Energy Transfer Equations Research Articles

Solid oxide fuel cell cathode diffusion polarization: materials and exergy study

The efficiency of the solid oxide fuel cell (SOFC) still can be enhanced by decreasing the cathode diffusion polarization, however to resolve this problem, band gap materials engineering analysis for the SOFC cathode must be well-thought-out. In this research, we investigated theoretically and analytically the effect of several materials constraints and functional conditions on the air electrode diffusion polarization and its influence on exergy efficiency of the fuel cell. The novel model of this paper is the second part of the model created on the SOFC to investigate the polarization on the electrodes and electrolyte. The model is divided into a number of units axially; for each unit, the thermal equations and the continuity equations of the electrochemical reactions are progressively solved with an iterative approach. The partial differential equations for mass and energy transfer through the cathode were spatially discretized using the finite differences method (FDM). The results shows that in order to maintain the peak exergy efficiency of the SOFCs in applied application environments, it is suggested to have the SOFC air cathode porosity between 0.35 and 0.38, the cathode pore mean radius in the range of 0.00015 to 0.0002 cm, the cathode material tortuosity of 2.9, the SOFC functional temperature between 850 °C and 950 °C, the thickness of the air electrode between 0.01 and 0.02 cm, and have the operating pressure in the range of 3 and 5 bar.

Модель Шустера – Шварцшильда для цилиндрически симметричных течений излучающего газа

The paper presents an algorithm for calculating radiative heat transfer in an emitting and absorbing gas based on the Schuster – Schwarzschild approximation for the case of a cylindrically symmetric flow. In this approximation, the radiation energy density is calculated as a quadrature of the absorption coefficient and the emissivity of the plasma. This quadrature is used to calculate the contribution of radiative heat transfer to the energy balance of the gas flow. An algorithm is described for constructing a one-dimensional adaptive difference mesh for the numerical calculation of the radiative energy transfer equation, which makes it possible to obtain a high-precision numerical solution on a mesh of a smaller dimension than the original “gas-dynamic” mesh. On test examples simulating the spatial inhomogeneity of the radiation field, a numerical comparison of the proposed method with some diffusion-type methods on the initial gas-dynamic and adapted computational meshes is carried out.

Application of the immersed boundary method in solution of radiative heat transfer problems

The immersed boundary method (IBM) was developed for the first time in the present study to simulate the radiative heat transfer problems in complex geometries. The pseudo time stepping technique was applied to solve the radiative transfer equation (RTE) using the finite volume method (FVM). Also, the direct forcing-sharp interface was used in IBM. This method was validated with some benchmark problems in two states of pure radiative and combined radiative-conductive heat transfer. Then, the effects of the conduction-radiation parameter and absorption coefficient on heat flux distribution and isotherms were investigated in an enclosure with inner complex bodies. The results indicated that this method acts well in solving the radiative heat transfer problems in complex geometries. Also, due to using unique Eulerian and Lagrangian grids for solving the energy and radiative transfer equations, production of a separate grid is not required in solving combined heat transfer problems. This is one of the most significant features of the presented method.

A method to solve the heat transfer problem for laminar flow of Phan-Tien-Tanner fluid in a flat channel

The paper presents a solution to the heat transfer problem for a flow of a four-mode viscoelastic linear Phan-Tien Tanner fluid in flat channels at a constant wall temperature. A special feature of this solution is the parametric representation of dependence of an axial velocity and temperature on coordinates. To obtain a solution to the energy transfer equation, we used the method of separating variables and decomposing the sought-for function into series by parameter. For a specific liquid, the profiles of dimensionless velocity and distribution of dimensionless temperature, Nusselt numbers, and other heat transfer characteristics for various Weissenberg numbers are presented.

Entropic and heat-transfer analysis of EMHD flows with temperature-dependent properties

This paper focusses on a theoretical analysis of the entropic generation and heat-transfer characteristics of electromagnetohydrodynamic (EMHD) flow in vertical hydrophobic microchannels. The flow viscosity, electrical conductivity, and thermal conductivity are assumed to be temperature variant. The fluid velocity and energy transfer equations associated with a system of coupled non-linear equations dealing with hydrophobic slip conditions are solved using a finite volume method associated with lubrication theory. The Debye–Hückel approximation is employed in an electrical double layer combined with the Poisson–Boltzmann equation to acquire an analytical solution for the electrical potential function. Slip velocities along with constant temperatures are provided to obtain numerical solutions for the case of a fully developed EMHD flow, in order to reveal the influence of fluid rheology. The results are presented for electromagnetic transport with variable viscosity over hydrophobic interfaces. Numerical and analytical validations are performed using the existing experimental results. In this study, we vizualize the significance of variable viscosity, electrical conductivity, and thermal conductivity on temperature distributions in the presence of a magnetic field. In this work, entropy generation is represented in terms of the Bejan number, which greatly impacts the normalized electroosmotic flow as well as the thermophysical parameters, leading to a minimization of the entropy-generation rate.

