Set Of Partial Differential Equations Research Articles

Modeling and simulation of non-Newtonian fluid flows and heat transfer in a non-isothermal coiled tubing to oil well operations

The coiled tubing system has become an attractive option for oil industry operations in deep-water environments. Composed of a long, continuous, and flexible steel pipe, the coiled tubing provides a quick and safe intervention in active wells reducing the process time and cost. The fluid flow in curved pipes is characterized by centrifugal forces that significantly affect the frictional head loss and fluid heat transfer. The temperature variation affects the fluid physicochemical properties, such as rheology, density, thermal properties, and the cement setting time. To avoid the risk of the cement slurry setting before reaching the desired location, set retarders are added to its formulation. Therefore, it is essential to predict the temperature profile in coiled tubing to optimize the fluid formulation and process control, avoiding increased process time and cost. This research aimed to develop a mathematical model to simulate the heat transfer in coiled tubing fluid flow based on momentum, energy, and mass conservation. The mathematical model is composed of a set of partial differential equations spatially discretized using the finite volume technique. A case study was used, providing actual oilfield data to simulate the non-isothermic flow through the coiled tubing. Temperature data from a Brazilian offshore oil well abandonment process in two different operating conditions were used to validate the mathematical model. The results obtained in this paper indicated a good agreement between the model predictions and the oil well experimental data, with average relative errors of up to 2.5%.

Double-Diffusive Nonlinear Buoyancy Force Significance on Free Convective Chemically Reacting Fluids Flow Past Vertical Riga Surface

In this research, double-diffusive nonlinear buoyancy force effects on free convective chemically reacting fluids flow past a vertical plate in a Riga surface have been addressed numerically. Cattaneo-Christov theories, viscous dissipation, thermal radiation, thermophoresis and the Soret-Dufour mechanism were modelled in this paper. Also, thermal radiative heat flux was considered based on the Rosseland approximation because the fluid is optically thin. The flow equations were represented using partial differential equations (PDEs). These sets of PDEs were transformed into ordinary differential equations by utilizing suitable similarity variables. The transformed flow equations were numerically solved by employing the spectral collocation method. The outcomes show that enhancement in the velocity and temperature are observed due to an increase in Eckert number, thermal radiation and heat generation parameters. An increase in the relaxation heat and mass flux is noticed to have effects on the concentration and temperature gradient due to the presence of Soret-Dufour mechanisms. The present outcomes were compared with previously published works and are found to be in good agreement.

A Model on Free Convective Casson Fluid Flow Past a Permeable Vertical Plate with Gyrotactic Microorganisms

This study focused on free convective MHD Casson fluid flow past a permeable vertical plate with microorganisms and Soret-Dufour mechanisms. The physical model is governed by system of partial differential equations (PDEs). The set of PDEs were overhaul into a nonlinear ordinary differential equation by considering acceptable similarity variable. The spectral relaxation method (SRM) was utilized to obtain solution to the overhaul nonlinear ordinary differential equations. This study examined critically the significant of the Gyrotactic microorganisms on Casson fluid locomotion Also, the significance of parameters such as thermal radiation, Soret-Dufour mechanisms, Brownian motion, Joule heating and so on are examined on velocity, concentration, motile microorganism and temperature profiles. The Casson parameter was found to intensify the velocity contour and the hydrodynamic boundary layer thickness. A higher magnetic parameter gives rise to Lorentz force which declines the motion of the fluid. There is higher temperature within the boundary layer because of the presence of thermal radiation.

Modeling Water Flow and Solute Transport in Unsaturated Soils Using Physics‐Informed Neural Networks Trained With Geoelectrical Data

AbstractAccurate modeling of water flow and solute transport in unsaturated soils is of significant importance for precision agriculture, environmental protection and aquifer management. Traditional modeling approaches are considerably challenging since they require well‐defined boundaries and initial conditions. Physics‐informed neural networks (PINNs) have recently been developed to learn and solve forward and inverse problems also constrained to a set of partial differential equations and are more flexible than traditional modeling approaches. However, hydrological applications of PINNs used so far spatial measurements of hydraulic head, water content and/or solute concentrations, which were well distributed in the subsurface for training the system. Such measurements are hard to obtain in real‐world applications. Here, we propose to train PINNs with non‐invasive geoelectrical tools for simulating two‐dimensional water flow and solute transport during infiltration and redistribution processes with unknown initial conditions. Two‐dimensional flow and transport numerical simulations were used as benchmarks to examine the suitability of the described approach. Results have shown that the trained PINNs system was able to reproduce the spatiotemporal distribution of both water content and pore‐water salinity during both processes with high accuracy, using five time‐lapse geoelectrical measurements and matric head measurements at a single location. The trained PINNs system reconstructed the initial conditions of both state parameters at both processes. It was also able to separate the measured electrical signal into its two components, that is, water content and pore‐water salinity. The subsurface geoelectrical tomograms were significantly improved compared to those obtained from a classical inversion of the raw geoelectrical data.

