scholarly journals On the Transient Atomic/Heat Diffusion in Cylinders and Spheres with Phase Change: A Method to Derive Closed-Form Solutions

2021 ◽  
Vol 2021 ◽  
pp. 1-19
Author(s):  
I. L. Ferreira ◽  
A. Garcia ◽  
A. L. S. Moreira
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Single Phase ◽  
Moving Boundary ◽  
Error Function ◽  
Solid State Diffusion ◽  
Two Phase ◽  
Closed Form Solutions ◽  
State Diffusion ◽  
Cylinders And Spheres

Analytical solutions for the transient single-phase and two-phase inward solid-state diffusion and solidification applied to the radial cylindrical and spherical geometries are proposed. Both solutions are developed from the differential equation that treats these phenomena in Cartesian coordinates, which are modified by geometric correlations and suitable changes of variables to achieve closed-form solutions. The modified differential equations are solved by using two well-known closed-form solutions based on the error function, and then equations are obtained to analyze the diffusion interface position as a function of time and position in cylinders and spheres. These exact correlations are inserted into the closed-form solutions for the slab and are used to update the roots for each radial position of the moving boundary interface. The predictions provided by the proposed analytical solutions are validated against the results of a numerical approach. Finally, a comparative study of diffusion in slabs, cylinders, and spheres is also presented for single-phase and two-phase solid-state diffusion and solidification, which shows the importance of the effects imposed by the radial cylindrical and spherical curvatures with respect to the Cartesian coordinate system in the process kinetics. The analytical model is proved to be general, as far as, a semi-infinite solution for diffusion problems with phase change exists in the form of the error function, which enables exact closed-form analytical solutions to be derived, by updating the root at each radial position the moving boundary interface.

Assessment of Aided-INS Performance

2011 ◽  
Vol 65 (1) ◽  
pp. 169-185 ◽  
Author(s):  
Itzik Klein ◽  
Sagi Filin ◽  
Tomer Toledo ◽  
Ilan Rusnak
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Navigation Systems ◽  
Inertial Navigation Systems ◽  
Closed Form Solutions ◽  
Land Vehicles ◽  
Error Covariance ◽  
Model Dependency ◽  
Error State ◽  
Insight Into

Aided Inertial Navigation Systems (INS) systems are commonly implemented in land vehicles for a variety of applications. Several methods have been reported in the literature for evaluating aided INS performance. Yet, the INS error-state-model dependency on time and trajectory implies that no closed-form solutions exist for such evaluation. In this paper, we derive analytical solutions to evaluate the fusion performance. We show that the derived analytical solutions manage to predict the error covariance behavior of the full aided INS error model. These solutions bring insight into the effect of the various parameters involved in the fusion of the INS and an aiding sensor.

Analysis of transient amplification for a torsional system passing through resonance

2014 ◽  
Vol 229 (13) ◽  
pp. 2341-2354 ◽  
Author(s):  
Laihang Li ◽  
Rajendra Singh
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Classical Problem ◽  
Peak Frequency ◽  
Empirical Formulas ◽  
Closed Form Solutions ◽  
Analytical Approximations ◽  
Numerical Predictions ◽  
Prior Literature ◽  
Run Up

The classical problem of vibration amplification of a linear torsional oscillator excited by an instantaneous sinusoidal torque is re-examined with focus on the development of new analytical solutions of the transient envelopes. First, a new analytical method in the instantaneous frequency (or speed) domain is proposed to directly find the closed-form solutions of transient displacement, velocity, and acceleration envelopes for passage through resonance during the run-up or run-down process. The proposed closed-form solutions are then successfully verified by comparing them with numerical predictions and limited analytical solutions as available in prior literature. Second, improved analytical approximations of maximum amplification and corresponding peak frequency are found, which are also verified by comparing them with prior analytical or empirical formulas. In addition, applicability of the proposed analytical solution is clarified, and their error bounds are identified. Finally, the utility of analytical solutions and approximations is demonstrated by application to the start-up process of a multi-degree-of-freedom vehicle driveline system.

Relaxation of Thick-Walled Cylinders and Spheres

1982 ◽  
Vol 49 (3) ◽  
pp. 487-491 ◽  
Author(s):  
N. S. Ottosen
Keyword(s):  
Closed Form ◽  
Plane Strain ◽  
Approximate Solutions ◽  
Nonlinear Creep ◽  
Closed Form Solutions ◽  
Cylinders And Spheres

Using the nonlinear creep law proposed by Soderberg, closed-form solutions are derived for the relaxation of incompressible thick-walled spheres and cylinders in plane strain. These solutions involve series expressions which, however, converge very quickly. By simply ignoring these series expressions, extremely simple approximate solutions are obtained. Despite their simplicity these approximations possess an accuracy that is superior to approximations currently in use. Finally, several physical aspects related to the relaxation of cylinders and spheres are discussed.

