Differential Equations Of Energy Research Articles

Extension of Gibbs–Duhem equation including influences of external fields

Gibbs–Duhem equation is one of the fundamental equations in thermodynamics, which describes the relation among changes in temperature, pressure and chemical potential. Thermodynamic system can be affected by external field, and this effect should be revealed by thermodynamic equations. Based on energy postulate and the first law of thermodynamics, the differential equation of internal energy is extended to include the properties of external fields. Then, with homogeneous function theorem and a redefinition of Gibbs energy, a generalized Gibbs–Duhem equation with influences of external fields is derived. As a demonstration of the application of this generalized equation, the influences of temperature and external electric field on surface tension, surface adsorption controlled by external electric field, and the derivation of a generalized chemical potential expression are discussed, which show that the extended Gibbs–Duhem equation developed in this paper is capable to capture the influences of external fields on a thermodynamic system.

Thermal and velocity slip effects on Casson nanofluid flow over an inclined permeable stretching cylinder via collocation method

The main emphasis of present work is to investigate the velocity and thermal slip effects on Casson nanofluid with heat and mass transfer phenomena over an inclined permeable stretching cylinder. The cylinder is subject to transverse magnetic field. Buongiorno’s model is adapted to study the Brownian motion and thermphoresis effects which play a dominant role in nanofluid. Governing set of equations are derived in terms of partial differential equations for Casson nanofluid model, consisting continuity, momentum, energy and concentration equation which are transformed into set of coupled nonlinear ordinary differential equations using similarity transformation. The numerical solution is obtained using collocation method. The literature survey shows that the present problem has not been studied before. Physical quantities of interest are nanofluid velocity, temperature, concentration, skin friction coefficient, Nusselt number and Sherwood number which are analyzed through graphs against the emerging physical parameters. It is found that Nb and Nt play a dominant role within the thermal and concentration boundary layer regions. In the same manner, suction parameter and both velocity and thermal slip parameters depicts the dynamic effects in the entire domain of stretching surface of the cylinder.

Theoretical Investigation of Thermal Effects in an Adiabatic Chromatographic Column Using a Lumped Kinetic Model Incorporating Heat Transfer Resistances

A nonequilibrium nonisothermal lumped kinetic model (LKM) is analytically and numerically investigated to evaluate the effects of inherent temperature fluctuations in an adiabatic chromatographic column. The model comprises convection–diffusion partial differential equations quantifying mass and energy balances in the mobile phase coupled with differential and algebraic equations for mass and energy in the stationary phase. Besides two mass transfer coefficients, two heat transfer coefficients are involved in the model equations. The properties of the considered model are investigated for linear concentration and temperature dependencies of the equilibrium loadings. The Laplace transformation and Eigen decomposition techniques are utilized to solve the set of equations. These solutions are helpful for understanding, analyzing, and interpreting the propagation speeds and shapes of both concentration and thermal fronts migrating in chromatographic columns. The moment generating property of the Laplace domain solutions is employed to obtain explicit analytical temporal moments of the concentration and energy profiles which provide instructive tools to analyze the properties of the model considered and to estimate unknown model parameters from measured transients. For illustration, several case studies are carried out by assuming realistic model parameters. The applicability range of the analytical solutions derived is assessed by comparing selected specific results with numerical results of a nonequilibrium and nonisothermal model by considering nonlinear adsorption isotherms.

Singular perturbed components of flows – linear precursors of shock waves

A comparative analysis of the infinitesimal symmetries of various well-known systems of governing equations used for mathematical descriptions of flows and waves in fluids has shown that only the basic system of equations, including the empirical equation of state and the partial differential equations of mass, momentum, energy and matter transport, is characterized by a ten-parameter Galilean transformation group. An analysis of the complete solutions of the linearized system of fundamental equations for weakly dissipating media reveals a wide class of previously unknown singularly perturbed solutions supplementing well investigated regular solutions describing propagating waves. Fine flow components, whose geometry is typical for internal boundary layers that supplement the wave fields exist both at the boundaries and inside the volume of the liquid, are classified as linear precursors of shock waves. The calculated pattern of periodic internal waves beams covered with high-gradient envelopes agrees with data from independently performed experiments on measurements and visualization of the fine structure of linear and nonlinear waves in continuously stratified media.

