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êMariya A. Vasilyeva, Stefan Voth
Multiphysical Model of Heterogeneous Flow Moving.
Electromechanics and Mechanical Engineering
UDC 004.942
MULTIPHYSICAL MODEL OF HETEROGENOUS FLOW MOVING ALONG А CHANNEL OF VARIABLE CROSS-SECTION
Mariya A. VASILYEVA1, Stefan VOTH2
1 Saint-Petersburg Mining University, Saint-Petersburg, Russia
2 Technische Hochschule Georg Agricola, Bochum, Germany
The article deals with the problem aimed at solving the fundamental problems of developing effective methods and tools for designing, controlling and managing the stream of fluid flowing in variable-section pipelines intended for the production of pumping equipment, medical devices and used in such areas of industry as mining, chemical, food production, etc. Execution of simulation modelling of flow motion according to the scheme of twisted paddle static mixer allows to estimate the efficiency of mixing by calculating the trajectory and velocities of the suspended particles going through the mixer, and also to estimate the pressure drop on the hydraulic flow resistance. The model examines the mixing of solids dissolved in a liquid at room temperature. To visualize the process of distributing the mixture particles over the cross-section and analyzing the mixing efficiency, the Poincareplot module of the COMSOL Multiphysics software environment was used.
For the first time, a multi-physical stream of heterogeneous flow model has been developed that describes in detail the physical state of the fluid at all points of the considered section at the initial time, takes into account the design parameters of the channel (orientation, dimensions, material, etc.), specifies the laws of variation of the parameters at the boundaries of the calculated section in conditions of the wave change in the internal section of the working chamber-channel of the inductive peristaltic pumping unit under the influence of the energy of the magnetic field.
Key words: heterogeneous flow, mathematical modelling, elastomer, mixer, peristaltic transportation
How to cite this article: Vasilyeva M.A., Voth S. Multiphysical Model of Heterogeneous Flow Moving Along a Channel of Variable Cross-section. Zapiski Gornogo instituta. 2017. Vol. 227. P. 558-562. DOI: 10.25515/PMI.2017.5.558
Introduction. Many problems of hydromechanics, hydraulics and the organization of technological processes due to the moderate velocities of medium displacement can be studied within the framework of model of an incompressible viscous fluid [1]. At the present time, there are various types of problems which complete investigation can be carried out only by means of a computational experiment or an accurately set up physical experiment [2, 3, 5, 10, 11, 14, 15]. However, phenomena and technological processes having a practical interest are often not available for comprehensive physical modeling and the costs of such experiments are excessively large.
The issues having practical interest, as a rule, are characterized by multidimensionality, nonsta-tionarity, nonlinearity, the presence of free boundaries and boundary layers, and are described by the Navier - Stokes equations [6-9, 12, 13]. The nonlinearity of the Navier - Stokes equations and the presence of a small parameter for the highest derivatives (especially for large Reynolds numbers) create serious difficulties both in their analytical investigation and in the numerical solution of these equations by means of a computer.
Formulation of the problem. The inductive peristaltic pumping unit is designed for movement of fluid with the help of a wave of local deformation of the working chamber-channel of incompressible materials through a pipe of variable geometry.
The working chamber-channel of inductive peristaltic pumping unit plays a role of a static mixer. The transported material is pumped through a pipe with stationary screws. This mixing method is especially efficient for mixing with laminar flow since it creates only small losses of flow pressure (Fig. 1).
The peristaltic pipeline can be viewed as a deformable body along which the forced waves of contraction and elongation deformation move. There is no transfer of pipeline particles during this process. Part of the sections of the pipeline is compressed, reducing the cross-sectional area, and some of the subsequent sections are expanded. This process sequentially goes along the pipeline,
êMariya A. Vasilyeva, Stefan Voth
Multiphysical Model of Heterogeneous Flow Moving.
0.05
-0.05
x105
I«
5 4 3 2 1
Fig. 1. Internal surface relief pattern of working chamber-channel
20
X
y z
30
20
40
60
100
Fig.2. Distribution of velocity vectors when passing of a section of a working chamber-channel of a mixer
2
0
forming a moving wave. The compression of the pipe allows to pump the mixture using a laminar flow with a relatively low shear stress. The relief of the internal surface of the pipeline ensures the structuring of the flow, as well as its mixing simultaneously with transportation, which prevents the delamination of non-homogeneous media [16, 17].
Methodology. When the media streams around the body along with head resistance there may appear the components of viscous forces, which are orthogonal to the velocity vector of the body relative to the moving media (the lifting force according to Zhukovsky).
Viscosity is a macroscopic manifestation of molecular motion and mixing. If macroscopic pulsations are formed in the fluid, this leads to mixing on a macroscopic scale and to changing the viscous forces, in particular, the dragging forces with which the flow acts on the streamlined body and the walls of the tubes. Therefore, the equation given for the pressure field p(x,t) and velocity field w(x,t)t
+ («(t, t), V)u
Vt
= -Vp-^Vu(x, t), (1)
is valid only if the velocity does not exceed the critical value determined by the Reynolds number, i.e. the current lines of real currents do not become «tortuous». The Prandtl formula for turbulent viscosity:
% =l:
Vu
Vy
(2)
where l - mixing length determined by experiments; u - average value of flow velocity.
