Generalized Integral Transform Technique Research Articles

New approach to solving the atmospheric pollutant dispersion equation using fractional derivatives

In atmospheric environments, traditional differential equations do not adequately describe the problem of turbulent diffusion because the usual derivatives are not well defined in the non-differentiable behaviour introduced by turbulence, where the fractional calculation has become a very useful tool for studying anomalous dispersion and other transportation processes. Considering a new direction, this paper presents an analytical series solution of a three-dimensional advection–diffusion equation of fractional order, in the Caputo sense, applied to the dispersion of atmospheric pollutants. The solution is obtained by applying the generalised integral transform technique (GITT), solving the transformed problem by the Laplace decomposition method (LDM), and considering the lateral and vertical turbulent diffusion dependence on the longitudinal distance from the source, as well as a fractional parameter. The fractional solution is more general than the traditional solution in the sense that consideration of the integer order of the fractional parameter yields the traditional solution. The solution considers the memory effect in eddy diffusivity and in the fractional derivative, and it is simple, easy to implement, and converges rapidly. Numerical simulations were conducted to compare the performance of the proposed fractional solution to the traditional solution using an experimental dataset and other models, which also made it possible to find a better parametrisation for use in Gaussian models. The best results are obtained with the fractional order of the derivative.

An exact GITT solution for static bending of clamped parallelogram plate resting on an elastic foundation

PurposeThe purpose of this study is to propose generalised integral transform technique (GITT) to obtain the exact solutions for bending of clamped parallelogram plate resting on elastic foundation.Design/methodology/approachThe GITT is used to solve the bending problem of the full clamped parallelogram plate under an elastic foundation. The auxiliary problem was developed and the corresponding eigenfunction and eigenvalue were calculated simultaneously. The original partial differential governed equation has been represented by the transformed ordinary differential equation system and solved by the subroutine DBVPFD from International Mathematics and Statistics Library.FindingsThe GITT has been proven to be an efficient approach to solve the bending problem of the plate with different loads, boundary conditions and elastic foundations. The parametric study indicates that the elastic foundation modulus has significant contribution in reducing the vertical deflections and moments for both rectangular and parallelogram plates. With the increasing of aspect ratio (a/b) and the elastic foundation modulus, the trends of the deflection and moment reduction decreased significantly.Originality/valueThe present hybrid analytical-numerical methodology was first used to solve the mechanics problem of the clamped parallelogram plate resting on elastic foundation. Excellent convergence and high accuracy was observed by comparing with the published results. It exhibits potential application to investigate the mechanics problem of the composite plate with different boundary conditions in the shipbuilding and civil engineering.

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

The Generalized Integral Transform Technique (GITT) is reviewed as a hybrid numerical–analytical approach for fluid flow problems, with or without heat and mass transfer, here with emphasis on the literature related to flow problems formulated through the full Navier–Stokes equations. A brief overview of the integral transform methodology is first provided for a general nonlinear convection–diffusion problem. Then, different alternatives of eigenfunction expansion strategies are discussed in the integral transformation of problems for which the fluid flow model is either based on the primitive variables or the streamfunction-only formulations, as applied to both steady and transient states. Representative test cases are selected to illustrate the different eigenfunction expansion approaches, with convergence being analyzed for each situation. In addition, fully converged integral transform results are critically compared to previously reported simulations obtained from traditional purely discrete methods.

Integral transform analysis of heat and mass diffusion in chemically reacting systems with Michaelis–Menten kinetics

A Generalized Integral Transform Technique (GITT) solution scheme for transient non-linear heat and mass diffusion in chemically reacting systems with Michaelis–Menten type kinetics has been developed. A model with two one-dimensional coupled PDEs was selected for illustrating the solution methodology, in which temperature and substrate concentration are the dependent variables. Although the model is one-dimensional, a generalized formulation valid for both plane and curved systems was employed. The main novelty of this investigation is the treatment of the reaction kinetics term, whose functional form is a major obstacle for GITT solutions. This term was resolved by an automatic scheme based on Taylor Series approximations for which the expansion point can be progressively updated with the numerical time-integration, using data from the previous time-step. The outcome of the solution implementation is then verified with data from a reference finite-volumes solution, exhibiting a very good agreement. Finally, illustrative results of averaged temperature and substrate concentration are presented, showing that different type of solutions are obtained within the considered range of dimensionless parameters.

