Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-18T22:41:42.807Z Has data issue: false hasContentIssue false
  • English
  • Français

On the cost efficiency of mixing optimization

Published online by Cambridge University Press:  16 December 2011

Oleg Gubanov  and
Luca Cortelezzi
Oleg Gubanov
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal, Quebec, H3A 2K6, Canada
Luca Cortelezzi*
Affiliation:
Department of Mechanical Engineering, McGill University, Montreal, Quebec, H3A 2K6, Canada
*
†Email address for correspondence: crtlz@cim.mcgill.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

In this study we discuss the cost efficiency of the optimization of a new prototypical mixing flow, the Fourier sine flow, an extension of the sine flow. The Fourier sine flow stirs a mixture on a two-dimensional torus by blinking, at prescribed switching times, two orthogonal velocity fields with profiles represented by a Fourier sine series. We derive a family of mixers of increasing complexity by truncating the series to one, two, three and four modes. We consider the optimization of the velocity profiles and the optimization of the stirring protocol. We implement the former by computing, at each iteration, the amplitudes and phase shifts of the Fourier modes synthesizing the velocity profiles that minimize the mix-norm, our cost function, i.e. maximize the quality of mixing. We implement the latter by selecting, at each iteration, the best performing of the two orthogonal stirring velocity fields, i.e. the velocity field that minimizes the mix-norm. To obtain a physically meaningful optimization problem, we constrain the kinetic energy of the flow to be the same among all mixers and use the viscous dissipation as an estimate of the power input needed to operate the mixers. We characterize the performance of the mixers using three cost functions: the homogenization time, the computational cost of optimization and the total energy consumption. We test the mixers on a range of admissible power inputs using two representative switching times. We report some surprising results. Mixers equipped with the velocity profile optimization and a periodic stirring protocol cannot be optimal, i.e. their performance depends on the switching time chosen, independently of the number of Fourier modes used in the optimization. Apparently, optimal mixers can be obtained only by coupling velocity profile and stirring protocol optimizations. The computational cost of the optimization depends only on the number of Fourier modes used and grows by about an order of magnitude for each Fourier mode added to the optimization. At low power inputs, the coupled optimizations allow us to obtain an attractive reduction of the homogenization time in combination with a reduction of the total energy required to produce it. However, increasing the power input does not guarantee a reduction of the homogenization time. Counter-intuitively, there are ranges of power inputs for which both the homogenization time and the total energy increase when increasing the power input. Finally, for large enough power inputs, optimizations with two, three and four Fourier modes perform similarly, making the former optimization the most cost-efficient.

