GE Cortes Garcia, KMP van Eeten, MM de Beer, JC Schouten, and J van der Schaaf (2018). On the bias in the Danckwerts’ plot method for the determination of the gas–liquid mass-transfer coefficient and interfacial area. Fluids 3, p. 18.
The Danckwerts’ plot method is a commonly used graphical technique to independently determine the interfacial area and mass-transfer coefficient in gas–liquid contactors. The method was derived in 1963 when computational capabilities were limited and intensified process equipment did not exist. A numerical analysis of the underlying assumptions of the method in this paper has shown a bias in the technique, especially for situations where mass-transfer rates are intensified, or where there is limited liquid holdup in the bulk compared to the film layers. In fact, systematic errors of up to 50% in the interfacial area, and as high as 90% in the mass-transfer coefficients, can be expected for modern, intensified gas–liquid contactors, even within the commonly accepted validity limits of a pseudo-first-order reaction and Hatta numbers in the range of 0.3 < Ha < 3. Given the current computational capabilities and the intensified mass-transfer rates in modern gas–liquid contactors, it is therefore imperative that the equations for reaction and diffusion in the liquid films are numerically solved and subsequently used to fit the interfacial area and mass-transfer coefficient to experimental data, which would traditionally be used in the graphical Danckwerts’ method.
AN Manzano Martínez, KMP van Eeten, JC Schouten, and J van der Schaaf (2017). Micromixing in a rotor-stator spinning disc reactor. Industrial and Engineering Chemistry Research 56, p. 13454.
This paper presents the micromixing times in a rotor–stator spinning disc reactor. Segregation indices are obtained at different rotational speeds performing the Villermaux–Dushman parallel-competitive reaction scheme. Consequently, the corresponding micromixing times are calculated using the engulfment model, while considering the self-engulfment effect. It was found that the segregation index decreases with an increasing disc speed. Furthermore, for the investigated operational conditions, the estimated micromixing times are in the range of 1.13 · 10–4 to 8.76 · 10–3 seconds, in agreement with the theoretical dependency on the energy dissipation rate of ε–0.5. In a rotor–stator spinning disc reactor it is thus possible to further continue the theoretical trend of decreasing micromixing times with very high levels of energy dissipation rates that are unattainable in traditional types of process equipment.
KMP van Eeten, DHJ Hülsman, J van der Schaaf, and JC Schouten (2015). Hydrodynamical particle containment in a rotor-stator spinning disc device. AIChE Journal 61, p. 3656.
A novel type of rotor-stator spinning disk device is proposed which allows for the entrapment of solid particles solely by hydrodynamic means. In this new configuration, the solid rotating disk is replaced with two conjoined rotors with a variable gap spacing. Liquid is fed through the top stator and can flow out again through the rotor-rotor interior and the hollow rotation axis. Moreover, the volume between the two rotors is optionally filled with a highly porous reticulated carbon foam. It was found that particle containment was strongly improved by the presence of this reticulated foam as it hinders the buildup of centripetal boundary layer flow near the disks in the interior of the rotor-rotor assembly. These centripetal boundary layers drag along particles resulting in a loss of containment. Experiments utilizing glass beads showed that particles with a diameter down to 17.8 µm can be completely entrapped when a carbon foam is placed between the two conjoined disks at rotor speeds up to the maximum investigated value of 178 rad s−1. Additionally, the rotor-rotor gap did not have an effect on the particle entrapment level when the reticulated carbon foam was omitted and can be ascribed to the build-up of boundary layers, which is independent of rotor-rotor distance.
KMP van Eeten, R Verzicco, J van der Schaaf, GJF van Heijst, and JC Schouten (2015). A numerical study on gas-liquid mass transfer in the rotor-stator spinning disc reactor. Chemical Engineering Science 129, p. 14.
The gas–liquid mass transfer coefficient was investigated in a novel multiphase reactor: the rotor–stator spinning disc reactor. Direct Numerical Simulations of the flow field around a single bubble in the reactor showed that vortex stretching invoked the presence of turbulence inside the thin liquid film surrounding the bubble. The Direct Numerical Simulations further provided a measure for the eddy diffusivity in the thin liquid film caused by this increase in vorticity. An expression was subsequently derived from a mass balance using these eddy diffusivities in order to estimate the order of magnitude of gas–liquid mass transfer coefficients. These estimates were found to lie more in line with experimental results in literature than previously used mass transfer models based on Higbie׳s penetration theory.
KMP van Eeten, HHH Houben, J van der Schaaf, and JC Schouten (2014). An experimental and theoretical study on the size of bubbles formed between a rotating disc and a stationary wall. Chemical Engineering Science 109, p. 251.
This paper comprises an experimental and theoretical study on gas bubble formation in a liquid in a spinning disc device. Measurements were done in a device with a rotor radius of 0.135 m and a distance of 2 · 10−3 m between the rotating disc and the stationary wall. Experiments have been performed at rotational velocities where the Von Kármán boundary layer at the rotor and the Bödewadt layer at the stationary wall interfere. It was found that the highest angular velocities resulted in the smallest average bubble diameters (3.32±0.662 mm), while at the highest gas mass flow rate and lowest rotational velocities, the largest bubbles were produced (15.3±1.89 mm). Variation of liquid density from 1000 to 1150 kg m−3 and liquid viscosity from 0.81 to 1.70 mPa s appeared to have a negligible effect on the bubble size. A model was derived from a mass and momentum balance, which incorporates the unsteady effects of added mass, gas momentum, bubble growth rate, drag force and centrifugal buoyancy. The general trends in calculated average bubble size are in agreement with the experimental results and the model calculations were able to simulate average bubble diameters within a single experimental standard deviation.
KMP van Eeten, J van der Schaaf, GJF van Heijst, and JC Schouten (2013). Lyapunov-stability of solution branches of rotating disk flow. Physics of Fluids 25, p. 073602.
This paper investigates the stability of solutions to the problem of viscous flow between an infinite rotating disk and an infinite stationary disk. A random perturbation, satisfying the Von Kármán similarity transformation, is applied to the steady velocity profiles for four solution branches, after which the disturbance propagation is tracked as a function of time. It was found that three of the four solution branches (including the Batchelor solution) were Lyapunov-stable and the development of the Lyapunov-coefficients as a function of the Reynolds number was determined. Stewartson-type of flow was found to be unstable and developed into a flow field corresponding to the Batchelor-solution. The mechanism with which the non-viscous core in this latter type of flow acquired its angular momentum was identified as being dominated by radial convection towards the axis of rotation.
KMP van Eeten, J van der Schaaf, GJF van Heijst, and JC Schouten (2012). Boundary layer development in the flow field between a rotating and a stationary disk. Physics of Fluids 24, p. 033601.
This paper discusses the development of boundary layers in the flow of a Newtonian fluid between two parallel, infinite disks. One of the disks is rotating at a constant angular velocity while the other remains stationary. An analytical series approximation and a numerical solution method are used to describe the velocity profiles of the flow. Both methods rely on the commonly used similarity transformation first proposed by Von Kármán [T. von Kármán, ZAMM 1, 233 (1921)]. For Reh < 18, the power series analytically describe the complete velocityprofile. With the numerical model a Batchelor type of flow was observed for Reh> 300, with two boundary layers near the disks and a non-viscous core in the middle. A remarkable conclusion of the current work is the coincidence of the power series’ radius of convergence, a somewhat abstract mathematical notion, with the physically tangible concept of the boundary layer thickness. The coincidence shows a small deviation of only 2% to 4%.