Single domain integral transform analysis of natural convection in cavities partially filled with heat generating porous medium

An analysis of natural convection within a rectangular cavity partially filled with a heat-generating porous medium is carried out through the Generalized Integral Transform Technique (GITT), in which the laminar flow and energy equations are solved with automatic error control. A single domain reformulation strategy is adopted to rewrite the governing equations within the fluid and the porous medium as a single heterogeneous medium formulation, with spatially variable physical properties and source terms that account for the abrupt transition of the two regions. This fundamental study is motivated by the analysis of wet storage of spent nuclear fuel elements with passive cooling of the pool and physical conclusions are drawn from the hybrid numerical-analytical solution. Increases in the Rayleigh number with constant internal heat generation are found to lower the maximum temperature within the cavity. Moreover, decreasing the aspect ratio also has positive effects on the cooling of the cavity.

An analytical study of the periodic laminar forced convection of non-Newtonian nanofluid flow inside an elliptical duct

In this study, an analytical solution was obtained for a laminar forced convection of non-Newtonian nanofluid flowing inside an elliptical duct, with inlet temperature varying periodically with the time. The solution was obtained using the Generalized Integral Transform Technique (GITT). The thermal behavior of Cu-water non-Newtonian nanofluid, described by the power-law model was investigated. Also, an accurate correlation was established to estimate the thermal length required to achieve 99% of the amplitude attenuation. The results show a significant effect of aspect ratio β/α and fluid behavior index on the temperature amplitude reduction. For instance, an elliptical duct with β/α = 0.25 reduces the thermal length Lth more than 50% compared with circular duct. Adding nanoparticles until 5% increases the heat transfer coefficient up to 27% for the cylindrical tube. Besides, the heat transfer coefficient is improved over 42% relatively to the cylindrical configuration by reducing the aspect ratio to 0.25. Therefore, adding 5% of nanoparticles both using an elliptical duct with β/α = 0.25, improves the mean heat transfer coefficient around 83% compared to the flow of water base fluid inside a cylindrical duct.

MODIFIED SERIES RESISTANCE MODEL - DETERMINATION OF MEAN CONCENTRATION BY INTEGRAL TRANSFORMATION

There are several mathematical models that describe permeate flow in membrane separation processes. Among these, the series resistance model plays a prominent role. It takes into account membrane strength, concentration polarization, polarized layer and fouling to describe the permeate flow over time. In this work, an analysis of the modified series resistance model was performed, in which the resistance by polarization of the concentration is defined as being directly proportional to the transmembrane pressure. The proportionality constant is given by the product of a specific coefficient of resistance – which is determined by means of experimental data – the thickness of the boundary layer of concentration and the mean concentration. Due to the inability to obtain experimentally the value of the average concentration within the boundary layer of concentration, its simulation is carried out from the conservation equation of the chemical species. Thus, the objective of the present work was to solve the equation of the conservation of chemical species using GITT (Generalized Integral Transform Technique) and apply the modified series resistance model to describe the permeate flow of a solution of dextran through a permeable tube under laminar flow. GITT provided satisfactory results for the mean concentration, verified by comparison with the permeate flow obtained by the series resistance model with experimental results reported in the literature.