Heat and mass transport of MHD viscoelastic fluid flow towards a permeable stretching cylinder

Heat and mass transport of MHD flow of viscoelastic Maxwell fluid due to a stretching cylinder on a porous medium with the impact of curvature parameter, Schmidt number, Deborah number and chemical reaction are investigated. Radiation, MHD, Brownian motion and thermophoresis effects are also considered for modelling. A set of (PDEs) partial differential equations was used to define the mathematical description of the flow. These equations were turned into nonlinear (ODEs) ordinary differential equations via similarity transformations. Since these PDEs are challenging to solve via the analytical method, we compute the solutions using the RK45 technique using inbuilt software bvp4c. This article's critical findings and novelty present the answers for the various physical parameters in two distinct geometries; among them, one represents the cylinder (curvature parameter αc = 1), and other represents the horizontal plate (curvature parameter αc = 0). The Maxwell fluid parameter elevates the temperature and concentration contour for both the plate and cylinder. Further, the velocity of the viscoelastic fluid displays inconsistent behaviour for the Deborah number and curvature parameter. In addition, we compared the acquired numerical result to existing work and discovered that the current numerical results are in remarkable agreement.

Front propagation in the interaction of gases to model a fuel tank inerting process with a nonlinear parabolic operator

PurposeThe objective of this study is to model the propagating front in the interaction of gases in an aircraft fuel tank. To this end, we introduce a nonlinear parabolic operator, for which solutions are shown to be regular.Design/methodology/approachThe authors provide an analytical expression for the propagating front, that shifts any combination of oxygen and nitrogen, in the tank airspace, into a safe condition to avoid potential explosions. The analytical exercise is validated with a real flight.FindingsAccording to the flight test data, the safe condition, of maximum 7% of oxygen, is given for a time t = 45.2 min since the beginning of the flight, while according to our analysis, such a safe level is obtained for t = 41.42 min. For other safe levels of oxygen, the error between the analytical assessment and the flight data was observed to be below 10%.Originality/valueThe interaction of gases in a fuel tank has been little explored in the literature. Our value consists of introducing a set of nonlinear partial differential equations to increase the accuracy in modeling the interaction of gasses, which has been typically done via algebraic equations.

Dynamics of ferromagnetic cross fluid flow capturing magnetic dipole and radiation aspects

Increasing population around the globe due to promotion of industries has created severe problems for human beings like shortage of existing fossil fuels, energy crisis, climate change which are major hurdles for human survival on the planet. To solve all these problems, researchers recommended two appropriate solutions, one of them is to promote thermal energy which is environmentally friendly source of energy and other solution is promotion of ferro-fluid usage as an effective medium for transmission of heat. In recent study, we considered the impacts of thermal energy along with magnetic effects and compute its results. This work is done by using appropriate similarity variables to obtain the system of nonlinear ordinary differential equations (ODEs) from set of partial differential equations (PDEs). We solved the resultant ODEs numerically by means of a famous numerical method termed as bvp4c. Here, effects of Weissenberg parameter, power law index, Curie temperature and radiation parameter are determined for velocity and temperature profiles. Furthermore, graphical representations of velocity as well as thermal gradients are computed. The effects of various parameters on Nusselt number are deliberated and expressed in tabular form. Velocity of magnetized cross ferro-fluid dwindles for [Formula: see text] and n. Moreover, it is detected that temperature of ferro-fluid deteriorates for [Formula: see text] and [Formula: see text].

A mathematical model of a twin-shaft parallel flow regenerative lime kiln

ABSTRACT Lime is an important flux used in the iron and steel-making processes. Tata Steel Jamshedpur uses a twin-shaft Parallel Flow Regenerative type kiln to calcine limestone. Previously, the kiln was operated based on the temperature measurement at only three locations. This led to suboptimal kiln performance and difficulty in maintaining lime quality. A thermal digital twin was developed using first principles to address these issues. A set of unsteady-state partial-differential equations representing the thermophysical phenomena is solved numerically using the finite volume method. The model was validated with the measured flue gas temperature and lime temperatures. The model is implemented online with the real-time operating parameters to estimate the temperature profile of solid and gas and the degree of conversion of limestone. This real-time visibility enables the operators to optimize the process and achieve lime conversion up to 94%, resulting in energy savings of 31.8 GJ/day and stable lime quality.

