Mass Transport Problem Research Articles

Design of well and groove microchannel bioreactors for cell culture

Microfluidic bioreactors have been shown valuable for various cellular applications. The use of micro-wells/grooves bioreactors, in which micro-topographical features are used to protect sensitive cells from the detrimental effects of fluidic shear stress, is a promising approach to culture sensitive cells in these perfusion microsystems. However, such devices exhibit substantially different fluid dynamics and mass transport characteristics compared to conventional planar microchannel reactors. In order to properly design and optimize these systems, fluid and mass transport issues playing a key role in microscale bioreactors should be adequately addressed. The present work is a parametric study of micro-groove/micro-well microchannel bioreactors. Operation conditions and design parameters were theoretically examined via a numerical model. The complex flow pattern obtained at grooves of various depths was studied and the shear protection factor compared to planar microchannels was evaluated. 3D flow simulations were preformed in order to examine the shear protection factor in micro-wells, which were found to have similar attributes as the grooves. The oxygen mass transport problem, which is coupled to the fluid mechanics problem, was solved for various groove geometries and for several cell types, assuming a defined shear stress limitation. It is shown that by optimizing the groove depth, the groove bioreactor may be used to effectively maximize the number of cells cultured within it or to minimize the oxygen gradient existing in such devices. Moreover, for sensitive cells having a high oxygen demand (e.g., hepatocytes) or low endurance to shear (e.g., human embryonic stem cells), results show that the use of grooves is an enabling technology, since under the same physical conditions the cells cannot be cultured for long periods of time in a planar microchannel. In addition to the theoretical model findings, the culture of human foreskin fibroblasts in groove (30 microm depth) and well bioreactors (35 microm depth) was experimentally examined at various flow rates of medium perfusion and compared to cell culture in regular flat microchannels. It was shown that the wells and the grooves enable a one order of magnitude increase in the maximum perfusion rate compared to planar microchannels. Altogether, the study demonstrates that the proper design and use of microgroove/well bioreactors may be highly beneficial for cell culture assays.

On a distance representation of Kantorovich potentials

We address the question of how to represent Kantorovich potentials in the mass transportation (or Monge–Kantorovich) problem as a signed distance function from a closed set. We discuss geometric conditions on the supports of the measure f + and f − in the Monge–Kantorovich problem which ensure such a representation. Finally, we obtain, as a by-product, the continuous differentiability of the potential on the transport set.

Non-negativity and stability analyses of lattice Boltzmann method for advection–diffusion equation

Stability is one of the main concerns in the lattice Boltzmann method (LBM). The objectives of this study are to investigate the linear stability of the lattice Boltzmann equation with the Bhatnagar–Gross–Krook collision operator (LBGK) for the advection–diffusion equation (ADE), and to understand the relationship between the stability of the LBGK and non-negativity of the equilibrium distribution functions (EDFs). This study conducted linear stability analysis on the LBGK, whose stability depends on the lattice Peclet number, the Courant number, the single relaxation time, and the flow direction. The von Neumann analysis was applied to delineate the stability domains by systematically varying these parameters. Moreover, the dimensionless EDFs were analyzed to identify the non-negative domains of the dimensionless EDFs. As a result, this study obtained linear stability and non-negativity domains for three different lattices with linear and second-order EDFs. It was found that the second-order EDFs have larger stability and non-negativity domains than the linear EDFs and outperform linear EDFs in terms of stability and numerical dispersion. Furthermore, the non-negativity of the EDFs is a sufficient condition for linear stability and becomes a necessary condition when the relaxation time is very close to 0.5. The stability and non-negativity domains provide useful information to guide the selection of dimensionless parameters to obtain stable LBM solutions. We use mass transport problems to demonstrate the consistency between the theoretical findings and LBM solutions.

ChemInform Abstract: Metal Supported Oxide Nanostructures: Model Systems for Advanced Catalysis

AbstractChemInform is a weekly Abstracting Service, delivering concise information at a glance that was extracted from about 200 leading journals. To access a ChemInform Abstract of an article which was published elsewhere, please select a “Full Text” option. The original article is trackable via the “References” option.