Graphical Interpretation of Wet Surface Evaporation Equations

AbstractPotential (wet surface) evaporation is the basis for many methods to estimate actual evaporation. Penman's (1948, https://doi.org/10.1098/rspa.1948.0037) combination of energy budget with mass and energy transfer equations can be depicted on temperature‐vapor pressure graphs. Key to this depiction is two straight lines on the graph representing constant enthalpy of the air at measurement height and at the surface skin, with the gap between the lines representing a combination of aerodynamic resistance variables and available energy. The equations on which Penman based his formula are easily solved numerically (without need for Penman's famous assumption) for T0w, the temperature the surface would have if it was saturated, keeping all other variables constant. Wet surface evaporation is proportional to the vapor pressure difference between the measurement height and the surface skin. Equilibrium evaporation, based on the slope of the vapor pressure curve at T0w, is also easily represented in the graph. The difference between the correct wet surface evaporation rate and Penman's approximation is immediately visible on the graphs. The different wet surface evaporation rates are compared using data from a tropical savanna in Australia. Implications for the classic two‐component interpretation of Penman's equation are discussed.

A Method to Solve the Heat Transfer Problem in a Circular Duct With an Internal Longitudinal Fin

Abstract The heat transfer problem in a circular duct with an internal longitudinal fin is investigated under constant wall temperature. Different fin heights are considered. The problem is solved in two stages. In the first stage, the distribution of axial velocity corresponding to the fully developed laminar viscous flow is obtained using comsolmultiphysics. In the second stage, the variable-separation method is implemented to reduce the solution of the energy transfer equation under constant wall temperature to an eigenvalue problem. Finally, the temperature distribution is obtained in the analytical form as a series expansion. The influence of fin height on the Nusselt number and the friction factor is discussed.

Energy transfer and red fluorescence properties of (Ce3+/Eu3+):Fluorophosphate glasses for lighting applications

Intense bluish-red color de-excitation by combining Ce3+ and Eu3+ ions has been observed in fluorophosphate (P2O5+K2O+Al2O3+BaF2+NaF2 (PKABfNf)) glasses which were fabricated by adopting conventional melting and rapid quench technique. Photoluminescence spectra, lifetimes, and energy transfer (ET) process in the Ce3+/Eu3+:PKABfNf glasses were studied. The fluorescence spectra of co-doped glasses exhibit broad band emission at blue corresponding to Ce3+ ions which can enhance the intensity of the red emission of Eu3+ via an efficient ET mechanism. The ET from Ce3+to Eu3+ can be explained based on the Blasse's equation and Dexter's theory of resonant ET. Furthermore, critical ET separation of distance (Rc), luminous efficiency due to energy transfer (ηET), CIE chromaticity co-ordinates (x,y), correlated color temperature (CCT) and color purity (CP) were calculated. The results reveal that the optimized Ce3+/Eu3+:PKABfNf glass can be a potential candidate for w-LEDs in lighting applications.

State-to-State Master Equation and Direct Molecular Simulation Study of Energy Transfer and Dissociation for the N2-N System.

We present a detailed comparison of two high-fidelity approaches for simulating non-equilibrium chemical processes in gases: the state-to-state master equation (StS-ME) and the direct molecular simulation (DMS) methods. The former is a deterministic method, which relies on the pre-computed kinetic database for the N2-N system based on the NASA Ames ab initio potential energy surface (PES) to describe the evolution of the molecules' internal energy states through a system of master equations. The latter is a stochastic interpretation of molecular dynamics relying exclusively on the same ab initio PES. It directly tracks the microscopic gas state through a particle ensemble undergoing a sequence of collisions. We study a mixture of nitrogen molecules and atoms forced into strong thermochemical non-equilibrium by sudden exposure of rovibrationally cold gas to a high-temperature heat bath. We observe excellent agreement between the DMS and StS-ME predictions for the transfer rates of translational into rotational and vibrational energy, as well as of dissociation rates across a wide range of temperatures. Both methods agree down to the microscopic scale, where they predict the same non-Boltzmann population distributions during quasi-steady-state dissociation. Beyond establishing the equivalence of both methods, this cross-validation helped in reinterpreting the NASA Ames kinetic database and resolve discrepancies observed in prior studies. The close agreement found between the StS-ME and DMS methods, whose sole model inputs are the PESs, lends confidence to their use as benchmark tools for studying high-temperature air chemistry.