Optimal classification and similarity solution for unsteady flow behind a shock wave in a perfectly conducting dusty gas with magnetic field using group invariance method

We have studied the one dimensional unsteady isothermal and adiabatic flows behind the shock waves in a non-ideal dusty gas under the effect of magnetic field using Lie group invariance method. This method is used to get the infinitesimal generators of the Lie algebra. The concept of optimal systems is used for minimizing the group-invariant solutions. The optimal system of subalgebra is used to obtain the similarity variables and similarity transformation which transform the set of partial differential equations (PDEs) into a set of ordinary differential equations (ODEs). Consequently, the system of ODEs in different cases is solved by using Runge–Kutta (R-K) method of forth order to derive the similarity solution in the cases of isothermal and adiabatic flows. The detailed discussions are illustrated by graph through numerical calculation in the case of power-law shock path. It is shown that the shock wave strength has decaying effects with an increase in the values of Alfven Mach number, adiabatic exponent of the gas, and gas non-idealness parameter. Also, the shock strength increases with an increment in the values of mass concentration of solid particles, magnetic field variation exponent, the ratio of density of solid particles to the gas initial density, and by changing the geometry from cylindrical to spherical.

Study on Sinusoidal Post-Buckling Deformation of Coiled Tubing in Horizontal Wells Based on the Separation Constant Method

In this paper, a set of partial differential equations considering the dynamic effects of the coiled tubing (CT) is established based on the bending theory of slender beams considering the axial loads. The analytical solution of sinusoidal deformation with the time term is obtained. The critical load of coiled tubing during sinusoidal buckling and the change of half-wave number during sinusoidal post-buckling are studied through the introduction of the necessary conditions, solution and the separation constant. The contact force of coiled tubing with sinusoidal post-buckling on the horizontal well wall is analyzed. The results show that the critical load for sinusoidal buckling of fixed-size CT is related to the section angular acceleration coefficient. The half-wave number produced by the sinusoidal post-buckling bending of the CT gradually decreases during compression. The contact force of the deformed CT to the borehole wall is related to the compression speed of the CT. By introducing the dynamic term and the separation constant, this research model can provide a theoretical basis for studying the transformation of the CT from sinusoidal buckling to helical buckling.

Parallel Routing in Wireless Sensor Network

The limitations of wireless sensor nodes are power, computational capabilities, and memory. This paper suggests a method to reduce the power consumption by a sensor node. This work is based on the analogy of the routing problem to distribute an electrical field in a physical media with a given density of charges. From this analogy a set of partial differential equations (Poisson's equation) is obtained. A finite difference method is utilized to solve this set numerically. Then a parallel implementation is presented. The parallel implementation is based on domain decomposition, where the original calculation domain is decomposed into several blocks, each of which given to a processing element. All nodes then execute computations in parallel, each node on its associated sub-domain. With this method power consumption by the central node which is responsible to compute routing in the network is reduced.

Free Convection of Viscoelastic Nanofluid Flow on a Horizontal Circular Cylinder with Constant Heat Flux

The current research study is focusing on a mathematical model for free convection boundary flow on a horizontal circular cylinder in a viscoelastic nanofluid has been constructed in this paper with boundary conditions constant heat flux. The Tiwari and Das Nanofluid model have been chosen in this analysis to investigate more nanofluid effects. A dimensionless set of partial differential equations is formed by using suitable non-dimensional variables. The transformed boundary layer equations are solved numerically using a finite difference scheme namely the Keller-box method. The effects of a few chosen factors on flow and heat transmission are investigated. Numerical solutions were obtained for the reduced skin friction coefficient, Nusselt number and Sherwood number as well as the velocity and temperature profiles. The features of the flow and heat transfer characteristics for various values of the viscoelastic parameter and nanoparticles volume fraction were analysed and discussed.

