Assessment of homotopy–perturbation and perturbation methods in heat radiation equations

One of the newest analytical methods to solve the nonlinear heat transfer equations is using both homotopy and perturbation methods in equations. Here, homotopy–perturbation method is applied to solve heat transfer problems with high nonlinearity order. The origin of using this method is the difficulties and limitations of perturbation or homotopy. It has been attempted to show the capabilities and wide-range applications of the homotopy–perturbation method in comparison with the previous ones in solving heat transfer problems. In this research, homotopy–perturbation method is used to solve an unsteady nonlinear convective-radiative equation and a nonlinear convective-radiative conduction equation containing two small parameters of ε₁ and ε₂.     

Assessment of homotopy analysis method and homotopy perturbation method in non-linear heat transfer equation

Two new analytical methods to solve nonlinear heat transfer equations are homotopy perturbation method and homotopy analysis method. Here, homotopy analysis method, which gives us a vast freedom to choose the answer type, is applied to solve nonlinear heat transfer differential equations and analytical results are compared with those of HPM and the numerical results. In this study, the procedure of HAM is applied to two cases in different ways according to the physics of the target problem. Comparing the two methods, our attention is focused on the results accuracy; and applicability of different methods in many cases with different limitation is studied. In the two examples of this paper, the effect of small parameter increaser on the accuracy of the analytical results of two methods also has been studied. The first differential equation is the modeling equation of a cooling lumped system with combined convection and radiation. The second one is the modeling equation of heat transfer with conduction in a slab of thermal dependent conductivity.     

Semi-exact solution of elastic non-uniform thickness and density rotating disks by homotopy perturbation and Adomian's decomposition methods. Part I: Elastic solution

In this work, two powerful analytical methods, namely homotopy perturbation method (HPM) and Adomian's decomposition method (ADM), are introduced to obtain distributions of stresses and displacements in rotating annular elastic disks with uniform and variable thicknesses and densities. The results obtained by these methods are then compared with the verified variational iteration method (VIM) solution. He's homotopy perturbation method which does not require a “small parameter” has been used and a homotopy with an imbedding parameter p ∈ [0,1] is constructed. The method takes the full advantage of the traditional perturbation methods and the homotopy techniques and yields a very rapid convergence of the solution. Adomian's decomposition method is an iterative method which provides analytical approximate solutions in the form of an infinite power series for nonlinear equations without linearization, perturbation or discretization. Variational iteration method, on the other hand, is based on the incorporation of a general Lagrange multiplier in the construction of correction functional for the equation. This study demonstrates the ability of the methods for the solution of those complicated rotating disk cases with either no or difficult to find fairly exact solutions without the need to use commercial finite element analysis software. The comparison among these methods shows that although the numerical results are almost the same, HPM is much easier, more convenient and efficient than ADM and VIM.     

Comparison of HAM and HPM solutions in heat radiation equations

In this paper it is proved that the perturbation and homotopy perturbation solutions for the two problems namely (i) unsteady convective–radiative equation and (ii) non-linear convective–radiative conduction equation are only valid for weak non-linearity. It is worth mentioning that homotopy perturbation method (HPM) does not provide exact and convergent solutions in most of the situations. Moreover, HPM have no criteria for establishing the convergence of the series solution. In the present study, it is shown and commented that the results in the two examples are valid only for small values of the parameters. The homotopy analysis method (HAM) is used in obtaining more meaningful and valid solutions.     

Homotopy analysis of unsteady heat transfer started impulsively from rest along a symmetric wedge

In this paper, the unsteady heat transfer over an impulsively started wedge is investigated. The mixed convection, namely, the thermal boundary-layer growth is caused by the sudden increase of the surface temperature as it is set into motion. The analytic approximations with high accuracy are given using the Homotopy–Padé technique, which agree well with the numerical results using an implicit finite-difference method known as Keller–Box method. It is found that the Homotopy–Padé approximations have no relation to the auxiliary parameters ?_f and ?_g, which play important roles in the homotopy analysis method.     

