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.     

A novel effective approach for solving nonlinear heat transfer equations

In this paper, a novel analytic technique, namely the Laplace transform new homotopy perturbation method (LTNHPM), is applied for solving the nonlinear differential equations arising in the field of heat transfer. This approach is a new modification to the homotopy perturbation method based on the Laplace transform. Unlike the previous approach implemented by the present authors for these problems, the present method does not consider the initial approximation as a power series. The nonlinear convective–radiative cooling equation and nonlinear equation of conduction heat transfer with the variable physical properties are chosen as illustrative examples. The exact solution has been found for the first case and for the others; results with remarkable accuracy have been achieved which verify the efficiency as well as accuracy of the presented approach. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20411     

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     

Adomian decomposition method for the MHD flow of a viscous fluid with the influence of dissipative heat energy

An investigation is carried out on the effect of dissipative heat energy on the flow of an electrically conducting viscous fluid past a shrinking sheet. Both viscous and Joule dissipation effects are considered along with heat generation/absorption for the enhancement of heat transfer properties. The governing nonlinear coupled partial differential equations are transformed into nonlinear ordinary differential equations by a suitable choice of similarity transformations. However, the complex transformed equations are solved by an approximate analytical method known as the Adomian decomposition method with a suitable initial guess solution assumed from the known initial conditions. Moreover, the behavior of several parameters characterizing the flow phenomena are studied via graphs and the numerical computations for the engineering coefficients are obtained and presented through tables. However, the major outcomes of the results are that a higher suction is required to resist the fluid temperature and sinks as well as the dissipative heat energy favors enhancing the fluid temperature at all points in the flow domain.     

On the effect of heat generation/absorption on magnetohydrodynamic free convective flow in a vertical annulus: An Adomian decomposition method

This article deliberates fully developed natural convective flow of heat-generating/absorbing fluid in open-ended vertical concentric annulus under a magnetic field. The momentum and energy equations which arise from the definition of velocity and temperature are written in the dimensional form and then recast into the nondimensional form. Approximate solutions are obtained by using the semi-analytical Adomian decomposition method. The solutions for the velocity, temperature, skin friction, mass flux and rate of heat transfer are obtained. The influence of physical parameters such as heat generation/absorption parameter $\left(\delta \right)$ and Hartmann number (M) are illustrated with the aid of graphs and tables. In the course of this investigation, it is found that an increase in the heat generation/absorption parameter increases the temperature. In addition, the magnitude of the temperature is higher for heat-generating fluid in comparison to heat-absorbing fluid. Furthermore, the temperature and velocity can be controlled by carefully selecting suitable values of the heat generation/absorption parameter and Hartmann number respectively.     

Adomian decomposition method for the effect of transpiration on natural convection MHD flow in a vertical annulus with heat generation/absorption

This article studies the effect of transpiration on a fully developed natural convection MHD flow in a vertical annulus with heat generation/absorption. The governing coupled differential equations are solved by an estimated analytical method identified as the Adomian decomposition method with initial conditions and an appropriate initial guess condition. Furthermore, the effect of several parameters illustrating the flow phenomena is considered through graphs and the numerical computations are obtained and presented through tables. However, it is observed that there is a decrease in velocity due to the effect of the suction of fluid on the heated porous wall with concurrent injection. Also, heat generation parameter enhances the temperature of the fluid on both the isothermal and constant heat flux.     

Investigation of dynamic heat transfer process through coaxial heat exchangers in the ground

Recently, researchers are focussing on using ground coupled heat pump systems as a heat source or sink rather than air source heat pumps for HVAC needs due to the stable temperature and the high thermal inertia of the soil. The investment cost of these systems is too expensive therefore the precise thermal analysis, design and parameter optimization are essential. For an accurate design, the maximum of physical phenomena such as: axial effects, seasonal effects, underground water flow and BHE dynamic behaviour must be accounted for in order to reflect exactly the real physical situation. In the present paper thermal interferences are investigated under seasonal effects and a dynamic heat flux for a vertical coaxial borehole heat exchangers field. This enables to avoid thermal interferences by predicting efficient period of operation corresponding to the beginning of the studied phenomena (interferences) for a given separation distance between two boreholes. To reach this purpose, as a first step, a transient 2D Finite volume method (FVM) for a single borehole heat exchanger was built using MATLAB, which accounts for accurate axial and seasonal effects and a dynamic heat flux that is function of depth and time. This model has been validated against the Finite Line Source (FLS) analytical solution and good agreement between analytical and numerical methods has been obtained. Then the model has been extended to a quasi-3D model in order to investigate thermal interferences between two neighbouring boreholes. After 500 h and at the mid-point of the separating distance (1.5 m) where interferences are the strongest, the temperature is 50% (6.64 °C) lower than the case where there are no interferences.     

