Numerical investigation of transient heat transfer to hydromagnetic channel flow with radiative heat and convective cooling

This present study consists of a numerical investigation of transient heat transfer in channel flow of an electrically conducting variable viscosity Boussinesq fluid in the presence of a magnetic field and thermal radiation. The temperature dependent nature of viscosity is assumed to follow an exponentially model and the system exchanges heat with the ambient following Newton’s law of cooling. The governing nonlinear equations of momentum and energy transport are solved numerically using a semi-implicit finite difference method. Solutions are presented in graphical form and given in terms of fluid velocity, fluid temperature, skin friction and heat transfer rate for various parametric values. Our results reveal that combined effect of thermal radiation, magnetic field, viscosity variation and convective cooling have significant impact in controlling the rate of heat transfer in the boundary layer region.     

Similarity analysis of a nonlinear fin equation

A nonlinear fin equation in which the thermal conductivity is an arbitrary function of the temperature and the heat transfer coefficient is an arbitrary function of a spatial variable is considered. Scaling, translational and spiral group symmetries of the equations are determined. Classification of the functions for which these symmetries exist is performed. In general, no useful symmetries exist for arbitrary thermal conductivity and heat transfer coefficients. However, for some restricted forms of the functions, useful symmetries exist. A similarity transformation is used to reduce the partial differential equation to an ordinary differential equation as an example.     

Heat and mass transfer in annular liquid jets: III. Combustion within the volume enclosed by the jet

The nonlinear dynamics of and heat and mass transfer processes in annular liquid jets are analyzed by means of a nonlinear system of integrodifferential equations which account for the liquid motion and the gases enclosed by the jet. Both linear and sinusoidal heat and mass addition sources are considered to take place homogeneously within the volume enclosed by the jet's inner interface in an attempt to simulate the combustion of hazardous wastes or materials within this volume. It is shown that the liquid's temperature at the jet's inner interface increases rapidly with linear heat addition, but drops also quickly to its initial value once heat addition is ended, whereas the pressure coefficient and the volume enclosed by the jet increase until they reach a maximum value and then decrease in an oscillatory manner towards their steady values. For the case of sinusoidal heat addition, it is shown that the pressure coefficient and interfacial concentration, temperature and heat and mass fluxes oscillate in a sinusoidal manner with the same frequency as that of the sinusoidal heat source. It is also shown that mass transfer phenomena are much slower than heat transfer ones. For the case of linear mass addition, it is shown that the temperature of the gases enclosed by the jet first decreases because of dilution and then it increases until it reaches a constant value that corresponds to the same temperature for the gases and the flowing liquid. The pressure of the gases enclosed by the jet first increases because of mass addition and then slowly decreases because of mass absorption by the jet.     

Analytical solutions to nonlinear equations arising in heat transfer by variational iteration,homotopy perturbation,and Adomian decomposition methods

As thermal conductivity plays an important role on fin efficiency, we tried to solve heat transfer equation with thermal conductivity as a function of temperature. In this research, some new analytical methods called homotopy perturbation method, variational iteration method, and Adomian decomposition method are introduced to be applied to solve the nonlinear heat transfer equations, and also the comparison of the applied methods (together) is shown graphically. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Comparison of symmetry and homotopy solutions for nonlinear heat transfer systems

This study deals with the analytic solutions for the two nonlinear problems arising in heat transfer. These problems are due to (i) temperature distribution in lumped system of combined convection–radiation and (ii) temperature distribution in a uniformly thick rectangular fin radiation to free space. Large symmetry algebras are obtained for the nonlinear ordinary differential equations (ODEs) describing the heat transfer. We use method of canonical variables to either linearize or transform the governing equations to integrable forms. Exact solutions are constructed. Finally, a comparison is given between the homotopy and symmetry solutions.     

A nonlinear nonlocal mixed problem for a second order pseudoparabolic equation

We study a nonlocal mixed problem for a nonlinear pseudoparabolic equation, which can, for example, model the heat conduction involving a certain thermodynamic temperature and a conductive temperature. We prove the existence, uniqueness and continuous dependence of a strong solution of the posed problem. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplicator. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem.     