Semi-exact solution of non-uniform thickness and density rotating disks. Part II: Elastic strain hardening solution

Analytical solutions for the elastic–plastic stress distribution in rotating annular disks with uniform and variable thicknesses and densities are obtained under plane stress assumption. The solution employs a technique called the homotopy perturbation method. A numerical solution of the governing differential equation is also presented based on the Runge–Kutta's method for both elastic and plastic regimes. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening. The results of the two methods are compared and generally show good agreement. It is shown that, depending on the boundary conditions used, the plastic core may contain one, two or three different plastic regions governed by different mathematical forms of the yield criterion. Four different stages of elastic–plastic deformation occur. The expansion of these plastic regions with increasing angular velocity is obtained together with the distributions of stress and displacement.     

An explicit solution for the combined heat and mass transfer by natural convection from a vertical wall in a non-Darcy porous medium

An analytic technique, namely the homotopy analysis method, is applied to solve the combined heat and mass transfer by natural convection adjacent to a vertical wall in a non-Darcy porous medium governed by a set of three fully coupled, highly nonlinear similarity equations. An explicit, totally analytic and uniformly valid solution is derived, which agrees well with numerical results.     

A note on the thermal stability of a reactive non-Newtonian flow in a cylindrical pipe

The effect of variable viscosity and viscous dissipation on the thermal stability of a one-step exothermic, reactive non-Newtonian flow in a cylindrical pipe was considered assuming negligible reactant consumption. The governing equations; momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by closed-form and approximate techniques respectively. We obtained closed-form solutions for the momentum equation via the Homotopy analysis method(HAM), while the resulting energy equation was analysed for thermal stability via the variational technique. Parametric analysis of the solutions were conducted on the dimensionless velocity (v(r)), the critical Frank-Kamenetskii parameter(δ_cr) and the critical temperature(θ_cr) and expressed graphically and in tabular forms.     

Application of the homotopy perturbation method for the solution of inverse heat conduction problem

This paper deals with an application of the homotopy perturbation method for the solution of inverse heat conduction problem. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of functions describing the temperature and heat flux on the boundary, when the temperature measurements in the domain are known. Examples illustrating discussed application and confirming utility of this method in such a type of problem was also presented.     

The application of the homotopy perturbation method to one-phase inverse Stefan problem

In this paper, the application of the homotopy perturbation method for solving the inverse Stefan problem is presented. This problem consists in the calculation of temperature distribution in the domain, as well as in the reconstruction of the functions describing temperature and heat flux on the boundary, when the position of the moving interface is known.     

Heat generation effects in natural convection inside a porous annulus

The interplay between internal heat generation and externally driven natural convection inside a porous medium annulus is studied in detail using numerical methods. The axisymmetric domain is bounded with adiabatic top and bottom walls and differentially heated side walls sustaining steady natural convection of a fluid with Prandtl number, Pr = 5, through a porous matrix of volumetric porosity, ? = 0.4. The generalized momentum equation with Brinkman–Darcy–Forchheimer terms and the local thermal non-equilibrium based two-energy equation model are solved to determine the flow and the temperature distribution. Beyond a critical heat generation value defined using an internal Rayleigh number, Ra_I,cr^?, the convection transits from unicellular to bicellular mode, as the annulus T_max becomes higher than the fixed hot-wall temperature. The Ra_I,cr^? increases proportionately when the permeability based external Rayleigh number Ra_E^? and the solid–fluid thermal conductivity ratio γ are independently increased. A correlation is proposed to predict the overall annulus Nu as a function of Ra_E^?, Ra_I^?, Da and γ. It predicts the results within ± 20% accuracy.     

