Damkohler Number Research Articles

NUMERICAL SIMULATION OF CARBONATE ROCKS DISSOLUTION NEAR THE WELLBORE

This paper examines the process of wormholes formation during hydrochloric acid treatment of well bottom-hole zone in carbonate formations. An algorithm for solving the problem of wormhole formation in a porous medium for a two-dimensional case was developed. Two-scale model (porescale and Darcy’s scale) taking into account convection, diffusion, and chemical reaction were used in this work in order to describe the dissolution of carbonates with hydrochloric acid (hydrochloric acid treatment). The initial distribution of the porosity field was generated as a distribution of random numbers around some mean value. Based on the distribution of the initial porosity field, the initial permeability field of the rock was calculated. The random distribution was used to describe the heterogeneity of the actual rock. The rest of the study parameters were taken from known experiments on the dissolution of carbonate cores. The numerical model was built for solving the system of equations for acid dissolution, and carbonate dissolution modes with hydrochloric acidwere obtained depending on the Damkohler number, Thiele modulus on the pore-scale and Darcy’s scale as a result of this research. Also, the optimal Damkohler numbers (injection rates) were found. The computer code for the problem of the development/growth of wormholes in a porous medium based on the developed algorithm was built using the C++ programming language.

Electrochemical Biosensor for Markers of Neurological Esterase Inhibition.

A novel, integrated experimental and modeling framework was applied to an inhibition-based bi-enzyme (IBE) electrochemical biosensor to detect acetylcholinesterase (AChE) inhibitors that may trigger neurological diseases. The biosensor was fabricated by co-immobilizing AChE and tyrosinase (Tyr) on the gold working electrode of a screen-printed electrode (SPE) array. The reaction chemistry included a redox-recycle amplification mechanism to improve the biosensor’s current output and sensitivity. A mechanistic mathematical model of the biosensor was used to simulate key diffusion and reaction steps, including diffusion of AChE’s reactant (phenylacetate) and inhibitor, the reaction kinetics of the two enzymes, and electrochemical reaction kinetics at the SPE’s working electrode. The model was validated by showing that it could reproduce a steady-state biosensor current as a function of the inhibitor (PMSF) concentration and unsteady-state dynamics of the biosensor current following the addition of a reactant (phenylacetate) and inhibitor phenylmethylsulfonylfluoride). The model’s utility for characterizing and optimizing biosensor performance was then demonstrated. It was used to calculate the sensitivity of the biosensor’s current output and the redox-recycle amplification factor as a function of experimental variables. It was used to calculate dimensionless Damkohler numbers and current-control coefficients that indicated the degree to which individual diffusion and reaction steps limited the biosensor’s output current. Finally, the model’s utility in designing IBE biosensors and operating conditions that achieve specific performance criteria was discussed.

Pore-scale investigation on reactive flow in non-uniform dissolved porous media considering immiscible phase by lattice Boltzmann method

The development of shale gas by injecting CO2 is complex, which results in the complicated dissolving process in porous formation. The components of underground formation are complicated and the non-uniform distribution greatly affects heat and mass transfer. In this study, the non-uniform porous media generated by quartet structure generation with dissolved solid, undissolved solid or immiscible phase is studied by lattice Boltzmann method, in which the flow, concentration and temperature fields are solved by Shan-Chen, mass transport and heat transfer lattice Boltzmann model, respectively. The results indicate that the reaction becomes further at the higher Damkohler number, while the diffusion becomes further at the higher Peclet number. The breakthrough time and the dissolution intersection points between the same solid distributions are affected by the solid dissolution rate non-uniformity. The breakthrough area is not affected by the vertical undissolved solid, but is considerably affected by the horizontal undissolved solid. But the breakthrough area is slightly affected by the vertical or horizontal immiscible gas, and the breakthrough time of the cases of immiscible gas is much more than those of undissolved solid.

