In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propos...In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propose Bernoulli(G'/G)-expansion method to study nonlinear fractional differential equations(NFDEs).Subsequently,closed form solutions of the density-independent FDR equation are acquired successfully.In order to better understand the dynamic behaviors of these solutions,3D,contour map and line plots are given by the computer simulation.The results show that the proposed method is a reliable and efficient approach.展开更多
In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant r...In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the exp(-φ(ε))-expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.展开更多
New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends ...New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.展开更多
In this paper,finite element method(FEM)is used to solve two-dimensional diffu-sion-reaction equations of boundary layer type.This kind of equations are usually too complicatedand diffcult to be solved by applying the...In this paper,finite element method(FEM)is used to solve two-dimensional diffu-sion-reaction equations of boundary layer type.This kind of equations are usually too complicatedand diffcult to be solved by applying the traditional methods used in chemical engineering becauseof the steep gradients of concentration and temperature.But,these difficulties are easy to be over-comed when the FEM is used.The integraded steps of solving this kind of problems by the FEMare presented in this paper.By applying the FEM to the two actual examples,the conclusion can bereached that the FEM has the advantages of simplicity and good accuracy.展开更多
Al-Ti diffusion couples were made by embedded technology and treated at the temperature between the melting points of Al and Ti. The microstructure evolution and growth mechanism of the Al-Ti DRZ were investigated. Th...Al-Ti diffusion couples were made by embedded technology and treated at the temperature between the melting points of Al and Ti. The microstructure evolution and growth mechanism of the Al-Ti DRZ were investigated. The result shows that the DRZ, the mixture of TiAl3 and Al, grows layer by layer along their chemical equilibrium zone. In the course, the growth interface moves toward the aluminum side. TiAl3 is the only new phase which forms earliest in the course of heat-treatment. The growth mechanism of the DRZ changes after the phase transition of titanium. Before the phase transition of titanium, the growth of the DRZ is controlled by the dissolution speed of the titanium to the molten aluminum, while after the phase transition of titanium, the growth is controlled by the chemical reaction speed of Al and Ti atoms, and consequently, its growth rate is greatly increased.展开更多
The macro-kinetics of benzene hydrogenation over porous Ni/γ-Al<sub>2</sub>O<sub>3</sub> catalyst was measured with asingle-pellet diffusion reactor.Based on the dusty-gas model,the differenti...The macro-kinetics of benzene hydrogenation over porous Ni/γ-Al<sub>2</sub>O<sub>3</sub> catalyst was measured with asingle-pellet diffusion reactor.Based on the dusty-gas model,the differential equations by taking into account ofthe variation of effective diffusivities with the composition were derived to describe the diffusion and reactionprocesses of reacting species over porous catalyst.Using the tortuosity factors determined under the condition ofsteady physical diffusion,the global reaction rates were calculated by solving the equations,and the results are ingood agreement with the experimental data obtained.展开更多
A new visualization method for studying the damage to gel structure caused by high salinity ions is explored by using the characteristics of suppression image signal of Mn^(2+) and nuclear magnetic resonance(NMR) imag...A new visualization method for studying the damage to gel structure caused by high salinity ions is explored by using the characteristics of suppression image signal of Mn^(2+) and nuclear magnetic resonance(NMR) imaging technique. The diffusion and distribution characteristics of Mn^(2+) in porous media-gel system were studied based on manganese chloride static diffusion and gel flooding experiments, and the gel's nuclear magnetic image and displacement pressure were tested. The results show that the diffusion of Mn^(2+)conforms to the Fick diffusion law in porous media-gel system, and the diffusion speed of Mn^(2+) increases and the area of gel image decreases gradually with the increase of concentration, and the image of gel decreases faster and the pressure drop of water drive is larger in flooding experiment of manganese chloride with higher concentration. Reaction-diffusion model with the reaction of Mn^(2+) with gel was established to study the concentration distribution characteristics of Mn^(2+). The model is validated by comparing the results with magnetic resonance imaging(MRI) experiments and the diffusion coefficient of Mn^(2+) equals 1.6 mm^2/h, and the minimum concentration of Mn^(2+) to impact gel NMR image signals is 2.5 g/L. The above results show that the diffusion of Mn^(2+) into the gel in the rock core inhibits the imaging signal of the gel and damages its strength, and the greater the concentration is, the greater the influence. Increase of adsorption amount of gel and reaction rate, reduction of diffusion time, and addition of ion adsorption isolator all can reduce the impact of Mn^(2+) on the gel.展开更多