Measurability of optimal transportation and convergence rate for Landau type interacting particle systems

In this paper, we consider nonlinear diffusion processes driven by space-time white noises, which have an interpretation in terms of partial differential equations. For a specific choice of coefficients, they correspond to the Landau equation arising in kinetic theory. The main goal of the paper is to construct an easily simulable diffusive interacting particle system, converging towards this nonlinear process and to obtain an explicit pathwise rate. This requires to find a significant coupling between finitely many Brownian motions and the infinite dimensional white noise process. The key idea will be to construct the right Brownian motions by pushing forward the white noise processes, through the Brenier map realizing the optimal transport between the law of the nonlinear process, and the empirical measure of independent copies of it. A crucial problem will then be to establish the joint measurability of this optimal transport map, with respect to the space variable and the parameter (time-randomness) that makes the marginals vary. To overcome this point, we shall prove a general measurability result for the mass transportation problem and for the supports of the optimal transfer plans, in the sense of set-valued mappings. This will allow us to construct the coupling and to obtain explicit convergence rates.

Optimal Transportation Problem by Stochastic Optimal Control

We address an optimal mass transportation problem by means of optimal stochastic control. We consider a stochastic control problem which is a natural extension of the Monge–Kantorovich problem. Using a vanishing viscosity argument we provide a probabilistic proof of two fundamental results in mass transportation: the Kantorovich duality and the graph property for the support of an optimal measure for the Monge–Kantorovich problem. Our key tool is a stochastic duality result involving solutions of the Hamilton–Jacobi–Bellman PDE.

On the Spheroidal Semiseparation for Stokes Flow

Many heat and mass transport problems involve particle‐fluid systems, where the assumption of Stokes flow provides a very good approximation for representing small particles embedded within a viscous, incompressible fluid characterizing the steady, creeping flow. The present work is concerned with some interesting practical aspects of the theoretical analysis of Stokes flow in spheroidal domains. The stream function ψ, for axisymmetric Stokes flow, satisfies the well‐known equation E4ψ = 0. Despite the fact that in spherical coordinates this equation admits separable solutions, this property is not preserved when one seeks solutions in the spheroidal geometry. Nevertheless, defining some kind of semiseparability, the complete solution for ψ in spheroidal coordinates has been obtained in the form of products combining Gegenbauer functions of different degrees. Thus, the general solution is represented in a full‐series expansion in terms of eigenfunctions, which are elements of the space kerE2 (separable solutions), and in terms of generalized eigenfunctions, which are elements of the space kerE4 (semiseparable solutions). In this work we revisit this aspect by introducing a different and simpler way of representing the aforementioned generalized eigenfunctions. Consequently, additional semiseparable solutions are provided in terms of the Gegenbauer functions, whereas the completeness is preserved and the full‐series expansion is rewritten in terms of these functions.

Electro-Mechano-Chemistry; Transport Problem in Four Time Scales

The mass transport in the presence of stress, electrical, mechanical and chemical potential gradients in multicomponent solid solution is analyzed. The method bases on the Darken concept and the calorimetric equation of state. We effectively coupled the conservation of the mass (continuity equations), energy, momentum and Gauss equations. The diffusion fluxes of the components are given by the Nernst-Planck formulae and take into account the electro-chemical and mechanical potentials. We simulate the deformation field during the diffusion caused by the gradients of the chemical potential of all elements in non-ideal Fe-Cu-Ni alloy. The simulations show that the model is compatible with experimental results, and can be effectively used for modelling the energy, momentum and mass transport problems in compressible multicomponent solid solutions. The numerical problems and methods of solution are presented.

A Mixed Formulation of the Monge-Kantorovich Equations

We introduce and analyse a mixed formulation of the Monge-Kantorovich equations, which express optimality conditions for the mass transportation problem with cost proportional to distance. Furthermore, we introduce and analyse the finite element approximation of this formulation using the lowest order Raviart-Thomas element. Finally, we present some numerical experiments, where both the optimal transport density and the associated Kantorovich potential are computed for a coupling problem and problems involving obstacles and regions of cheap transportation.