CO2-permselective membrane reactor for steam reforming of methane

Steam reforming of methane (SRM) for hydrogen production with simultaneous CO2 removal in a CO2-permselective membrane reactor is studied experimentally and analyzed by a mathematical model. The membrane consists of a thick porous BYS (Bi1.5Y0.3Sm0.2O3-δ)-SDC (Sm0.2Ce0.8O2-δ) support layer and a thin, dense SDC-carbonate dual-phase layer (∼150 μm) for CO2 separation. Experimental results of CO2 permeation in the BYS-SDC membrane and SRM reactions in a fixed-bed reactor were conducted to obtain the CO2 permeation and SRM reaction rate parameters for the model. The model's reliability is confirmed by comparing the simulation results with atmospheric pressure experimental data for SRM in the BYS-SDC membrane reactor. Simulation results are presented to show the effects of permeation number (θ) and Damkohler number (Da) (which respectively represent membrane CO2 permeance and SRM reaction rate), feed pressure in the reaction side, and sweep side conditions on the performance of SRM in the membrane reactor. Overall, increasing θ results in an increase of H2 yield, retentate H2 concentration, CO2 recovery, and a decrease of retentate CO concentration, with such effects being enhanced at higher Da and feed pressure or vacuum in the sweep side. CO2 permeance affects CH4 conversion only at high feed pressures. At feed pressure of 5 atm and applying vacuum at the sweep side of the membrane reactor, the simulation results show that the membrane reactor can achieve over 99% CH4 conversion, H2 yield, and CO2 recovery and produce an essentially pure H2 stream with zero CO concentration at Da >10,000 and θ > 1.

Turbulent burning velocity and its unified scaling of butanol isomers/air mixtures

Turbulent expanding flames of butanol isomers/air mixtures are investigated in a wide range of conditions (T = 393 K, P = 1, 2 bar, ϕ = 0.7, 1.0, 1.3 and u’=0, 0.89, 1.77, 2.66 m/s). Turbulent burning velocity (ST) of four butanol isomers are determined, compared, and analyzed in detail. A unified scaling of ST is presented to consider the effects of molecular transport, during which different mathematical correlations are evaluated and modified. Results show that ST is increased with u’ and P. The maximum ST will shift to richer side compared to laminar condition, and the shifting trend will be more evident at high u’. The order of ST between four butanol isomers is similar to that of laminar burning velocity (SL) and it is: n-butanol > i-butanol≈s-butanol > t-butanol. The discrepancy in ST is mainly induced by SL, while turbulence stretch and flame intrinsic instability play an amplifying role. Turbulent Reynolds number (ReT), Karlovitz number (Ka), and Kobayashi model can all give a reasonable correlation of current experimental data while an acceptable scatter exists in Damkohler number (Da) correlation. The primary ST models are modified by Len and the value of n is unified to −0.35 for different correlations, which ensures the molecular transport effects at the quantitative level. It appears that these correlations are applicable for hydrocarbons and alcohols, which can be more simplified and suitable ST correlations for a wide range of conditions.

Monomer Transport in Emulsion Polymerization II: Copolymerization

AbstractThe method for evaluating the Damkohler number for monomer transport during emulsion homopolymerization is extended to copolymerization. It is shown that monomers that are not monomer‐transport limited during homopolymerization may become more so during binary copolymerization, and monomers that are monomer‐transport limited during homopolymerization may become less so during binary copolymerization.

Optimization Packed Bed Column Reactor Parameters for Enzymatic Hydrolyzing of Lactose

The purpose of the study to increase the conversion ratio of lactose hydrolysis to glucose and galactose via packed bed reactors containing immobilised enzyme and design a reactor that will achieve a high conversion ratio in continuous systems. For this purpose, the dynamic properties of a packed bed reactor containing immobilised β-galactosidase enzyme on Duolite A 568 resin were determined by a stimulus-response technique and moment analysis. The relation between kinetic behaviour and The dynamic response was determined via Peclet number (Pe) and Damkohler number (Da). This study was completed with the development of a mathematical model using parameters such as flow dynamics, column geometry, enzyme immobilization performance and the behaviour of the enzyme in the column. According to the results, the particle size, L/D ratio and flow rate appropriate for this system were determined as 390 m, 10 and 4 mL/min, respectively. Furthermore, 55% conversion of lactose from milk was obtained using the enzyme immobilized packed bed reactor.