The high price and toxicity of ionic liquids(ILs) have limited the design and application of supported ionic liquid membranes(SILMs) for CO_2 separation in both academic and industrial fields. In this work, [Choline][...The high price and toxicity of ionic liquids(ILs) have limited the design and application of supported ionic liquid membranes(SILMs) for CO_2 separation in both academic and industrial fields. In this work, [Choline][Pro]/polyethylene glycol 200(PEG200) mixtures were selected to prepare novel SILMs because of their green and costeffective characterization, and the CO_2/N_2 separation with the prepared SILMs was investigated experimentally at temperatures from 308.15 to 343.15 K. The temperature effect on the permeability, solubility and diffusivity of CO_2 was modeled with the Arrhenius equation. A competitive performance of the prepared SILMs was observed with high CO_2 permeability ranged in 343.3–1798.6 barrer and high CO_2/N_2 selectivity from 7.9 to 34.8.It was also found that the CO_2 permeability increased 3 times by decreasing the viscosity of liquids from 370 to38 m Pa·s. In addition, the inherent mechanism behind the significant permeability enhancement was revealed based on the diffusion-reaction theory, i.e. with the addition of PEG200, the overall resistance was substantially decreased and the SILMs process was switched from diffusion-control to reaction-control.展开更多
Diffusion-Reaction (DR) equation has been used to model a large number of phenomena in nature. It may be mentioned that a linear diffusion equation does not exhibit any traveling wave solution. But there are a vast nu...Diffusion-Reaction (DR) equation has been used to model a large number of phenomena in nature. It may be mentioned that a linear diffusion equation does not exhibit any traveling wave solution. But there are a vast number of phenomena in different branches not only of science but also of social sciences where diffusion plays an important role and the underlying dynamical system exhibits traveling wave features. In contrast to the simple diffusion when the reaction kinetics is combined with diffusion, traveling waves of chemical concentration are found to exist. This can affect a biochemical change, very much faster than straight diffusional processes. This kind of coupling results into a nonlinear (NL) DR equation. In recent years, memory effect in DR equation has been found to play an important role in many branches of science. The effect of memory enters into the dynamics of NL DR equation through its influence on the speed of the travelling wavefront. In the present work, chemotaxis equation with source term is studied in the presence of finite memory and its solution is compared with the corresponding chemotaxis equation without finite memory. Also, a comparison is made between Fisher-Burger equation and chemotaxis equation in the presence of finite memory. We have shown that nonlinear diffusion-reaction-convection equation is equivalent to chemotaxis equation.展开更多
Multi-component flow with chemical reactions is a common problem in different industrial applications:the mixing chamber of a reaction injection molding(RIM)machine;the dynamics of diesel soot particles interacting wi...Multi-component flow with chemical reactions is a common problem in different industrial applications:the mixing chamber of a reaction injection molding(RIM)machine;the dynamics of diesel soot particles interacting with a porous-ceramic particulate filter;reactive transport in porous media;bio-chemical processes involving enzyme-catalyzed kinetics.In all these cases,mass diffusion/convection and wall or volume chemical interactions among components play an important role.In the present paper we underline the importance of diffusion/convection/reaction mechanisms in bio-chemical processes using the Lattice Boltzmann(LB)technique.The bio-application where we studied diffusion/convection/reaction mechanisms is the quorum-sensing pathway for the bio-synthesis of the AI-2,a molecule that allows the bacteria to launch a coordinated attack on a host immune system(see[9,10]for more details of the bio-application).The overall goal is to create a micro-device to screen potential drugs that inhibit AI-2 bio-synthesis.The Michaelis-Menten saturation kinetic model is implemented at the reactive surface and the results are shown in terms of two dimensionless numbers:Damkohler(Da)and Peclet(Pe)number.For high Pe number a small conversion of reactants into products is obtained at the reactive surface,but the overall flux of products is high;moreover,a fast saturation of the conversion of reactants to products is obtained for high Da numbers.The trade-off for setting the Pe and Da numbers depends on the specific application and the technologies used in the micro-device(e.g.,sensitivity of the detector,cost of reactants).展开更多
基金Supported by the National Natural Science Foundation of China(11901111)Guangzhou Science and Technology Plan Project(202201011602)。
文摘In this paper,the density-independent fractional diffusion-reaction(FDR)equation involving quadratic nonlinearity is investigated.The fractional derivative is illustrated in the beta derivative sense.We firstly propose Bernoulli(G'/G)-expansion method to study nonlinear fractional differential equations(NFDEs).Subsequently,closed form solutions of the density-independent FDR equation are acquired successfully.In order to better understand the dynamic behaviors of these solutions,3D,contour map and line plots are given by the computer simulation.The results show that the proposed method is a reliable and efficient approach.