Rigorous solution for drained analysis of spherical cavity expansion in soils of finite radial extent

This paper revisits the expansion problem of a spherical cavity in an elastic–plastic soil mass with a finite radial extent under drained conditions, which proves to be non-self-similar. Rigorous semi-analytical solutions for the analysis of this problem are developed adopting several well-known soil models. With the small strain assumption, analytical expressions for both elastic stresses and displacements are obtained considering the stress dependency of soil moduli and influences of the outer boundary. In the plastic zone, the non-self-similar expansion process is formulated into a set of first-order partial differential equations (PDEs) based on the innovative combination use of the Lagrangian and Eulerian descriptions, and an efficient algorithm for solving these equations is developed. The solution method is quite general to be extended to many other constitutive models and is validated by comparing with existing solutions in some special cases. Then the cavity expansion behaviours in both clay and sand are discussed with a particular focus on the boundary effect. It is shown that, when the ratio of the outer to inner radius is small, the boundary effect may significantly influence the cavity expansion response. The present solution may provide a useful theoretical method for the validation of relevant finite element calculations and interpretation of some typical geotechnical problems such as cone penetration tests in small-size chambers, considering the boundary effect.

Interfacial properties of fast chemical reactions occurring in demixed ternary mixtures

A new theoretical framework regarding the interfacial properties of reactive liquid mixtures forming a liquid-liquid phase split has been introduced. Based on the kinetic rate equations, which rely on activities rather than concentrations, and the incompressible density gradient theory, which provides an expression for the Gibbs energy of a nonuniform system, a method is developed to study the dynamics of chemical reactions taking place in the coexisting liquid bulk phases as well as in the interface between them. The assumption that the chemical reaction is much faster than the diffusion leads to a set of partial differential equations describing the spatial and temporal evolution of the mole fractions of all the involved components in the interface and the temporal evolution of the mole fractions in the bulk phases. The obtained equations were used to investigate the dynamics of the interfacial concentration profiles of a ternary mixture with a large miscibility gap, applied to a simple example. Due to the fast chemical reaction and the slow diffusion, the mixture will not remain in phase equilibrium anymore. It was found that the chemical reaction reaches chemical equilibrium first in both bulk phases and afterward in the interface existing between them, which is in contrast to the classical two-film theory often used for mass transport calculations in chemical engineering.

Mathematical Modelling of Magnetohydrodynamic Nanofluid Flow with Chemically Reactive Species and Outer Velocity Towards Stretching Cylinder

This article investigate the impact of magnetohydrodynamic nanofluid past a stretching cylinder with chemical reactive species. The momentum, energy and concentration equations are represented by a set of partial differential equations which are moulded into a system of ordinary differential equations using mathematical modelling of the physical problem. After adopting the Runge Kutta Fehlberg approach, the moulded equations are solved using the shooting procedure. To study the effects of various fluid parameters, a parametric analysis was performed. Brownian motion and thermophoresis were investigated in the appealing pattern. The effects of important fluid characteristics, such as outer velocity, chemical reaction, thermophoresis, Lewis number, Brownian motion on concentration, temperature, and velocity have been investigated and shown in graphically and tabulated forms. The core findings of this work is that concentration of the nanofluid decreasing with more reacting species and rate of heat transfer is significantly controlled by outer velocity parameter and magnetic parameter which is very useful in manufacturing processes.

An efficient model for capturing gas transients in the energy flow of integrated electricity–gas systems

Gas-fired units provide great flexibility to accommodate the intermittent nature of renewable energy sources and are also considered to be an ideal substitute for retiring coal-fired generations. However, the increasing deployment of gas-fired units deepens interdependency between electric power and natural gas, calling for efficient modeling of the heterogeneous energy flow in the integrated electricity–gas systems (IEGS). Plenty of research works show the necessity of considering gas flow transients in the IEGS, which can be described by a set of partial differential equations (PDEs). To ensure a tractable optimization, the PDEs are often discretized using the finite/infinite difference method, which may result in a large computational scale for achieving good precision. To this end, this paper proposes an efficient model for capturing gas flow transients in the IEGS, which can be taken as the operational constraints in the optimal energy flow model for IEGS coordinated planning, scheduling, and control. For improving the computational efficiency, the PDE-based transient gas flow model is discretized using a space–time orthogonal collocation (OC) method, which provides high-quality results using fewer spatial–temporal discrete points than differential methods. Further, aiming at the difficulty in the determination of the number of space–time collocation points, an adaptive strategy is proposed and embedded in the proposed model utilizing the local fitting errors of state variables related to pipelines. Case studies show the superiority in computational accuracy and efficiency of the proposed model over the existing models based on Wendorff difference and global OC. And an illustrative case based on the modified IEGS benchmark system composed of the IEEE 118-bus power system and the Belgian 20-node gas network exhibits the application value of the proposed model in the optimization and control of the IEGS.

A 3D Approach Using a Control Algorithm to Minimize the Effects on the Healthy Tissue in the Hyperthermia for Cancer Treatment.

