Power Characteristics of a Screw Agitator in a Tube

Screw agitators rotating in tubes are very efficient tools for mixing and pumping viscous liquids. They are also suitable for cases where the viscosity of the liquid changes during operation. The power characteristic of the agitator in the tube must be known to enable its power consumption in a given configuration to be calculated. An estimate of power consumption P from the power characteristic is schematically shown in Fig. 1, where the specific energy e, for which the power is determined, is given by the intersection of the pumping characteristic of the screw and the hydraulic characteristic of the system (dependencies of flow rate Q on specific energy).


Introduction
Screw agitators rotating in tubes are very efficient tools for mixing and pumping viscous liquids. They are also suitable for cases where the viscosity of the liquid changes during operation. The power characteristic of the agitator in the tube must be known to enable its power consumption in a given configuration to be calculated. An estimate of power consumption P from the power characteristic is schematically shown in Fig. 1, where the specific energy e, for which the power is determined, is given by the intersection of the pumping characteristic of the screw and the hydraulic characteristic of the system (dependencies of flow rate Q on specific energy).
The information available in the literature on the power consumption of screws concerns mainly the creeping flow regime (screws in extruders). However, screw agitators are often used in transition and eventually turbulent regimes. A method for calculating the power characteristics of screw agitators in a creeping flow regime was proposed in [1]. The power characteristics at higher Reynolds number values must be determined experimentally. An experimental method based on measurements in several configurations was reported previously [2]. However, it should be noted that it is difficult to keep the same geometrical parameters (especially clearance between tube and screw) in different configurations. Therefore, a new method based on measurement in a single configuration was used [3], [4].

Theoretical
The power characteristic is the dependence of power consumption P on specific energy e. Applying inspection analysis of the governing equations, the following relationship for the dimensionless power characteristic was proposed in [2] ( ) dimensionless specific energy e e N * = n (3) and Reynolds number Re = ND 2 n (4) In the creeping flow regime Eq.(1) reduces to ( ) P P e * * * = (5) and the dimensionless power characteristic is linear P c ae * * = + (6) At higher Reynolds number values the power characteristics are generally non-linear. However, the dependence of dimensionless power on dimensionless specific energy is not very strong and decreases with increasing Reynolds number, as it is shown Fig. 2, based on the experimental results

F. Rieger
Screw agitators rotating in tubes are very efficient tools for mixing and pumping viscous liquids. The power characteristic of the agitator-tube assembly must be known to enable its power consumption in a given configuration to be calculated. The dimensionless power characteristic is described by Eq. (6). An estimate of power consumption from the power characteristic is schematically shown in Fig. 1 [3]. For this reason we may approximate the dependence of dimensionless power on dimensionless specific energy at given Reynolds number values also by a straight line (6), with coefficients c and a dependent on the Reynolds number. Two pairs of P* and e* values are sufficient for the design of these straight lines. These two points can be represented by the asymptotic power characteristics at minimum and maximum specific energy presented in [4]. Inserting in Eq.(6) we can write P c ae The value of the dimensionless minimum specific energy e min * was calculated (in the creeping flow region -see [5], in the turbulent region -see [6]), the value of the dimensionless maximum specific energy e max * was determined experimentally (see [7] or [3]).
Substracting equations above we obtain the following equation for Coefficient c can be expressed from (8) c P ae = max * max * (10)

Results and discussion
The procedure proposed above was applied for a screw agitator with the following parameters: pitch s = 2D, root diameter d = 0.2 D and length L = 1.4 D -see Fig. 3.
The asymptotic power characteristics for this agitator obtained experimentally and reported in [4] are presented in Fig. 4.
The characteristic at minimum specific energy was measured in a relatively large vessel with low hydraulic resistance. The characteristic at maximum specific energy (zero pumping capacity) was attained covering the draught tube. Using the values of dimensionless power P Po * Re = obtained from Fig. 4, the values of coefficients a and c were calculated from Eqs. (9)  where c t = c/Re. Equation (11) is recommended for standard pumps used for pumping low viscosity liquids. Fig. 5 also shows that the value of a in a creeping flow regime is greater than the corresponding value in a turbulent region, which means that the dependence of dimensionless power on specific energy (hydraulic resistance of the system) is less pronounced in a turbulent region. Using the coefficients Acta Polytechnica Vol. 41 No. 6/2001 Power characteristics in the form of Eq. (11) for Re = 1000 and a turbulent region are shown in Fig. 7.
Using the pumping characteristics presented in [8] the power characteristics can be expressed in an alternative form, more frequently used in literature on pumps (se e.g. [9]), with dimensionless flow rate Q* on the axis of the abscissas shown in Figs. 8 and 9.