Unsteady Convection-diffusion Research Articles

Effect of Reaction Conditions on Nitrogen Conversion During Char-O2/NO Combustion

ABSTRACT Nitrogen conversion during char-O2 combustion with a high initial-NO concentration (char-O2/NO combustion) is an important issue for low-NOx technologies and oxy-fuel combustion technologies, such as air/fuel-staged and flue gas recycling combustion. This study employed both experimental and numerical methods to investigate the mechanisms of N conversion and formation during char-O2/NO combustion. In the numerical model, the mass and heat transfer processes are governed by unsteady convection-diffusion partial differential equations. The numerical results agree well with the experimental findings, assuming NO is the sole primary N-containing product. Based on the numerical results, the effects of reaction conditions on N conversion were decoupled. As the O2/fuel stoichiometric ratio increases, the char-N release rate substantially rises, and the shrinkage of pore surface area becomes significant, leading to poorer NO reduction performance. With an increase in initial-NO concentration, the apparent conversion of char-N to NO decreases, while the NO reduction efficiency remains almost unchanged. Due to the lower temperature sensitivity of mass transfer limitations in the char-NO reaction compared to the char-O2 reaction, NO reduction performance is enhanced at higher reaction temperatures.

Streamline Diffusion Weak Galerkin Finite Element Methods for Linear Unsteady State Convection Diffusion Equations and Error Analysis

In this paper, the streamline diffusion weak Galerkin finite element method is proposed and analyzed for solving unsteady time convection diffusion problem in two dimension. The v-elliptic property and the stability of this scheme are proved in terms of some conditions. We derive an error estimate in L2(μ) and H1(μ) norm. Numerical experiments have demonstrated the effectiveness of the method in solving convection propagation problems, and the theoretical analysis has been validated.

On the BEM solution of convection–diffusion type equations involving variable convective coefficients

In this study, numerical solutions of the unsteady convection–diffusion type equations of variable convective coefficients are investigated by two effective techniques alternative to direct boundary element method (BEM). That is, the domain boundary element and the dual reciprocity BEMs with the fundamental solution of convection–diffusion equation are used in space to transform the governing differential equations into equivalent integral equations while a backward finite difference scheme is utilized in time discretization. The fruitfulness of these combined techniques are shown by the implementation of the techniques for some widely investigated fluid dynamics problems. In fact, the lid-driven cavity flow is solved, and precise results are attained as benchmarks for assessing the accuracy of the aforementioned methods. Further, the application of the methods is extended for the natural convection flow which is governed mainly by the convection–diffusion type equations accompanied by the velocity components as variable convective coefficients. The obtained numerical results reveal that the domain BEM which uses the fundamental solution of convection–diffusion equation captures the characteristics and physical advancement of the fluid flow quite well at various combined values of the problem physical parameters.

Transient dispersion of reactive solute transport in electrokinetic microchannel flow

Motivated by emerging applications in bio-microfluidic devices, the present study rigorously examines the generalized Taylor–Gill hydrodynamic dispersion of a point source solute injected into a microchannel, influenced by a constant axial static electric field along the channel and charged surface with different wall potentials. The solute engages in a first-order irreversible chemical reaction at both the microchannel walls. By incorporating different wall potentials and absorptive coefficients at the lower and upper walls, the current transport model for electro-osmotic flows is extended to encompass a wider range of applications. The solute transport phenomenon is intricately modeled using the unsteady convective diffusion equation. Employing Gill's generalized dispersion model, a concentration decomposition technique, up to the third-order accuracy, we meticulously analyze the transport process. Furthermore, a comprehensive comparison between analytical outcomes and numerical simulations using the Brownian Dynamics method is undertaken, enhancing the robustness of the analytical approach. The scattering process is mainly analyzed with the help of exchange, convection, dispersion, and asymmetry coefficients, along with the mean concentration profile. The effect of initial solute release at various vertical locations in the microchannel is shown to exert a considerable impact on all the transport coefficients at initial times.

