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International audience; An original dynamic model for substrate uptake under transient conditions is established and used to simulate a variety of biological responses to external perturbations. The actual uptake and growth rates, treated as cell properties, are part of the model variables as well as the substrate concentration at the cell-liquid interface. Several regulatory loops inspired by the structure of the glycolytic chain are considered to establish a set of ordinary differential equations. The uptake rate evolves so as to reach an equilibrium between the cell demand and the environmental supply. This model does not contain any of the usual algebraic closure laws relating to the instantaneous uptake, growth rates, and the substrate concentration, nor does it enforce the continuity of mass fluxes at the liquid-cell interface. However, these relationships are found in the steady-state solution. Previously unexplained experimental observations are well reproduced by this model. Also, the model structure is suitable for further coupling with flux-based metabolic models and fluid-flow equations.

An open-source CFD software OpenFOAM® is used to simulate two multiphase stirred-tank reactors relevant to industrial processes such as slurry polymerization and fuel production. Gas-liquid simulations are first performed in a single-impeller stirred-tank reactor, studied experimentally by Ford, J. J., T. J. Heindel, T. C. Jensen, and J. B. Drake. 2008. “X-Ray Computed Tomography of a Gas-Sparged Stirred-Tank Reactor.” Chemical Engineering Science 63: 2075–85. Three impeller rotation speeds (200, 350 and 700 rpm) with three different bubble diameters (0.5, 1.5 and 2.5 mm) are investigated. Flow patterns compared qualitatively to those from experiments. Compared to the experimental data, the simulations are in relatively good agreement for gas holdup in the reactor. The second multiphase system is a multi-impeller stirred-tank reactor, studied experimentally by Shewale, S. D., and A. B. Pandit. 2006. “Studies in Multiple Impeller Agitated Gas-Liquid Contractors.” Chemical Engineering Science 61: 486–504. Gas-liquid simulations are performed at two impeller rotation speeds (3.75 and 5.08 RPS). The simulated flow patterns agree with published pictures from the experiments. Gas-liquid-solid simulations of the multi-impeller stirred-tank reactor are also carried out at impeller rotation speed 5.08 RPS. The addition of solid particles with a volume fraction characteristic of slurry reactors changes the flow pattern significantly. The bottom Rushton turbine becomes flooded, while the upper pitched-blade downflow turbines present a radial-pumping flow pattern instead of down-pumping. Nonetheless, the solid phase has a similar flow pattern to the liquid phase, indicating that the particles modify the effective density of the fluid.

Two solution algorithms are developed for the conditional moment closure (CMC) using quadrature-based moment methods (QBMM). Their primary purpose is to eliminate the necessity of the additional grid for the conditioning variable (e.g., mixture fraction). As in almost every probability-density-function (PDF)-based description, the main technical hurdle is the solution of the molecular-mixing term. Here the focus is on solving this term coupled with chemistry, as opposed to spatial transport. In the first algorithm (referred to as SA-CMC), this is done by semi-analytically solving an equation for the deviation variable of the conditional scalar mean in terms of Jacobi polynomials. The mixture-fraction space is represented by the Gauss–Lobatto quadrature rule and a β-PDF. In the second algorithm (referred to as QBMM-CMC), a closed set of differential equations is written for the joint moments of a scalar and the mixture fraction, thereby eliminating the need to assume a form for the mixture-fraction PDF. Both solution algorithms are tested for multi-step H2 combustion, and the expected results are achieved with at most six quadrature nodes. Since in the proposed algorithms the scalar mean is ensured to be constant during molecular mixing, it is further concluded that the proposed algorithms are accurate, computationally cost-effective, and straightforward alternatives for traditional CMC solution methods.

