Modernization of the mechanical mixing power numerical calculation method

The modernized numerical method for the mechanical mixing power calculation is proposed. Proposed calculation method allows to decrease the dependence of a stirred vessel mixing power numerical CFD calculation result on an empirical data by using additional term for considering own inertia of a mechanical mixing device. Proposed method bases on the mixing power consumption calculation technique as the function of a mixing device shaft torque. The verification of proposed method was carried out using the experimental measured and simulated values of the circulation-type reactor pump power. The results of proposed method verification show that non-modernized and modernized methods calculation errors are 3.9 % and 4.8 % respectively. Thus, calculation errors are comparable for modernized and non-modernized methods both. Wherein, modernized method unlike non-modernized does not use error correction coefficient that may be determined experimental only. Thus, proposed modernized method of the mechanical mixing power calculation may be used for engineering design calculations and simulations.


Introduction
Mechanical mixing using rotating devices is the most common way to organize mixing for the chemical productions [1 -4].Moreover, design of the mixing processes with rotating devices is an important task of the centrifugal pump's development and selection, especially for the non-standard pumps for the circulation in circulation-type chemical reactors [5].One of the most important characteristics of the mixing device regime is the mixing power consumption [1 -4, 6 -10].The classical mechanical power calculation technique bases on the hydrodynamic similarity theory using and often may be concluded to the use of the equation [1 -4]: ( There Nuseful is a useful mixing power, W; Eu is a centrifugal Euler number; ρ is a fluid density, kg/m 3 ; n is a mixing device rotational velocity, s -1 ; d is the mixing device characteristic size (often, impeller diameter), m.
It should be noted that in a general case, the useful mixing power is determined not only by the centrifugal Euler criterion, but also by the Froude criterion, however, in the absence of wave formation on the surface of the stirred liquid, in apparatuses of relatively small sizes and for values of the centrifugal Reynolds number greater than 300, such an effect can be neglected [11].
Dependencies of the Euler number from the Reynolds number traditionally may be defined experimentally.With the development of a computer technology and computational fluid dynamics (CFD), such dependencies are obtained by numerical simulation [9,10,12,13].A significant drawback of the classical technique for calculating power is the dependence on purely empirical data.The situation is aggravated by the fact that for the geometrically similar apparatuses equipped with the geometrically similar mixers of the same type, with the same values of the Reynolds number, different researchers present different values of the Euler number, especially for low Reynolds flows.For example, authors [8,14,15] show the following values of the Euler number for the six-blade turbine mixers for a Reynolds number of 100: 5 [8], 4 [14], 8 [15].
One of the inconsistencies of the different researchers' experimental data reasons is incomplete scalability of mechanical mixing processes [11].In addition, even when power consumption estimation takes place using the direct measuring of torque on the mixing device shaft measured power is not precisely useful power.The reason for this is the power consumption to overcome the own inertia of the mixing device.On laboratory-scale apparatus with a volume of up to tens of liters, on which experimental studies are often carried out, the contribution of power costs to overcome inertia to the total mixing power consumption can be relatively small.However, for industrial-scale vessels and apparatus, these costs cannot be neglected.
According to our practical experience, the classical calculation technique allows obtaining correct results for mixing in apparatuses with diameters in the range of 400-1500 mm.For the apparatuses with smaller or larger diameters the classical technique gives underestimated and overestimated values respectively.
In our previous works [5,16,17] we proposed the mixing power consumption method based on the CFD simulation results post-processing.By the proposed method mechanical mixing power consumption may be calculated by the equations: There ω is the rotor angular velocity, rad/s; i is the index of the rotor area unit; pdyn is the dynamic pressure, Pa; r is the radius vector, m; A is the rotor working surface area, m 2 ; x, y are the coordinates, m; N is the total power consumption, W; kδ is the empirical error coefficient; η is the drive efficiency; ηj is the drive element efficiency.
The error coefficient kδ use to be about 1.12 -1.15 according to the practical experience [5,16].This coefficient considers, among other things, the power costs to overcome the inertia of the mixing device and a calculation error of a numerical simulation.Although the proposed technique makes it possible to reduce the dependence of the calculation results on some empirical factors (such as vessel and mixing device geometry mutual influence and scale factor), the experimental error coefficient is a still factor depended on the experimental data and numerical calculation errors (finite element mesh density).
This work proposes modernization of our numerical mixing power consumption method.