A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.

A dispersion model for initial consequence analysis based on diffusion equations

Most factories use toxic or flammable chemicals in their industrial processes; this poses a risk of leakage due to accidents, which can sometimes result in mass casualties and considerable property damage. Therefore, quantitative risk prediction and assessment are necessary to protect the public and minimize losses in the event of a chemical release. Several methods can be used to evaluate chemical dispersion in the atmosphere, but most are based on the assumption of neutral buoyancy and far-field wind conditions. With this assumption, a model is valid only after a significant amount of time has elapsed from the moment chemicals are released or dispersed from a source. However, the most dangerous locations are typically close to a leak source. Therefore, a dispersion model for the initial phase of a leak is required to quantify the risk and predict the extent of damage. This study developed a dispersion model for initial consequence analysis using a three-dimensional unsteady convective diffusion equation. A continuous point source was assumed, and ethane was used as the leak material to minimize the effects of buoyancy. The unsteady concentration field developed rapidly as the diffusion coefficient and wind velocity increased, but the steady-state concentration field was not affected by wind velocity. This dispersion model also allows for the simple consideration of ground adsorption, making it scalable to real-world situations. The time to reach a steady state required for an effective emergency response was predicted, and the results were interpreted in terms of mathematical methods and physical characteristics.

Error estimates of characteristic finite elements for bilinear convection–diffusion optimal control problems

This paper investigates a fully discrete characteristic finite element approximation of bilinear unsteady convection–diffusion optimal control problems. The characteristic line method is used to treat the convection term and the finite element method is adopted to treat the diffusion term. The state and adjoint state are discretized by piecewise linear functions, the control is approximated by piecewise constant functions. A priori error estimates are derived for the state, adjoint state and control variables. Some numerical examples are provided to confirm our theoretical findings.

Analyzing the effects of suction and injection Reynolds number on the transport process in a hydromagnetic flow through a channel of reactive porous walls

The present paper analyzes the effects of suction and injection Reynolds number on the dispersion process of solute in an incompressible magneto-hydrodynamics flow through a channel. The walls of the channel are permeable and a first order heterogeneous reaction is applied on it. Also, an inclined magnetic field is employed at both walls of the channel. The unsteady convection-diffusion equation is solved semi-analytically to find out the coefficient of dispersion, skewness and kurtosis of the solute distribution for all time period. Using Hermite polynomial representation, the distribution of mean concentration is obtained. The effects of various parameters such as suction Reynolds number (R), injection Reynolds number (R′), boundary absorption parameter (β), inclination angle of the magnetic field (α), Hartmann number (M) and dispersion time (t) on dispersion process of the solute are examined. It is shown that with the increment of suction Reynolds number, the peak of the mean concentration of solute enhances prominently. But, when injection Reynolds number increases, the amplitude of the mean concentration decreases. The current research gives an insight to understand the complex behavior of transport process of the solute in a hydro-magnetic channel flow with reactive porous walls.

Application of Fixed Point Theory and Solitary Wave Solutions for the Time-Fractional Nonlinear Unsteady Convection-Diffusion System

Application of Fixed Point Theory and Solitary Wave Solutions for the Time-Fractional Nonlinear Unsteady Convection-Diffusion System

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

Fourth order compact scheme for the Navier–Stokes equations on time deformable domains