General exact solution of the fin problem with the power law temperature‐dependent thermal conductivity

In this paper, the general exact implicit solution of the second‐order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo‐geometric fin parameter is obtained for any value of the mode of heat transfer n and the constant β. Copyright © 2015 John Wiley & Sons, Ltd.     

Optimal design of a fin in steady-state

A fin serves as an extended surface to enhance the heat transfer from a larger heated mass to which it is attached. We consider a fin in steady-state and find its shape that maximizes the efficiency. Our approach is based on a piecewise linear approximation of design. We develop the corresponding algorithm and conduct numerical experiments. Our optimization problem is similar but not identical to the problem studied by L. Hanin. We discuss the applicability of Schmidt’s hypothesis for our model. This paper is an extension of the previous authors’ paper where we considered the shape of the fin that minimizes the cooling time of a heated mass.     

Some exact solutions of the fin problem with a power law temperature-dependent thermal conductivity

This study investigates the exact solutions of a nonlinear fin problem with temperature-dependent thermal conductivity and the heat transfer coefficient. Both the conduction and heat transfer terms are given by the same power law in one case and the distinct power law in the other. Classical Lie symmetry techniques are employed to construct the exact solutions which satisfy the realistic boundary conditions. The effects of the physical applicable parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.     

Inverse shape and surface heat transfer coefficient identification

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin is considered. This problem which consists of determining an unknown inner boundary of an annular domain and possibly its surface heat transfer coefficient from one or two pairs of boundary Cauchy data (boundary temperature and heat flux) is solved numerically using the meshless method of fundamental solutions (MFS). A nonlinear unconstrained minimisation of the objective function is regularised when noise is added to the input boundary data. The stability of the numerical results is investigated for several test examples with respect to noise in the input data and various values of the regularisation parameters.     

Multi-objective optimization of pin fin to determine the optimal fin geometry using genetic algorithm

In this study, one dimensional heat transfer in a pin fin is modeled and optimized. We used Bezier curves to determine the best geometry of the fin. The model equations are solved to analyze the heat transfer. Total heat transfer rate and fin efficiency factor are considered as two objective functions and multi-objective optimization carried out to maximize heat transfer rate and fin efficiency simultaneously. Fast and elitist non-dominated sorting genetic algorithm (NSGA-II) is used to determine a set of multiple optimum solutions, called ‘Pareto optimal solutions. The optimized results are presented with Pareto front which demonstrate conflict between two objective functions in the optimized point, both energy conservation and thermal analysis are carried out to verify the solution method and the results shows good precision.     

Monotone iterative method of upper and lower solutions applied to a multilayer combustion model in porous media

This paper presents a new nonlinear reaction–diffusion–convection system coupled with a system of ordinary differential equations that models a combustion front in a multilayer porous medium. The model includes heat transfer between the layers and heat loss to the external environment. A few assumptions are made to simplify the model, such as incompressibility; then, the unknowns are determined to be the temperature and fuel concentration in each layer. When the fuel concentration in each layer is a known function, we prove the existence and uniqueness of a classical solution for the initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. We construct monotone iterations of upper and lower solutions and prove that these iterations converge to a unique solution for the problem, first locally and then, in time, globally.     

Investigation of phase change in a spiral-fin heat exchanger

In this study the effect of a spiral fin on the melting process of a phase change material in a heat exchanger is investigated. Water as the heat transfer fluid enters the tube at 348 K and the phase change material, RT50, the melting temperature of which is in the range of 318 K to 324 K, fills the space between the tube and the shell. This study focuses on the effect of geometrical parameters of the fin including the fin thickness and fin pitch on the melting process. In addition, the effect of changing the heat exchanger angle with respect to the horizontal is examined. Results indicate that fin shape is a critical parameter in every modification. In fact, for constant fin thickness, provided the fin pitch varies from 10 mm to 20 mm, due to 58% increase in fin height, the melting time reduces about 35%. Moreover, considering constant fin pitch, when the fin thickness increases from 1.5 mm to 2.5 mm, because of the 28% reduction of fin height, the melting time increases 59%. Also, an increase in the angle of the heat exchanger from zero to 90°, affects the melting process considerably.     