Modeling of Particle Deposition in a Two-Fluid Flow Environment

Particle deposition occurs in many engineering multiphase flows. A model for particle deposition in two-fluid flow is presented in this article. The two immiscible fluids with one carrying particles are model using incompressible Navier-Stokes equations. Particles are assumed to deposit onto surfaces as a first order reaction. The evolving interfaces: fluid-fluid interface and fluid-deposit front, are captured using the level-set method. A finite volume method is employed to solve the governing conservation equations. Model verifications are made against limiting cases with known solutions. The model is then used to investigate particle deposition in a stratified two-fluid flow and a cavity with a rising bubble. For a stratified two-fluid flow, deposition occurs more rapidly for a higher Damkholer number but a lower viscosity ratio (fluid without particle to that with particles). For a cavity with a rising bubble, deposition is faster for a higher Damkholer number and a higher initial particle concentration, but is less affected by viscosity ratio.

Viscous propulsion of a two-dimensional Marangoni boat driven by reaction and diffusion of insoluble surfactant

A simple model of viscous propulsion of a two-dimensional Marangoni boat caused by surface diffusion and sublimation (reaction) of insoluble surfactant is formulated and studied analytically. The speed of the boat is determined explicitly as a function of a Damkohler number by fully resolving the Marangoni-induced bulk flow caused by the release of surfactants onto the surface of a semi-infinite viscous fluid. A complementary general integral formula for the boat speed is also derived using the reciprocal theorem.

Effect of first order chemical reactions through tissue-blood interface on the partial pressure distribution of inhaled gas

Gas exchange is an essential process to get fresh oxygenated air from the environment. In the human respiratory system, partial pressure is responsible for exchanging gas between tissue-blood capillary (inter capillary). However, the mechanisms of partial pressure distribution in the human respiratory system remain incompletely understood in terms of inter-capillary transmission with tissue porosity and reactive boundary conditions. In this paper, we worked on the spatial (radial) and temporal variations of inhaled gas partial pressure through inter capillary. We assumed that the tissue of alveoli is porous and the material of blood capillary is absorptive and reactive and gas could bear linear first-order kinetic reactions, one is reversible among the material of blood capillary and the other is irreversible into the surrounding tissue. Mathematical modeling is done by using diffusion equation; and the effect of various dimensionless parameters e.g. the Damkohler number (DA), phase partitioning number (α), dimensionless absorption number (Γ) are analyzed. Numerical simulation shows that an increment in porosity does not change convection speed but the diffusion of gas increases in alveolar tissue, resultantly, partial pressure gradient of the gas decreases in tissue and increases in blood capillary. However, by increasing the breathing rate, the partial pressure of the gas inside the blood first decreases, and after some time it increases gradually with the breathing rate. Additionally, the dispersion coefficient advanced toward its steady-state in a short time at absorption rate 0.1 and Damkohler number while long-time dispersion is achieved at porosity ϵ = 0.9, absorption rate Γ = 1, and phase exchange rate Ultimately, the findings of this study can be helpful for a better understanding of dispersion through human lung affected by aging and various lung diseases.

Contributions of flame thickening and extinctions to a heat release rate marker of intensely turbulent premixed hydrogen-enriched methane-air flames

Influences of the reaction zone thickening and extinctions on a heat release rate marker of extremely turbulent hydrogen-enriched methane-air flames are investigated using simultaneous planar laser-induced fluorescence of formaldehyde molecule and hydroxyl radical as well as separate stereoscopic particle image velocimetry techniques. The fuel-air equivalence ratio is set to 0.7, and the amount of hydrogen-enrichment varies from 0% to 70% by volume. Mean bulk flow velocities ranging from 5 to 35 m/s with Reynolds and Karlovitz numbers of 18 to 2729 and 0.1 to 76.0, respectively, are examined. It is shown that, by increasing the turbulence intensity, the preheat and reaction zone thicknesses can increase to values that are, respectively, 6.3 and 4.9 of the corresponding laminar flames, which were measured experimentally. Broadening of these zones for intensely turbulent hydrogen-enriched methane-air flames is shown experimentally for the first time. Broadening of the reaction zone suggests that the flamelet assumption used for development of the turbulent flame speed formulations may not hold. Thus, a new formulation, which does not utilize the flamelet assumption, is developed and used to calculate a heat release rate marker of the tested flames. It is shown that, at small turbulence intensities, the heat release rate marker values follow those of the local consumption speed, which is developed in the literature based on the flamelet assumption. However, at large turbulence intensities, the estimated heat release rate marker features large values, and the ratio of this parameter to the local consumption speed is consistent with the ratio of the global and local consumption speeds reported in the literature. It is shown that the ratio of the normalized heat release rate marker to the normalized local consumption speed is correlated with the broadening of reaction zone, suggesting that the disparity between the values of the heat release rate marker and the local consumption speed is linked to the reaction zone thickening. It is shown, although the flame thickening increases the heat release rate marker, extinctions decrease this parameter, which may contribute to the characterized bending behavior. It is argued that the amount of bending is related to the inverse of the Damköhler number. Using this, a mathematical formulation that allows for estimation of the heat release rate marker is developed. The predictions of the proposed formulation agree well with the normalized global consumption speeds of past experimental investigations.

