The intermolecular interactions in the pseudo-potential lattice Boltzmann (PPLB) method can readily be extended to more than two components. We report about a three-component PPLB approach to explore whether the effect of a surfactant could be included in describing droplet behaviour in (liquid–liquid) emulsions. The two main liquid components are taken to follow the Carnahan-Starling equation of state (EoS), while the surfactant obeys an ideal EoS. We investigate the nature of the phases present at equilibrium and the dependence of the interfacial tension between the two liquid phases on the amount of surfactant. The response of a droplet subjected to simple shear is investigated in the absence and the presence of a surfactant. Our exploratory simulations show how during droplet deformation the surfactant re-distributes itself due to the action of the shear and flows towards the far ends of the deformed droplet, up to the moment the droplet breaks up. This inhomogeneous surfactant distribution along the interface increases the shear rate that is needed for droplet breakup such that the critical capillary number for breakup increases and the breakup process is delayed. The simulations also reveal the detailed flow fields inside and outside the deforming droplet.

A treatment of the transport and transfer processes of heat, mass and momentum in terms of their analogy. The processes are described with the help of macro and micro balances which in many cases lead to differential equations. This way, the textbook also prepares for Computational Fluid Dynamics techniques. The topics of the five chapters of the textbook are: • Balances: shape and recipe, mass balance, residence time distribution, energy and heat balances, Bernoulli equation, momentum balances • Molecular transport, dimensional analysis, forces on immersed objects • Heat transport: steady-state and unsteady conduction, the general heat transport equation, forced and free convective heat transport, radiant heat transport • Mass transport: steady-state and unsteady diffusion, the general mass transport equation, mass transfer across a phase interface, convective mass transport, wet bulb temperature • Fluid mechanics: flow meters, pressure drop, packed beds, laminar flow of Newtonian and non-Newtonian fluids, Navier-Stokes equations The leading idea behind this textbook is to train students in solving problems where transport phenomena are key. To this end, the textbook comprises almost 80 problems with solutions. • Retaining the analogy between mass, heat and momentum transport • Using the technique of drawing up balances throughout the book • Suitable for a single term course.

The development of a novel numerical model for droplet drying is the topic of this paper. The three main stages of droplet drying are distinguished, viz. unhindered evaporation of a ’wet’ particle (the droplet), restricted drying at a falling rate due to the formation of a crust around a wet core, and inert heating of the dry porous particle. Each stage is mathematically detailed to replicate all phenomena occurring throughout the drying process. The focus, however, is on the falling rate drying regime which is described in terms of Stefan diffusion of water vapour through the pores of a thickening crust. To this end, the model needs the material properties. This permits the droplet characteristics to be determined by composition rather than through single-droplet drying experiments. Finally, the model is validated against five of such experiments from literature using skim milk. Good agreement is found at each comparative case for the particle mass and temperature throughout the various drying regimes providing that for good reasons in three cases a lower drying air temperature is applied than reported for the experiments. The model is capable of predicting the entire drying process at low computational cost and without requiring empirical input.

We report the findings of a detailed assessment of various options for computational two-fluid RANS simulations of an aerated agitated 2-L bioreactor equipped with a single baffle and several dip tubes. The simulations were carried out by using the commercial flow solver ANSYS/Fluent. Our focus was on (1) the outlet condition at the liquid's surface (i.e., including an air head space in the simulation yes or no); (2) the choice between the steady-state Multiple Reference Frames (MRF) approach for modelling the impeller rotation and the dynamic Sliding Mesh (SM) option; (3) the choice between two computational meshes (mosaic or polyhedral); and (4) the effect of using either the realizable k-ε model or the SST k-ω model for dealing with the turbulence in combination with different values for the fixed bubble size (either 1.8 or 2.8 mm). The final conclusion is that the SM impeller model in combination with a polyhedral computational mesh and the SST k-ω turbulence model is to be preferred. All simulations suffer from the occurrence of spurious velocities larger than the impeller tip velocity.

We carried out 3D simulations of monodisperse particle suspensions subjected to a constant shear rate with the view to investigate the effect of electrical double layers around the particles on apparent suspension viscosities. To this end, expressions for Debye length, zeta potential, and ionic strength (pH) of the liquid were incorporated into our in-house lattice Boltzmann code that uses the immersed boundary method and includes subgrid lubrication models. We varied the solids concentration and particle radius, keeping the particle Reynolds number equal to 0.1. We report on results with respect to the effect of pH in the range 9 through 12 and of Debye length on apparent viscosity and spatial suspension structures, particularly at higher solids volume fractions, and on the effect of flow reversals.

This contribution to the journal issue commemorating Professor Robert Byron Bird focuses on his Dutch connection, which dates back to as early as 1950. Bird twice spent a semester-long period at (the current) Delft University of Technology. The development over time of two different schools of teaching transport phenomena and their mutual influencing are reviewed in quite some detail. The cornerstone in both schools is the analogy between the transport modes for mass, momentum, and energy. However, the role of fluid mechanics and its treatment is different. In addition, the didactic concepts underpinning the textbooks from the two schools are rather different as well, both having their pros and cons. This is illustrated for the mechanical energy balance and its derivation.