Temperature Increase Research of Particles Formed by Ultrasonic Grinding of Spinach Leaves

The researchers chose spinach leaves as the research object for obtaining a natural dye based on chlorophyll because the vegetable is a food raw material rich in this pigment, and biologically active substances source with variety of its functional properties. To solve the problem of more complete chlorophyll extraction from crushed spinach leaves, they are to subject to a preliminary short-term soaking in a water/organic reagent emulsion under ultrasonic influence. Given that the water-based spinach suspension is thermolabile, a man must observe temperature restrictions, since when a certain critical temperature value is exceeded, the chlorophyll decomposition probability with its consumer properties loss is high. As it is practically impossible to determine the temperature inside a fine particle suspended in an aqueous emulsion experimentally, the authors considered to model internal and external heat transfer by solving the differential equation of heat energy transfer under appropriate boundary conditions using the numerical method of finite differences. The research determined structural-mechanical and thermophysical characteristics of crushed spinach leaves experimentally using the complex calculation method of thermophysical parameters based on the thermal inertia properties of a thermocouple. A man verified the solution adequacy by empirically determined suspension temperature after completion of the technological procedure. According to the developed algorithm, the authors compiled a program for the temperature fields evolution in the object of study considering the incident ultrasonic flow and an internal heat source with the new surface formation when grinding spinach particles in order to comply with thermal restrictions and maximize the quality parameters preservation. The article considers the rational duration of the ultrasonic grinding process without exceeding the recommended threshold temperatures.

Williamson nanofluid flow through porous medium in the presence of melting heat transfer boundary condition: semi-analytical approach

PurposePresent investigation based on the flow of electrically conducting Williamson nanofluid embedded in a porous medium past a linearly horizontal stretching sheet. In addition to that, the combined effect of thermophoresis, Brownian motion, thermal radiation and chemical reaction is considered in both energy and solutal transfer equation, respectively.Design/methodology/approachWith suitable choice of nondimensional variables the governing equations for the velocity, temperature, species concentration fields, as well as rate shear stress at the plate, rate of heat and mass transfer are expressed in the nondimensional form. These transformed coupled nonlinear differential equations are solved semi-analytically using variation parameter method.FindingsThe behavior of characterizing parameters such as magnetic parameter, melting parameter, porous matrix, Brownian motion, thermophoretic parameter, radiation, Lewis number and chemical particular case present result validates with earlier established results and found to be in good agreement. Finally reaction parameter is demonstrated via graphs and numerical results are presented in tabular form.Originality/valueThe said work is an original work of the authors.

Circulation cells topology and their effect on migration pattern of different multi-bend meandering rivers

In meandering rivers, a cross-stream flow, referred to as a secondary current, has important effects on broad spectra of hydraulic/environmental characteristics, running the gamut from river hydrodynamics and geomorphology to stream ecology. The transport equation for vorticity and kinetic energy transfer should be analyzed to specify terms involved in generation of secondary currents. However, there is limited research on scrutinizing these terms in meandering rivers. On the other hand, while rivers are mostly multi-bend, previous studies have been limited to single bends. In the current paper, three physical multi-bend channels representing a strongly curved bend, a mild bend and an elongated symmetrical meander loop are designed in order to unravel mechanisms responsible for forming circulation cells in cross sections. Experiments are carried out in the middle bend of these models. Cross-stream turbulence anisotropy considerably strengthens almost all near bank cells. Moreover, contrary to single sharp bends, multi bend effects hinder the transfer of the kinetic energy in both directions in the entrance section of the strongly curved bend.

Three-dimensional thermal calculations of the radiation chamber of a cylindrical heating tube furnace

The paper briefly describes the method of thermal calculations of radiation chambers of tube furnaces on the basis of modelling interconnected processes taking place in the furnace by three-dimensional differential energy equations, the transfer of momentum components, mass concentrations of methane and air, continuity, a two-parameter model of turbulence and the equation of energy transfer by radiation. A software package using this method is applied to carry out numerical analysis of the spatial temperature distribution, the averaged values of velocities of the combustion products turbulent flow in the combustion chamber of a cylindrical heating tube furnace of the oil-processing industry. A furnace diagram with diffusion burners on the radiation chamber floor and differential equations describing physical processes taking place in the furnace are given. Methods for solution of the initial equations are described. The obtained distributions of the surface densities of heat fluxes to the coils along the chamber perimeter at different heights of the furnace are demonstrated. Graphs of the temperature change of the combustion products and the tubular reactor walls for different values of the angular coordinate are given.

