Investigation of reacting flow fields in miscible viscous fingering by a novel experimental method

The reacting flow fields in reactive miscible viscous fingering in a Hele‐Shaw cell studied by Nagatsu and Ueda had not been completely elucidated, mainly because one cannot exactly recognize where and when the reaction takes place in the reactive fingering pattern. We developed a novel experimental method that allowed us to identify the reaction region in the fingering pattern employed in the previous studies. The novel method involves switching of the less‐viscous liquid injected in both the nonreactive and reactive experiments. By using the novel method, we succeeded in showing how the reaction region in the fingering pattern was affected by the initial reactant concentrations, the Péclet number, and time. We propose physical models of the reacting flow field in the cell's gap direction that can explain the obtained experimental results. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Effect of Korteweg stress on viscous fingering of solute plugs in a porous medium

In this work we first present a unified framework for the analysis of miscible and immiscible viscous fingering. To establish this we introduce the Korteweg stresses for miscible fluids which arise due to concentration gradients and are analogous to surface tension between two immiscible fluids. For the immiscible fingering we use the continuum representation of surface tension which treats surface tension as a body force. This analogy between the two types of fingering helps us obtain a better insight into the role of Korteweg stresses in miscible fluids and enables in resolving ambiguities in the role of these stresses in a dynamic system. Finally we examine the effect of Korteweg stresses to see how they can be a stabilizing influence for miscible viscous fingering of finite slices in a chromatography column and discuss if this can give an extra control option to improve the performance of the separation.     

Numerical simulation of viscous fingering of shear‐thinning fluids

The viscous fingering instability of miscible shear‐thinning fluids has been examined using a pseudo‐spectral numerical technique based on the Hartley transform. The instability was studied for a flow in a rectilinear Hele‐Shaw cell, and the shear‐thinning character of the fluids has been modelied using the Carreau equation. New mechanisms of viscous fingering not previously observed in the case of similar Newtonian flow displacements have been identified. These mechanisms, which are reminiscent of the fractal patterns observed in experimental studies, were interpreted in terms of the velocity‐dependent mobility of the flow.     

The Simulation of Single Phase Tubular Reactors with Incomplete Reactant Mixing

The influence of reactant segregation and of the rate of reactant mixing on fast homogeneous chemical reaction rates is considered and a computer model presented to simulate the effect of mixing within a tubular reactor fed by two segregated but miscible streams. Results obtained from use of the model are discussed.     

An investigation of mass transfer in a countercurrent three-phase fluidized bed

The rate of absorption with chemical reaction has been measured in a bed of solid spheres fluidized by upward flowing gas and irrigated by downward flowing liquid for different reactant concentrations in the liquid phase and different values of the liquid superficial velocity.Values of the effective inerfacial area per unit volume of a static bed, a, and the gas-film mass-transfer coefficient, k_g, are reported as functions of the liquid superficial velocity.     

Hydrodynamical instability of spatially extended bistable chemical systems

Hydrodynamical density fingering of chemical fronts separating two miscible, stable steady states of different chemical composition, and hence density, can lead to complex spatio-temporal dynamics. The most striking feature of such dynamics is the disconnection of droplets of one stable steady state from fingers invading the other stable steady state. Such disconnected droplets do not exist in pure density fingering and are thus the result of the bistable kinetics. We study such dynamics by direct numerical simulations of Darcy's law for flow in Hele-Shaw cells coupled to the kinetic equation for the concentration of a chemically reacting solute controlling the density of the miscible solutions. The concentration of this solute obeys a simple cubic model leading to bistability. Experimental realization of such dynamics in spatially extended Hele-Shaw cells calls for the use of the concept of spatial bistability which implies construction of new continuously fed open reactors.     

Concentration profiles of reactant in a viscous finger formed during the interfacially reactive immiscible displacements in porous media

A simple unsteady-state convective-diffusion model was employed to simulate reactant concentration profiles in a viscous finger, which is created during interfacially reactive displacement of two immiscible fluids in porous media. A two-dimensional system was considered. It was assumed that the finger grew in one direction at a constant flow rate and that a fast chemical reaction was taking place at the oil–water interface layer. Interaction with reservoir rock was also taken into account. Two types of convective flow were studied: steady-state laminar flow with a parabolic velocity profile and plug flow with a flat velocity profile. The numerical method of lines was used to solve the model equation. A significant decrease of reactant concentration in the water phase, especially in the fingertip area was observed. This can cause important changes in interfacial behavior and influence the efficiency of fluid displacement. The model numerically confirmed previous visual observations of displacing patterns during the displacement of acidified oil by alkaline solutions.     

