A Moving Boundary Problem in a Finite Domain

1990 ◽  
Vol 57 (1) ◽  
pp. 50-56 ◽  
Author(s):  
Z. Dursunkaya ◽  
S. Nair
Keyword(s):  
Temperature Distribution ◽  
Differential Equations ◽  
Moving Boundary ◽  
Moving Boundary Problem ◽  
Fourier Series Expansion ◽  
Finite Domain ◽  
Finite Region ◽  
Interface Location ◽  
Solid Liquid Interface ◽  
Solid Liquid

The heat conduction and the moving solid-liquid interface in a finite region is studied numerically. A Fourier series expansion is used in both phases for spatial temperature distribution, and the differential equations are converted to an infinite number of ordinary differential equations in time. These equations are solved iteratively for the interface location as well as for the temperature distribution. The results are compared with existing solutions for low Stefan numbers. New results are presented for higher Stefan numbers for which solutions are unavailable.

Analytical Modeling of Outward Solidification With Convective Boundary in Cylindrical Coordinates

2020 ◽  
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Ahmad F. Zueter ◽  
Mahmoud A. Alzoubi ◽  
Agus P. Sasmito
Keyword(s):  
Moving Boundary ◽  
Two Phase ◽  
Convective Boundary ◽  
Interface Motion ◽  
Stefan Number ◽  
Melting Phenomenon ◽  
Approximate Analytical Solutions ◽  
Highly Nonlinear ◽  
Solid Liquid Interface ◽  
Solid Liquid

Abstract Solidification consists of three stages at macroscale: subcooling, freezing and cooling. Classical two-phase Stefan problems describe freezing (or melting) phenomenon initially not at the fusion temperature. Since these problems only define subcooling and freezing stages, an extension to characterize the cooling stage is required to complete solidification. However, the moving boundary in solid-liquid interface is highly nonlinear, and thus exact solution is restricted to certain domains and boundary conditions. It is therefore vital to develop approximate analytical solutions based on physically tangible assumptions, like a small Stefan number. This paper proposes an asymptotic solution for a Stefan-like problem subject to a convective boundary for outward solidification in a hollow cylinder. By assuming a small Stefan number, three temporal regimes and four spatial layers are considered in the asymptotic analysis. The results are compared with numerical method. Further, effects of Biot numbers are also investigated regarding interface motion and temperature profile.

scholarly journals Numerical Solution of Heat Transfer Process in PCM Storage Using Tau Method

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
B. Heydari ◽  
F. Talati
Keyword(s):  
Phase Change ◽  
Phase Change Materials ◽  
Numerical Approach ◽  
Heat Transfer Process ◽  
Liquid Interface ◽  
Tau Method ◽  
Two Dimensional ◽  
Interface Location ◽  
Solid Liquid Interface ◽  
Solid Liquid

Thermal energy storage units that utilize phase change materials have been widely employed to balance temporary temperature alternations and store energy in many engineering systems. In the present paper, an operational approach is proposed to the Tau method with standard polynomial bases to simulate the phase change problems in latent heat thermal storage systems, that is, the two-dimensional solidification process in rectangular finned storage with a constant end-wall temperature. In order to illustrate the efficiency and accuracy of the present method, the solid-liquid interface location and the temperature distribution of the fin for three test cases with different geometries are obtained and compared to simplified analytical results in the published literature. The results indicate that using a two-dimensional numerical approach can predict the solid-liquid interface location more accurately than the simplified analytical model in all cases, especially at the corners.

Numerical and Experimental Investigation of Phase Change Heat Transfer in the Presence of Rayleigh–Benard Convection

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Mohammad Parsazadeh ◽  
Xili Duan
Keyword(s):  
Nusselt Number ◽  
Phase Change ◽  
Local Heat ◽  
Liquid Interface ◽  
Interface Location ◽  
Local Heat Flux ◽  
Solid Liquid Interface ◽  
Solid Liquid ◽  
Rayleigh Benard Convection ◽  
Rayleigh Benard

Abstract This research investigates the melting rate of a phase change material (PCM) in the presence of Rayleigh–Benard convection. A scaling analysis is conducted for the first time for such a problem, which is useful to identify the parameters affecting the phase change rate and to develop correlations for the solid–liquid interface location and the Nusselt number. The solid–liquid interface and flow patterns in the liquid region are analyzed for PCM in a rectangular enclosure heated from bottom. Numerical and experimental results both reveal that the number of Benard cells is proportional to the ratio of the length of the rectangular enclosure over the solid–liquid interface location (i.e.,, the liquified region aspect ratio). Their effect on the local heat flux is also analyzed as the local heat flux profile changes with the solid–liquid interface moving upward. The variations of average Nusselt number are obtained in terms of the Stefan number, Fourier number, and Rayleigh number. Eventually, the experimental and numerical data are used to develop correlations for the solid–liquid interface location and average Nusselt number for this type of melting problems.

