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Position control of charged spherical particles suspended in laminar flow within a channel

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Abstract

In this paper, steering a set of charged particles suspended in fluid flow within a plane channel is considered. In this way, the linear-state feedback control methodology is used to determine the external electric field. In order to design the proposed controller, at first, the governing mathematical ordinary differential equations are derived by the combination of Newton’s second law, Coulomb’s law and Navier–Stokes equations. Then, a linear quadratic regulator, which is one of the optimal control methods, is used to design the state-feedback control gains. For this purpose, continuous-time and discrete-time control methods are utilized. Simulation studies demonstrate the efficiency of the implemented control strategy for controlling the particles position and also tracking desired trajectories within the fluid flow. These results can contribute substantially to the development of related industrial processes.

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Abbreviations

\({\bar{u}}_b\) :

Average velocity, \(\text {m s}^{-1}\)

\(\forall \) :

Particles volume, \(\text {m}^{3}\)

\(\kappa \) :

Dielectric constant

\({\textbf{E}}_\textrm{c}\) :

Electric field (control input), \(\text {V m}^{-1}\)

\({\textbf{g}}\) :

Gravitational acceleration, \(\text {m s}^{-2}\)

\({\textbf{r}}_{ij}\) :

Distance between particles i and j, m

\({\textbf{V}}\) :

Fluid velocity vector, \(\text {m s}^{-1}\)

\({\textbf{X}}\) :

State vector

h :

Channel width, m

\({{\textbf{F}}_\textrm{E}}\) :

Electric field force vector, N

\({{\textbf{F}}_\textrm{W}}\) :

Weight force vector, N

\({{\textbf{P}}}\) :

Pressure vector, Pa

\({\varepsilon _0}\) :

Permittivity of free space

\({d_\textrm{P}}\) :

Particles diameter, m

\({E_\textrm{ex}}\) :

External electric field, \(\text {V m}^{-1}\)

k :

Coulomb’s constant

m :

Particles mass, kg

q :

Electric bar, C

u :

Fluid velocity in x direction, \(\text {m s}^{-1}\)

v :

Fluid velocity in y direction, \(\text {m s}^{-1}\)

x :

Streamwise direction

y :

Spanwise direction

\({{\textbf{F}}_\textrm{B}}\) :

Buoyancy force vector, N

\({{\textbf{F}}_\textrm{C}}\) :

Coulomb force vector, N

\({{\textbf{F}}_\textrm{D}}\) :

Drag force vector, N

\({{{\textbf{V}}_\textrm{p}}}\) :

Particles velocity vector, \(\text {m s}^{-1}\)

\({{{\textbf{x}}_\textrm{P}}}\) :

Particles position vector, m

\({{\mu _\textrm{f}}}\) :

Fluid dynamic viscosity, \(\text {kg m}^{-1}~\text {s}^{-1}\)

\({{\rho _\textrm{f}}}\) :

Fluid density, \(\text {kg m}^{-3}\)

\({{\rho _\textrm{p}}}\) :

Particles density, \(\text {kg m}^{-3}\)

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The authors did not receive support from any organization for the submitted work.

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Authors and Affiliations

  1. Control Research Laboratory, Department of Mechanical Engineering, Yasouj University, Yasouj, 75918-74934, Iran

    Ehsan Motamedi & Behrooz Rahmani

  2. Turbulence Research Laboratory, Department of Mechanical Engineering, Yasouj University, Yasouj, 75918-74934, Iran

    Amin Moosaie

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  1. Ehsan Motamedi
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  2. Behrooz Rahmani
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This manuscript is prepared based on the M.Sc. thesis of E. Motamedi, B. Rahmani and A. Moosaie are his supervisors in this research.

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Correspondence to Behrooz Rahmani.

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Motamedi, E., Rahmani, B. & Moosaie, A. Position control of charged spherical particles suspended in laminar flow within a channel. Comp. Part. Mech. 10, 853–864 (2023). https://doi.org/10.1007/s40571-022-00537-y

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