Dissipative heat energy on Cu and Al2O3 ethylene–glycol-based nanofluid flow over a heated semi-infinite vertical plate

The present analysis describes the effect of dissipative heat energy transfer in Ethylene–Glycol (EG) based on conducting nanofluid over a heated semi-infinite vertical plate past through a porous medium. Uniform magnetic field, heat source/sink, and the effect of particle concentration also have been discussed by incorporating in the energy and solutal transfer equations, respectively. In addition to that, the thermal properties of the nanofluid are affected by the thermal slip boundary condition since; the temperature slip is favorable for the reduction in the heat transfer. Assuming self-similar transformations, the governing PDEs are transformed into non-linear coupled ODEs. These transformed equations are solved by using semi-analytical techniques such as Adomian Decomposition Method (ADM). The characteristics of different parameters on the flow phenomena are obtained and presented via graphs. The numerical values of the thermophysical properties of both the nanoparticles and the base fluid are shown in the table. Validation of the present work is obtained by comparing our result with the earlier established result and it is found that both the results are coinciding with each other. However, the main quantified results are the following: due to heavy density of the Cu nanoparticles, increasing volume fraction in ethylene–glycol base fluid resists the fluid motion and the inclusion of dissipative heat energy is favorable to enhance the nanofluid temperature.

MATHEMATIC MODEL OF AND METHOD FOR SOLVING THE DIRICHLET HEAT-EXCHANGE PROBLEM FOR ONE-SHEET ROTARY HYPERBOLOID

It is the first generalized 3D mathematical model developed for calculating temperature fields in the thin-wall one-sheet rotary hyperboloid, which rotates with constant angular velocity around the axis OZ, ; the model was created with the help of known equations of generating lines in cylindrical coordinate system with taking into account finite velocity of heat conductivity and in the form of the Dirichlet boundary problem for the hyperbolic equation of heat conduction under condition that heat-conduction properties of the body were constant, and no internal sources of the heat were available. At initial moment of time, the body temperature was constant; values of temperature on outside surfaces of the body were known and presented continuous function of coordinate.The hyperbolic heat-conductivity equation was derived from the generalized energy transfer equation for the moving element of continuous medium with taking into account finiteness of the heat conductivity velocity.In order to solve the boundary problem, the desired temperature field was represented as a complex Fourier series. The obtained boundary problems for the Fourier coefficients were found with the help of Laplace integral transformations and the new integral transformation for two-dimensional finite space. Intrinsic values and intrinsic functions for the integral transformation kernel were found by method of finite element and the Galerkin methods. Besides, the domain was divided into simplex element.As a result, the temperature field in the thin-wall one-sheet rotary hyperboloid was found in the form of convergent series in Fourier functions.

Conservative finite volume strategy for investigation of solution crystal growth techniques

The article presents a finite volume method for the numerical study of solution crystal growth. The mathematical model is based on the time-dependent Navier-Stokes equations in the stream function/vorticity form, energy and mass transfer equations in solid and liquid phases, and thermodynamics equilibrium conditions at the phase transition interface. The problem is axisymmetric and studied in a cylindrical body-fitted coordinate system. The equations for temperature and composition in the solid and liquid phases, and interface velocity are solved simultaneously. The approach ensures global conservation of kinetic and thermal energy and conservation of the solute. The proposed strategy allows performing a full-scale numerical simulation of a variety of technological regimes with a reversal in the crystal dissolution and growth. Sample calculations are reported for prototypes of liquid phase diffusion growth process and Bridgman-Stockbarger method.

Investigation of the influence of the gas pipeline tee geometry on hydraulic energy loss of gas pipeline systems

CFD simulation investigated turbulent flows in equal gas pipeline tees, in which the gas flow completely moves from the main line to the branch. The study was performed for tees of different geometry – stamped with different bending radii of the transition from the branch to the main line and weld, where the main line and branch connection is made at right angles. The outer diameter of the tees varied from 219 mm to 1,420 mm, the bending radius of the transition from the branch to the main line from the minimum permissible to the maximum possible, the pressure in the gas pipeline at the tee location from 3 MPa to 7 MPa. The mathematical model is based on the solution of the Navier-Stokes and energy transfer equations closed by a two-parameter high-Reynolds k – e Launder-Sharma turbulence model. To describe the processes occurring at the wall, the wall function was used. It was found that the bending of the transition from the branch to the main line, the increase in the bending radius lead to a decrease in the intensity of flow separation at the bending point and a decrease in turbulence kinetic energy in recirculation areas. The velocity field of the gas flow after it moves from the main line to the branch becomes more uniform. All this greatly affects the magnitude of hydraulic energy loss of the gas flow in the tees. In this case, the greatest energy losses were observed in the tees located at the lowest pressure points in the gas pipeline system. An analysis of the results showed that if the ratio of the bending radius of the main line and branch connection to the outer diameter is more than 0.25, then the influence of such a tee on the energy loss of the gas pipeline system is minimal. Local resistance coefficients of equal gas pipeline tees are calculated and the resulting equation for their calculation will be useful for specialists designing gas pipeline systems