Exact solution for unsteady free convection flow of Casson fluid in vertical channel

The influence of free convection flow on unsteady Casson fluid over a vertical channel is studied in this paper. The governing equation consists of partial differential momentum and energy equations have been transformed into non-dimensional equations by using the appropriate nondimensional variables. The expression of exact solutions for nondimensional velocity and temperature profiles are obtained by using the Laplace transform method. The obtained exact solutions are found satisfied all the initial and boundary conditions. The behaviour of velocity and temperature profiles influenced by Casson fluid parameter, Grashof number, Prandtl number and time are plotted graphically and presented in tables. It can be observed that, the velocity profile and boundary layer thickness decrease when the values of Casson fluid parameter and Prandtl number increase. This is due to the increase of viscous force and retard the movement of the velocity. The present result is validated when the obtained exact solution of velocity is compared with published result and found in a good agreement.

Effect of Flowing Seawater on Supercritical CO2 - Superheated Water Mixture Flow in an Offshore Oil Well Considering the distribution of heat generated by the work of friction

The modeling of thermal fluid (mostly reported for saturated steam) flow in wells involving steady-state heat transfer inside wellbores and transient heat transfer in sea water or formation is reviewed and inherited. It is a good addition of modeling of supercritical/superheated fluid to the existing body of literature. The flow of supercritical CO2 coupled with superheated water (SHW) in offshore wells is described by means of the differential mass, energy and momentum balance equations along the vertical wellbores. The effect of supercritical CO2 on pressure and temperature of the multi-component thermal fluid is expressed in terms of the real gas model (the S-R-K model). The differential equations are solved with finite difference method on space involving the constant injection parameters at wellhead. It is found out that: (a). while seawater has a significant influence on temperature drop in wellbores, its effect on pressure profiles is weak. (b). both temperature and superheat degree decrease with increasing of the content of supercritical CO2. Besides, study of the effect of injection rate and pressure is conducted.

Компьютерная модель теплообмена в установке непрерывного литья и прессования цветных металлов

Based on the software complex Ansys CFX, a three-dimensional computer model of complex heat transfer in a continuous casting and pressing of non-ferrous metals has been developed. The features of constructing a computational grid and solving the system of differential energy equations for the processed metal and the elements of the installation are considered. The dynamics of the process of casting and pressing of aluminum alloy on an experimental laboratory installation is studied. Calculated temperatures of the metal and elements of the prototype of the installation are determined. An experimental estimation of reliability of modeling results is carried out and adequacy of the developed computer model is shown.

Finite time thermodynamic analysis of small alpha-type Stirling engine in non-ideal polytropic conditions for recovery of LNG cryogenic exergy

Abstract In the paper, a second order thermodynamic analysis of a small scale alpha-type Stirling engine is presented. The developed mathematical model is based on the Finite Time Thermodynamics (FFT) approach and it is derived from the differential time dependent equations of energy and mass conservation. The engine model consists of three spaces: compression, expansion and regenerator. In contrast to available models, the model presented in this paper assumes polytropic processes in the compression and expansion spaces, which corresponds to the non-ideal heat transfer in these spaces. The results, obtained using the developed second order model, were compared with the results obtained using the Computational Fluid Dynamics (CFD) analysis of the same engine. A very good agreement between the second order model and the CFD model was achieved. Finally, the developed model was applied to analyse potential to recover cryogenic exergy of the Liquefied Natural Gas (LNG). The paper presents the results of the dimensional upscaling of the engine and the influence of the average working pressure. It was revealed that increasing the size of the engine or the average working pressure moves the engine to practically unachievable working conditions.

Impact of nanoparticle volume fraction on squeezed MHD water based Cu, Al2O3 and SWCNTs flow over a porous sensor surface

The squeezed MHD flow of water based metallic nanoparticles over a porous sensor surface in the presence of heat source has been investigated. The physical significance of the problem is water based on the geometry and the interaction of copper (Cu), aluminum oxide (Al2O3) and SWCNTs. The governing partial differential equations of momentum and energy are converted into ODEs for assured groups of the controlling parameters. The numerical and analytical results of the ODEs are determined utilizing fourth or fifth order Fehlberg method with shooting technique and OHAM and it is analyzed that there is no import difference between them. It is investigated that in squeezing flow phenomena, the effect of nanoparticle volume fraction on SWCNTs–water in the presence of magnetic field with thermal radiation energy plays a dominant role on heat transfer as compared to the other mixtures in the flow regime.

Double slip effects of Magnetohydrodynamic (MHD) boundary layer flow over an exponentially stretching sheet with radiation, heat source and chemical reaction

The double slips effect on the magnetohydrodynamic boundary layer flow over an exponentially stretching sheet with suction/blowing, radiation, chemical reaction and heat source is presented in this analysis. By using the similarity transformation, the governing partial differential equations of momentum, energy and concentration are transformed into the non-linear ordinary equations. These equations are solved using Runge-Kutta-Fehlberg method with shooting technique in MAPLE software environment. The effects of the various parameter on the velocity, temperature and concentration profiles are graphically presented and discussed.

