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Register a new userSimple approximate analytical solution for non-isothermal single-step transformations: kinetic analysis
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In this paper, we develop a method for obtaining the approximate solution for the evolution of single-step transformations under non-isothermal conditions. We have applied it to many reaction models and obtained very simple analytical expressions for the shape of the corresponding transformation rate peaks. These analytical solutions represent a significant simplification of the systems description allowing easy curve fitting to experiment. A remarkable property is that the evolutions of the transformed fraction obtained at different heating rates are identical when time is scaled by a time constant. The accuracy achieved with our method is checked against several reaction models and different temperature dependencies of the transformation rate constant. It is shown that its accuracy is closely related with that of the Kissinger method.
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Avramis model describes the kinetics of phase transformation under the assumption of spatially random nucleation. In this paper we provide a quasi-exact analytical solution of Avramis model when the transformation takes place under continuous heating. This solution has been obtained with different activation energies for both nucleation and growth rates. The relation obtained is also a solution of the so-called Kolmogorov-Johnson-Mehl-Avrami transformation rate equation. The corresponding non-isothermal Kolmogorov-Johnson-Mehl-Avrami transformation rate equation only differs from the one obtained under isothermal conditions by a constant parameter, which only depends on the ratio between nucleation and growth rate activation energies. Consequently, a minor correction allows us to extend the Kolmogorov-Johnson-Mehl-Avrami transformation rate equation to continuous heating conditions.
We apply moment methods to obtaining an approximate analytical solution to Knudsen layers. Based on the hyperbolic regularized moment system for the Boltzmann equation with the Shakhov collision model, we derive a linearized hyperbolic moment system to model the scenario with the Knudsen layer vicinity to a solid wall with Maxwell boundary condition. We find that the reduced system is in an even-odd parity form that the reduced system proves to be well-posed under all accommodation coefficients. We show that the system may capture the temperature jump coefficient and the thermal Knudsen layer well with only a few moments. With the increasing number of moments used, qualitative convergence of the approximate solution is observed.
In this work we obtain an approximate solution of the strongly nonlinear second order differential equation $frac{d^{2}u}{dt^{2}}+omega ^{2}u+alpha u^{2}frac{d^{2}u}{dt^{2}}+alpha uleft( frac{du}{dt}right)^{2}+beta omega ^{2}u^{3}=0$, describing the large amplitude free vibrations of a uniform cantilever beam, by using a method based on the Laplace transform, and the convolution theorem. By reformulating the initial differential equation as an integral equation, with the use of an iterative procedure, an approximate solution of the nonlinear vibration equation can be obtained in any order of approximation. The iterative approximate solutions are compared with the exact numerical solution of the vibration equation.
Non-planar solar-cell devices have been promoted as a means to enhance current collection in absorber materials with charge-transport limitations. This work presents an analytical framework for assessing the ultimate performance of non-planar solar-cells based on materials and geometry. Herein, the physics of the p-n junction is analyzed for low-injection conditions, when the junction can be considered spatially separable into quasi-neutral and space-charge regions. For the conventional planar solar cell architecture, previously established one-dimensional expressions governing charge carrier transport are recovered from the framework established herein. Space-charge region recombination statistics are compared for planar and non-planar geometries, showing variations in recombination current produced from the space-charge region. In addition, planar and non-planar solar cell performance are simulated, based on a semi-empirical expression for short-circuit current, detailing variations in charge carrier transport and efficiency as a function of geometry, thereby yielding insights into design criteria for solar cell architectures. For the conditions considered here, the expressions for generation rate and total current are shown to universally govern any solar cell geometry, while recombination within the space-charge region is shown to be directly dependent on the geometrical orientation of the p-n junction.
A simple numerical model which calculates the kinetics of crystallization involving randomly distributed nucleation and isotropic growth is presented. The model can be applied to different thermal histories and no restrictions are imposed on the time and the temperature dependencies of the nucleation and growth rates. We also develop an algorithm which evaluates the corresponding emerging grain size distribution. The algorithm is easy to implement and particularly flexible making it possible to simulate several experimental conditions. Its simplicity and minimal computer requirements allow high accuracy for two- and three-dimensional growth simulations. The algorithm is applied to explore the grain morphology development during isothermal treatments for several nucleation regimes. In particular, thermal nucleation, pre-existing nuclei and the combination of both nucleation mechanisms are analyzed. For the first two cases, the universal grain size distribution is obtained. The high accuracy of the model is stated from its comparison to analytical predictions. Finally, the validity of the Kolmogorov-Johnson-Mehl-Avrami model is verified for all the cases studied.
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