negligible.Measurement without tempering can lead to a significant temperature rise and unacceptable error of measurement.

Bi
Biot number, 1 c specific heat capacity, J×kg

Fig. 5 :
Fig. 4: Dependence of dimensionless temperature on dimensionless time where viscosity was calculated at the mean temperature ( ) T T + f 2.