Isoconversional kinetic analysis of co-combustion of sewage sludge with straw and coal

In this paper, the co-combustion behaviour of sewage sludge with straw and coal were carried out in a thermogravimetric analyzer under different atmospheres and at different heating rates (10, 20 and 40 K min⁻¹) in the temperature range from ambient temperature to 1000 °C. TG and DTG curves were analysed. The Flynn–Wall–Ozawa’s (FWO) and Kissinger–Akahira–Sunose’s (KAS) isoconversional methods were used to yield dependency of the activation energy of reduction process on the degree of conversion. The values of E_α were obtained. The results indicate that: with the increase of heating rate, the maximum weight loss rate of samples increase obviously. The activation energy is practically constant in the 0.2 ? α ? 0.9 range, with the average values of E_α = 137.27 and 132.38 kJ mol⁻¹ calculated by FWO and KAS methods, respectively.     

Numerical evaluations of transcendental equations for transient experiments

Transcendental equations have to be solved in order to extract surface exchange and diffusion coefficients from transient techniques. In this work, numerical evaluations of the equation β_ntanβ_n = A, frequently used for a thin plane sheet geometry, have been studied in detail. Based on the Newton–Raphson method and the characteristics of the tangent function, a fast and stable algorithm was suggested to solve this equation for all values of A, which has so far not been the case in the literature. Moreover, a numerical approach to the transcendental equation used in a spherical geometry was also brought up.     

He's homotopy perturbation method for two‐dimensional heat conduction equation: Comparison with finite element method

Heat conduction appears in almost all natural and industrial processes. In the current study, a two‐dimensional heat conduction equation with different complex Dirichlet boundary conditions has been studied. An analytical solution for the temperature distribution and gradient is derived using the homotopy perturbation method (HPM). Unlike most of previous studies in the field of analytical solution with homotopy‐based methods which investigate the ODEs, we focus on the partial differential equation (PDE). Employing the Taylor series, the gained series has been converted to an exact expression describing the temperature distribution in the computational domain. Problems were also solved numerically employing the finite element method (FEM). Analytical and numerical results were compared with each other and excellent agreement was obtained. The present investigation shows the effectiveness of the HPM for the solution of PDEs and represents an exact solution for a practical problem. The mathematical procedure proves that the present mathematical method is much simpler than other analytical techniques due to using a combination of homotopy analysis and classic perturbation method. The current mathematical solution can be used in further analytical and numerical surveys as well as related natural and industrial applications even with complex boundary conditions as a simple accurate technique. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20292     

A coupled volume-of-fluid and level set (VOSET) method for computing incompressible two-phase flows

A coupled volume-of-fluid and level set (VOSET) method, which combines the advantages and overcomes the disadvantages of VOF and LS methods, is presented for computing incompressible two-phase flows. In this method VOF method is used to capture interfaces, which can conserve the mass and overcome the disadvantage of nonconservation of mass in LS method. An iterative geometric operation proposed by author is used to calculate the level set function ? near interfaces, which can be applied to compute the accurate curvature κ and smooth the discontinuous physical quantities near interfaces. By using the level set function ? the disadvantages of VOF method, inaccuracy of curvature and bad smoothness of discontinuous physical quantities near interfaces, can be overcome. Finally the computing results made with VOSET method are compared with those made with VOF and LS methods.     

Electrical and dielectric properties of lithium manganate nanomaterials doped with rare-earth elements

Substituted LiR_xMn_2 − xO₄ (R = La³⁺, Ce³⁺_, Pr³⁺ and x = 0.00 − 0.20) nanoparticles are prepared by the sol–gel method and the consequent changes in their lattice structure, dielectric and electrical parameters are determined by XRD, ED-XRF, SEM, LCR meter bridge and dc electrical resistivity measurements. Diffraction data show that the samples are single-phase spinel materials with crystallites sizes between 21 and 38 nm. The lattice parameter, cell volume and X-ray density are found to be affected by doping the Li-manganate with the rare-earth elements. The ED-XRF analysis confirms the stoichiometric composition of the synthesized samples and SEM reveals their morphology. Calculated values of the dielectric constant (?) and the dielectric loss (tan δ) decrease with the frequency of the applied field. This is attributed to Maxwell–Wagner polarization. Replacement of manganese by the rare-earth elements results in an improvement in the structural stability of the material, which is considered to be useful for enhancement of the cycleability of the compounds when used in lithium rechargeable batteries, and increases significantly the values of ? and tan δ (except for Ce). Lithium manganate nanomaterials with high ? and low tan δ may be attractive for application in memory storage devices.     

Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body

An analytic approach based on the homotopy analysis method is proposed to solve a nonlinear model of combined convective and radiative cooling of a spherical body. An explicit series solution is given, which agrees well with the exact or numerical solutions. Our series solutions indicate that, for the nonlinear model of combined convective and radiative cooling of a spherical body, the temperature on the surface of the body decays more quickly for larger values of the Biot number Bi and/or the radiation–conduction parameter N_rc. Different from traditional analytic techniques based on eigenfunctions and eigenvalues for linear problems, our approach is independent of the concepts of eigenfunctions and eigenvalues, and besides is valid for nonlinear problems in general. This analytic method provides us with a new way to obtain series solutions of unsteady nonlinear heat conduction problems, which are valid for all dimensionless times 0 ⩽ τ < +∞.     

Modification of ZnS1−x−0.5yOx(OH)y–ZnO photocatalyst with NiS for enhanced visible-light-driven hydrogen generation from seawater

NiS was loaded on ZnS_1−x−0.5yO_x(OH)_y–ZnO photocatalyst by two methods (depositing NiS on the photocatalyst via impregnation with a NiS sol and via a precipitation reaction). The activity of the NiS-modified photocatalyst for hydrogen evolution from pure water and seawater was investigated using mixed electron donors (sulfide and sulfite). The activity (in pure water-electron donor solution) is dependant of the modification method and loading content of NiS. When Na₂S·9H₂O and Na₂SO₃ are added into seawater, a mixture precipitate (Mg(OH)₂ and CaSO₃) produces. The precipitate is detrimental to the photocatalytic hydrogen evolution over the unmodified ZnS_1−x−0.5yO_x(OH)_y–ZnO. However, for the NiS-modified photocatalyst, the detrimental effect decreases notably. Interestingly, the activity of the modified photocatalyst in seawater-electron donor solution (with the precipitate) is higher than that in the pure water-electron-donor solution (without the precipitate). The possible mechanism was discussed.     

Ab initio study of hydrogen adsorption on Zn2(NDC)2(diPyTz) metal-organic framework decorated with alkali and alkaline earth metal cations

We have studied effect of alkali and alkaline earth metal cations (Li⁺, Na⁺, K⁺, Be²⁺, Mg²⁺) decoration on hydrogen adsorption of the organic linker of Zn₂(NDC)₂(diPyTz) by employing three cluster models: diPyTz:mLi⁺ (m = 1–4), diPyTz:mLi⁺:nH₂ (m = 0,1,2 and n = 1–5) and diPyTz:1M⁺:1H₂ (M⁺ = Na⁺, K⁺, Be²⁺, Mg²⁺) complexes, using density functional theory (DFT) and second-order Moller–Plesset perturbation theory (MP2). The calculated binding energies show that decoration of the organic linker with alkali and alkaline earth metal cations enhanced H₂ interaction with diPyTz when compared with the pristine diPyTz. The atomic charges were derived by Mulliken, ChelpG and ESP methods. Finally, the atoms in molecules theory (AIM) were also applied to get more insight into the nature of the interaction of diPyTz and Li⁺. Results of AIM analysis show that N–Li⁺ bond in diPyTz organic linker's complex appears as shared electron interaction.     

Steady flow of an Eyring Powell fluid over a moving surface with convective boundary conditions

In this article, the flow and heat transfer of Eyring Powell fluid over a continuously moving surface in the presence of a free stream velocity are investigated. Convective boundary conditions have been used in the problem formulation. The solution for velocity and temperature are computed by applying the homotopy analysis method (HAM). The effects of emerging fluid parameters (?), (δ) and Prandtl number (Pr) on the velocity and temperature are illustrated through graphs and tables for different values of λ. It is found that the boundary layer thickness is an increasing function of (?) and decreasing function of (δ). However the temperature and thermal boundary layer thickness decrease when the values of (?) and (δ) are increased.