Starting from a generalized population balance equation and the Boltzmann–Enskog collision model for hard spheres, a kinetic theory model for polydisperse gas–particle flows is presented. Here, polydispersity results from spherical particles with the same material density but different diameters. The particle size distribution (PSD) of the particles is allowed to evolve in space and time due to physical processes such as mixing. In order to treat a continuous PSD, the particle-phase model is formulated in terms of the moments of the PSD, and velocity moments conditioned on the particle size. Velocity moments up to second order are included, resulting in transport equations for the mass, momentum and granular temperature, all conditioned on the particle size. In the numerical implementation, the PSD is represented using quadrature-based moment methods (QBMM). With QBMM, a continuous PSD can be treated using a relatively small number of moments as compared to class or sectional methods. Here, a realizable numerical algorithm for solving the moment system in a finite-volume code is proposed, valid for dilute systems wherein frictional forces are negligible. The ability of the proposed model to describe polydisperse gas–particle systems is demonstrated using cluster-induced turbulence and riser flow.

The Eulerian–Eulerian two-fluid model is widely used for computational fluid dynamics simulations of gas–solid flows. For non-isothermal flows, the averaged conservation equations solved in the two-fluid model require closures for drag, gas–solid heat transfer, pseudo-turbulent velocity fluctuations and the pseudo-turbulent heat flux (PTHF). However, the pseudo-turbulence terms are usually neglected in two-fluid simulations due to the lack of accurate correlations. With the increase in computational power, closures for these terms are now available from particle-resolved direct-numerical simulation (PR-DNS). Here, the PTHF closure as well as the heat-transfer closure (i.e., the Nusselt number) extracted from PR-DNS are implemented in the two-fluid thermal energy equation in OpenFOAM. The implementation is validated by comparing the simulation results with the PR-DNS data for the temperature profiles. Based on the analysis of the thermal energy budget, the PTHF can have a significant contribution and neglecting it can lead to large errors.

Droplet dispersion in liquid–liquid systems is a crucial step in many unit operations throughout the chemical, food, and pharmaceutic industries, where improper operation causes billions of dollars of loss annually. A theoretical background for the description of droplet breakup has been established, but many assumptions are still unconfirmed by experimental observations. In this investigation, a von Kármán swirling flow device was used to produce homogeneous, low-intensity turbulence suitable for carrying out droplet breakage experiments using optical image analysis. Individual droplets of known, adjustable, and repeatable sizes were introduced into an isotropic turbulent flow field providing novel control over two of the most important factors impacting droplet breakage: turbulence dissipation rate and parent droplet size. Introducing droplets one at a time, large data sets were gathered using canola, safflower, and sesame oils for the droplet phase and water as the continuous phase. Automated image analysis was used to determine breakage time, breakage probability, and child droplet size distribution for various turbulence intensities. Breakage time and breakage probability were observed to increase with increasing parent droplet size, consistent with the classic and widely used Coulaloglou–Tavlarides breakage model (C–T model). The shape of the child drop size distribution function was found to depend upon the size of the parent droplet.

Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces involving spatial gradients (e.g. lift). In the derivation, the particles in a given phase have identical mass and volume, and have no internal degrees of freedom (i.e. the particles are adiabatic). The ‘hard-sphere-fluid’ phase is obtained in the limit where the particle diameter in one phase tends to zero with fixed phase density so that the number of fluid particles tends to infinity. The moment system resulting from the two BE kinetic equations is closed at second order by invoking the anisotropic Gaussian closure. The resulting two-fluid model for a binary, hard-sphere mixture therefore consists (for each phase $\unicode[STIX]{x1D6FC}=1,2$) of transport equations for the mass $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}$, mean momentum $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ (where $\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}$ is the velocity) and a symmetric, second-order, kinetic energy tensor $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}}=\frac{1}{2}\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}(\boldsymbol{u}_{\unicode[STIX]{x1D6FC}}\otimes \boldsymbol{u}_{\unicode[STIX]{x1D6FC}}+\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})$. The trace of the fluctuating energy tensor $\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}}$ is $\text{tr}(\unicode[STIX]{x1D748}_{\unicode[STIX]{x1D6FC}})=3\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ where $\unicode[STIX]{x1D6E9}_{\unicode[STIX]{x1D6FC}}$ is the phase temperature (or granular temperature). Thus, $\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}E_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D71A}_{\unicode[STIX]{x1D6FC}}\text{tr}(\unicode[STIX]{x1D640}_{\unicode[STIX]{x1D6FC}})$ is the total kinetic energy, the sum over $\unicode[STIX]{x1D6FC}$ of which is the total kinetic energy of the system, a conserved quantity. From the analysis, it is found that the BE finite-size correction leads to exact phase pressure (or stress) tensors that depend on the mean-slip velocity $\boldsymbol{u}_{12}=\boldsymbol{u}_{1}-\boldsymbol{u}_{2}$, as well as the phase temperatures for both phases. These pressure tensors also appear in the momentum-exchange terms in the mean momentum equations that produce the Archimedes force, as well as drag contributions due to fluid compressibility and a lift force due to mean fluid-velocity gradients. The closed BE energy flux tensors show a similar dependence on the mean-slip velocity. The characteristic polynomial of the flux matrix from the one-dimensional model is computed symbolically and depends on five parameters: the particle volume fractions $\unicode[STIX]{x1D711}_{1}$, $\unicode[STIX]{x1D711}_{2}$, the phase density ratio ${\mathcal{Z}}=\unicode[STIX]{x1D70C}_{f}/\unicode[STIX]{x1D70C}_{p}$, the phase temperature ratio $\unicode[STIX]{x1D6E9}_{r}=\unicode[STIX]{x1D6E9}_{2}/\unicode[STIX]{x1D6E9}_{1}$ and the mean-slip Mach number $Ma_{s}=\boldsymbol{u}_{12}/\sqrt{5\unicode[STIX]{x1D6E9}_{1}/3}$. By applying Sturm’s Theorem to the characteristic polynomial, it is demonstrated that the model is hyperbolic over a wide range of these parameters, in particular, for the physically most relevant values.