Characteristics of exothermic reaction fronts in the gasless combustion system

Gasless combustion model of the self-propagating high-temperature synthesis process was numerically studied in the non-adiabatic cylindrical sample. The model equations, which are very stiff in the dimension of length as well as time, were solved using finite difference method on adaptive meshes. Travelling waves with constant pattern were observed for adiabatic systems. For higher values of heat of reaction and activation energy, combustion fronts started to oscillate for adiabatic and non-adiabatic systems. Simple and complex oscillatory fronts were observed. Multi-peak and irregular oscillations were also detected to presumably result in the gasless chaotic combustion. In oscillatory fronts the temperature can overshoot the adiabatic reaction temperature to result in the complete conversion of solid reactant. In the two dimensional non-adiabatic cylindrical sample in the domain of longitudinal and angular directions, oscillatory piston waves were observed. In addition asymmetrical fingering as well as rotating waves were detected for an asymmetrical perturbation. For the adiabatic annulus cylindrical sample, the velocity of propagating fronts increased with time and the temperature overshooted the adiabatic reaction temperature if the sample were ignited from the inside. If the sample were ignited from the outside, the velocity of propagating fronts decreased with time and the temperature again overshooted the adiabatic reaction temperature. For smaller diameter of sample, the temperature increased very slowly with time for inside ignition. The temperature after ignition increased very fast overshooting the adiabatic reaction temperature for outside ignition. After several oscillations, the reaction rate decreased and the region with very slow reaction was established.     

Non-linear interactions of dynamic reactive interfaces in porous media

Viscous fingering of reactive miscible flow displacements in a homogeneous porous media is examined. A general model where the two reactants and the chemical product have different viscosities is adopted. The problem is formulated using the continuity equation, Darcy's law, and volume-averaged forms of convection–diffusion–reaction equations for mass balance, and is solved using a pseudo-spectral method. A parametric study was performed to examine the effect of the Peclet number and the log-mobility ratios between the chemical product and the reactants. It is shown that the development and growth of the instability as well as the efficiency of the reaction expressed in terms of the amount of chemical product can be predicted based on the mobility ratio at the initial front between the two reactants and effective mobility ratios between the chemical product and either one of the two reactants. Furthermore, it is reported that larger Peclet numbers lead to slower rates of chemical production.     

甲烷水蒸气重整制氢反应及其影响因素的数值分析

本文通过建立包含动量、能量、质量以及化学反应的多物理场耦合数值模型, 以多孔介质模型表征催化剂层, 对工业转化炉管中的甲烷水蒸气重整制氢过程进行了详细分析。计算得到了转化炉管内甲烷重整过程反应物及产物气体的速度、温度及浓度场分布, 以此分析了甲烷重整制氢过程的反应特性, 并阐明了转化炉管的壁面温度、原料气入口水碳比以及入口速度对甲烷转化率的影响。结果表明:水蒸气重整在转化炉管的入口区域反应迅速, 沿着气体流动方向, 反应速率由于反应物浓度的不断降低而减小, 导致混合气体流动速度和温度也逐渐趋于稳定;水碳比和转化管壁面温度的增加以及原料气体入口流速的降低, 都会提高甲烷的转化率。本文所得到的结论对于优化实际生产中甲烷水蒸气重整制氢反应的工况条件具有一定的参考和借鉴意义。     

Mass transfer with chemical reaction inside single droplets and gas bubbles: Mathematical mechanisms

The effect of a first order homogeneous chemical reaction on the rate of mass transfer inside liquid droplets and gas bubbles was determined. One of the models presented assumes the non-oscillating dispersed phase is moving in the creeping flow region and the Hadamard stream functions are applicable. The results of the numerical solution of this model are presented graphically as a function of the Peclet number, reaction number, and dimensionless contact time. An oscillating droplet model is also presented which considers the effect of reaction within the dispersed phase.     