文摘In this article, a special type of fractional differential equations(FDEs) named the density-dependent conformable fractional diffusion-reaction(DDCFDR) equation is studied. Aforementioned equation has a significant role in the modelling of some phenomena arising in the applied science. The well-organized methods, including the exp(-φ(ε))-expansion and modified Kudryashov methods are exerted to generate the exact solutions of this equation such that some of the solutions are new and have been reported for the first time. Results illustrate that both methods have a great performance in handling the DDCFDR equation.
基金supported by the National Natural Science Foundation of China under Grant No. 10671156the Program for New Century Excellent Talents in Universities under Grant No. NCET-04-0968
文摘New classes of exact solutions of the quasi-linear diffusion-reaction equations are obtained by seeking for the high-order conditional Lie-Baeklund symmetries of the considered equations. The method used here extends the approaches of derivative-dependent functional separation of variables and the invariant subspace. Behavior to some solutions such as blow-up and quenching is also described.
基金Project financially supported by scientific research foundation coferring to Ph.D.
文摘In this paper,finite element method(FEM)is used to solve two-dimensional diffu-sion-reaction equations of boundary layer type.This kind of equations are usually too complicatedand diffcult to be solved by applying the traditional methods used in chemical engineering becauseof the steep gradients of concentration and temperature.But,these difficulties are easy to be over-comed when the FEM is used.The integraded steps of solving this kind of problems by the FEMare presented in this paper.By applying the FEM to the two actual examples,the conclusion can bereached that the FEM has the advantages of simplicity and good accuracy.
基金Project (ZR2011EL023) supported by the Natural Science Foundation of Shandong Province,ChinaProject (12CX04057A) supported by the Fundamental Research Funds for the Central Universities,China
文摘Al-Ti diffusion couples were made by embedded technology and treated at the temperature between the melting points of Al and Ti. The microstructure evolution and growth mechanism of the Al-Ti DRZ were investigated. The result shows that the DRZ, the mixture of TiAl3 and Al, grows layer by layer along their chemical equilibrium zone. In the course, the growth interface moves toward the aluminum side. TiAl3 is the only new phase which forms earliest in the course of heat-treatment. The growth mechanism of the DRZ changes after the phase transition of titanium. Before the phase transition of titanium, the growth of the DRZ is controlled by the dissolution speed of the titanium to the molten aluminum, while after the phase transition of titanium, the growth is controlled by the chemical reaction speed of Al and Ti atoms, and consequently, its growth rate is greatly increased.
基金Supported by the National Natural Science Foundation of China
文摘The macro-kinetics of benzene hydrogenation over porous Ni/γ-Al<sub>2</sub>O<sub>3</sub> catalyst was measured with asingle-pellet diffusion reactor.Based on the dusty-gas model,the differential equations by taking into account ofthe variation of effective diffusivities with the composition were derived to describe the diffusion and reactionprocesses of reacting species over porous catalyst.Using the tortuosity factors determined under the condition ofsteady physical diffusion,the global reaction rates were calculated by solving the equations,and the results are ingood agreement with the experimental data obtained.
基金Supported by the National Natural Science Foundation of China(51274136 51704190)
文摘A new visualization method for studying the damage to gel structure caused by high salinity ions is explored by using the characteristics of suppression image signal of Mn^(2+) and nuclear magnetic resonance(NMR) imaging technique. The diffusion and distribution characteristics of Mn^(2+) in porous media-gel system were studied based on manganese chloride static diffusion and gel flooding experiments, and the gel's nuclear magnetic image and displacement pressure were tested. The results show that the diffusion of Mn^(2+)conforms to the Fick diffusion law in porous media-gel system, and the diffusion speed of Mn^(2+) increases and the area of gel image decreases gradually with the increase of concentration, and the image of gel decreases faster and the pressure drop of water drive is larger in flooding experiment of manganese chloride with higher concentration. Reaction-diffusion model with the reaction of Mn^(2+) with gel was established to study the concentration distribution characteristics of Mn^(2+). The model is validated by comparing the results with magnetic resonance imaging(MRI) experiments and the diffusion coefficient of Mn^(2+) equals 1.6 mm^2/h, and the minimum concentration of Mn^(2+) to impact gel NMR image signals is 2.5 g/L. The above results show that the diffusion of Mn^(2+) into the gel in the rock core inhibits the imaging signal of the gel and damages its strength, and the greater the concentration is, the greater the influence. Increase of adsorption amount of gel and reaction rate, reduction of diffusion time, and addition of ion adsorption isolator all can reduce the impact of Mn^(2+) on the gel.