Inertial particles in turbulent flows are characterised by preferential concentration and segregation and, at sufficient mass loading, dense particle clusters may spontaneously arise due to momentum coupling between the phases. These clusters, in turn, can generate and sustain turbulence in the fluid phase, which we refer to as cluster-induced turbulence (CIT). In the present work, we tackle the problem of developing a framework for the stochastic modelling of moderately dense particle-laden flows, based on a Lagrangian probability-density-function formalism. This framework includes the Eulerian approach, and hence can be useful also for the development of two-fluid models. A rigorous formalism and a general model have been put forward focusing, in particular, on the two ingredients that are key in moderately dense flows, namely, two-way coupling in the carrier phase, and the decomposition of the particle-phase velocity into its spatially correlated and uncorrelated components. Specifically, this last contribution allows us to identify in the stochastic model the contributions due to the correlated fluctuating energy and to the granular temperature of the particle phase, which determine the time scale for particle–particle collisions. The model is then validated and assessed against direct-numerical-simulation data for homogeneous configurations of increasing difficulty: (i) homogeneous isotropic turbulence, (ii) decaying and shear turbulence and (iii) CIT.

The objective of the presented work is to verify a delayed detached eddy simulation (DDES) model for simulating transitional swirling flow in a micro-scale multi-inlet vortex reactor (MIVR). The DDES model is a k- ω based turbulence model with a low Reynolds number correction applied to the standard k- ω model such that the Reynolds-averaged Navier-Stokes (RANS) component of the DDES model is able to account for low Reynolds number flow. By limiting the dissipation rate in the k-equation, the large-eddy simulation (LES) part of the DDES model behaves similarly to a one-equation sub-grid model. The turbulent Reynolds number is redefined to represent both modeled and resolved turbulence level so that underestimation of the RANS length scale in the LES range can be reduced. Applying the DDES model to simulate both laminar and transitional flow in the micro-scale MIVR produces an accurate prediction of mean velocity and turbulent intensity compared with experimental data. It is demonstrated that the proposed DDES model is capable of simulating transitional flow in the complex geometry of the micro-scale MIVR. These simulation results also help to understand the flow and mixing patterns in the micro-scale MIVR and provide guidances to optimize the reactor for the application of producing functional nanoparticles.