Experimental and numerical study on heat transfer and flow characteristics in an alternating cross‐section flattened tube

An experimental and numerical study on convection heat transfer of water flowing through an alternating cross‐section flattened (ACF) tube are investigated in this paper. The thermal‐fluid characteristics were evaluated by numerical simulation. The test run conditions covered a mass flux of 200 to 800 kg m^?2 s^?1, a heat flux of 10 kW/m², and an inlet temperature of 40°C. The results showed that the Nusselt number increased with the increase in mass flux. Moreover, the heat transfer was also affected by the flow characteristics. Vortices were formed at the curved wall, and their intensities were increased along the flow direction. It was also found that the heat transfer and pressure drop were larger than that of the circular tube. However, the thermal performance was greater than the pressure loss penalty. The comparison results showed that the ACF tube had better performance than the circular tube. Further, the details of heat transfer, flow resistance, and fluid behavior were investigated and discussed in this study.     

Investigation of conjugate heat transfer in a photovoltaic wall

In this paper, the results of an experimental and theoretical investigation of combined heat transfer in a photovoltaic wall have been reported. The photovoltaic wall is a prototype, which is composed of two pieces of BP PV panels and a Styrofoam board, and part of the light of radiation energy from the indoor lamps can be converted into electricity. Through experiments, the performance of such a photovoltaic wall has been studied. For the convenience of the treatment of heat radiation, a model in terms of the integration of the absolute temperature has been proposed for the numerical simulation of the combined heat transfer in the test wall. By comparison, it is found that with regard to the thermal radiation of lamp surface, good agreement between the results of simulation and experimental data is obtained. © 2004 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(2): 117–128, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10136     

Exergy transfer characteristics of forced convective heat transfer through a duct with constant wall heat flux

The examination of exergy transfer characteristics caused by forced convective heat transfer through a duct with constant wall heat flux for thermally and hydrodynamic fully developed laminar and turbulent flows has been presented. The exergy transfer Nusselt number is put forward and the dependence relationships of the exergy transfer Nusselt number on the heat transfer Nusselt number, Reynolds number and Prandtl number are obtained. Expressions involving relevant variables for the local and mean convective exergy transfer coefficient, non-dimensional exergy flux and exergy transfer rate, etc. have been derived. By reference to a smooth duct, the numerical results of exergy transfer characteristics for fluids with different Prandtl number are obtained and the effect of the Reynolds number and non-dimensional cross-sectional position on exergy transfer characteristics is analyzed. In addition, the results corresponding to the exergy transfer and energy transfer are compared.     

微槽群平板热管散热器传热性能的研究

分别选择不同的翅片间距和高度,对一种新型微槽群平板热管散热器的翅片结构进行优化,得到了热管散热器的最佳整体结构。结果表明:翅片的间距为14mm、高度为60mm时,平板热管散热器的传热性能最好。将热管、管脚以及翅片的温度与实验结果进行对比,结果吻合良好。     

Shock wave boundary layer interactions in hypersonic flows over a double wedge geometry by using conjugate heat transfer

Shock wave boundary layer interaction phenomena play a critical role in the design of supersonic and hypersonic vehicles. Consequently, this paper mainly focuses on hypersonic flow over a double wedge model, flow fields around concave corners are relatively complicated, and produce several classical viscous flow features depending on the combination of the first and second wedge, and the important characteristic phenomena are mainly the shock‐boundary layer and shock‐shock interaction. For these interactions, aerodynamic heating and pressure loads increase greatly when interactions are present. The conjugate heat transfer (CHT) technique is expected to exactly predict the separation bubble length, heat flux, skin friction coefficient, and pressure distributions in double wedge studies in hypersonic applications. In the present CHT studies, the different wall materials used are thermal insulation, Macor, and SiC, it is clearly shown that while using Macor and thermal insulctation wall material in CHT studies, the interface temperature, skin friction coefficient, heat flux distribution along the length change significantly with increase in simulation time. In comparing the CHT results with the fluid flow solver with the wall, considering isothermal and adiabatic boundary results, it is clearly indicated that the fluid flow solver results are either underpredicting or overpredicting the interface properties, but CHT studies give an accurate prediction of the separation length and interface properties.     

