2 results
On the cost efficiency of mixing optimization
- Oleg Gubanov, Luca Cortelezzi
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- Journal:
- Journal of Fluid Mechanics / Volume 692 / 10 February 2012
- Published online by Cambridge University Press:
- 16 December 2011, pp. 112-136
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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.
Towards the design of an optimal mixer
- OLEG GUBANOV, LUCA CORTELEZZI
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- Journal:
- Journal of Fluid Mechanics / Volume 651 / 25 May 2010
- Published online by Cambridge University Press:
- 22 March 2010, pp. 27-53
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We define as an optimal mixer a mixing device able to deliver a uniformly optimal mixing performance over a wide range of operating and initial conditions. We consider the conceptual problem of designing an optimal mixer starting from a well-known reference mixer, the sine flow. We characterize the mixing performance of the reference mixer, and show that it performs poorly and erratically over a wide range of operating conditions and is quite sensitive to the geometry of the initial concentration field. We define as a target performance the best mixing performance the reference mixer is able to achieve. In steps we modify the design of the reference mixer. First, we optimize the time sequence of the switching protocols and show that the mixing performance of the time-optimized mixer, although substantially improved with respect to the reference mixer, is still far from achieving the target performance and being insensitive to the geometry of the initial concentration field. The analysis of the performance of the time-optimized mixer brings to light the deficiency of the actuating system used, which delivers always the same amount of shear at the same locations. We modify the actuating system by allowing the stirring velocity fields to shift along their coordinate axes. A new mixer, the space-optimized mixer, is created by equipping the reference mixer with the new actuating system and optimizing the shift of the stirring velocity field at each iteration. The space-optimized mixer is able to deliver the target performance over the upper two-thirds of the operating range. In the lower one-third, the performance of the space-optimized mixer deteriorates because of the use of a periodic protocol. A optimal mixer is finally obtained using the actuating system of the space-optimized mixer and coupling the time and shift optimizations. The resulting optimal mixer is able to deliver a uniform target performance, insensitive to the geometry of the initial conditions, over the entire operating range.