Solution Of Transfer Equation Research Articles

Computational modeling of petroleum purification for removal of sulfur compounds: Process analysis for reduction of environmental impacts and material costs

In the course of this investigation, three machine learning models (Gaussian Process Regression, Decision Tree Regression, and Kernel Ridge Regression) were examined for determining the correlation between the input variables (x and y) which are spatial coordinates of model’s geometry, and the content of species in adsorption for sulfur capture. For the process modeling, mass transfer was analyzed, and the concentration distribution of sulfur compound was obtained via numerical solution of mass transfer equations, and then used for machine learning models. The machine learning models were trained using a dataset of 19,000 observations, and their performance was assessed through metrics including R2 score, MAE, and RMSE. Analysis of the results reveals that Decision Tree Regression surpassed the other two models in performance, with an R2 score of 0.9989, MAE of 6.64405E-01, and RMSE of 1.1277E+00. Gaussian Process Regression had an R2 score of 0.97106, MAE of 3.65541E+00, and RMSE of 5.6821E+00, while Kernel Ridge Regression had an R2 score of 0.86347, MAE of 8.26121E+00, and RMSE of 1.1330E+01. The Clonal Selection Algorithm was used for hyper-parameter optimization for all models. These findings demonstrate the potential of machine learning techniques for accurately and reliably predicting the concentration of chemical species and highlight the importance of considering the choice of model and hyper-parameter optimization for optimal performance.

Theoretical investigations on the purification of petroleum using desulfurization process: Analysis and optimization of process

This paper introduces a comprehensive approach for predicting chemical concentrations (C) of sulfur compound in a two-dimensional space (x, y) by numerical solution of mass transfer equation and integration of machine learning. Data of training/validation are obtained from mass transfer modeling on adsorption separation utilizing porous adsorbent for petroleum desulfurization. Mass transfer modeling data were utilized for machine learning models which included three different regression models, namely Support Vector Machine (SVM), Decision Tree Regression (DT), and Multilayer Perceptron (MLP). The hyper-parameters of these base models were optimized using the Grey Wolf Optimization (GWO) algorithm. The ensemble models, denoted as ADA-DT, ADA-MLP, and ADA-SVM were assessed based on key performance metrics, including R2, MAE, RMSE, and MAPE. Results demonstrated the efficacy of the ensemble models in capturing complex relationships within the dataset. ADA-DT obtained an exceptional R2 score of 0.99031, highlighting its outstanding predictive accuracy. Similarly, ADA-MLP and ADA-SVM showed great accuracy, achieving R2 of 0.88272 and 0.96842, respectively. In this study, we uncover valuable insights into the application of ensemble methods and hybridized optimization methods for accurate and robust regression modeling in chemical concentration prediction scenarios applicable for petroleum engineering.

Analytical propagation model for underwater free-space optical communication through realistic levels of oceanic absorption and scattering

Optical communications (OC) through water bodies is an attractive technology for a variety of applications. Thanks to current single-photon detection capabilities, OC receiver systems can reliably decode very weak transmitted signals. This is the regime where pulse position modulation is an ideal scheme. However, there has to be at least one photon that goes through the pupil of the fore optics and lands in the assigned time bin. We estimate the detectable photon budget as a function of range for propagation through ocean water, both open and coastal. We make realistic assumptions about the water’s inherent optical properties, specifically, absorption and scattering coefficients, as well as the strong directionality of the scattering phase function for typical hydrosol populations. We adopt an analytical (hence very fast) path-integral small-angle solution of the radiative transfer equation for multiple forward-peaked scattering across intermediate to large optical distances. Integrals are performed both along the directly transmitted beam (whether or not it is still populated) and radially away from it. We use this modeling framework to estimate transmission of a 1 J pulse of 532 nm light through open ocean and coastal waters. Thresholds for single-photon detection per time bin are a few km and a few 100 m. These are indicative estimates that will be reduced in practice due to sensor noise, background light, turbulence, bubbles, and so on, to be included in future work.