Research on the influence of foaming gas in compressed air/nitrogen foam on extinguishing the n-heptane tank fire

Tank fires threaten the lives of people and pollute the environment for their intense radiant heat, rapid fire spread and explosion hazard. Compressed air/nitrogen foam (CAF/CNF), a cleaner fire extinguishing technique used for the tank fire suppression because halogen-based agents were prohibited for environmental reasons. In this work, the influence of foaming gas in CAF/CNF on extinguishing the n-heptane tank fire was firstly investigated. Firstly, it was found that CNF spreads faster with rapid increase in foam thickness, mainly due to its better stability and less evaporation. Secondly, after foam was discharged, there existed a short increase of the combustion intensity, associated with three monotonous regions and two time delays in the whole extinguishing time. The two time delays were caused by Rayleigh–Taylor instability and flame sheet shift, respectively, and the shift distance was larger for CNF. Finally, the influential factors contributing to flame extinction were exhibited to be mainly related to the decrease in liquid burning rate and gas-phase Damkohler number. Among these factors, foam spreading rate and thickness dominated due to coupled chemical and physical extinguishing effects. Resulted from some competitive effects, CNF was slightly more efficient at extinguishing tank fires than CAF.

Numerical analysis supported with experimental measurements of premixed oxy‐propane flames in a fuel‐flex gas turbine combustor

The combustion, exhaust-gas concentrations, and stability characteristics of premixed CO2-diluted oxy-propane (C3H8/O2/CO2) flames were investigated computationally using large eddy simulations (LES) in a swirl-stabilized model gas-turbine combustor. The simulations were carried out for ranges of equivalence ratio (Φ: 0.26-0.80) and oxygen fraction (OF: 35%-60%) at a fixed bulk inlet velocity of 5.2 m/s. The results indicate that the reaction rates increase with the increase of Φ and OF. Flames of the same adiabatic flame temperature (Tad) show similar macro-and flowfield-structures, in terms of flame shape, flow circulation, and temperature, and species distributions, irrespective of the values of Φ and OF. At higher Tad, the flames were observed to be more compact with reduced flame thickness and higher Damkohler number (Da). In all cases, Da > 1 were observed, indicating the dominance of reaction rate in oxy-propane flames than the diffusion rate. A secondary IRZ was also observed near the outlet of the combustor. Mixture composition and combustion temperature influence the CO concentration at the combustor exit. The highest CO emission was observed at 0.17 ppm for the flame with highest Φ and Tad, meanwhile no CO emission was seen for flames with high OF and low Φ among the studied cases. Highlights Adiabatic flame temperature is a quantifying parameter for flame characterization. The outer recirculation zone plays a vital role in flame anchoring/stabilization. Primary inner recirculation zone has several eddies controlled by the equivalence ratio. Secondary inner recirculation zone was also observed near the combustor outlet. Elevated CO emission was observed at high equivalence ratio and low oxygen fraction.