Simulation modeling of flow motion according to the scheme of a twisted paddle static mixer allows estimating the mixing efficiency by calculating the trajectory and velocities of the suspended particles through the mixer, and also estimating pressure losses due to the hydraulic flow resistance (Fig.2).
The model examines the mixing of solids dissolved in a liquid at room temperature. The geometry of the channel consists of a tube with three twisted screws of variable rotation.
The flow at the inlet is laminar and fully developed at an average speed of 1 cm/s. At the end of the process, the model defines a constant head pressure of 0 Pa. Laminar flow is described by the following equation
p (uV)u = V- pI + + (Vu )r )], (3)
Vu = 0.
The particle trajectories are calculated from the Newtonian model using the Stokes drag force law [4]:
A MariyaA. Vasilyeva, Stefan Vöth DOI: 10.25515/PMI.2017.5.558
Multiphysical Model of Heterogeneous Flow Moving...
(mpv)= — mp (u - v), (4)
d
— \m„ dr p
where v - particle velocity; tp - particle velocity response time,
1 p =-
2
P"P .
P pd
18^
(5)
;
Fig.3. Trajectory maps of particles in different Poincare sections
0.05 -0.05
0.05
;
Fig.4. The path of particle trajectories inside the section of the mixing chamber-channel
pp - density; dp - particle diameter.
The particle density is normalized in accordance with the velocity of the liquid at the input. This means that the particles are larger at the entrance to the mixer, where the speed is the highest, and smaller, where the speed is small.
Discussion. Since the particles have mass, they do not necessarily all reach the outlet; some particles get stuck in the wall of the working chamber-channel of the mixer. The probability of transmission is defined as the ratio of the number of particles that reach the output to the total number
of particles. For this particular configuration, the transmission probability is about 0.80. This means that about 20 % of the particles remain trapped in a mixer.
Modeling the stream of heterogeneous flow allows visualizing the distribution of the solid phase and evaluating the efficiency of the mixer.
To visualize the process of distribution of the particles in the mixture over the cross-section and to analyze the efficiency of mixing, the Poincare plot module of the COMSOL Multiphysics software environment was used (Fig.3). Poincare plot sets a color point for each of the particles at the place where the particle passes through the cross-section plane (the Poincare section).
The color indicates the location of the particle in its original position. Thus, the particles marked in red had an initial position x < 0. The particles marked in blue had an initial position x > 0. The operator at (x) is used to designate the particles in their initial position. The first Poincare section (rightmost in Fig.3) shows where the particles with coordinates x <0 are placed. As the particles start to follow the stream, they begin to mix. By the end of the working chamber-channel, the particles are not completely mixed, there are still significant focal areas where only red and only blue particles are concentrated.
Based on the known properties of the chamber-channel material and the parameters of the transported medium, the model makes it possible to evaluate the expanding forces on the pipe walls, as well as the force necessary to move the portion of the substance (Fig.4).
The sum of all the forces acting on the impurity in the chamber-channel is
F = {ptS -Pig -5B)>
(6)
where pT - density of mechanical impurities; pi - liquid density; g - free fall acceleration; 5 - current density; B - magnetic field induction.
p
2
0
êMariya A. Vasilyeva, Stefan Voth
Multiphysical Model of Heterogeneous Flow Moving.
Let us imagine that in some coordinate system the electromagnetic field is characterized by the vectors of the electric field strength and the magnetic induction B of the magnetic field. In relation to this coordinate system, a particle with charge q moves in the gap of the magnetic device with a velocity v. From the side of the electric and magnetic fields, the following force acts on this particle
f = qE + q[vB ], (7)
where qE - a force of electric field; q[vB] - a force of magnetic field.
The particle will be acted upon by electric fe and magnetic fm forces, which coincide in direction. In addition to the forces fe and fm, the flow force fp of the fluid system moving along the direction of velocity v will also act on the particle, as well as the forcefem from the interaction of Bp and the current flowing in the aqueous medium. Thus, the resultant force will be
f = fe + fm + fp + fem. (8)
Under the action of these forces, the particle in the gap of the working chamber-channel will move along a spiral around the magnetic apparatus. For particles having the same charge sign (+), the direction of the spiral-like rotation will be the same, for particles of charge (-), the direction of the spiral-shaped rotation will be opposite to the particles of the sign (+).
Conclusion. The developed multi-physical model of the stream of heterogeneous flow allows detailed description of the physical state of the substance at all points of the considered volume at specific instants of time, takes into account the design parameters of the channel (orientation, dimensions, material, etc.), and also specifies the laws of variation of parameters at the boundaries of the calculated region in conditions of the wave change of the internal section of the working chamber-channel under the influence of the energy of the magnetic field.