Thermodynamics of non-isothermal diffusion at the extraction from cheese whey by Lupin

In many cases, extraction is accompanied by thermal phenomena. We have established the possibility of intensifying the process through the use of heated cheese whey. Lupine has a geometric shape (sphere, cylinder, plate) loaded into an extractor filled with cheese whey. Due to the temperature difference between the solid and the liquid, temperature gradients are observed. As the body warms up, the temperature gradient decreases and then disappears. For example, an organized step temperature mode. However, such a regime should be technologically and energetically justified. Thus, during extraction there is a periodic non-stationarity. The emergence of this period is noted in the main works. The expression for the increase in entropy per unit time is written. Given the changes in entropy, the Gibbs equation is written. The basics of equations are the second laws of thermodynamics. As a result, the results obtained as a result of thermodynamic driving forces were obtained. The equations of energy (heat) and mass transfer of substances are written. Thermodynamic forces contribute to the formation of heat flux and mass flow of substances. The consumption of a substance depends not only on the gradient (diffusion), but also on the temperature gradient (thermal diffusion). Air temperature is defined as a temperature gradient. The differential equations of heat and mass transfer of Lykov were rewritten taking into account the extraction process. The numerical values of the coefficients Dт and aс they relate to the assessment of the effect of superposition effects (thermal diffusion and diffusion thermal conductivity). The overlay effect can be neglected, since the relatively small gradients of temperatures and concentrations arising in the lupine. It is noted that the possibility of simplified differential equations is associated with small values of the Lykov criterion. Because of this, there should be little.

On the dendrite growth simulation during multitrack selective laser melting process

The evolution of the microstructure in the selective laser melting (SLM) process is investigated in the framework of a multiscale approach. The kinetic equation for the phase-field (order parameter) coupled with the energy transfer equation is used to describe the crystallization processes. To study the effect of technological parameters on the microstructure of the synthesized product, and to establish a connection between the models of macro- and micro-scale through the temperature gradient, numerical studies of macro processes in the formation of a series of parallel SLM tracks were carried out. It is shown that the adjustment of the laser power level changes the temperature dynamics in the test areas, which, in turn, through the changed temperature gradients affects the evolution of the microstructure in the micro-scale model. The calculations showed that the implemented numerical model qualitatively describes the kinetics of crystallization and formation of microstructures in the SLM process.

Mathematical modeling of heat transfer in a closed two- phase thermosyphon

We suggested a new approach for describing heat transfer in thermosyphons and determining the characteristic temperatures. The processes of thermogravitation convection in the coolant layer at the lower cap, phase transitions in the evaporation zone, heat transfer as a result of conduction in the lower cap are described at the problem statement. The main assumption, which was used during the problem formulation, is that the characteristic times of steam motion through the thermosyphon channel are much less than the characteristic times of thermal conductivity and free convection in the coolant layer at the lower cap of the thermosyphon. For this reason, the processes of steam motion in the thermosyphon channel, the condensate film on the upper cap and the vertical walls were not considered. The problem solution domain is a thermosyphon through which heat is removed from the energy-saturated equipment. The ranges of heat flow changes were chosen based on experimental data. The geometric parameters of thermosyphon and the fill factors were chosen the same as in the experiments (height is 161 mm, diameter is 42 mm, wall thickness is 1.5 mm, ε=4-16%) for subsequent comparison of numerical simulation results and experimental data. In the numerical analysis it was assumed that the thermophysical properties of thermosyphon and coolant caps do not depend on temperature; laminar flow regime was considered. The dimensionless equations of vortex, Poisson and energy transfer for the liquid coolant under natural convection and the equations of thermal conductivity for the lower cap wall are solved by the method of finite differences. Numerical simulation results showed the relationship between the characteristic temperatures and the heat flow supplied to the bottom cap of thermosyphon. The results of the theoretical analysis are in satisfactory agreement with the known experimental data.