Dynamic response of free span pipelines via linear and nonlinear stability analyses

A thorough stability analysis of submerged pipeline dynamics has been performed. The problem of current-induced vibrations in a free spanning pipeline was analyzed using both linear and non-linear methodologies. While most investigations dedicated to the analysis of this type of problem use either approximate analytical methods or traditional numerical methods, two alternative approaches are presented: Linear Stability Analysis (LSA) and the Generalized Integral Transform Technique (GITT). Although strictly valid when the pipeline is subject to small amplitude disturbances, LSA is simple to apply and yields accurate either explicit or transcendental analytical relations that predict dominant pipeline vibration frequencies and wavelengths as well as the onset of buckling. On the other hand, the GITT can predict pipeline behavior for arbitrary disturbance amplitudes more efficiently than traditional numerical methods. These features are illustrated by considering the problem of a submerged pipeline subjected to soil conditions and ocean currents. This problem was solved using LSA, the GITT and the Finite Element Method (FEM) for comparison purposes. Moreover, the DNV-RP-F105 standard was also used for verifying the proposed methods. The alternative methods offered in this paper can overcome limitations associated with traditional analytical and numerical approaches, and are employed to produce previously unreported results.

Semi-analytical solution for soil-constrained vibration of subsea free-spanning pipelines

ABSTRACTApart from being induced by internal and external flow, vibration of free-spanning pipelines is also constrained by the soil that supports the span on both ends. A wake oscillator model is chosen to describe the vortex shedding acting on the pipeline, and the soil spring is adopted to describe the pipe–soil interaction at the span shoulders. A semi-analytical solution – the generalised integral transform technique (GITT) – is used to study the dynamic response of free-spanning submarine pipelines. With this method, the coupled fluid-structure system of partial differential equations is transformed into a system of second-order ordinary differential equations in temporal variable. Parametric studies are carried out to analyse the influence of different internal and external flow velocities, and different soil stiffness. GITT is proved to be a fast and accurate approach for analysing the dynamic response of free-spanning submarine pipelines considering the influence of the current and the internal flow, as well as the effect of the soil.

On the mass transport in membraneless flow batteries with flow-by configuration

Progress on the analysis of the mass transport phenomena in a membraneless redox flow battery with porous electrodes in flow-by configuration is here reported. A species transport model for a typical redox reactant interacting with a porous electrode is proposed. A hybrid numerical-analytical solution is obtained through the Generalized Integral Transform Technique (GITT), adopting a single domain formulation that includes both the porous and pure fluid regions. The influence of the Reynolds number and the thickness of the electrode on the parameters of interest is theoretically examined. The importance of considering the simultaneous development of flow and mass transport is analyzed, and the presence of the transversal convective flux of species proves to have a significant role on the generation of current inside the electrode. A scaling of the limiting current density with ∼Re0.41 is demonstrated and some physical conclusions are drawn. Guidelines for the prevention of crossover are also offered, with increasing Reynolds number and decreasing relative thickness of the electrode having a positive effect, as far as avoiding the mixed potentials effects is concerned. The physical insights attained through the present analysis should add to the efforts in achieving membraneless redox flow batteries with performances comparable to membrane-based devices of similar size.

Integral transform solution for the analysis of hydrodynamic lubrication of thrust bearings

PurposeThe aim of this paper is related to an analysis of hydrodynamic lubrication of circular thrust bearings.Design/methodology/approachThe modified Reynolds equation was treated to obtain a hybrid numerical-analytical solution through the generalized integral transform technique (GITT) for the problem.FindingsNumerical results for the engineering parameters such as pressure field, load capacity and power consumption were thus produced as functions of the radial and circumferential directions. These parameters depend on the geometry of sector-shaped used: Rayleigh pad with 4, 8 and 16 steps. Comparing among them, on the numerical point of view, the Rayleigh pad geometry with N = 16 steps has a better satisfactory performance because it has a lower power consumption.Originality/valueThe present GITT results and those obtained by the finite volume method (FVM) from previous works in the literature were confronted to verify whether the results are consistent and to demonstrate the capacity of the GITT approach in handling thrust bearing problems.