According to the World Health Organization, cancer is a worldwide health problem. Its high mortality rate motivates scientists to study new treatments. One of these new treatments is hyperthermia using magnetic nanoparticles. This treatment consists in submitting the target region with a low-frequency magnetic field to increase its temperature over 43 °C, as the threshold for tissue damage and leading the cells to necrosis. This paper uses an in silico three-dimensional Pennes' model described by a set of partial differential equations (PDEs) to estimate the percentage of tissue damage due to hyperthermia. Differential evolution, an optimization method, suggests the best locations to inject the nanoparticles to maximize tumor cell death and minimize damage to healthy tissue. Three different scenarios were performed to evaluate the suggestions obtained by the optimization method. The results indicate the positive impact of the proposed technique: a reduction in the percentage of healthy tissue damage and the complete damage of the tumors were observed. In the best scenario, the optimization method was responsible for decreasing the healthy tissue damage by 59% when the nanoparticles injection sites were located in the non-intuitive points indicated by the optimization method. The numerical solution of the PDEs is computationally expensive. This work also describes the implemented parallel strategy based on CUDA to reduce the computational costs involved in the PDEs resolution. Compared to the sequential version executed on the CPU, the proposed parallel implementation was able to speed the execution time up to 84.4 times.

Qualitative modeling of solar panel cooling by nanofluid jets: Heat transfer and second law analysis

The present paper highlights heat transfer and entropy generation due to mixed convection for an inclined channel. The channel is heated by its upper wall and cooled by two jets of nanofluids penetrating through its lower wall. The studied configuration is chosen to model, in qualitative way, a cooling system for photovoltaic panel. The set of partial differential equations that governing the flow was numerically solved using COMSOL software. Effects of the inclination angle of the channel, Reynolds number and nanoparticles fraction, on heat transfer, hydrodynamic and created entropy are examined. The inclination angle, the Reynolds number and the nanoparticle fraction are ranging from 0° to 30°, from 50 to 150 and from 0% to 8% respectively. It was found that the effect of the angle of inclination, on heat transfer and thermal irreversibility, is weak and that it cannot exceed 2%. Whereas the nanofluid concentration and the Reynolds numbers alter at once the created entropy and the heat transfer. Results show an increase of nearly 15% of Nusselt number and thermal irreversibility when the nanoparticle concentration reaches 8%. The local irreversibility maps reveal that the created entropy is significantly localized at the impact locations of the cooling jets. Since irreversibility is synonymous with the aging of the system, which naturally leads to usury, it can be concluded that the channel may be damaged at these locations.

Simulation of CO2 capture from natural gas by cyclic pressure swing adsorption process using activated carbon

This work presented modeling and simulation of CO2 from natural gas. One of the most promising technologies is Pressure Swing Adsorption (PSA), which is an energy-efficient and cost-effective process for separating and capturing CO2 from industrial processes and power plants. This paper provides an overview of the PSA process and its application for CO2 capture, along with a discussion of its advantages, limitations, and future research directions. This process is pressure swing adsorption (PSA) with four adsorption beds. The adsorption bed columns fill with activated carbon as adsorbent. In this simulation momentum, mass and energy balance are solved simultaneously. The process was designed with two beds in adsorption conditions and the other two beds in desorption conditions. The desorption cycle includes blow-down and purge steps. The linear driving force (LDF) estimates the adsorption rate in modeling this process. The extended Langmuir isotherm is used for the equilibrium between solid and gas phases. The temperature changes by heat transfer from the gas phase to solid and axial heat dispersion. The set of partial differential equations is solved using implicit finite difference.

Transient MHD Thermally Radiating Free Convective Flow Past an Exponentially Moving and Rotating Isothermal Vertical Plate with Heat Generation, Reacting Species and Fluctuating Mass Diffusion Effects

The problem of transient MHD thermally radiating free convective flow past an exponentially moving and rotating isothermal vertical plate in the presence of heat generation reacting species, and fluctuating mass diffusion is examined. The flow is governed by a set of non-linear partial differential equations of the Boussinesq approximation type. In particular, the momentum equations are simplified using the 2-D fluid flow analysis in the complex plane, and the governing equations are solved using the time-dependent Homotopy Perturbation Method. Expressions for the concentration, temperature, and velocity are obtained and presented graphically. The results amidst others depict that an increase in the chemical reaction rate decreases the concentration, but causes fluctuation in the flow velocity structure; an increase in the Heat generation/absorption parameter increases the fluid temperature but causes fluctuation in the flow velocity structure.