Unsteady magnetohydrodynamics flow about a stagnation point on a stretching plate embedded in porous medium

This paper presents an unsteady two-dimensional laminar boundary layer flow of an incompressible, viscous and electrically conducting fluid in a vicinity of a stagnation point towards a stretching plate embedded in porous medium with variable free stream velocity. The governing nonlinear partial differential equations of mass, momentum and energy are reduced to a system of nonlinear ordinary differential equations by using suitable similarity variables. The ordinary differential equations are tackled numerically by using Runge-Kutta Fehlberg fourth-fifth order method with shooting technique. The effect of the significant parameters on the fluid temperature and velocity are graphically presented and discussed. The numerically computed results are in good agreement with the available published results in the literature. It is found that the velocity increases with the increasing unsteadiness parameter where the velocity ratio parameter λ = 0.5 but it decreases for λ = 1.5.

Magnetohydrodynamic (MHD) boundary layer stagnation point flow and heat transfer of a nanofluid past a stretching sheet with melting

The paper examines the melting heat transfer in magnetohydrodynamic (MHD) stagnation point flow of a nanofluid past a stretching sheet. Boundary layer theory is employed to simplify the system of partial differential equations of motion, energy and concentration to three coupled non-linear ordinary differential equations. The non-linear ordinary differential equations and their boundary conditions are changed into dimensionless form by using suitable similarity variables before numerically solved using fourth order Runge-Kutta-Fehlberg method along with shooting technique. The effect of pertinent parameters on different flow fields are determined and discussed in detail through several plots and tables.The numerical results are obtained for velocity, temperature and concentration profiles. It is found that the skin friction coefficient and Sherwood number decrease with an increase in B and M parameters. However, the local Nusselt number -θ′(0) increases with an increase in B and Nt. Then, the results are compared and found to be in good agreement with the previously published results in limiting cases of the problem.

Unique Superposition Solution of Multiple Plumes’ Flow via Mean Kinetic Energy Fluxes

AbstractThis paper simplifies the partial differential equation (PDE) of total mean energy in the plumelike region of turbulent buoyant jets under the well-known distributions in the longitudinal a...

Numerical analysis of Al2O3/water nano-fluids natural convection and entropy generation in enclosures

The aim of this work is to analyze the natural convection phenomena and entropy generation of water-based Al 2 O 3 nanofluids in square enclosure. The simulated domain corresponds to a square cavity heated from below and cooled from the top. The left and right walls are heated up to a height H = (3/4 W) and are adiabatic in the remaining part (1- H ). Numerical investigations have been carried out based on coupled partial differential equations of momentum and energy which are solved using finite volume method. The effective thermal conductivity of the nanofluid was expressed by the Maxwell-Garnetts model however the dynamic viscosity was calculated according to the Brinkman formula. The obtained results were presented by average Nusselt number, streamlines, isotherms and entropy generation with various pertinent parameters, namely, Rayleigh number (10 0 ≤ Ra ≤ 10 6 ), volumetric fraction of nanoparticles (1% ≤ ϕ ≤ 4% ). It was found that the heat transfer increases with the increase of Rayleigh number and volume fraction. The choice of these parameters is important to obtain maximum enhancement of heat transfer with minimum entropy generation.

Electrothermal transport of nanofluids via peristaltic pumping in a finite micro-channel: Effects of Joule heating and Helmholtz-Smoluchowski velocity

The present article studies theoretically the electrokinetic pumping of nanofluids with heat and mass transfer in a micro-channel under peristaltic waves, a topic of some interest in medical nano-scale electro-osmotic devices. The microchannel walls are deformable and transmit periodic waves. The Chakraborty-Roy nanofluid electrokinetic formulation is adopted in which Joule heating effects are incorporated. Soret and Dufour cross-diffusion effects are also considered. Under low Reynolds number (negligible inertial effects), long wavelength and Debye linearization approximations, the governing partial differential equations for mass, momentum, energy and solute concentration conservation are derived with appropriate boundary conditions at the micro-channel walls. The merging model features a number of important thermo-physical, electrical and nanoscale parameter, namely thermal and solutal Grashof numbers, the Helmholtz-Smoluchowski velocity (maximum electro-osmotic velocity) and Joule heating to surface heat flux ratio. Closed-form solutions are derived for the solute concentration, temperature, axial velocity, averaged volumetric flow rate, pressure difference across one wavelength, and stream function distribution in the wave frame. Additionally expressions are presented for the surface shear stress function at the wall (skin friction coefficient), wall heat transfer rate (Nusselt number) and wall solute mass transfer rate (Sherwood number). The influence of selected parameters on these flow variables is studied with the aid of graphs. Bolus formation is also visualized and analyzed in detail.