SP2D: Simplified pseudo 2D battery model by piecewise sinusoidal/quadratic functions of potential curves

The pseudo 2D battery model is a widely used tool for analysing the electrochemical behaviour of lithium-ion batteries. However, the high computational expense of solving the fully coupled governing equations is a significant challenge. In this study, a simplified pseudo 2D battery model has been developed to address this bottleneck. The model gets the electrolyte potential with relative potential plus reference potential, which are described by piecewise quadratic or sinusoidal function and determined by boundary conditions. Two electrodes are defined with uniform potentials. This approach provides a simple solution of charge transfer equations and decouples them from the governing equations of the pseudo 2D model. It shows excellent agreement with the original pseudo 2D battery model for a wide range of C-rates, drive cycles and is applicable to electrolytes with different conductivities. The proposed approach provides a simpler and faster solution than the pseudo 2D battery model, with negligible accuracy sacrifice. Consequently, this work is crucial for applying physics-based models to real-time battery control and management systems.

Numerical study of microalgae cultivation process in a batch liquid foam-bed photobioreactor using CFD approach

Microalgae have different uses as feedstock of beneficial chemicals such as polysaccharides, carotenoids, and polyunsaturated fatty acids. Furthermore, they have been applied in the renewable energy, food, agriculture and feed industries, and in wastewater treatment. The growth and cultivation of microalgae are highly regarded by researchers worldwide. So, a variety of systems and designs have been used to cultivate microalgae. A research team recently proposed a new experimental design of photobioreactors based on a liquid foam flow at Wageningen University in the Netherlands. In this study, a numerical study of hydrodynamics and mass transfer of this novel system was conducted using computational fluid dynamics (CFD). This method is based on the simultaneous solution of transfer equations including hydrodynamic equations, mass transfer, cell growth, and radiation transport equation. The simulation was performed in an unsteady and three-dimensional state in COMSOL 5.5 software. The flow regime was considered laminar and mixture model and transport dilution species physics were used to model the behavior of two-phase fluid. Manual geometry grid generation was performed after defining the velocity and concentration boundaries. The number of optimal meshes was determined to be approximately 1.2 × 105 using the mesh independence test. The simulation results were validated according to the concentration-time curve. The cell density, 22 g l−1, is minimal in the center. The maximum biomass is 30.9 g l−1 at the bed wall after 80 h at 1446 μmol photon. m−2. s−1 light flux for the minimal cell density.

Radiative Transfer Equation Solution for Many Scattering Types

The radiative transfer equation is a mathematical equation that describes the changes in the number of photons within a specified volume of a medium over time, taking into account phenomena such as scattering, absorption, and re-emission resulting from photon interactions with the medium. In this study, the radiative transfer equation is considered for a finite slab which anisotropic scattering in a homogeneous medium. The equation solution is done by Legendre polynomials for linear anisotropic, pure quadratic and Rayleigh scattering types. The numerical results are displayed in the tables up to the 13th iteration of the Legendre polynomials. Tables are obtained using different scattering coefficients and single scattering albedo values. The results contain a wide range of data obtained from the method of solving the Legendre polynomial of the radiative transfer equation. Thus, with this study, the effect of different scattering types on the solution of the radiative transfer equation has been demonstrated.

Nonlinear thermal analysis of two-dimensional materials with memory

A nonlinear hyperbolic heat transport equation has been proposed based on the Cattaneo model without mechanical effects. We analyze the two-dimensional Maxwell-Cattaneo-Vernotte heat equation in a medium subjected to homogeneous and non-homogeneous boundary conditions and with thermal conductivity and relaxation time linearly dependent on temperature. Since these nonlinearities are essential from an experimental point of view, it is necessary to establish an effective and reliable way to solve the system of partial differential equations and study the behavior of temperature evolution. A numerical scheme of finite differences for the solution of the two-dimensional non-Fourier heat transfer equation is introduced and studied. We also investigate the attributes of the numerical method from the aspects of stability, dissipation and dispersive errors.