INFLUENCE OF SYNTHESISING ROUTES ON RESISTIVITY AND AGING CHARACTERISTICS OF Bi-CUPRATE SUPERCONDUCTORS

1992 ◽  
Vol 06 (24) ◽  
pp. 1535-1540
Author(s):  
V. VIDYALAL ◽  
K. RAJASREE ◽  
C.P.G. VALLABHAN
Keyword(s):  
Solid State ◽  
Single Phase ◽  
Cuprate Superconductors ◽  
Diffusion Reaction ◽  
Solid State Diffusion ◽  
Degradation Behavior ◽  
High Quality ◽  
State Diffusion ◽  
The Matrix ◽  
Reaction Routes

Lead-doped Bi-2223 superconductors were prepared by two main popular solid state reaction routes, viz. solid state diffusion reaction and the matrix method. Even though both routes produce single-phase materials superconducting at 110 K, the resistivity behaviors above Tc are found to be different. Aging/degradation behavior was studied in terms of Tc on both sets of samples stored in a desiccator for eight months. Our studies indicate that the synthesising routes play a major role in the preparation of high quality bulk Pb-doped Bi-2223 superconductors which are more resistant to degradation when exposed to humid conditions.

scholarly journals Some recent results for heat-diffusion moving boundary problems

1985 ◽  
Vol 32 (3) ◽  
pp. 437-460
Author(s):  
James M. Hall ◽  
Jeffrey N. Dewynne
Keyword(s):  
Stefan Problem ◽  
Moving Boundary ◽  
Formal Series ◽  
Heat Diffusion ◽  
Upper And Lower Bounds ◽  
Two Phase ◽  
Boundary Problems ◽  
Stefan Problems ◽  
Analytical Approximations ◽  
Cylinders And Spheres

Integral formulations for the three classical single phase Stefan problems involving the infinite slab and inward solidifying cylinders and spheres are utilized to generate standard analytical approximations. These approximations include the pseudo steady state estimate, large Stefan number expansions, upper and lower bounds, approximations based on integral iteration and related results such as formal series solutions. In order to demonstrate the applicability and limitations of the integral formulations three generalizations of the classical stefan problem are considered briefly. These problems are diffusion with two simultaneous chemical reactions, a Stefan problem with two moving boundaries and the genuine two phase Stefan problem.

scholarly journals Analytical Solutions for Flow of a Dusty Fluid Between Two Porous Flat Plates

1994 ◽  
Vol 116 (2) ◽  
pp. 354-356 ◽  
Author(s):  
Ali J. Chamkha
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Skin Friction ◽  
Analytical Solutions ◽  
Velocity Profiles ◽  
Flat Plates ◽  
Friction Coefficients ◽  
Dusty Fluid ◽  
Closed Form Solutions

Equations governing flow of a dusty fluid between two porous flat plates with suction and injection are developed and closed-form solutions for the velocity profiles, displacement thicknesses, and skin friction coefficients for both phases are obtained. Graphical results of the exact solutions are presented and discussed.

Analytical solutions to a fractional generalized two phase Lame-Clapeyron-Stefan problem

2014 ◽  
Vol 24 (6) ◽  
pp. 1251-1259 ◽  
Author(s):  
Xicheng Li
Keyword(s):  
Heat Conduction ◽  
Liquid Phase ◽  
Stefan Problem ◽  
Analytical Solutions ◽  
Moving Boundary ◽  
Solid Phase ◽  
Point Of View ◽  
Two Phase ◽  
Content Type ◽  
Advection Term

Purpose – The mathematical model of a two-phase Lamé-Clapeyron-Stefan problem for a semi-infinite material with a density jump is considered. The purpose of this paper is to study the analytical solutions of the models and show the performance of several parameters. Design/methodology/approach – To describe the heat conduction, the Caputo type time fractional heat conduction equation is used and a convective term is included since the changes in density give rise to motion of the liquid phase. The similarity variables are used to simplify the models. Findings – The analytical solutions describing the changes of temperature in both liquid and solid phases are obtained. For the solid phase, the solution is given in the Wright function form. While for the liquid phase, since the appearance of the advection term, an approximate solution in series form is given. Based on the solutions, the performance of the parameters is discussed in detail. Originality/value – From the point of view of mathematics, the moving boundary problems are nonlinear, so barely any analytical solutions for these problems can be obtained. Furthermore, there are many applications in which a material undergoes phase change, such as in melting, freezing, casting and cryosurgery.

scholarly journals Analytical Solutions of Carbonate Acidizing in Radial Flow

2021 ◽  
Author(s):  
Polyneikis Strongylis ◽  
Euripides Papamichos
Keyword(s):  
Analytical Solutions ◽  
Moving Boundary ◽  
Radial Flow ◽  
Fluid Velocity ◽  
Boundary Problems ◽  
Reactive Infiltration ◽  
Closed Form Solutions ◽  
Engineered Systems ◽  
Carbonate Acidizing ◽  
Reactive Fluids

AbstractThe flow of reactive fluids into porous media, a phenomenon known as reactive infiltration, is important in natural and engineered systems. While most of the studies in this area cover theoretical and experimental analyses in linear acid flow, the present work concentrates on radial flow conditions from a wellbore in the field and on finding exact analytical solutions to moving boundary problems of the uniform dissolution front. Closed-form solutions are obtained for the transient convection–diffusion which allow the demarcation of the range of applicability of the quasi-static limit. The fluid velocity dependency of the diffusion–dispersion coefficient is also examined by comparing results from analytical solutions from constant and velocity-dependent coefficients. These contributions form the basis for linear stability analyses to describe acid fingering encountered in reservoir stimulation.

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