Performance of PEM Fuel Cells at Sub-Freezing Temperatures

The performance of single-PEM fuel cells at sub-freezing temperatures has been characterized. Fully humidified fuel cells that were subjected to multiple freeze/thaw cycles down to -40 oC showed degradation in performance associated with mass transport problems at the gas diffusion layer. The performance loss was negligible for cloth GDLs and slow cooling rates while fast cooling rates and paper GDLs showed significant degradation. When a single-fuel cell was operated at -10 oC at a constant voltage, the current first increased to a maximum value, then started decaying due to ice formation. This ice formation was monitored using neutron radiography and was found to be concentrated near the inlets, outlets and cell edges resulting in a significant increase in mass transport resistance as revealed by AC impedance spectroscopy. Cyclic voltammograms at -10 oC show no change in catalyst electrochemical surface area associated with this decreased performance. Finally the cell did not exhibit any degradation in performance at 80 oC even after 5 separate operations at -10 oC, where the cell was allowed to completely clog with ice each time until no further current was produced.

Metal supported oxide nanostructures: model systems for advanced catalysis

Metal supported oxide nanostructures are discussed within the framework of the "inverse model catalyst" concept. We show that oxide nanostructures on metal surfaces may be regarded as artificial oxide materials, which display novel properties as compared to bulk oxide compounds and are stabilised by interfacial interactions and two-dimensional confinement effects. This is illustrated for prototypical examples of vanadium oxide overlayers on Rh(111) and Pd(111) surfaces. Structure and morphological changes of the oxide phase on V-oxide/Rh and V-oxide/Pd inverse catalyst surfaces are discussed, and the mass transport problem in catalyst systems during oxidation-reduction cycles is addressed. We demonstrate that the diffusion of oxide cluster over the metal surface provides a effective means of mass transport. The role of metal-oxide interface in determining the oxide nanolayer structure on particular substrate surfaces is investigated, and interfacial chemistry and interfacial strain effects are identified as important parameters.

On action minimizing measures for the Monge-Kantorovich problem

In recent years different authors ([4, 16, 17]) have noticed and investigated some analogy between Mather’s theory of minimal measures in Lagrangian dynamic and the mass transportation (or Monge-Kantorovich) problem. We replace the closure and homological constraints of Mather’s problem by boundary terms and we investigate the equivalence with the mass transportation problem. An Hamiltonian duality formula for the mass transportation and the equivalence with Brenier’s formulation are also established.

Analysis of pH Gradients Resulting from Mass Transport Limitations in Engineered Heart Tissue

Transport limitations of critical nutrients are a major obstacle in the construction of engineered heart tissues (EHTs), and the importance of oxygen in this regard is well-documented throughout the literature. An indirect effect of cellular hypoxia is the shunt to the less-efficient glycolytic metabolism, which is accompanied by a reduction in extracellular pH. Image analysis of phenol red coloration in an experimental model of EHTs demonstrated pH gradients towards the center of the construct, which were dependent on experimental variables. Based on these observations, a four-species, 2-D diffusion-reaction mathematical model was developed to predict pH in a radial-diffusion model. The mathematical model predicted lethal values of pH (<6.5) in EHTs comprised of a nominal cell density of 10(6) cells/cm(3). pH predictions were moderately dependent on O(2) concentration, and strongly dependent on cell density, CO(2) concentration, and diffusion path length. It can be concluded from this study that hypoxia-induced acidosis is an important element in the mass transport problem, and future experiments measuring pH with more sensitive methods is expected to further elucidate the extent of this effect.

A variational approach to the reconstruction of galactic peculiar velocities

A variational method for determining the peculiar velocities of galaxies using a catalog of their positions and masses is suggested. The method is based on an approximate reconstruction of the dynamical history of the patch of the Universe covered by the catalog through the optimization of a suitable integral action functional and is an improvement on the method of the Monge-Kantorovich mass transportation problem developed in [1].

On the sufficiency of c-cyclical monotonicity for optimality of transport plans

We consider a mass transport problem among Polish spaces, and we show that a transport plan concentrated in a c-cyclical monotone set is optimal if the cost function is continuous and possibly + ∞ valued; the same result is proved for a generic l.s.c. cost function in the case when the measures are purely atomic. This generalizes the previously known results.