In this work, we report the development of a spatially fourth order temporally second order compact scheme for incompressible Navier–Stokes (N–S) equations in time-varying domain. Sen (2013) put forward an implicit compact finite difference scheme for the unsteady convection–diffusion equation. It is now further extended to simulate fluid flow problems on deformable surfaces using curvilinear moving grids. The formulation is conceptualized in conjunction with recent advances in numerical grid deformations techniques such as inverse distance weighting (IDW) interpolation and its hybrid implementation. Adequate emphasis is provided to approximate grid metrics up to the desired level of accuracy and freestream preserving property has been numerically examined. As we discretize the non-conservative form of the N–S equation, the importance of accurate satisfaction of geometric conservation law (GCL) is investigated. To the best of our knowledge, this is the first higher order compact method that can directly tackle the non-conservative form of the N–S equation in single and multi-block time dependent complex regions. Several numerical verification and validation studies are carried out to illustrate the flexibility of the approach to handle high-order approximations on evolving geometries. The method presented here is found to be effective in modeling incompressible flow on deformable domains, including situations of fluid–structure interaction (FSI). Additionally, the necessity of GCL preserving metric computation is eased as a result of the scheme’s adaptation for non-conservative formulation.

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

The laminar seabed thermal boundary layer forced by propagating and standing free-surface waves

A mathematical model is developed to investigate seabed heat transfer processes under long-crested ocean waves. The unsteady convection–diffusion equation for water temperature includes terms depending on the velocity field in the laminar boundary layer, analogous to mass transfer near the seabed. Here we consider regular progressive waves and standing waves reflected from a vertical structure, which complicate the convective term in the governing equation. Rectangular and Gaussian distributions of seabed temperature and heat flux are considered. Approximate analytical solutions are derived for uniform and trapezoidal currents, and compared against predictions from a numerical solver of the full equations. The effects of heat source profile, location and strength on heat transfer dynamics in the thermal boundary layer are explained, providing insights into seabed temperature forced convection mechanisms enhanced by free-surface waves.

SIMULATION OF UNSTEADY TRANSPORT PHENOMENA USING NEW FINITE VOLUME METHOD

Based on the finite volume method (FVM), a numerical scheme is constructed to simulate the unsteady convection–diffusion transport problem. New expressions are obtained for interface approximation of the field variable, subsequently, these newly obtained interface expressions are used to develop the numerical scheme. Convection-dominant and diffusion-dominant phenomena are simulated by taking different values of convective velocity [Formula: see text] and diffusion coefficient [Formula: see text]. This newly proposed numerical scheme gives second order of convergence along space and time. Experiments are carried out to test the new proposed upwind approach. Numerical results produced by the proposed approach are compared with the conventional finite volume method, step-wise approach FVM and quadratic upwind interpolation finite volume approach. This comparative study indicates that for different cases for convection-dominant and diffusion-dominant problems, our proposed approach gives highly accurate and stable solution. The conventional finite volume method and other approaches result solution with non-physical oscillations. Our obtained numerical results are consistent and support our theoretical approach.

Effect of Catheter and Stenosis on Solute Diffusion in Non-Newtonian Blood Flow through a Catheterized Stenosed Artery

The presence of stenosis at the wall of the artery lead to further cardiovascular diseases such as heart attack, stroke and many more. Treatment of a stenosed artery includes the insertion of a catheter through the artery which affects the blood flow and solute dispersion. This present study focuses on the effect of catheter radius and stenosis height on the blood flow and solute dispersion behavior. The problem is modelled using the Herschel-Bulkley fluid to represent the blood rheology, with catheter and stenosis as the boundary conditions. Analytical solutions in integral form are obtained by solving the momentum equation and Herschel-Bulkley constitutive equation. The integrals are numerically evaluated using the Simpson’s 3/8 rule and Regula-Falsi method to obtain the blood velocity. The obtained velocity is employed into the unsteady convective-diffusion equation and solved using the generalized dispersion model (GDM) to analyse the behaviour of solute diffusion. The influence of catheter radius and stenosis height on the diffusion coefficient and mean concentration of solute are observed. Results show that the diffusion coefficient decreases as the catheter radius and stenosis height increases. A decrease in diffusion coefficient simultaneously increases the solute mean concentration.