Elektromanyetik Alana Maruz Kalmış Kas Dokusunun Termal Analizi

The interaction between electromagnetic fields and biological masses is increasing day by day. There are thousands of studies in the literature on this interaction. Until now, most of the studies on bio heat transfer have focused on damage caused by electromagnetic fields. However, the therapeutic side of electromagnetic fields has been neglected. This study aimed to bring a new perspective. Using the bio heat transfer equation, the temperature effect of different parameters due to the electromagnetic field was investigated. In the present study, the analytical solution of Pennes' bioheat transfer equation was evaluated together with real muscle tissue parameters. By trying different parameter variations, the effect of parameters such as perfusion constant and thermal conductivity coefficient on tissue warming was observed. In the light of this study, device designs for magnetic heating processes applied in physiotherapy will be possible. After entering the tissue parameters and electromagnetic field values, it was possible to find out what kind of temperature change can be observed as a result of the formulation obtained in this study. The effects of thermal conductivity coefficient, blood perfusion value and heat transfer coefficient on temperature distribution were graphically analyzed. Effect of randomly selected thermal parameters are evaluated within the study. This study has shown to what extent the increasing thermal parameters of the tissue will be affected by the electromagnetic field.

An extrapolation method for improving the quality of tomographic images using multiple short-pulse irradiations

Abstract This paper investigates the inverse problem for the non-stationary radiation transfer equation, which involves finding the attenuation coefficient using the data of serial irradiation of the medium with pulses of various durations. In the framework of single and double scattering approximations, we obtain asymptotic estimates of the scattered radiation flux density for a short duration of the probing pulse. We propose extrapolation procedures for the ballistic component of the radiation transfer equation solution using the data of multiple irradiations of the medium by pulsed radiation sources, which allows us to obtain approximate formulas for finding the attenuation coefficient. The results of numerical experiments with a well-known digital phantom confirm the effectiveness of the extrapolation algorithm for improving the quality of tomographic images of scattering media.

Numerical Analysis of Time-Fractional Porous Media and Heat Transfer Equations Using a Semi-Analytical Approach

In nature, symmetry is all around us. The symmetry framework represents integer partial differential equations and their fractional order in the sense of Caputo derivatives. This article suggests a semi-analytical approach based on Aboodh transform (AT) and the homotopy perturbation scheme (HPS) for achieving the approximate solution of time-fractional porous media and heat transfer equations. The AT converts the fractional problems into simple ones and obtains the recurrence relation without any discretization or assumption. This nonlinear recurrence relation can be decomposed via the use of the HPS to obtain the iterations in terms of series solutions. The initial conditions play an important role in determining the successive iterations and yields towards the exact solution. We provide some numerical applications to analyze the accuracy of this proposed scheme and show that the performance of our scheme has strong agreement with the exact results.

Numerical simulation of ozonation in hollow-fiber membranes for wastewater treatment

In this study, we developed a comprehensive modeling framework for simulation of ozonation process using combination of artificial intelligence and computational fluid dynamics (CFD). The process is carried out in a hollow-fiber membrane contactor in which the concentration data of ozone was obtained by solution of mass transfer equations, and then the results were used for artificial intelligence modeling. We used three different machine learning models to predict concentration of ozone (C) in a system based on its coordinates, i.e., r and z. The models were optimized using the Bat Algorithm (BA) and were trained on a dataset consisting of over 10,000 data points. The three models developed were Support Vector Regression (SVR), Decision Tree Regressor, and Orthogonal Matching Pursuit (OMP). These models were evaluated using three common metrics — Mean Squared Error (MSE), R-squared (R2), and Mean Absolute Error (MAE). Our results indicated that the SVR model overperformed the other two models in terms of all evaluation metrics. Specifically, the SVR model achieved an MSE of 0.003, an R2 of 0.998, and an MAE of 0.046. The Decision Tree Regressor and OMP models achieved less favorable results with MSEs of 0.007 and 0.221, R2 scores of 0.996 and 0.878, and MAEs of 0.056 and 0.359, respectively.

Approximate Solution of Sub diffusion Bio heat Transfer Equation

In this paper, author’s study sub diffusion bio heat transfer model and developed explicit finite difference scheme for time fractional sub diffusion bio heat transfer equation by using caputo fabrizio fractional derivative. Also discussed conditional stability and convergence of developed scheme. Furthermore numerical solution of time fractional sub diffusion bio heat transfer equation is obtained and it is represented graphically by Python.