Recent advances in computational-analytical integral transforms for convection-diffusion problems

An unifying overview of the Generalized Integral Transform Technique (GITT) as a computational-analytical approach for solving convection-diffusion problems is presented. This work is aimed at bringing together some of the most recent developments on both accuracy and convergence improvements on this well-established hybrid numerical-analytical methodology for partial differential equations. Special emphasis is given to novel algorithm implementations, all directly connected to enhancing the eigenfunction expansion basis, such as a single domain reformulation strategy for handling complex geometries, an integral balance scheme in dealing with multiscale problems, the adoption of convective eigenvalue problems in formulations with significant convection effects, and the direct integral transformation of nonlinear convection-diffusion problems based on nonlinear eigenvalue problems. Then, selected examples are presented that illustrate the improvement achieved in each class of extension, in terms of convergence acceleration and accuracy gain, which are related to conjugated heat transfer in complex or multiscale microchannel-substrate geometries, multidimensional Burgers equation model, and diffusive metal extraction through polymeric hollow fiber membranes. Numerical results are reported for each application and, where appropriate, critically compared against the traditional GITT scheme without convergence enhancement schemes and commercial or dedicated purely numerical approaches.

Semi-analytical solution of land-derived solute transport under tidal fluctuation in a confined aquifer

A one-dimensional semi-analytical solution of land-derived solute transport, subject to tidal fluctuation in a coastal confined aquifer, was derived using the generalized integral-transform technique (GITT). To investigate the plume migration of land-derived contaminants within a tidally influenced aquifer, both spatially and temporally varying expressions of the Darcy velocity and dispersion coefficients obtained from the analytical solution of the groundwater head response, which were subject to sinusoidal boundary conditions due to tidal fluctuation, were considered. This new semi-analytical solution was verified against a numerical solution, as well as the peak location trajectory obtained using the Predictor-Corrector method. Sensitivity analyses of tidal amplitude, hydraulic conductivity, and storage coefficient using the proposed solution were performed to understand plume behavior with regard to plume shape, plume spatial moments, and macrodispersion coefficients to gain a better understanding of the transport mechanisms. As the tidal amplitude, hydraulic conductivity, and storage coefficient were increased, the peaks were travelled faster, and peak concentrations were decreased. In addition, an increase in tidal amplitude, hydraulic conductivity, and storage coefficient caused an increase in variance as well as the macrodispersion coefficient. It was observed that negative macrodispersion appeared when the storage coefficient was largest, as well as when the difference between landward-directed advective velocity at the leading and trailing edges of the plume was greatest. This newly developed semi-analytical solution provides a useful mathematical tool for validating numerical models and understanding the physical mechanism of the migration of plume discharge to the sea or estuaries within a tidally influenced aquifer.

Convective Eigenvalue Problems for Convergence Enhancement of Eigenfunction Expansions in Convection–Diffusion Problems

The present work considers the application of the generalized integral transform technique (GITT) in the solution of a class of linear or nonlinear convection–diffusion problems, by fully or partially incorporating the convective effects into the chosen eigenvalue problem that forms the basis of the proposed eigenfunction expansion. The aim is to improve convergence behavior of the eigenfunction expansions, especially in the case of formulations with significant convective effects, by simultaneously accounting for the relative importance of convective and diffusive effects within the eigenfunctions themselves, in comparison against the more traditional GITT solution path, which adopts a purely diffusive eigenvalue problem, and the convective effects are fully incorporated into the problem source term. After identifying a characteristic convective operator, and through a straightforward algebraic transformation of the original convection–diffusion problem, basically by redefining the coefficients associated with the transient and diffusive terms, the characteristic convective term is merged into a generalized diffusion operator with a space-variable diffusion coefficient. The generalized diffusion problem then naturally leads to the eigenvalue problem to be chosen in proposing the eigenfunction expansion for the linear situation, as well as for the appropriate linearized version in the case of a nonlinear application. The resulting eigenvalue problem with space variable coefficients is then solved through the GITT itself, yielding the corresponding algebraic eigenvalue problem, upon selection of a simple auxiliary eigenvalue problem of known analytical solution. The GITT is also employed in the solution of the generalized diffusion problem, and the resulting transformed ordinary differential equations (ODE) system is solved either analytically, for the linear case, or numerically, for the general nonlinear formulation. The developed methodology is illustrated for linear and nonlinear applications, both in one-dimensional (1D) and multidimensional formulations, as represented by test cases based on Burgers' equation.

