The analytical solution of the one alloy solidification problem

In this paper we obtain the analytical solution for a semi-infinite solidifying alloy. Thus, a three-phase problem including solid, solid–liquid, and liquid phases is analytically solved. Linearization of the heat conduction equation for an alloy is based on the method proposed in our recent papers.Note that the method does not allow one to solve the problem of solidification of an alloy with the given function λ(T) (liquid fraction). The dependence λ(T) is determined from the condition of linearization of the heat conduction equation within the mush zone.The analytical solution presented is an important test example for analysis of the numerical schemes used for systems with moving boundaries, e.g., for programs simulating vacuum arc remelting.     

An approximate analytical solution for the nonlinear inverse axial dispersion model

An analytical solution for diffusion with homogenous n the order nonlinear chemical reaction in laminar flow of a Newtonian fluid through tubular reactor with unknown boundary conditions is presented. The results of the new solution have been compared with previously published numerical solutions and shown to be accurate.     

A Least-Squares Solution to Nonlinear Steady-State Multi-Dimensional IHCP

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Exact analytical solution of forced convection through a porous-saturated duct

In the current paper, the effect of a magnetic field on the fully developed forced convective flow and heat transfer is studied. An exact solution is extracted when the flow in the porous medium is governed by the Brinkman–Forchheimer Extended Darcy model. First, the problem formulation is explained to obtain a new system of mathematical formulation. Then, by utilizing the properties which are imposed into the problem, the exact closed-form analytical solution of the problem is explored. Finally, the main results are illustrated to show the impact of the porous media-shaped parameter, magnetic parameter, Forchheimer number, and viscosity ratio. It should be mentioned that the asymptotic results achieved in this study were compared with the exact results and it is found that they are in good agreement.     

Exact analytical solution of a two diode circuit model for organic solar cells showing S-shape using Lambert W-functions

Organic solar cells (OSCs) often show a kink, also called S-shape, in the current–voltage (I–V) characteristics, that has been attributed to different physical phenomena such as poor quality of cathode-active layer interface or unbalance charge carrier mobilities. This non-ideal behaviour can be electrically modelled including a second diode, in reverse bias, together with an extra shunt resistance (R_P2) in the traditional solar cell equivalent circuit. In this paper, we solve without approximations the transcendental equation system derived from this modified circuit. We have obtained an analytical expression for I–V curves decoupling the voltage drop in each diode using Lambert W function. This expression has been fitted to experimental data in order to obtain circuital parameters. Simulations varying saturation current of reverse diode (I₀₂) and R_P2 have been performed in order to study the dependence of S-shape with these parameters.     

Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates

This study presents an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer. Here, it is supposed that the fibers are winded around the cylinder and their direction can be changed in each lamina. Laplace transformation is applied to change the domain of the solutions from time into the frequency. An appropriate Fourier transformation has been derived using the Sturm–Liouville theorem. Here, a set of equations for Fourier coefficients are obtained based on the boundary conditions both inside and outside the cylinder, and the continuity of temperature and heat flux at boundaries between adjacent layers. The exact solution of this set of equations is obtained using Thomas algorithm and Fourier coefficients are expressed by recessive relations. Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate. Some industrial examples are presented to investigate the ability of current solution for solving the wide range of applied steady and unsteady problems.     

The analytical solution for a model of direct contact evaporation in spray columns

Reference [1] presented a simple model on direct contact heat transfer between two immiscible liquids in a countercurrent spray column and got the numerical solution with the variable step Runge-Kutta method. This paper obtains the analytical solution of this model, from which the exact relationships between target quantities and influence quantities are given. In this solution the explicit formulae for the column height required for complete evaporation and the temperature of continuous phase at the end of evaporation are given, which will be very useful for the initial design of spray columns in engineering. The two formulae have the following forms     

SOFC mathematic model for systems simulations—Part 2: definition of an analytical model

In Part 1 of the present study it was shown that the equations governing SOFC electrochemical behavior must be locally considered, since a zero-dimensional approximation can lead to inaccurate results. Nevertheless, it was also emphasized that a SOFC model must be as simple as possible, when the whole system is simulated and parametric analyses are conducted.In the second part of the study, an analytical, one-dimensional model is developed integrating the local equations previously defined.The results are expected to be more accurate then the ones obtained in part one, since the fuel cell is considered as a reactor where, for example, hydrogen, carbon monoxide and oxygen react while passing through the cell itself. Consequently, the equations take into account the concentration variation of both the anodic and cathodic gas components along the cell. Moreover, for the anodic gas, the water-shift reaction is also considered.     