An extended Reynolds analogy for excited wavy instabilities of developing streamwise vortices with applications to scalar mixing intensification

Studies are presented to elucidate the role of steady streamwise vortex structures, initiated upstream from weak Görtler vortices in the absence of explicit vortex generators, and their excited nonlinear wavy instabilities in the intensification of scalar mixing in a spatially developing mixing region. While steady streamwise vortex flow gives rise to significant mixing enhancement, the excited nonlinear wavy instabilities, which in turn modify the basic three-dimensional streamwise vortices, give rise to further mixing intensification which is quantitatively assessed by a mixedness parameter. Possibility of similarity between the dimensionless streamwise momentum and scalar transport problems leading to an extended Reynolds analogy is sought. This similarity is shown earlier to hold for the steady streamwise vortex flow in the absence of nonlinear wavy instabilities (Liu &amp; Sabry 1991 Proc. R. Soc. A 432 , 1–12). In this paper, the momentum conservation equations for the nonlinear wavy or secondary instabilities together with the advected fluctuation scalar problems are examined in detail. The presence of the streamwise fluctuation pressure gradient, which prevents the similarity, is estimated in terms of the fluctuation dynamical pressure and its relative importance to advective transport. It is found from scaling that the fluctuating streamwise pressure gradient, though not completely negligible, is sufficiently unimportant so as to render similarity between fluctuation streamwise velocity and fluctuation temperature and concentration a distinct possibility. The scalar fluctuations are then inferable from the fluctuation streamwise velocity and that the Reynolds stresses of the nonlinear fluctuations and the scalar fluxes are also similar. The nonlinear instability-modified mean streamwise momentum and the modified mean heat and mass transport problems are also similar, thus providing a complete ‘Reynolds analogy’, rendering possible the interpretation of the scalar mixedness for a gaseous medium for which the Prandtl and Schmidt numbers are near unity. It is found that the nonlinearity of the wavy instability, which induces scalar fluxes modifying the mean scalar transport, further intensifies scalar mixedness over a significant streamwise region which is well above that achieved by the steady, unmodified streamwise vortices alone for the numerical example corresponding to the most amplified wavy-sinuous mode.

Multiscale modeling of particle–solidification front dynamics, Part I: Methodology

The interaction between an advancing solidification front and a micron-size particle is an inherently multiscale heat and mass transport problem. Transport at the micro-scale (i.e. the scale of the particle dimension) couples with intermolecular interactions and lubrication forces in a thin layer of melt between the particle and the front to determine the overall dynamics of the interaction. A multiscale model is developed to simulate such front–particle interactions. The solution to the lubrication equations in the melt layer is coupled to the solution of the Navier–Stokes equations for the overall particle–front system. Techniques are developed for coupling the dynamics at the two disparate scales at a common “matching plane”. All interfaces are represented and tracked using the level-set approach. A sharp-interface technique is employed for solution of the governing equations in the resulting moving boundary problem. Validation of the coupling strategy and results for the particle–front interaction phenomenon with the multiscale approach are presented.

A Delivery System for Precursor Vapors Based on Sublimation in a Fluidized Bed

AbstractA fluidized‐bed, solid‐precursor delivery system is found to overcome the mass and heat transport problems encountered in a fixed‐bed system. To achieve ideal fluidization behavior, a combination of solid precursor (minor component) and inert particles such as alumina or silica (major component) are charged to the vessel. Using [Al(acac)3] as a test precursor and operating at 423K, the rate of mass transport is measured as a function of the carrier gas flow rate. When operating at carrier gas flow rates above the minimum value required to fluidize the bed, the precursor flow rate significantly exceeds the performance of the fixed‐bed system. A simple model of a solid‐precursor delivery system is developed to quantify the relationship between operating conditions and gas‐phase precursor flow rate. The concept of the fluidized bed, solid‐precursor delivery system is validated by directing the precursor and carrier gas mixture into a CVD reactor where the flow is combined with a source of water vapor and oxygen to deposit amorphous Al2O3 films on coupons of a Ti6242 alloy.

A note on scenario reduction for two-stage stochastic programs

We extend earlier work on scenario reduction by relying directly on Fortet–Mourier metrics instead of using upper bounds given in terms of mass transportation problems. The importance of Fortet–Mourier metrics for quantitative stability of two-stage models is reviewed and some numerical results are also provided.

On the equality between Monge's infimum and Kantorovich's minimum in optimal mass transportation

This paper concerns the Monge's transport problem in a general Polish space. We find optimal conditions to establish the equality between the infimum of Monge's problem and the minimum of the Kantorovich's relaxed version of the problem. A preliminary version of the results of this paper is contained in the Ph.D. thesis [A. Pratelli, Existence of optimal transport maps and regularity of the transport density in mass transportation problems, Ph.D. Thesis, Scuola Normale Superiore, Pisa, Italy, 2003. Available on http://cvgmt.sns.it/].