Analytical model for advective-dispersive transport involving flexible boundary inputs, initial distributions and zero-order productions

A novel solution method is presented which leads to an analytical model for the advective-dispersive transport in a semi-infinite domain involving a wide spectrum of boundary inputs, initial distributions, and zero-order productions. The novel solution method applies the Laplace transform in combination with the generalized integral transform technique (GITT) to obtain the generalized analytical solution. Based on this generalized analytical expression, we derive a comprehensive set of special-case solutions for some time-dependent boundary distributions and zero-order productions, described by the Dirac delta, constant, Heaviside, exponentially-decaying, or periodically sinusoidal functions as well as some position-dependent initial conditions and zero-order productions specified by the Dirac delta, constant, Heaviside, or exponentially-decaying functions. The developed solutions are tested against an analytical solution from the literature. The excellent agreement between the analytical solutions confirms that the new model can serve as an effective tool for investigating transport behaviors under different scenarios. Several examples of applications, are given to explore transport behaviors which are rarely noted in the literature. The results show that the concentration waves resulting from the periodically sinusoidal input are sensitive to dispersion coefficient. The implication of this new finding is that a tracer test with a periodic input may provide additional information when for identifying the dispersion coefficients. Moreover, the solution strategy presented in this study can be extended to derive analytical models for handling more complicated problems of solute transport in multi-dimensional media subjected to sequential decay chain reactions, for which analytical solutions are not currently available.

Dynamic response of pipes conveying two-phase flow based on Timoshenko beam model

The dynamic behavior of pipes subjected to internal gas–liquid two-phase flow has been studied using the Timoshenko beam model and the slip-ratio factor model. In this paper, the governing equations were carried out using the generalized integral transform technique (GITT) by transforming the governing partial differential equations into a set of second-order ordinary differential equations. The comparison between Timoshenko beam model and Euler–Bernoulli beam model has been conducted through parametric study on dimensionless frequencies and amplitudes over various aspect ratios, internal fluid flow rates, and volumetric gas fractions. The results show that the frequencies of Timoshenko beam model are less and the amplitude is larger than that of Euler–Bernoulli beam model at low aspect ratio. In addition, the amplitude for Timoshenko beam model increases more dramatically than that of Euler–Bernoulli beam model when the pipe is about to lose stability. The high flow rate leads to the divergence of the dynamic system, as well as the two-phase flow accelerates the instability and has significant influence on the dynamic response when the pipe is long and the internal liquid flows fast.

Integral transform solution of micropolar magnetohydrodynamic oscillatory flow with heat and mass transfer over a plate in a porous medium subjected to chemical reactions

The main goal of the present work is to show the procedure, application and main features of the hybrid numerical-analytical approach known as GITT (Generalized Integral Transform Technique) by solving an unsteady, one-dimensional magnetohydrodynamic (MHD) oscillatory flow of a micropolar and incompressible fluid with heat and mass transfer through a permeable vertical plate embedded in a porous medium in the presence of chemical reaction. The mathematical formulation of the studied model was obtained from the equation of motion and the mass and energy balances by considering laminar and incompressible flow subjected to a constant transverse magnetic field with constant physical properties. Convergence analysis was performed and presented to illustrate the consistency of the integral transform technique. Linear and angular velocities distribution, temperature and concentration profiles were generated and numerically verified with an approximate solution found in the literature and with the results of the method of lines (MOL) with good agreement. The effects of some governing parameters, namely, dimensionless time, magnetic field parameter, Schmidt and Prandtl numbers, permeability and chemical reaction parameters, on these fields were presented. The effects of these parameters on the local skin friction coefficient, the couple stress coefficient, the local Nusselt number and the local Sherwood number were also critically evaluated. Therefore, results show that the linear velocity decreases with increasing magnetic field parameter, while the angular velocity increases with increasing the same and the linear and angular velocities and the concentration field decrease as the Schmidt number increases while the temperature field decreases with increasing Prandtl number.