Parametric model for the analytical determination of the solidification and cooling times of semi-crystalline polymers

This work presents an efficient method to quickly calculate with good accuracy (to 5%) the solidification time of an injected semi-crystalline polymer slab. Under some hypotheses this polymer can be considered as a phase change material with a constant phase change temperature. We use a noteworthy property established as the ratio between the thickness of a solidifying phase change finite medium and the solidified thickness in a semi-infinite medium. The knowledge of this ratio enables to predict analytically the solidification time in a 1D finite medium. This ratio can be parameterized as a function of characteristic numbers in phase change problems: Stefan numbers and the ratio of thermal diffusivities of both phases. The results are compared with those given by a complete model integrating the physics of the coupling between heat transfer and crystallization kinetics. The solidification times computed from both models are very close, demonstrating the relevance of the simplified model. Finally, we also get a very good accuracy in calculating the total cooling time, from injection to ejection.     

Approximate analytical model for PCM solidification in a rectangular finned container with convective cooling boundaries

Thermal energy storage units that utilize latent heat storage materials have received increased attention in the recent years because of their relatively large heat storage capacities and isothermal behavior during charging and discharging. In this study, an analytical approach is presented for the prediction of temperature during the solidification in a two-dimensional rectangular latent heat storage using a phase change material (PCM) with internal plate fins. The basic energy equation is formulated accounting for the presence of a heat thermal fluid (HTF) on the walls. A two-dimensional numerical model is developed based on the enthalpy method to predict the distribution temperature of the fin and solid–liquid interface in storage. Results from the analytical solution and numerical model show a good agreement. The developed analytical model estimates satisfactorily the solidification time of PCM in storage, which is useful in the design of PCM-based thermal energy storages and cooling systems.     

The approximate solution of nonlinear heat equation

Nowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.     

Analytical solution for solidification of close-celled metal foams

This paper introduces an analytical model capable of predicting the location of solidification front as well as the full solidification time for heterogeneous materials such as close-celled metallic foams. Full numerical simulations with the method of finite difference are separately conducted to validate the analytical model. The model predicts that an increase in porosity causes significant retardation of full solidification as a result of decreased effective thermal conductivity and diffusivity of the porous medium. Effects of pore shape and cooling temperature on overall solidification behavior were also studied.     

The Analytical Approximate Solution of the 2D Thermal Displacement

The 2D plane gas flow under heating (with nonentity boundary condition) has been discussed by the analytical approach in this paper. The approximate analytical solutions have been obtained for the flow passing various kinds of heat sources. Solutions demonstrate the thermal displacement phenomena are strongly depend on the heating intensity.     

Exact solution for freezing of humid porous half-space

New Stephan-like problem is defined and exact solutions for temperature and moisture distributions as well as the position of the moving freezing front in a humid porous half-space are obtained. The analytical solution is programmed in BASIC and some illustrative examples are plotted.     

Exact analytical solution of a convective diffusion from a wedge to a flow with a first order chemical reaction at the surface

The problem of mass transfer in a fluid flow past a wedge with a chemical reaction of the first order at the surface is investigated. The approach suggested previously by Apelblat [1] for the solution of a mass transfer problem with a first order chemical reaction at the interface in boundary layer flow is generalized for the case of flow past a wedge. The solution of the problem is derived in a closed analytical form.     

Transient analytical solution of solar energy devices

This note presents a transient analytical solution of fluid temperature variation in solar energy devices, viz. suspended plate solar air heater [Fitzgibbons and Kline (1978), Fig. 1], parallel flat plate collector (Fig. 2), solar stills [Tiwari and Malik (1982), Fig. 3] and pebble bed storage [Duffie and Beckman, (1980)] etc. as a function of time and space coordinate. Till now no such analytical solutions are available for such types of devices, except periodic and numerical analysis.     

The mathematical model of solidification latent heat under high cooling rate

Treatment of solidification latent heat is a key point in solidification simulation by the finite difference method. When latent heat is dealt with in a traditional method of equivalent latent heat, it was found that heat was increased when casting with a high cooling rate, and then the simulation result was distorted. In this paper, a new method is proposed to deal with solidification latent heat. Moreover, a mathematical model was suggested, in which the latent heat can be dealt with accurately under high or normal cooling rates. By contrasting the simulation results from this new method with the traditional one, it was indicated that this new model can obtain more accurate simulation results than the traditional model under high or normal cooling rates. © 2006 Wiley Periodicals, Inc. Heat Trans Asian Res, 35(2): 115–121, 2006; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20104     

Filling-box process during solidification of a liquid hypoeutectic binary solution

Solidification of binary solutions often occurs in many industrial applications, including the casting of binary alloys. In this study we consider the effect of a side cooling wall on the development of double-diffusive convection during solidification of a hypoeutectic aqueous ammonium chloride (NH₄Cl–H₂O) solution. To study flow development during solidification of this solution, we used the shadowgraph technique, particle image velocimetry, and a thermochromic-liquid-crystal slurry. In addition, the transient temperature distribution within the test cell was measured by type-T thermocouples. The results of these experiments revealed that the filling-box process originated from the bottom of the test cell to the top. This process induced several double-diffusive layers and counterclockwise roll cells in the melt, mainly caused by double-diffusive convection. Consequently, the filling-box process may cause serious V-segregates and material defects in solidified ingots.