A simple model for precisely describing liquid–liquid equilibrium and better understanding extraction mechanism

Describing distribution of solute equilibrium concentrations between two solvents is important for liquid–liquid equilibrium (LLE). Due to non-ideality of solutions, the distribution usually possesses non-linear characteristic. In this work, the non-ideality was simply represented by pseudo reaction (or chemical interaction) between the solute and the solvent without recourse to the complicated activity coefficient models. On this basis, a simple model, which combined physical partition and pseudo reaction equilibriums, was proposed and can precisely correlate various LLE systems, for example, the immiscible, the partially miscible, the organic, and even the electrolyte. More interestingly, the model can accurately reflect intrinsic mechanism by fitted parameters, that is, stoichiometric ratio of the pseudo reaction and total equilibrium constant for the interaction intensity. Besides, influence of the solute, the extractant, and temperature on the parameters was discussed deeply. This simple model is promising for precisely describing the LLE systems and better understanding their intrinsic extraction mechanism.     

MASS TRANSFER WITH CHEMICAL REACTION IN PARTIALLY WETTED FLAT PLATE CATALYST PELLETS: RIVULET FLOW ANALYSIS

Mass transfer with chemical reaction is analyzed in a system formed by a flat plate solid catalyst, partially wetted by a flowing rivulet of a liquid in contact with a stagnant pure gas. The paper solves the fluid dynamic problem of the liquid phase first, and afterwards incorporates the mass transfer and the chemical reaction. The system is assumed to be isothermal and at steady state, with a first order kinetics whose limiting reactant is in the gas phase. This work studies the influence of the gas-liquid surface tension, the liquid reactant flow rate, the liquid viscosity and the angle of inclination of the solid, upon the wetting factor. The model proposed also predicts the effect of these parameters and the Thiele modulus on the overall effectiveness factor and the molar flux of the limiting gaseous reactant at the catalytic solid-liquid interface in a direct way. This approach makes the wetting factor a non-manipulated variable.     

Investigation of concentration‐dependent diffusion on frontal instabilities and mass transfer in homogeneous porous media

Mass transfer plays an important role in influencing the efficiency of miscible displacements in solvent‐based processes in enhanced oil recovery. The mass transfer rate because of the pure molecular diffusion is very slow. However, this process can be greatly enhanced by the appearance of frontal instabilities called viscous fingering mechanisms, which are beneficial for improving the mixing and mass transfer between the injected solvent and oil. Instead of a piston‐like displacement, the interface between solvent and oil is very convoluted with intricate finger‐like patterns of the less viscous solvent intruding into the highly viscous oil. This intrusion significantly increases the surface area of contact of the two fluids and leads to more efficient mass transfer and mixing. Experimental measurements on the diffusion coefficients of two miscible fluids indicate that, instead of a constant diffusion coefficient (CDC), a concentration‐dependent diffusion coefficient (CDDC) is more realistic. In the present study, a CDDC relation in which the diffusion coefficient is exponentially proportional to concentration is adopted. Its effect on the development of frontal instabilities is examined through highly accurate nonlinear numerical simulations. The differences between the CDDC case and the widely assumed CDC case are discussed. Furthermore, the enhancement of frontal instabilities on mass transfer when the CDDC is considered is investigated at various mobility ratios and Peclet numbers. The special characteristics for the CDDC case indicate its important role in miscible displacements. Eventually, the relation of breakthrough time to parameters is correlated to accurately predict the breakthrough time in any CDDC scenario.     

Linear and non-linear analyses on the onset of miscible viscous fingering in a porous medium

The onset of miscible viscous fingering in porous media was analyzed theoretically. The linear stability equations were derived in the self-similar domain, and solved through the modal and non-modal analyses. In the non-modal analysis, adjoint equations were derived using the Lagrangian multiplier technique. Through the non-modal analysis, we show that initially the system is unconditionally stable even in the unfavorable viscosity distribution, and there exists the most unstable initial disturbance. To relate the theoretical predictions with the experimental work, nonlinear direct numerical simulations were also conducted. The present stability condition explains the system more reasonably than the previous results based on the conventional quasi-steady state approximation.     