Velocity fluctuations in the carrier phase and dispersed phase of a dispersed multiphase flow are studied using particle-resolved direct numerical simulation. The simulations correspond to a statistically homogeneous problem with an imposed mean pressure gradient and are presented for $${Re}_{m}=20$$ and a wide range of dispersed phase volume fractions $$\left( 0.1 \le \phi \le 0.4 \right) $$ and density ratios of the dispersed phase to the carrier phase $$\left( 0.001 \le \rho _\mathrm{p}/\rho _\mathrm{f} \le 1000 \right) $$ . The velocity fluctuations in the fluid and dispersed phase at the statistically stationary state are quantified by the turbulent kinetic energy (TKE) and granular temperature, respectively. It is found that the granular temperature increases with a decrease in the density ratio and then reaches an asymptotic value. The qualitative trend of the behavior is explained by the added mass effect, but the value of the coefficient that yields quantitative agreement is non-physical. It is also shown that the TKE has a similar dependence on the density ratio for all volume fractions studied here other than $$\phi =0.1$$ . The anomalous behavior for $$\phi =0.1$$ is hypothesized to arise from the interaction of particle wakes at higher volume fractions. The study of mixture kinetic energy for different cases indicates that low-density ratio cases are less efficient in extracting energy from mean flow to fluctuations.

In this work, model closures of the multiphase Reynolds-averaged Navier–Stokes (RANS) equations are developed for homogeneous, fully developed gas–particle flows. To date, the majority of RANS closures are based on extensions of single-phase turbulence models, which fail to capture complex two-phase flow dynamics across dilute and dense regimes, especially when two-way coupling between the phases is important. In the present study, particles settle under gravity in an unbounded viscous fluid. At sufficient mass loadings, interphase momentum exchange between the phases results in the spontaneous generation of particle clusters that sustain velocity fluctuations in the fluid. Data generated from Eulerian–Lagrangian simulations are used in a sparse regression method for model closure that ensures form invariance. Particular attention is paid to modelling the unclosed terms unique to the multiphase RANS equations (drag production, drag exchange, pressure strain and viscous dissipation). A minimal set of tensors is presented that serve as the basis for modelling. It is found that sparse regression identifies compact, algebraic models that are accurate across flow conditions and robust to sparse training data.

International audience; This work reports an experimental investigation of a liquid-solid fluidized bed involving inertial particles at a large Reynolds number. Owing to optical techniques and index matching, the statistics of the velocity fluctuations of both the particles and the liquid are measured for a wide range of the particle volume fraction αp. The dynamics of the fluctuations suggests that the flow possesses the following three properties: (1) The liquid volume involves a wake region in which vertical fluctuations are negative and an interstitial region where they are positive. (2) The statistics of the horizontal fluctuations are similar to vertical ones, except that they are symmetric. (3) Local instant particle fluctuations are proportional to liquid ones. Assuming these properties are true allows us to derive a model for the probability density functions (PDFs) of the two components of the velocity fluctuations of the two phases. This model involves a single reference PDF that is independent of αp and one weighting parameter for each phase. The weighting parameter of the liquid phase is an affine function of αp, which characterizes the volume of the wakes relative to that of the interstices. That of the particle phase depends on the preferential concentration of the particles, which tend to avoid the wakes at low αp. This model accurately describes the experimental PDFs up to the third-order moment and reproduces all their peculiar features: the skewness of the vertical fluctuations which reverses at a given volume fraction, the presence of exponential tails corresponding to rare intense events, and the symmetry between low and large volume fractions.

Modeling particle-laden turbulent flows at high volume fractions requires accounting for the coupling between phases. The latter is often a sensitive point, and proper closure of the exchange and production terms due to the presence of particles is not straightforward. In the present work, a Lagrangian probability-density-function model developed for homogeneous cluster-induced turbulence is extended to a channel flow. The key features are consistent two-way coupling and the decomposition of the particle velocity into spatially correlated and uncorrelated components, which is crucial for dense flows and which allows dealing with collisions from a statistical point of view. A numerical scheme for the coupled solution of the stochastic differential equations for the particles and a Reynolds-stress model for the fluid is developed. Tests with tracer particles without two-way coupling are done to assess the validity and the consistency of the numerical scheme. Finally, two sets of numerical simulations with particles with different diameters in a turbulent channel flow at a shear Reynolds of R e τ = 300 are reported. The effect of two-way coupling by varying the mass loading of the dispersed phase in the mass-loading range φ = 0–2 is analyzed, and the results are compared to previous Eulerian–Lagrangian and Eulerian–Eulerian direct-numerical simulation (DNS) studies. Mean velocities and turbulent kinetic energy show good agreement with DNS, especially regarding the trend with respect to mass loading. Consistent with prior work, increased mass loading causes a drastic reduction of turbulent kinetic energy in the range φ = 0–2.