Thermal analysis of a longitudinal trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient

A homotopy analysis method (HAM) is used to develop analytical solution for the thermal performance of a straight fin of trapezoidal profile when both the thermal conductivity and the heat transfer coefficient are temperature dependent. Results are presented for the temperature distribution, heat transfer rate, and fin efficiency for a range of values of parameters appearing in the mathematical model. Since the HAM algorithm contains a parameter that controls the convergence and accuracy of the solution, its results can be verified internally by calculating the residual error. The HAM results were also found to be accurate to at least three places of decimal compared with the direct numerical solution of the mathematical model generated using a fourth–fifth-order Runge–Kutta–Fehlberg method. The HAM solution appears in terms of algebraic expressions which are not only easy to compute but also give highly accurate results covering a wide range of values of the parameters rather than the small values dictated by the perturbation solution.     

变热特性参数条件下肋片温度周期性变化的传热研究*

本文研究了变热特性参数下,根部温度作周期性变化的肋片传热情况.应用摄动法求解控制微分方程;并且采用打靶法和叠加原理进行数值计算,求解过程是嵌进的、非迭代的.对某种形状的肋片而言,当肋片根部温度作周期性变化时,其传热过程受以下几个参数的影响:E──导热系数的温度系数;N──肋片传热的特性参数;ε──温度波动的幅度参数;B──温度波动的频率;以及对流系数的变化模式等.文中给出了这些参数变化时对肋片的温度分布及热流率、肋效率等的影响情况.所得结果,不但具有理论价值,而且对工程设计也有现实指导意义.     

Steady heat transfer through a radial fin with rectangular and hyperbolic profiles

We construct some exact solutions for thermal diffusion in a fin with a rectangular profile and another with a hyperbolic profile. Both the thermal conductivity and the heat transfer coefficient are assumed to be temperature dependent. Moreover, the thermal conductivity and the heat transfer terms are given by the same power law in one case and distinct power laws in the other. A point transformation is introduced to linearize the problem when the power laws are equal. In the other case, classical Lie symmetry techniques are employed to analyze the problem. The exact solutions obtained satisfy the realistic boundary conditions. The effects of applicable physical parameters such as the thermo-geometric fin parameter and the fin efficiency are analyzed.     

Applications of He’s principles to partial differential equations

This paper applies the variational iteration method (VIM) and semi-inverse variational principle to obtain solutions of linear and nonlinear partial differential equations. The nonlinear model is considered from gas dynamics, fluid dynamics and Burgers equation. The linear model is the heat transfer (diffusion) equation. Results show that variational iteration method is a powerful mathematical tool for solving linear and nonlinear partial differential equations, and therefore, can be widely applied to engineering problems.     

Analytic solutions for a rotating radial fin of rectangular and various convex parabolic profiles

The homotopy analysis method (HAM) is used to develop an analytical solution for the thermal performance of a radial fin of rectangular and various convex parabolic profiles mounted on a rotating shaft and losing heat by convection to its surroundings. The convection heat transfer coefficient is assumed to be a function of both the radial coordinate and the angular speed of the shaft. Results are presented for the temperature distribution, heat transfer rate, and the fin efficiency illustrating the effect of thickness profile, the ratio of outer to inner radius, and the angular speed of the shaft. Comparison of HAM results with the direct numerical solutions shows that the analytic results produced by HAM are highly accurate over a wide range of parameters that are likely to be encountered in practice.     

An approximation of the analytic solution of some nonlinear heat transfer in fin and 3D diffusion equations using HAM

In this article, the approximate solution of nonlinear heat diffusion and heat transfer equation are developed via homotopy analysis method (HAM). This method is a strong and easy‐to‐use analytic tool for investigating nonlinear problems, which does not need small parameters. HAM contains the auxiliary parameter ?, which provides us with a simple way to adjust and control the convergence region of solution series. By suitable choice of the auxiliary parameter ?, we can obtain reasonable solutions for large modulus. In this study, we compare HAM results, with those of homotopy perturbation method and the exact solutions. The first differential equation to be solved is a straight fin with a temperature‐dependent thermal conductivity and the second one is the two‐ and three‐dimensional unsteady diffusion problems. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010