3D thermoelastoplastic study of FML structure under the action of a moving Gaussian-distributed laser heat source

Fiber metal laminated (FML) structures are used widely, however, studies on the three-dimensional thermoelastoplastic behaviors of FML structure subjected to a moving Gaussian-distributed laser heat source have not yet been developed. In present work, the semi-analytical solution of three-dimensional transient heat transfer equation for FML structure under the action of a moving Gaussian-distributed laser heat source is obtained through the separate variable method (SVM) and the Newton Cotes method (NCM). A yield criterion related to spherical tensor of stress is proposed to describe the mixed hardening of the orthotropic FML structure. Based on classical nonlinear laminated plate theory, the incremental nonlinear governing equations are obtained, and the equations are solved by the combination of finite difference method, Newmark method and iterative method. Numerical results show that the temperature field, moving velocity of laser, number of FML layers, thickness ratio and boundary condition have great influences on thermoelastoplastic behaviors of the FML structure.

Thermo-large deflection coupled dynamic characteristics of rotating thickness-varying plates subjected to thermal shock

Thermal shock load has an important influence on dynamic characteristics of blades in turbomachines. Due to higher level operating thermal shock for improving the efficiency, the damage to the blades has greatly increased. Considering the major effect of variable thickness and large deflection on thermal dynamic behavior of blades, their thermo-large deflection-variable thickness coupled dynamic characteristics are unknown. This paper develops a thermo-large deflection coupled semi-analytical model based on a rotating variable thickness plate under thermal shock. Geometric nonlinear governing equation and an analytical solution of heat transfer equation are derived. Numerical examples of natural frequencies are investigated to validate the proposed model. The non-uniform temperature distribution features of the rotating variable thickness plate are found. Besides, when the thermal shock is serious, the large deflection theory must be considered to analyze thermo-large deflection coupled dynamic characteristics. Based on theoretical and experimental results, deformation characteristic of the cantilever plate subjected to thermal shock is obtained, which is helpful to find the optimal thickness distribution. Finally, the effect of variable thickness on thermo-large deflection coupled dynamics of the rotating plate is demonstrated. The vibration induced by thermal shock can be reduced by thickening the plate on its free side or both fixed–free sides. However, when thickening other positions, the vibration is either intensified or unchanged. Results are helpful to reduce blade’s vibrations under thermal shock.

Numerical modeling of atmospheric pollution in the approaches of random selection of discrete dusting sites and interval distribution of dust size

The results of numerical simulation of atmospheric pollution in Apatity are presented with variations in the dusting area of discrete spatially spaced areas selected randomly and the wind flow velocity. CFD modeling in the volumetric formulation was performed using the COMSOL program. To calculate the aerodynamic characteristics, an incompressible fluid approximation was used using the standard ( k -ε) turbulence model. The process of fine dust propagation is modeled by numerical solution of the convective-diffusion impurity transfer equation taking into account the deposition rate. Numerical experiments (with a total number of more than 1,400) were carried out with a variation of the wind flow velocity from 5 to 23 m/s and a dusting area from 2 to 10 ha with a random selection of 20 discrete sites. Dynamic velocity distributions for specific areas of dusting, interval and total spatial distributions of dust pollution (dust particles with a diameter from 0 to 70 microns in increments of 10 microns) are obtained. The peculiarities of the influence on the levels of atmospheric pollution of specific areas of Apatity depending on the location of dusty areas on the surface of the beach of the tailings dump are noted. The calculated levels of atmospheric pollution in the center of Apatity averaged by the number of combinations of dusting sites are analyzed and generalized to functional dependencies. The calculated dependences of the dust concentration on the dusting area at a fixed wind speed are described by linear functions. The dependence of the dust concentration on the wind flow velocity at a fixed dusting area can be approximated by a power function. The generalized functional dependence makes it possible to predict the dust concentration in Apatity depending on the dusting area of randomly selected discrete areas on the surface of the tailings dump and the wind flow velocity. The obtained dependence permits to make a forecast of the critical dusting area at which the level of atmospheric pollution reaches the maximum permissible concentrations, depending on the speed of the wind flow.