Exact Analysis of Unsteady Convective Diffusion in Herschel-Bulkley Fluid Flow- Application to Catheterised Stenosed Artery

One of the major causes of cardiovascular disease is atherosclerosis or stenosis. This study is designed to improve the current body of knowledge regarding the condition by inserting a long thin tube called a catheter to widen the narrow part in the artery. The study reviewed the effects of catheter radius, yield stress, and power law index on the velocity distribution, and transport coefficients of solute. A mathematical model is deployed to investigate the dispersion of solute in the flow of a Herschel-Bulkley (H-B) fluid in an annulus, whereas the dispersion process is studied using the generalised dispersion model (GDM) by solving the convective diffusion equation. Resultan tly, the velocity reduces following an increase in the yield stress, catheter size, and power law index. Meanwhile, the dispersion coefficient exhibits a same behaviour as the aforementioned parameters ascend considerably. The dispersion coefficient alterations occurred rapidly for small values of time and became significantly constant following an increase in the time values. Conclusively, this study can be useful in dispersion of a drug to the affected artery where an abnormal plaque was formed.

Analysis of Unsteady Solute Dispersion in a Blood Flow of Herschel-Bulkley through a Catheterized Stenosed Artery

Blockage of blood flow due to cholesterol deposits at the arterial wall, known as stenosis, can lead to conditions such as heart attack and stroke. Treatment such as balloon angioplasty involves the catheterization of an artery where a stented catheter is inflated at the stenosis site to open the narrowed artery. The catheterization of the stenosed artery affects the surrounding blood flow and dispersion process. The present study analyses the effect of catheter radius and stenosis height on the blood velocity and solute dispersion behavior. Herschel-Bulkley fluid is used to model the problem with stenosis as the boundary condition. The momentum equation and Herschel-Bulkley constitutive equation are solved analytically into integral forms. Simpson’s 3/8 rule and Regula-Falsi method were used to evaluate the integral numerically to obtain the velocity. The velocity was utilized to solve the unsteady convective-diffusion equation using the generalized dispersion model (GDM) to obtain the dispersion function. This present research can potentially help the medical field and industry in determining the suitable catheter radius for patients, calculating drug dosage and improving stent catheter design. Results show that the velocity decreases as the catheter radius and stenosis height increase. A decrease in velocity simultaneously increases the solute dispersion function.

An analytical approximate method for solving unsteady state two-dimensional convection-diffusion equations

In this paper, an analytic approximate method for solving the unsteady two-dimensional convection-diffusion equations is introduced. Also, the convergence of the approximate methods is analyzed. Three test examples are presented, two have exact and one has not exacted solutions. The results obtained show that these methods are powerful mathematical tools for solving linear and nonlinear partial differential equations, moreover, new analytic Taylor method (NATM), reduced differential transform method (RDTM), and homotopy perturbation method (HPM), are more accurate and have less CPU time than the other methods.

Numerical Methods for Unsteady Convection-diffusion Problems Based on Combining Compact Difference Schemes with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in space and time. The one-dimensional unsteady convection-diffusion equation was solved by combing a compact difference scheme with the Runge-Kutta method. The combined method has fourth-order accuracy in space and time. To check the accuracy of the combined method, numerical experiments were carried out and comparisons were performed with the Crank-Nicolson method. The analysis results indicated that the combined method is numerically stable at low wave numbers and small Courant-Friedrichs-Lewy numbers. The combined method has higher accuracy than the Crank-Nicolson method.

A new implicit finite difference method with a compact correction term for solving unsteady convection diffusion equations

A new implicit finite difference method with a compact correction term is established for solving unsteady convection-diffusion equations. The correction terms by connecting classical and compact finite difference formulas are proposed for improving the accuracies of numerical solutions. This new method with fourth order accuracy can be used for the numerical calculations of convection diffusion equation on both uniform and nonuniform grid systems. It can be extended not only from one-dimensional to multi-dimensional equations, but also from steady to unsteady convection-diffusion equations. Several convection-diffusion problems are stimulated for verifying the flexibility and high accuracy of this method.