Investigation of the effect of DamKohler and Stanton non-dimensional numbers on the stability of continuous stirred tank reactors

The fully mixed continuous stirred tank reactor is an important type of industrial reactors mainly used to produce high volume products such as petrochemicals, detergents, sanitary products and products that are in demand in the market. Knowing the dynamic behavior of chemical reactors is of great importance in setting up, designing, controlling and stopping reactors. In this paper, the effect of non-dimensional numbers Damkohler and Stanton on the stability of a continuous stirred tank reactor in which a first-order exothermic reaction takes place is investigated. First, a mathematical model of the system's dynamic behavior was presented. Then, by simultaneous solving of the equations of mass and energy around the fixed point in MATLAB software, the effect of the mentioned numbers was investigated. The results show that the continuous stirred tank reactor shows different behaviors in different ranges of Damkohler and Stanton numbers. This reactor behaves unstable in small and large ranges of Damkohler and Stanton numbers due to the presence of mixed or positive and negative eigenvalues. The best range for Damkohler and Stanton numbers is close to 1, because in this range the reactor shows stable behavior due to having two negative eigenvalues. In this range, in addition to the stability, the conversion is also 100%. Finaly the ratio of Stanton to DamKohler was investigated as St / Da˃1 and St / Da = 1. If St / Da = 1, the system is in steady state, but in St / Da˃1, the system moves away from steady state.

Upscaling bimolecular reactive transport in highly heterogeneous porous media with the LAgrangian Transport Eulerian Reaction Spatial (LATERS) Markov model

The LAgrangian Transport Eulerian Reaction Spatial (LATERS) Markov model was developed to predict upscaled bimolecular reactive transport in a flow around an array of solid cylinders. This method combines the stochastic Lagrangian Spatial Markov model (SMM) to predict transport and a volume averaged reaction rate equation to predict reactions of the form $$A+B\rightarrow \emptyset$$ . Here, we extend the LATERS Markov model to upscale bimolecular reactive transport in a Darcy flow through an idealized heterogeneous porous medium. In agreement with previous literature, the accuracy of the prediction is a function of the Damkohler (Da) numbers, i.e., high Da are more challenging because of incomplete mixing. It was found that a key component which must be incorporated into the upscaled model in these high Da systems is the idea that nearby A and B particles should be more likely to react than those that are farther apart. This is here achieved by appropriately reducing the resolution of the spatial grid employed to resolve the reactive process.

A pore–scale investigation of surface roughness on the evolution of natural fractures during acid dissolution using DBS method

Natural fractures are the essential pathways by which water, hydraulic fracturing fluids, oil and gas, reactants, and contaminants can travel in the porous media. To fully understand subsurface processes relevant to the fluid transport in natural fractures, the Darcy–Brinkmann–Stokes (DBS) pore–scale method was employed to investigate the effects of fracture surface roughness on the reactive transport and dissolution mechanisms of porous media with natural fractures. The advection–diffusion equation described the transport of reactants, and the incorporated mass balance equation dynamically tracked the fracture evolution during acid dissolution. The model has been developed based on the OpenFOAM platform, a CFD open–source code. The partial differential equations were discretized by the Finite Volume Method (FVM) on the structured grids. We studied the dissolution mechanisms, effective surface area, and normalized porosity–permeability relationship on different surface roughness. An Indiana limestone sample with different calcite content was investigated to elucidate the impacts of unreacted minerals on acidizing process. We categorized the solid dissolution of fractures into three dissolution patterns regarding the second Damkohler number and the Peclet number—face dissolution, front dissolution, and edge dissolution—, where the front dissolution pattern and the face dissolution pattern had the shorter time to consume up the rock in the fractured system. The quantified pore–scale parameters obtained from this study are expected to improve the accuracy of Darcy–scale models. The microscopic insights from the present study can guide effective operating parameters and enhance hydrocarbon production by optimizing the acidization process. • Pore–scale reactive transport and dissolution model in fractured porous media by DBS method. • Three dissolution and transport patterns in fractures were investigated. • The fracture evolution, dissolution rate, porosity–permeability relationship, effective surface area were elucidated. • μCT scan image model with different mineralogical content were analyzed.

