Improving the design of depth filters: A model‐based method using optimal control theory

Kühn, Michael; Kirse, Christoph; Briesen, Heiko

doi:10.1002/aic.15866

Cited by 5 publications

(14 citation statements)

References 29 publications

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“…t and z are the independent variables time and space, respectively. The described approach is widely used in literature and has proven successful for many different instances of depth filtration . Assuming stationary behavior and a constant filter coefficient λ , Equations can be simplified to the well‐known Iwasaki equation:

\frac{dc}{dz} = - λ \cdot c .

…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…Solving Equation by separation of variables leads to

c_{L} = c_{0} \cdot \exp ((), - \int_{0}^{L} λ ((), z) italicdz) = c_{0} \cdot e^{- \overset{λ}{true} \cdot L},

where c 0 and c L are the concentration values at the filter inlet ( z = 0) and outlet ( z = L ), respectively, and

\overset{λ}{true}

is the mean filter coefficient. As usually λ changes with deposition, it is often expressed in the literature as a function of σ :

λ = λ_{0} \cdot F ((),, σ, P),

where F is a functional expression that modifies the initial filter coefficient, that is, of the unclogged filter, depending on the deposition and a number of model parameters contained in the parameter vector P . Thus, only for short filtration times, the assumption λ = λ 0 can be made, which also implies F = 1.…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…The objective is to achieve homogeneous deposition within the filter media using filter coefficient λ as the control variable. Following Kuhn et al, the goal of homogeneous deposition can be formulated using the objective function

J_{OC} ((), λ ((), z)) = \int_{0}^{L} {[\overset{false¯}{σ} - σ (z, λ (z))]}^{2} italicdz

from which, applying Equations , the following trajectory of the filter coefficient can be derived:

λ ((), z) = - {[L \cdot (\frac{c_{0}}{Δ c} + \frac{z}{L})]}^{- 1} .

L is filter media's height and known form the microscale simulations. Δ c = c L − c 0 corresponds to the desired overall reduction in impurities.…”

Section: Computational Strategymentioning

confidence: 99%

“…The present work is based on two previous studies conducted by separate research groups . Both studies observed the lack of suitable design strategies for depth filters and made, coming from different directions, a very similar proposal to optimize filter design, namely the use of computationally determined graded structures.…”

Section: Introductionmentioning

confidence: 99%

“…Azimian et al studied different designs of fibrous filters by pore‐scale (PS) simulations and proposed a structure with an exponentially increasing fine fiber fraction from fluid inlet to outlet as a suitable setup. Kuhn et al presented a design method for depth filters using an optimal‐control approach based on continuum models. In the present study, both research groups bring together their findings to develop a novel multiscale method for optimizing depth filter design.…”

Section: Introductionmentioning

confidence: 99%

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Designing optimally‐graded depth filter media using a novel multiscale method

Geerling

Azimian²,

Wiegmann³

et al. 2019

AIChE Journal

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Even though current manufacturing methods allow to build precisely-defined graded depth filter media, it remains unclear which local filtration properties are desirable.We, therefore, introduce a multiscale approach which links pore and continuum scale (PS and CS) to address this question. Based on data from PS simulations, local filtration performance, as described by the filter coefficient, is predicted on the CS by an optimal control solution and the obtained trajectory is translated back to the PS where it is validated. Two case studies are presented: a bidisperse fibrous medium and a granular depth filter composed of bidisperse particles. Both media are optimized to achieve a homogenous deposition of separated impurities along the filter depth. It is found that the presented method allows to reach this goal reasonably well in both cases. We claim that our method forms a good basis for further developments for which promising possibilities are highlighted.

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\frac{dc}{dz} = - λ \cdot c .

…”

Section: Theoretical Backgroundmentioning

confidence: 99%

“…Solving Equation by separation of variables leads to

c_{L} = c_{0} \cdot \exp ((), - \int_{0}^{L} λ ((), z) italicdz) = c_{0} \cdot e^{- \overset{λ}{true} \cdot L},

where c 0 and c L are the concentration values at the filter inlet ( z = 0) and outlet ( z = L ), respectively, and

\overset{λ}{true}

is the mean filter coefficient. As usually λ changes with deposition, it is often expressed in the literature as a function of σ :

λ = λ_{0} \cdot F ((),, σ, P),

Section: Theoretical Backgroundmentioning

confidence: 99%

J_{OC} ((), λ ((), z)) = \int_{0}^{L} {[\overset{false¯}{σ} - σ (z, λ (z))]}^{2} italicdz

from which, applying Equations , the following trajectory of the filter coefficient can be derived:

λ ((), z) = - {[L \cdot (\frac{c_{0}}{Δ c} + \frac{z}{L})]}^{- 1} .

L is filter media's height and known form the microscale simulations. Δ c = c L − c 0 corresponds to the desired overall reduction in impurities.…”

Section: Computational Strategymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Designing optimally‐graded depth filter media using a novel multiscale method

Geerling

Azimian²,

Wiegmann³

et al. 2019

AIChE Journal

Self Cite

View full text Add to dashboard Cite

show abstract

Model‐based design of stratified packings for enhanced mass transfer using optimal control theory

Eppink,

Kuhn,

Briesen

2023

AIChE Journal

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Various examples show that stratified columns can improve the transport performance of particle packings. However, to date, there is no universal strategy to design these packings to yield optimal performance. This study proposes a model‐based approach for designing particle packings in which mass transfer occurs between a liquid phase and a stationary phase using optimal control theory. The primary objective is to provide a general design strategy that is applicable across different unit operations in chemical, pharmaceutical, and food applications. Optimal control is utilized to determine the optimal particle diameter as a function of the axial position within the column. We demonstrate the approach using two case studies and three different optimization criteria. Numerical results indicate that the proposed method is highly effective, for example, the solvent demand is reduced by up to . Moreover, the optimally graded packing yields a significantly sharper breakthrough curve of an adsorption column.

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