Influence of heat transfer on the nonorthogonal stagnation point flow of a third‐order fluid towards a stretching surface

The present article looks at the theoretical analysis of a steady stagnation‐point flow with heat transfer of a third‐order fluid towards a stretching surface. The formulation of the problem has been carried out for a third order fluid and constructed partial differential equations are rehabilitated into ordinary differential equations. The consequential ordinary differential equations are solved analytically using the homotopy analysis method (HAM). Graphical illustrations are shown for various parameters involved in the flow equations. Numerical values of skin friction coefficients and heat flux are computed and presented through tables. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21042     

Effect of two immiscible multifluid flow on internal heat generation or absorption in a vertical channel in the presence of concentration

The effect of multifluid flow on internal heat generation or absorption in a vertical channel in the presence of concentration is investigated in this paper. The fluids are incompressible in both regions, that is, Region-I and Region-II, and it is assumed the transport properties of fluid flow are constant. With the help of the analytical method, all the basic equations transformed into governing coupled nonlinear ordinary differential equations are solved, and the solutions obtained for fluid velocity, temperature, and concentration. These results are illustrated by plotting graphs and for various physical parameters. Here, we can control the results by means of the heat absorption coefficient, width ratio, and viscosity ratio.     

Effect of thermophoresis and Brownian motion on the melting heat transfer of a Jeffrey fluid near a stagnation point towards a stretching surface: Buongiorno's model

This analysis intends to address the coupled effect of phase change heat transfer, thermal radiation, and viscous heating on the MHD flow of an incompressible chemically reactive nanofluid in the vicinity of the stagnation point toward the stretching surface, taking a Jeffrey fluid as the base fluid. Convergent analytical solutions for the nonlinear boundary layer equations are obtained by the successive application of scaling variables and the highly efficacious homotopy analysis method. Error analysis is implemented to endorse the convergence of the solutions. Through parametric examination, influence of various physical parameters occurring in analysis of the profiles of velocity, temperature, and nanoparticle concentration, coefficient of surface drag, rates of mass and heat transfer is explored pictorially. The Deborah number and the melting parameter are found to enhance velocity, and the associated momentum boundary layers are thicker, whereas the magnetic field depreciates the flow rate. Temperature is observed to enhance with the thermophoresis parameter, Prandtl number and Eckert number, whereas a reduction is seen with the thermal radiation parameter and Brownian motion parameter. Nanoparticle concentration is depleted by the chemical reaction parameter, the thermophoresis parameter, and the Lewis number.     

Effect of thermal radiation on magnetohydrodynamic three‐dimensional motion of a nanofluid past a shrinking surface under the influence of a heat source

An analytical technique known as the homotopy analysis method is used to acquire solutions for magnetohydrodynamic 3‐D motion of a viscous nanofluid over a saturated porous medium with a heat source and thermal radiation. The governing nonlinear partial differential equations are changed to ordinary differential equations employing appropriate transformations. Validation of the present result is done with the help of error analysis for flow and temperature. The influences of pertinent parameters on momentum, energy, and Nusselt number are studied and discussed. The major findings are: the velocity of the nanofluid is affected by the nanoparticle volume fraction and the thickness of the thermal boundary layer becomes thinner and thinner subject to sink, whereas the effect is revered in case of the source.     

MHD conjugate heat transfer and entropy generation analysis of MWCNT/water nanofluid in a partially heated divided medium

The aim of this article is to conduct the lattice Boltzmann simulation of the magnetohydrodynamic (MHD) natural conjugate heat transfer in an apportioned cavity loaded with a multiwalled carbon nanotube/water nanofluid. The divided cavity is, to some extent, heated and cooled at the upright walls, whereas the horizontal walls are adiabatic. The nanofluid properties are evaluated on the basis of experimental correlations. The parameters ranges in the study are as follows: nanoparticles' volume fraction (%): 0 ≤ ? ≤ 0.5, temperature (°C): T = 27, Rayleigh number (Ra): 10³ ≤ Ra ≤ 10⁵, Hartmann number (Ha): 0 ≤ Ha ≤ 90, and the magnetic field inclination angle (γ): 0 ≤ γ ≤ π/2. The current outcomes are observed to be in great concurrence with the numerical results introduced in the literature. The impacts of the aforesaid parameters on local and average heat transfer, entropy generation, and Bejan number (Be) are explored and discussed. Indeed, the transfer of heat increases linearly with ? for a low Ra. As Ra increases, the average Nusselt number decreases for a high value of ?. The increase of nanoparticles' volume fraction leads to a reduction in the entropy generation and an increase in the Bejan number for a high Ra, but at low Ra, these functions remain constant. As the Ha increases, the transfer of heat and the entropy generation decreases, whereas there is an increase in Be. The transfer of heat, total entropy generation, and the Be depends strongly on the direction of the magnetic field. The increase of heater and cooler size has a great influence on the transfer of heat, entropy generation, and Be.