Monomer transport in emulsion polymerization

AbstractA simple mathematical model of the transport of monomer from the monomer droplets to the polymerizing polymer particles during emulsion polymerization is developed. A Damkohler number for monomer transport in emulsion polymerization is proposed as the ratio of the maximum rate of polymerization (propagation) divided by the maximum rate of monomer transport out of the monomer droplets. This Damkohler number is calculated from literature values for a number of common monomers. Following standard practise for the use of such a Damkohler number in other reaction‐with‐transport systems, monomers with a Damkohler number above 0.1 are considered to be transport limited. Results of these calculations indicate that, while most monomers are not monomer transport limited, the higher molecular weight acrylates and methacrylates are.

Pore-Scale Study to Analyze the Impacts of Porous Media Heterogeneity on Mineral Dissolution and Acid Transport Using Darcy–Brinkmann–Stokes Method

Inherent pore structure of rocks has a significant impact on the acid–rock interaction during the acidizing process. In this study, a new pore-scale reactive transport model applying Darcy–Brinkmann–Stokes method has been developed based on the open-source computational fluid dynamics platform, OpenFOAM, and validated with experimental results. By solving the mass balance equation and advection–diffusion equation, the developed numerical simulation model could capture the evolving pore structure and track the distribution of transported acid. The impacts of different grain sizes and distributions on the dissolution rate of mineral (calcite), evolving porosity, and the transport characteristics of acid have been investigated. In addition, sensitivity analysis has been conducted with respect to the various Damkohler numbers and Peclet numbers in the digital rock image model of Niobrara formation. The changing volume fraction of pores could be categorized into three patterns with different dimensionless numbers—linear growth, pseudo-linear growth, and a flat S-curve growth. The simulation cases of Estaillades carbonate, Massangis Jaune carbonate models, and Sample X segmentation showed that their inherent grain distribution was highly influential on the mineral dissolution rate, where 3-D simulation showed the same results. The developed pore-scale reactive transport model can be applied to various systems with complex pore structures, in order to provide a guidance to predict the system responses such as dissolution rates, surface area, and porosity changes during acidizing in large-scale continuum model.

Premixed flame ignition in high-speed flows over a backward facing step

Ignition plays a major role in combustion science and the ignition sequences are a crucial element in many engines. The current study focuses on ignition in subsonic, high-speed flows of premixed methane/air mixture. The Minimum Ignition Energy (MIE) is first evaluated for quiescent and constant speed flows to analyze its dependency on the flow speed and the value of this speed beyond which ignition can not be achieved anymore. To be able to obtain flame ignition and stabilization at higher flow velocities, ignition by a spark placed in the recirculation zone of a backward facing step is studied. For a range of inlet velocities, Direct Numerical Simulation (DNS) is used to measure the energy required for flame stabilization and analyze ignition sequences. Results show that a flame kernel is first created in the low-speed recirculation zone for all speeds. However, the subsequent flame propagation away from the recirculation zone depends on the flow speed: while low-speed cases produce a global ignition, high-speed cases can lead to quenching when the flame tries to leave the recirculation zone and enter the high-speed region, leading to global ignition failure. DNS also allows to propose and verify criteria for ignition in terms of flow speed and spark energies: following scaling arguments by Shanbhogue et al. (2009) for flame stabilization over obstacles, a simple Damkohler number is shown to control the success of ignition. Finally, DNS results are used to investigate mechanisms controlling the success or failure of the ignition sequence: they show that heat losses to the combustor walls play a limited role while flame stretch on the flame elements leaving the recirculation zone have a major influence.

Improving Blowout Performance of the Conical Swirler Combustor by Employing Two Parts of Fuel at Low Operating Condition

Aiming at the problem of the narrow combustion stability boundary, a conical swirler was designed and constructed based on the concept of fuel distribution. The blowout performance was studied at specified low operating conditions by a combination of experimental testing and numerical simulations. Research results indicate that the technique of the fuel distribution can enhance the combustion stability and widen the boundary of flameout within the range of testing conditions. The increase of the fuel distribution ratio improves the combustion stability but leads to an increase in NOx emission simultaneously. The simulation results show the increase of the fuel distribution ratio causes contact ratio increase in the area of lower reference velocity and gas temperature increase. The increased contact ratio and temperature contribute to the blowout performance enhancement, which is identical to the analysis result of the Damkohler number. The reported work in this paper has potential application value for the development of an industrial burner and combustor with high stability and low NOx emission, especially when the combustion system is required to be stable and efficient at low working conditions.