JFM classification

Mixing: Chaotic advection Mixing: Mixing Flow Control: Mixing enhancement
Type
Papers
Information
Journal of Fluid Mechanics , Volume 692 , 10 February 2012 , pp. 112 - 136
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alvarez, M. M., Muzzio, F. J., Cerbelli, S., Adrover, A. & Giona, M. 1998 Self-similar spatiotemporal structure of intermaterial boundaries in chaotic flows. Phys. Rev. Lett. 81, 3395–3398.CrossRefGoogle Scholar
2.Antonsen, T. M. Jr., Fan, Z., Ott, E. & Garcia-Lopez, E. 1996 The role of chaotic orbits in the determination of power spectra of passive scalars. Phys. Fluids 8, 3094–3104.CrossRefGoogle Scholar
3.Bigio, D. I. & Conner, J. H. 1995 Principal directions as a basis for the evaluation of mixing. Polym. Engng Sci. 35, 1527–1534.CrossRefGoogle Scholar
4.Cortelezzi, L., Adrover, A. & Giona, M. 2008 Feasibility, efficiency and transportability of short horizon optimal mixing protocols. J. Fluid Mech. 597, 199–231.CrossRefGoogle Scholar
5.D’Alessandro, D., Dahleh, M. & Mezić, I. 1999 Control of mixing in fluid flow: a maximum entropy approach. IEEE Trans. Autom. Control 44, 1852–1863.CrossRefGoogle Scholar
6.Gubanov, O. & Cortelezzi, L. 2009 Sensitivity of mixing optimization to the geometry of the initial scalar field. In Analysis and Control of Mixing with an Application to Micro and Macro Flow Processes (ed. Cortelezzi, L. & Mezić, I. ). CISM Courses and Lectures , vol. 510. Springer.CrossRefGoogle Scholar
7.Gubanov, O. & Cortelezzi, L. 2010 Toward the design of an optimal mixer. J. Fluid Mech. 651, 27–53.CrossRefGoogle Scholar
8.Khakhar, D. V., Rising, H. & Ottino, J. M. 1986 Analysis of chaotic mixing in two model systems. J. Fluid Mech. 172, 419–451.CrossRefGoogle Scholar
9.Kolda, T. G. 2005 Revisiting asynchronous parallel pattern search for nonlinear optimization. SIAM J. Optim. 16 (2), 563–586.CrossRefGoogle Scholar
10.Lamberto, D. J., Alvarez, M. M. & Muzzio, F. J. 2001 Computational analysis of regular and chaotic mixing in a stirred tank reactor. Chem. Engng Sci. 56, 4887–4899.CrossRefGoogle Scholar
11.Lewis, R. M. & Torczon, V. 2000 Pattern search methods for linearly constrained minimization. SIAM J. Optim. 10, 917–941.CrossRefGoogle Scholar
12.Lin, Z., Thiffeault, J.-L. & Doering, C. R. 2011 Optimal stirring strategies for passive scalar mixing. J. Fluid Mech. 675, 465–476.CrossRefGoogle Scholar
13.Liu, M., Muzzio, F. J. & Peskin, R. L. 1994 Quantification of mixing in aperiodic chaotic flows. Chaos, Solitons Fractals 4, 869–893.CrossRefGoogle Scholar
14.Malvern, L. E. 1969 Introduction to the Mechanics of a Continuous Medium. Prentice-Hall.Google Scholar
15.Mathew, G., Mezić, I., Grivopoulos, S., Vaidya, U. & Petzold, L. 2007a Optimal control of mixing in Stokes fluid flows. J. Fluid Mech. 580, 261–281.CrossRefGoogle Scholar
16.Mathew, G., Mezić, I. & Petzold, L. 2005 A multiscale measure for mixing and its applications. Physica D 211, 23–46.CrossRefGoogle Scholar
17.Mathew, G., Mezić, I., Serban, R. & Petzold, L. 2007b Optimization of mixing in an active micromixing device. In Technical Proceedings of the 2004 NSTI Nanotechnology Conference and Trade Show, Vol. 1. chapter 7. pp. 300–303. Nano Science and Technology Institute, Cambridge, MA. 6/8/2007.Google Scholar
18.Mathew, G., Petzold, L. & Serban, R. 2002 Computational techniques for quantification and optimization of mixing in microfluidic devices. Tech. Rep. UCRL-JC-148966. Lawrence Livermore National Laboratory, July 2002.Google Scholar
19.Muzzio, F. J., Alvarez, M. M., Cerbelli, S., Giona, M. & Adrover, A. 2000 The intermaterial area density generated by time- and spatially periodic 2D chaotic flows. Chem. Engng Sci. 55, 1497–1508.CrossRefGoogle Scholar
20.Ottino, J. M. 1989 The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge University Press.Google Scholar
21.Phelps, J. H. & Tucker, C. L. 2006 Lagrangian particle calculations of distributive mixing: limitations and applications. Chem. Engng Sci. 61, 6826–6836.CrossRefGoogle Scholar
22.Pierrehumbert, R. T. 1994 Tracer microstructure in the large-eddy dominated regime. Chaos, Solitons Fractals 4, 1091–1110.CrossRefGoogle Scholar
23.Szalai, E. S., Kukura, J., Arratia, P. E. & Muzzio, F. J. 2003 Effect of hydrodynamics on reactive mixing in laminar flows. Am. Inst. Chem. Engrs J. 49, 168–179.CrossRefGoogle Scholar
24.Thiffeault, J.-L., Doering, C. R. & Gibbon, J. D. 2004 A bound on mixing efficiency for the advection–diffusion equation. J. Fluid Mech. 521, 105–114.CrossRefGoogle Scholar
25.Vikhansky, A. 2002a Control of stretching rate in time-periodic chaotic flows. Phys. Fluids 14, 2752–2756.CrossRefGoogle Scholar
26.Vikhansky, A. 2002b Enhancement of laminar mixing by optimal control methods. Chem. Engng Sci. 57, 2719–2725.CrossRefGoogle Scholar