Numerical simulation and modernized post-processing method
Density and velocity distributions for a mixing power calculation may be calculated by a CFD simulation using any accessible or convenient mathematical models.In our research we used steadystate isothermal forms of Navier-Stokes equations [18] and standard k-epsilon turbulence model [19].The numerical simulations were carried out using ANSYS Fluent CFD code with pressure-based solver program.The 10 5 polyhedral elements mesh was used.Polyhedral mesh was obtained by the original tetrahedral mesh converting.The interaction of the rotor and stator mesh zones was set using multiple reference frames and relative velocities conditions.Modernized mechanical mixing power calculation method bases on the mixing power consumption calculation technique as the function of the mixing device shaft torque: There T is a torque on the mixing device shaft (shaft useful torque), N•m.The shaft useful torque may be calculated numerically based on CFD simulation results as the sum of the specific torques of all mixing device frontal surface points: There Fhdr is a hydrodynamic resistance force, N; r is a radius, m; avg is the index of a mixing device geometrical average parameters; ti is a specific torque for the mixing device frontal surface point, N•m/m 2 (N/m); Ai is the area of the site with the center in the mixing device frontal surface point, m 2 .
For Cartesian coordinate system and a reference frame with the z-axis coinciding with the mixing device shaft axis equation (7) takes form: Thus, equation ( 5) takes form: Power consumption for the inertia forces overcoming may be calculated as: There Ninertia is a power consumption for the inertia forces overcoming, W; Tinertia is a inertia torque on the mixing device shaft (shaft useful torque), N•m; m is the mixing device mass, kg; g is the gravity force acceleration, m/s 2 .
The shaft useful torque may be calculated numerically based on CFD simulation results as the sum of the specific torques of all mixing device frontal surface points.For Cartesian coordinate system and a reference frame with the z-axis coinciding with the mixing device shaft axis equation (11) takes form: There mi is the mass of the site with the center in the mixing device frontal surface point, kg; si is the thickness of the site with the center in the mixing device frontal surface point, m; ρsm is the mixing device structural material density, kg/m 3 .
Thus, equation (10) takes form: Full mixing power consumption be calculated as sum of useful and inertia power consumptions considering mechanical and electrical power loses of the mixing device drive elements:

Experimental data
The verification of proposed mixing power calculation method was carried out using experimental data on the circulation-type reactor pump power consumption that described in our previous work [5] in detail.Figure 1 shows the circulation-type reactor circulation pump scheme.The circulation type reactor pump consists of three zones with different geometric and flow parameters.The first zone is located at the pump inlet; the rotor of this zone has the shape of a small pitch auger.The second rotor zone is located on the cylindrical part of the pump.The rotor of the second zone has an auger with a large pitch.The first and second zones form predominantly an axial flow of liquid and operate on the principle of a screw mixer.The third pumping zone is located at the pump outlet.The third zone rotor has the shape of a common centrifugal pump impeller.In the third pumping zone, predominantly radial and tangential flows are formed.
The experimental results [5] includes the dependences of the full power consumption (full mixing power) on the angular velocity of rotation in the range of 750, 1000, 1250, 1410 and 1500 rpm.Table 1 shows the measurement full power consumption values [5].Total estimated efficiency of the mixing device drive was taken as 0.82 for all calculations.The drive elements efficiencies values were taken from the handbooks data and vendor catalogs data.

Results and discussion
Table 2 shows measured and calculated values of full mixing power and full mixing power relative calculation error distribution on the rotor angular velocity for modernized and non-modernized calculation methods.
The non-modernized and modernized methods average calculation relative errors are 3.9 % and 4.8 % respectively.Maximal values of the calculation relative error are about 7 -13 % and take places for the rotor angular velocity of 750 -1000 rpm.The calculation relative error decreases with the rotor angular velocity rising and takes value about 0.5 -2.5 % for the rotor angular velocities higher than 1300 rpm.The calculation errors of the modernized and non-modernized method are comparable for the entire considered range of the pump rotor angular velocities.For the rotor angular velocities of 750 -1200 rpm the modernized method error higher than the error of the non-modernized method.For the rotor angular velocities of 1200 -1500 rpm the modernized method error lower than the error of the non-modernized method.The errors of both variants of the mixing power calculation method are in the acceptable range of values.Proposed modernized and non-modernized method of the mechanical mixing power calculation may be used for engineering design calculations and simulations both.Wherein, modernized method unlike non-modernized does not use error empirical correction coefficient.
Thus, the use of empirical data with the modernized method of the mechanical mixing power calculation is limited only by using of reference data on mixing device drive parts mechanical and electrical efficiencies.This circumstance significantly expands the possibilities of applying the proposed modernized method of the mechanical mixing power calculation for engineering design calculations and simulations especially in cases of non-standard mechanical mixers, high-speed rotary machines, and large volume vessels.

Conclusion
The modernized numerical method of the mechanical mixing power calculation is proposed.The method bases on the mixing power consumption calculation technique as the function of the mixing device shaft torque which we suggested earlier in our previous works.Proposed calculation method allows to decrease the dependence of a mixing power numerical calculation result on an empirical data due to consideration of the own inertia of a mechanical mixing device.The use of empirical data with the modernized method is limited only to the use of reference data on mixing device drive parts mechanical and electrical efficiencies.
The errors of the considered modernized and non-modernized methods are quite acceptable for the engineering design calculations and simulations.
The verification of proposed method was carried out using experimental data on the circulation-type reactor pump power consumption.The non-modernized and modernized methods average calculation relative errors are 3.9 % and 4.8 % respectively.The calculation relative error decreases with the rotor angular velocity rising for the modernized and non-modernized methods both.Wherein, modernized method unlike non-modernized does not use error empirical correction coefficient.

Table 1 .
Measured circulation-type reactor pump full power consumption

Table 2 .
Measured and calculated full mixing power