Analysis of Entropy Generation Due to Micropolar Fluid Flow in a Rectangular Duct Subjected to Slip and Convective Boundary Conditions

The entropy generation due to steady, incompressible micropolar fluid flow in a rectangular duct with slip and convective boundary conditions (CBCs) is calculated. An external uniform magnetic field is applied which is directed arbitrarily in a plane perpendicular to the flow direction. The governing partial differential equations of momentum, angular momentum, and energy are solved numerically using finite-difference method. The obtained velocity, microrotation, and temperature distributions are then used to evaluate the entropy generation and Bejan number. The effects of various parameters on the entropy generation and Bejan number are discussed through graphs.

Numerical study of radiative Maxwell viscoelastic magnetized flow from a stretching permeable sheet with the Cattaneo–Christov heat flux model

In this article, the Cattaneo–Christov heat flux model is implemented to study non-Fourier heat and mass transfer in the magnetohydrodynamic flow of an upper-convected Maxwell fluid over a permeable stretching sheet under a transverse constant magnetic field. Thermal radiation and chemical reaction effects are also considered. The nonlinear partial differential conservation equations for mass, momentum, energy and species conservation are transformed with appropriate similarity variables into a system of coupled, highly nonlinear ordinary differential equations with appropriate boundary conditions. Numerical solutions have been presented for the influence of elasticity parameter (α), magnetic parameter (M 2), suction/injection parameter $$(\lambda ),$$ Prandtl number (Pr), conduction–radiation parameter (R d ), sheet stretching parameter (A), Schmidt number (Sc), chemical reaction parameter $$\left( {\gamma_{c} } \right)$$ , modified Deborah number with respect to relaxation time of heat flux (i.e., non-Fourier Deborah number) on velocity components, temperature and concentration profiles using the successive Taylor series linearization method (STSLM) utilizing Chebyshev interpolating polynomials and Gauss–Lobatto collocation. The effects of selected parameters on skin friction coefficient, Nusselt number and Sherwood number are also presented with the help of tables. Verification of the STSLM solutions is achieved with existing published results demonstrating close agreement. Further validation of skin friction coefficient, Nusselt number and Sherwood number values computed with STSLM is included using Mathematica software shooting quadrature.

Unsteady flow of a Maxwell nanofluid over a stretching surface in the presence of magnetohydrodynamic and thermal radiation effects

The problem of unsteady magnetohydrodynamic (MHD) boundary layer flow of a non-Newtonian Maxwell nanofluid over a stretching surface with thermal radiation is considered. The Maxwell model is used to characterize the non-Newtonian fluid behaviour. An appropriate similarity transformation is employed to transform the governing partial differential equations of mass, momentum, energy and nanoparticle concentration into ordinary differential equations. The coupled non-linear ordinary differential equations are solved by using the variational finite element method. The flow features and the heat transfer characteristics and nanoparticle volume fraction are analyzed and discussed in detail for several sets of values of the governing flow parameters. The results for the skin-friction coefficient, local Nusselt number and the local Sherwood number are presented in tables for various values of the flow controlling parameters.

MHD Boundary Layer Flow of Nanofluid over a Continuously Moving Stretching Surface

The present investigation provides an insight in the steady, incompressible and electrically conducting boundary layer flow of viscoelastic nanofluid flowing due to a moving, linearly stretched surface. The governing system of nonlinear partial differential equations is simplified by considering Boussinesq and boundary layer approximations. An analytical solution of the resulting nonlinear ordinary differential equations for momentum, energy and concentration profiles is obtained using the homotopy analysis method (HAM).

Mathematical modelling of the working process of a hot gas pressure feed system

The paper discusses improving the weight and size characteristics of propulsion systems with liquid thrusters due to the replacement of conventional compressed-gas systems by hot gas systems where a gas generator operating on propellants is used for hot gas production. Possible algorithms of the operation of such feed systems are discussed. A mathematical model has been created for the purpose of computational and theoretical study of operating processes in the considered feed system. The model represents system units as lumped volumes. Differential equations of weight, internal energy, and combustion gases concentration are written for every lumped volume. The differential equation of unit structure temperature is also added. The differential equations of mass and internal energy are converted into equations of pressure and temperature. The model is expressed through mass flow ratios and heat exchanges between the system units, dependences of thermo-physical properties on the temperature, combustion gas concentrations and the ratio of fuel components in the gas generator. The temperature, pressure, concentration of combustion gases in the system units and the unit structure temperature are specified as input parameters. The simulation results showed satisfactory agreement with the experimental data, which makes it possible to use the developed model for computational and theoretical studies of the hot gas pressure feed systems.