A numerical investigation is performed on buoyancy-driven homogeneous and heterogeneous bubbly flows to compare the bulk gas–liquid heat transfer effectiveness for Prandtl (Pr) numbers 0.2–20 and void fractions [Formula: see text] 0.3–0.5. For this purpose, transient two-fluid model simulations of bubbles rising in a stagnant pool of liquid are conducted in a rectangular box by applying periodic boundary conditions to all the sides. The temperature difference ([Formula: see text]) between gas and liquid phase is averaged over the rectangular box and monitored with respect to time, the heat transfer rate is studied based on the time at which the [Formula: see text] tends to zero. The results of numerical study show that at low Pr numbers, faster decay of [Formula: see text] is observed for homogeneous flow of bubbles indicating higher heat transfer rate in comparison with the heterogeneous flow of bubbles for the same void fraction. On the contrary, for high Pr numbers, higher heat transfer rate is observed in heterogeneous flow compared to the homogeneous. The comparison of heat transfer behavior between different void fractions for heterogeneous flow show that, for low Pr numbers higher heat transfer rate is achieved for void fraction 0.4 in comparison with void fraction 0.5. And for high Pr numbers, higher heat transfer is observed for void fraction 0.5 in comparison with void fraction 0.4.

We simulated particle-laden flow in a vertical channel with a Reynolds-averaged Navier--Stokes (RANS) two-fluid model including a Reynolds-stress model (RSM). Primary and turbulent statistics from counterpart Eulerian--Lagrangian (EL) and Eulerian--Eulerian Anisotropic-Gaussian (EE-AG) simulations were used to inform parameters and closures in the RSM. Flow in the channel center, including transitions, compared well with other simulations as mass loading increased. For small-Stokes particles, the RSM at greater mass loadings reproduced the transient clustering observed in the other models.

International audience; A hyperbolic two-fluid model for gas-particle flow derived using the Boltzmann-Enskog kinetic theory is generalized to include added mass. In place of the virtual-mass force, to guarantee indifference to an accelerating frame of reference, the added mass is included in the mass, momentum and energy balances for the particle phase, augmented to include the portion of the particle wake moving with the particle velocity. The resulting compressible two-fluid model contains seven balance equations (mass, momentum, and energy for each phase, plus added mass) and employs a stiffened-gas model for the equation of state for the fluid. Using Sturm's theorem, the model is shown to be globally hyperbolic for arbitrary ratios of the material densities Z = ρ f /ρ p. An eight-equation extension to include the pseudo-turbulent kinetic energy (PTKE) in the fluid phase is also proposed; however, PTKE has no effect on hyperbolicity. In addition to the added mass, the key physics needed to ensure hyperbolicity for arbitrary Z is a fluid-mediated contribution to the particle-phase pressure tensor that is taken to be proportional to the volume fraction of the added mass. A numerical solver for hyperbolic equations is developed for the 1-D model, and numerical examples are employed to illustrate the behaviour of solutions to Riemann problems for different material-density ratios. The relation between the proposed two-fluid model and prior work on effective-field models is discussed, as well as possible extensions to include viscous stresses and the formulation of the model in the limit of an incompressible continuous phase.

The hyperbolicity condition of the system of partial differential equations (PDEs) of the incompressible two-fluid model, applied to gas–liquid flows, is investigated. It is shown that the addition of a dispersion term, which depends on the drag coefficient and the gradient of the gas volume fraction, ensures the hyperbolicity of the PDEs, and prevents the nonphysical onset of instabilities in the predicted multiphase flows upon grid refinement. A constraint to be satisfied by the coefficient of the dispersion term to ensure hyperbolicity is obtained. The effect of the dispersion term on the numerical solution and on its grid convergence is then illustrated with numerical experiments in a one-dimensional shock tube, in a column with a falling fluid, and in a two-dimensional bubble column.