The understanding of the rheological behaviour of suspensions in aqueous electrolytes is necessary for the optimal design of hydraulic transport lines. In these applications, particle size is at least 10 micron, and the particle Reynolds number, Re_p, is finite: O(10⁻¹). Although there are some experimental and numerical data on the rheology of such suspensions, the number of detailed analyses is limited. Therefore, 3-D direct numerical simulations of dense suspensions in aqueous electrolytes are conducted to assess the dynamics of the relative apparent viscosity and particle structures. The solid–liquid interfaces are resolved, and the flow is simulated, employing an in-house immersed boundary-lattice Boltzmann method code. In addition to the hydrodynamics resolved in the computational grid, our simulations include unresolved sub-grid scale lubrication corrections and non-contact electric double layer (EDL) and Van der Waals forces for a wide range of particle volume fractions, ϕ_v, at a single Re_p=0.1. Under these conditions, the contribution of the Van der Waals force was found to be weak. With an increase in ϕ_v, the effect of EDL forces decreased the relative apparent viscosity. Particle layering and structural arrangements were analysed for ϕ_v=43 and 52%. As the Debye length (i.e., the thickness of EDL) decreases, the particle layers near the walls weakened. The analyses reveal how at these high volume fractions, chain-like assemblies are transformed into hexagonal arrangements.

This paper describes the effects of uniform and non-uniform liquid co-flow on the bubbly flow in a rectangular column (with two inlets) deliberately aerated unevenly. The two vertical bubbly streams, comprising uniform bubbles, started interacting downstream of the trailing edge of a splitter plate. This study quantifies the emergence of buoyancy driven flow patterns as a function of the degree of a-symmetric gas sparging and (non-)uniform liquid co-flow by using Bubble Image Velocimetry (BIV) and dual-tip optical fibre probes. Without liquid co-flow, small differences in the gas fraction of the left and right inlet had a large effect on the mixing pattern, whereas a liquid co-flow stabilized a homogeneous flow regime and the flow pattern was less sensitive to gas fraction differences. Void fractions, bubble velocities and chord lengths were measured at two fixed position in the flow channel, whereas BIV provided a global overview of the flow structures. A correlation was developed to predict (a-symmetric) operating conditions for which the gas fraction of the left and right inlet are balanced, such that the bubble motion is governed by advection and no buoyancy driven flow structures arise. The data obtained is highly valuable for CFD validation and development purposes.

A spatially resolved one dimensional pressure filtration model was developed for a slurry of edible fat crystals. The model focuses on the expression step in which a cake is compressed to force the liquid through a filter cloth. The model describes the local oil flow in the shrinking cake modeled as a porous nonlinear elastic medium existing of two phases, viz. porous aggregates and interaggregate liquid. Conservation equations lead to a set of two differential equations (vs. time and vs. a material coordinate ω) for two void ratios, which are solved numerically by exploiting a finite-difference scheme. A simulation with this model results in a spatially resolved cake composition and in the outflow velocity, both as a function of time, as well as the final solid fat contents of the cake. Simulation results for various filtration conditions are compared with experimental data collected in a pilot-plant scale filter press.

Unique experiments were performed in a homogeneously sparged rectangular 400×200×2630 mm (W×D×H) bubble column with and without liquid co-flow. Bubbles in the range 4–7 mm were produced by needle spargers, which resulted in a very uniform bubble size. Dual-tip optical fibre probes were used to measure horizontal profiles of gas fractions, bubble velocities and bubble chord lengths for superficial gas velocities U_sg in the range 0.63–6.25 cm/s and superficial liquid velocities U_sl up to 20 cm/s. Images of the bubble column were captured and a Bubble Image Velocimetry technique was adopted to calculate bubble (parcel) velocities. For low gas fractions, when a homogeneous flow regime occurred, both methods agreed very well and the optical fibre probes were found to be rather accurate for our bubbles. A liquid co-flow was found to have a calming effect and to stabilize a homogeneous bubbly flow regime, with less spatial variation in gas fractions and bubble velocities. Bubble chord lengths were almost normally distributed and do not exhibit the theoretical triangular probability density functions. The mean cord lengths were in the range 1.9–3.5 mm and found to increase with U_sg and to decrease slightly with increasing U_sl, while a liquid co-flow significantly reduced the standard deviation of the chord length distribution.

The option of varying the molecular mass in multicomponent lattice Boltzmann simulations is being explored. First, results are presented for droplet formation at an aperture in a second immiscible liquid medium in which the difference in density between the two media is achieved by introducing asymmetry in the EOS, via adding particularly intra-component interaction forces in a pseudo-potential LB model. The second application for models with variable molecular masses is a single-phase heterogeneous laminar-flow tubular chemical reactor, where the molecular masses of reactants and products differ. In this application, tuning the molecular mass requires modification of the standard equilibrium distribution function as well as the use of an extended velocity set, in our case D2Q13. The method is validated against analytical solutions for canonical 1-D diffusion-reaction cases. In both the droplet formation study and the chemical reactors, the results of the exploratory 2-D simulations look qualitatively correct.