基金Supported by the National Basic Research Program of China(2013CB733501)the National Natural Science Foundation of China(21136004,21176112,21476106,and21428601)+1 种基金Specialized Research Fund for the Doctoral Program of Higher Education(No.20133221110001)the Project of Priority Academic Program Development of Jiangsu Higher Education Institutions(PAPD)
文摘The high price and toxicity of ionic liquids(ILs) have limited the design and application of supported ionic liquid membranes(SILMs) for CO_2 separation in both academic and industrial fields. In this work, [Choline][Pro]/polyethylene glycol 200(PEG200) mixtures were selected to prepare novel SILMs because of their green and costeffective characterization, and the CO_2/N_2 separation with the prepared SILMs was investigated experimentally at temperatures from 308.15 to 343.15 K. The temperature effect on the permeability, solubility and diffusivity of CO_2 was modeled with the Arrhenius equation. A competitive performance of the prepared SILMs was observed with high CO_2 permeability ranged in 343.3–1798.6 barrer and high CO_2/N_2 selectivity from 7.9 to 34.8.It was also found that the CO_2 permeability increased 3 times by decreasing the viscosity of liquids from 370 to38 m Pa·s. In addition, the inherent mechanism behind the significant permeability enhancement was revealed based on the diffusion-reaction theory, i.e. with the addition of PEG200, the overall resistance was substantially decreased and the SILMs process was switched from diffusion-control to reaction-control.
文摘Diffusion-Reaction (DR) equation has been used to model a large number of phenomena in nature. It may be mentioned that a linear diffusion equation does not exhibit any traveling wave solution. But there are a vast number of phenomena in different branches not only of science but also of social sciences where diffusion plays an important role and the underlying dynamical system exhibits traveling wave features. In contrast to the simple diffusion when the reaction kinetics is combined with diffusion, traveling waves of chemical concentration are found to exist. This can affect a biochemical change, very much faster than straight diffusional processes. This kind of coupling results into a nonlinear (NL) DR equation. In recent years, memory effect in DR equation has been found to play an important role in many branches of science. The effect of memory enters into the dynamics of NL DR equation through its influence on the speed of the travelling wavefront. In the present work, chemotaxis equation with source term is studied in the presence of finite memory and its solution is compared with the corresponding chemotaxis equation without finite memory. Also, a comparison is made between Fisher-Burger equation and chemotaxis equation in the presence of finite memory. We have shown that nonlinear diffusion-reaction-convection equation is equivalent to chemotaxis equation.
文摘Multi-component flow with chemical reactions is a common problem in different industrial applications:the mixing chamber of a reaction injection molding(RIM)machine;the dynamics of diesel soot particles interacting with a porous-ceramic particulate filter;reactive transport in porous media;bio-chemical processes involving enzyme-catalyzed kinetics.In all these cases,mass diffusion/convection and wall or volume chemical interactions among components play an important role.In the present paper we underline the importance of diffusion/convection/reaction mechanisms in bio-chemical processes using the Lattice Boltzmann(LB)technique.The bio-application where we studied diffusion/convection/reaction mechanisms is the quorum-sensing pathway for the bio-synthesis of the AI-2,a molecule that allows the bacteria to launch a coordinated attack on a host immune system(see[9,10]for more details of the bio-application).The overall goal is to create a micro-device to screen potential drugs that inhibit AI-2 bio-synthesis.The Michaelis-Menten saturation kinetic model is implemented at the reactive surface and the results are shown in terms of two dimensionless numbers:Damkohler(Da)and Peclet(Pe)number.For high Pe number a small conversion of reactants into products is obtained at the reactive surface,but the overall flux of products is high;moreover,a fast saturation of the conversion of reactants to products is obtained for high Da numbers.The trade-off for setting the Pe and Da numbers depends on the specific application and the technologies used in the micro-device(e.g.,sensitivity of the detector,cost of reactants).