The comparison of a discrete-continuous approach and a method of finite differences in solving the problem of unsteady-state heat and moisture transfer in the building envelope

This article is devoted to the development of methods for calculating heat and humidity regime in the building envelope. The equation of steady-state thermal conductivity with boundary conditions of the third kind and the formula for calculating heat losses of a building based on the heat transfer equation have been considered. The equation of unsteady-state thermal conductivity as well as its solution using the discrete-continual approach has also been studied. The solution of the unsteady-state heat conductivity problem with invariable over time boundary conditions using the discrete-continuous approach was proposed by A.B. Zolotov and P.A. Akimov. The subsequent modernization of the solution was conducted by V.N. Sidorov and S.M. Matskevich. The unsteady-state equation of moisture transfer based on Fick’s second law using the theory of moisture potential is derived. The solution of the unsteady-state moisture transfer equation using the finite difference method according to an explicit difference scheme as well as the solution of the unsteady-state moisture transfer equation using the discrete-continuous approach is demonstrated. To prove the effectiveness of using the discrete-continuous approach in the area of the unsteady-state humidity conditions we compared the calculation results of the distribution of moisture in a single-layer enclosing structure made of aerated concrete using two methods of moisture potential theory. It was found that the difference in the results of calculation by the discrete-continual formula and by the method of finite differences does not exceed 3.2%.

On numerical solution of nonlinear parabolic multicomponent diffusion-reaction problems

Abstract This work continues our previous analysis concerning the numerical solution of the multi-component mass transfer equations. The present test problems are two-dimensional, parabolic, non-linear, diffusion- reaction equations. An implicit finite difference method was used to discretize the mathematical model equations. The algorithm used to solve the non-linear system resulted for each time step is the modified Picard iteration. The numerical performances of the preconditioned conjugate gradient algorithms (BICGSTAB and GMRES) in solving the linear systems of the modified Picard iteration were analysed in detail. The numerical results obtained show good numerical performances.

Numerical solution of stochastic time fractional heat transfer equation with additive noise

In the present paper, we developed an explicit finite difference scheme for stochastic time fractional heat transfer equation. Furthermore, we proved the conditional stability of the solution of the scheme and analyze the effect of the multiplier of the random noise in mean square stability. Also, we discuss the convergence of the explicit scheme in the mean square sense. The approximate solution of the practical problem is obtain by developed scheme and it is represented graphically by Mathematica software.

Tangent stiffness for nonlinear heat transfer and its application to additive manufacturing

Prediction-correction method is commonly employed in the solution of nonlinear heat transfer equation with temperature dependent thermophysical parameters arising in various engineering problems such as additive manufacturing of metal components. However, the stiffness of such conventional method is secant rather than tangent and thus it only results in suboptimal rate of convergence in Newton's iteration. In this paper, tangent stiffness for nonlinear heat transfer is proposed through the linearization of the weak form and an additional term presents which disappears in the conventional prediction-correction method. Numerical results show that the solution with the proposed tangent stiffness converges much faster than that with secant stiffness and approximately only half of the iteration steps are needed. Especially, in the case that the thermal conductivity varies dramatically with the temperature, the conventional method diverges whereas the tangent stiffness still shows quite good convergence. Application of the proposed method to the modeling of heat transfer in additive manufacturing (AM) process is presented and the obtained temperature results agree quite well with the existing experimental studies. It is worth mentioning that almost half of CPU time is saved due to the application of the proposed tangent stiffness. This saving is of great significance for modeling the AM process of complicated components, which is usually very time-consuming.

Characterization of Ferromagnetic Composite Implants for Tumor Bed Hyperthermia

Hyperthermia therapy (HT) is becoming a well-recognized method for the treatment of cancer when combined with radiation or chemotherapy. There are many ways to heat a tumor and the optimum approach depends on the treatment site. This study investigates a composite ferromagnetic surgical implant inserted in a tumor bed for the delivery of local HT. Heating of the implant is achieved by inductively coupling energy from an external magnetic field of sub-megahertz frequency. Implants are formed by mechanically filling a resected tumor bed with self-polymerizing plastic mass mixed with small ferromagnetic thermoseeds. Model implants were manufactured and then heated in a 35 cm diameter induction coil of our own design. Experimental results showed that implants were easily heated to temperatures that allow either traditional HT (39-45°C) or thermal ablation therapy (>50°C) in an external magnetic field with a frequency of 90 kHz and amplitude not exceeding 4 kA/m. These results agreed well with a numerical solution of combined electromagnetic and heat transfer equations solved using the finite element method.