Using an immersed boundary-lattice Boltzmann method, we investigated the response of dense granular suspensions to time-varying shear rates and flow reversals. The evolution of the relative apparent viscosity and particle structures was analysed. The concentration of solids (ϕ_v) and particle Reynolds numbers (Re_p) were varied over the ranges 6%≤ϕ_v≤47% and 0.105≤Re_p≤0.529. The simulations included sub-grid scale corrections for unresolved lubrication forces and torques (normal and tangential). When ϕ_v surpasses 30%, the contribution of the tangential lubrication corrections to the shear stress is dominant. While for intermediate solids fractions we find weak shear-thinning, we see weak shear-thickening for ϕ_v>40%. We show how the structure and apparent viscosity of a suspension evolves after a reversal of the shear direction. For 47% solids, simulations with step changes in the shear rate show the effects of the previous shear history on the viscosity of the suspension.

Three-dimensional direct numerical simulations of dense suspensions of monodisperse spherical particles in simple shear flow have been performed at particle Reynolds numbers between 0.1 and 0.6. The particles translate and rotate under the influence of the applied shear. The lattice Boltzmann method was used to solve the flow of the interstitial Newtonian liquid, and an immersed boundary method was used to enforce the no-slip boundary condition at the surface of each particle. Short range spring forces were applied between colliding particles over sub-grid scale distances to prevent overlap. We computed the relative apparent viscosity for solids volume fractions up to 38% for several shear rates and particle concentrations and discuss the effects of these variables on particle rotation and cluster formations. The apparent viscosities increase with increasing particle Reynolds number (shear thickening) and solids fraction. As long as the particle Reynolds number is low (0.1), the computed viscosities are in good agreement with experimental measurements, as well as theoretical and empirical equations. For higher Reynolds numbers, we find much higher viscosities, which we relate to slower particle rotation and clustering. Simulations with a sudden change in shear rate also reveal a history (or hysteresis) effect due to the formation of clusters. We quantify the changes in particle rotation and clustering as a function of the Reynolds number and volume fraction.

We report experiments on bubble formation from needles with and without liquid co-flow, carried out with needles in the range of 0.79<d _n <2.06 mm, for gas flow rates up to 4.5 cm ³ /s per needle, and with liquid co-flow velocities up to 0.4 m/s. Bubble sizes and frequencies were obtained by means measuring an acoustic signal in the pressurized chamber upstream, which is validated by high-speed imaging analysis. Bubble contours, bubble growth curves and time return plots were obtained to analyse the bubble formation process. Different bubbling regimes are distinguished and a novel dimensionless pressure ratio is proposed to forecast the emergence of weeping and the transition from constant flow rate bubbling to constant chamber pressure bubbling. A single correlation for the non-dimensional bubble size with and without liquid co-flow was developed and validated with the experimental data obtained in the present study.

We perform direct numerical simulations (DNS) of emulsions in homogeneous isotropic turbulence using a pseudopotential lattice-Boltzmann (PP-LB) method. Improving on previous literature by minimizing droplet dissolution and spurious currents, we show that the PP-LB technique is capable of long stable simulations in certain parameter regions. Varying the dispersed-phase volume fraction , we demonstrate that droplet breakup extracts kinetic energy from the larger scales while injecting energy into the smaller scales, increasingly with higher , with approximately the Hinze scale (Hinze, AIChE J., vol. 1 (3), 1955, pp. 289-295) separating the two effects. A generalization of the Hinze scale is proposed, which applies both to dense and dilute suspensions, including cases where there is a deviation from the inertial range scaling and where coalescence becomes dominant. This is done using the Weber number spectrum , constructed from the multiphase kinetic energy spectrum , which indicates the critical droplet scale at which . This scale roughly separates coalescence and breakup dynamics as it closely corresponds to the transition of the droplet size distribution into a scaling (Garrett et al., J. Phys. Oceanogr., vol. 30 (9), 2000, pp. 2163-2171; Deane & Stokes, Nature, vol. 418 (6900), 2002, p. 839). We show the need to maintain a separation of the turbulence forcing scale and domain size to prevent the formation of large connected regions of the dispersed phase. For the first time, we show that turbulent emulsions evolve into a quasi-equilibrium cycle of alternating coalescence and breakup dominated processes. Studying the system in its state-space comprising kinetic energy , enstrophy and the droplet number density , we find that their dynamics resemble limit cycles with a time delay. Extreme values in the evolution of are manifested in the evolution of and with a delay of and respectively (with the large eddy timescale). Lastly, we also show that flow topology of turbulence in an emulsion is significantly more different from single-phase turbulence than previously thought. In particular, vortex compression and axial straining mechanisms increase in the droplet phase.