Analysis of magnetohydrodynamic natural convection in closed cavities through integral transforms

A hybrid numerical-analytical solution is proposed to analyze MHD (magnetohydrodynamic) natural convection of an electrically-conducting fluid within a square cavity, differentially heated at the sidewalls and subjected to an inclined external magnetic field. The first goal is to expand the spectrum of application of the so called Generalized Integral Transform Technique (GITT), dealing with a multiphysics formulation, while further demonstrating the relative merits of the proposed eigenfunction expansion approach in handling highly nonlinear and coupled systems of partial differential equations. The second goal is to provide a set of benchmark results in this important application for quantities of practical interest in determining the heat transfer rates, such as the average Nusselt number. The two-dimensional steady state equations are written in dimensionless form using the streamfunction-only formulation and are subsequently solved with the GITT approach, under automatic relative error control. Critical comparisons are performed against previous work reported in the literature, both computational and experimental, together with the corresponding physical interpretations, for different values of the governing parameters, such as Grashof number, Hartmann number, Prandtl number, and magnetic field inclination angle.

Three-dimensional reaction-convection-diffusion analysis with temperature influence for biodiesel synthesis in micro-reactors

A three-dimensional nonlinear reaction-convection-diffusion model is proposed to analyze mass transfer with temperature effects, within micro-reactors of rectangular cross-section, towards the simulation of the transesterification process in biodiesel synthesis. The coupled nonlinear partial differential equations that govern the concentration of the species involved in the synthesis, is obtained from the general three-dimensional species balance equations for an isothermal and steady state system with constant physical properties. A kinetic model is considered to describe the variation of the species due to the transesterification process, assuming second order homogeneous and reversible chemical reactions. The velocity profile is analytically derived from the Navier-Stokes equations, considering fully developed stratified laminar flow of two immiscible Newtonian fluids in a rectangular cross section micro-reactor, with a plane interface between them and without field forces, and obtained by the Classical Integral Transform Technique (CITT). The nonlinear mass transport equations are then solved by the hybrid numerical-analytical approach known as the Generalized Integral Transform Technique (GITT). In the present GITT solution, two alternative eigenvalue problems are proposed, accounting or not accounting for the space variable velocity profile influence in the eigenfunction expansion basis, in order to investigate the differences in the convergence behavior of the solutions obtained through the two different auxiliary problems. The results for the concentration profiles of the biodiesel and triglyceride species are critically compared with literature data for the limiting case of parallel plates, showing a good agreement among them. The analysis also indicates that improved convergence rates are achievable through the eigenvalue problem with variable coefficients. The effects of governing parameters such as residence time, micro-reactor dimensions, and especially temperature, are investigated. It is then inspected the relative gains in triglycerides conversion rates at higher temperatures and residence times and lower micro-reactor hydraulic diameter.

Hybrid solution for the flow and heat transfer of a second-grade viscoelastic fluid on a stretching sheet with injection or suction

This paper analyzes the steady, laminar, two-dimensional flow and heat transfer of a second-grade viscoelastic fluid over a flat plate subjected to continuous stretching. Injection and suction effects are also investigated. The mathematical model of the problem was derived from the constitutive tensor for an incompressible fluid of the second grade, followed by thermodynamic considerations and usual boundary layer assumptions. The generalized integral transform technique (GITT) was used to transform the governing partial differential equations (continuity, conservation of momentum and energy) within the boundary layer hypotheses on a coupled system of ordinary differential equations. The numerical solution of the system of equations was performed using the appropriate subroutine DIVPAG from the IMSL Library, which it is well-indicated for solving initial value problems with stiff characteristics. Convergence behaviors of the numerical results for the velocity and temperature fields were discussed as functions of parameters of interest. Investigations on the influence of viscoelastic parameter (K), injection or suction parameter (F) and Prandtl number (Pr) were performed in order to evaluate the effects of such governing parameters on the velocity and temperature fields. Comparison with previous results from the literature were also carried out in order to verify the computational code developed in this work.