The significance of the results lies in the study of the effect of the factors that complicate calculation, namely the nonstationarity of the three-dimensional flow process in the channel with changing geometry, the presence of volume forces, the heterogeneity of the flow, the dependence of the thermophysical properties on the flow state parameters, the influence of the internal viscosity of the flow, possible phase transitions, various hydrodynamic flow regimes, turbulence, influence of magnetic fields, stress arising in the pipe material.
Acknowledgements. The research has been made under financial support of Russian Foundation of Basic Research, grant # 16-38-00169/16.
REFERENCES
1. Zaitsev A.V., Pelenko F.V. Modeling of viscous fluid flow in a pipe. Nauchnyi elektronnyi zhurnal NIUITMO. Seriya: Prot-sessy i apparaty pishchevykh proizvodstv. 2012. N 1, p. 163-168. Rezhim dostupa: http://processes.open-mechanics.com/
2. Jin Y., Li X., Zhao M.Y., Liu X.H., Li H. A mathematical model of fluid flow in tight porous media based on fractal assumptions. International Journal of Heat and Mass Transfer. 2017. N 108, p. 1078-1088.
3. Afrasiabikia P., Rizi A.P., Javan M. Scenarios for improvement of water distribution in Doroodzan irrigation network based on hydraulic simulation. Computers and Electronics in Agriculture. 2017. N 135, p. 312-320.
4. Alexandrov V., Vasilyeva M. Turbulent Transport of Solid Phase in Hydrotransport. Papers presented at the 15 th International Freight Pipeline Society Symposium; The Institute of Hydrodynamics AS CR, Academy of Sciences of the Czech Republic. Prague, 2014, p. 3-9.
5. Ates S. Hydraulic modelling of control devices in loop equations of water distribution networks. Flow Measurement and Instrumentation. 2017. N 35, p. 243-260.
6. Hirata M., Kurima S., Mizukami M., Yokota T. Boundedness and stabilization in a two-dimensional two-species chemo-taxis-Navier-Stokes system with competitive kinetics. Journal of Differential Equations. 2017. N 263(1), p. 470-490.
7. Bruno O.P., Cubillos M. On the quasi-unconditional stability of bdf-adi solvers for the compressible Navier-Stokes equations and related linear problems. Siam Journal on Numerical Analysis. 2017. N 55(2), p. 892-922. DOI: 10.1137/15M1042279.
8. Fang F., Ge M.W. Mathematical Constraints in Multiscale Subgrid-Scale Modeling of Nonlinear Systems. Chinese Physics Letters. 2017. № 34(3), N 030501. D0I:10.1088/0256-307X/34/3/030501.
9. Lai M.C., Seol Y. A short note on Navier-Stokes flows with an incompressible interface and its approximations. Applied Mathematics Letters. 2017. N 65, p. 1-6. D0I:10.1016/j.aml.2016.09.016.
10. Fathi-Moghadam M., Kiani S., Asiaban P., Behrozi-Rad R. Modeling of Perforated Sill-Controlled Hydraulic Jump. International Journal of Civil Engineering. 2017. N 15(4A), p. 689-695.
é Manya A. Vasilyeva, Stefan Voth
Multiphysical Model of Heterogeneous Flow Moving.
11. Wang L., Han Q.S., Chen D.Y., Wu C.Z., Wang X.Y. Non-linear modelling and stability analysis of the PTGS at pump mode. IETRenewable Power Generation. 2017. N 11(6), p. 827-836.
12. Nouri A.Z. Mathematical modeling of concrete pipes reinforced with CNTs conveying fluid for vibration and stability analyses. Computers and Concrete. 2017. N 19(3), p. 325-331.
13. Chen W.C., Dolguntseva I., Savin A., Zhang Y.L., Li W., Svensson E., Leijon M. Numerical modelling of a point-absorbing wave energy converter in irregular and extreme waves. Applied Ocean Research. 2017. N 63, p. 90-105. DOI: 10.1016/j.apor.2017.01.004.
14. Olsen J.E., Skjetne P., Johansen S.T. VLES turbulence model for an Eulerian-Lagrangian modeling concept for bubble plumes. Applied Mathematical Modelling. 2017. N 44, p. 61-71.
15. Ottosson A., Nilsson L., Berghel J. A mathematical model of heat and mass transfer in Yankee drying of tissue. Drying Technology. 2017. N 35(3), p. 323-334.
16. Vasilyeva M. Justification of the Choice Matrix Material of the Magnetoactive Elastomer for Working Camera-Channel Peristaltic Unit. Materials Science Forum. 2016. N 870, p. 13-19.
17. Vasilyeva M. Perspectives of Application of 3D Shape Memory Composite Materials for Peristaltic Transportation of Slurries. Key Engineering Materials. 2015. N 685, p. 291-294.
Authors: Mariya A. Vasilyeva, Candidate of Engineering Sciences, Associate Professor, [email protected] (Saint-Petersburg Mining University, Saint-Petersburg, Russia), Stefan Voth, Doctor of Engineering, Professor, Director of PROLab Centre, Stefan. [email protected] (Technische Hochschule Georg Agricola, Bochum, Germany). The paper was accepted for publication on 7.07.2017.