The effect of liquid flow distribution on the behaviour of a trickle bed reactor

A mathematical model has been formulated of the effect of flow distribution of the liquid phase carrying a dissolved reactant on the progress of an nth order, irreversible, catalytic reaction with heat effects in an adiabatic trickle bed reactor. The model has been stated in terms of the density of irrigation, temperature and concentration of the reactant in the liquid, all treated as spatially distributed variables. Provisions have been made to account for the existence of the flow down the surface of the wall, which has no catalytic effect.Local concentration and temperature have been proven to be coupled by the invariant T + Uγc = γU. The same invariant governs also local concentration and temperature of the wall flow. Mathematically, the model is represented by a coupled set of nonlinear parabolic partial differential equations enabling concentration and temperature fields to be obtained for an arbitrary type of liquid distribution and intensity of the wall flow.Numerical solutions have been obtained by the finite-difference method simulating reactors irrigated by liquid distributors as central discs of different radii, or a central annulus, and strongly exothermic reactions with the reaction order ranging between 0.1 and 2. Numerical results have shown the effect of liquid distribution on the overall reaction conversion to be very complex. Optimum initial distribution varies depending on the reaction order as well as the required degree of conversion. In general, however, the entrance region flow pattern may play a significant role in affecting especially reactions exhibiting kinetics close to zero order (hydrogenations). The effect of the wall flow has been found unambigously adverse to reaching high conversions and of increasing importance for low order reactions.     

CONVECTIVE DIFFUSION IN HETEROGENEOUS CATALYTIC DUCT REACTORS WITH NONCIRCULAR CROSS SECTION AND NONUNIFORM CATALYST ACTIVITY

Steady-state simulations of convection, diffusion and first-order irreversible heterogeneous chemical reaction are presented for catalytic channels with rectangular cross section and nonuniform catalyst activity. Finite difference results from the microscopic three-dimensional mass transfer equation also satisfy the cross-section-averaged one-dimensional form of the same equation. Comparisons between viscous flow and plug flow in square cross-section channels suggest how previous inferences of surface-averaged reaction velocity constants from plug flow simulations should be modified when convective diffusion in the mass transfer boundary layer adjacent to the catalytic surface is modeled correctly. Over the following range of Damkohler numbers (i.e., 20 to 10³), viscous flow in rectangular ducts with aspect ratios between two and three can be approximated by the corresponding problem in tubes with the same effective diameter. For Damkohler numbers between 0.5 and 10³, aspect ratios greater than 20 are required to simulate viscous flow between two parallel plates with catalyst coated on both walls. At low Damkohler numbers where reactant diffusion toward the catalytic surface is not the rate-limiting step, nonuniform activity profiles suggest that most of the catalyst should be deposited in regions that are easily accessible to the reactants. However, this strategy for converting reactants to products is not more effective than uniform deposition in the diffusion-limited regime.     

微尺度下液-液流动与传质特性的研究进展

微化学工程是现代化学工程学科的前沿,主要研究微时空尺度下流体流动、传热、传质现象与反应规律。着重介绍近十年来微通道内单相流体流动、互溶液-液两相流体流动与混合、互不相溶液-液两相流体流动与传质的理论和实验的最新研究进展。     

Direct production of crystalline boric acid through heterogeneous reaction of solid borax with propionic acid: Operation and simulation

The production of boric acid through reaction of borax crystals with propionic acid was investigated in batch mode. It was found that the product boric acid precipitates on the solid borax reactant. An increase in the coefficient of variation of feed crystals resulted in an increase in the time of completion of the reaction. A sharp interface model with variable bulk fluid concentration of liquid reactant was developed for simulation of the process. The analytical solution of the series of rate equations involving liquid film, solid boric acid layer and chemical reaction resistances was obtained, and a new method for simultaneous determination of kinetic parameters was established. The chemical reaction and diffusion through the solid layer of boric acid were the rate controlling mechanisms. A numerical algorithm was also developed for predicting the fractional conversion of multisize borax crystals using the mono size kinetic parameters determined according to the proposed analytical method.