Study of Solid/Liquid Interface of Ti46Al0.5W0.5Si Ingot Directionally Solidified by a Near-Rectangular Cold Crucible

2012 ◽  
Vol 472-475 ◽  
pp. 767-770
Author(s):  
Rui Run Chen ◽  
Jie Ren Yang ◽  
Hong Sheng Ding ◽  
Jing Jie Guo ◽  
Yan Qing Su ◽  
...  
Keyword(s):  
Temperature Distribution ◽  
Directional Solidification ◽  
Process Parameters ◽  
Electromagnetic Stirring ◽  
Cold Crucible ◽  
Interface Morphology ◽  
Directionally Solidified ◽  
Solid Liquid Interface ◽  
Solid Liquid ◽  
Future Work

In this study, Ti46Al0.5W0.5Si ingots were directionally solidified by a near-rectangular cold crucible under different process parameters. These process parameters include the electromagnetic stirring, the crucible configuration and the molten drop, all of them have important effect on the S/L interface. The effects of the parameters on the solid/liquid (S/L) interface morphology were investigated, and the mechanisms of the parameters influencing the S/L interface were discussed and revealed. Results showed that the typical S/L interface of the ingots was presented as a curved ‘W-type’ shape. The uneven temperature distribution in the front of the solidified interface is the main reason for a curved S/L interface. Further, the requirements for obtaining a planar S/L interface in the process of cold crucible directional solidification were given, which provided a guide for the future work.

Thermal calculation of submersible motors

2020 ◽  
Author(s):  
R.R. Gizatullin ◽  
S.N. Pescherenko ◽  
A.V. Shiverskiy
Keyword(s):  
Temperature Distribution ◽  
Cross Section ◽  
Annular Channel ◽  
Thermal Calculation ◽  
Ansys Fluent ◽  
Coefficient Of Thermal Conductivity ◽  
The Cross ◽  
Prandtl Numbers ◽  
Solid Liquid Interface ◽  
Solid Liquid

The paper proposes a method of thermal calculation of submersible electric motors intended for use at the stage of their conceptual design. The model is based on the complete system of hydrodynamic equations averaged over the motor cross section and the annular channel through which coolant is pumped. All geometric dimensions and properties of the substance are set. The temperature distribution over the cross section and along the length of the motor is calculated. Two approximations are used. In the first one, the temperature distribution in the cross section of the electric motor was averaged over the corners, which requires setting an effective coefficient of thermal conductivity inside the stator grooves filled with winding wires and electrical insulation. In the second approximation, the heat transfer at the solid – liquid interface was specified through the empirical dependence of the Nusselt number on the Reynolds and Prandtl numbers. To verify the model, the results obtained were compared with calculations by the method of computational fluid dynamics in the ANSYS Fluent software package. The error in calculating the insulation temperature was not more than 5%.

Non-Fourier Melting of a Semi-Infinite Solid

1977 ◽  
Vol 99 (1) ◽  
pp. 25-28 ◽  
Author(s):  
M. H. Sadd ◽  
J. E. Didlake
Keyword(s):  
Boundary Surface ◽  
Thermal Relaxation ◽  
Heat Propagation ◽  
Melting Phenomenon ◽  
Thermal Disturbance ◽  
Interface Location ◽  
Fourier Theory ◽  
Fourier Heat Conduction ◽  
Solid Liquid Interface ◽  
Solid Liquid

The melting of a semi-infinite solid subjected to a step change in temperature is solved according to a non-Fourier heat conduction law postulated by Cattaneo and Vernotte. Unlike the classical Fourier theory which predicts an infinite speed of heat propagation, the non-Fourier theory implies that the speed of a thermal disturbance is finite. The effect of this finite thermal wave speed on the melting phenomenon is determined. The problem is solved by following a similar method as used by Carslaw and Jaeger for the corresponding Fourier problem. Non-Fourier results differ from Fourier theory only for small values of time. Comparing the temperature profiles and the solid-liquid interface location for aluminum, differences between the two theories were significant only for times on the order of 10−9–10−11 s and in a region within approximately 10−4–10−5 cm from the boundary surface. However, these results are based on an approximate value of the thermal relaxation time.

scholarly journals Control of systems on spatial domains with moving boundaries: 3D printing and traffic

2019 ◽  
Vol 32 (4) ◽  
pp. 503-512
Author(s):  
Miroslav Krstic
Keyword(s):  
3D Printing ◽  
Differential Equations ◽  
Moving Boundary ◽  
Freeway Traffic ◽  
Control Laws ◽  
Infinite Dimensional ◽  
Spatial Domains ◽  
And Performance ◽  
Solid Liquid ◽  
Year 2000

Until roughly the year 2000, control algorithms (of the kind that can be physically implemented and provided guarantees of stability and performance) were mostly available only for systems modeled by ordinary differential equations. In other words, while controllers were available for finite-dimensional systems, such as robotic manipulators of vehicles, they were not available for systems like fluid flows. With the emergence of the ?backstepping? approach, it became possible to design control laws for systems modeled by partial differential equations (PDEs), i.e., for infinite dimensional systems, and with inputs at the boundaries of spatial domains. But, until recently, such backstepping controllers for PDEs were available only for systems evolving on fixed spatial PDE domains, not for systems whose boundaries are also dynamical and move, such as in systems undergoing transition of phase of matter (like the solid-liquid transition, i.e., melting or crystallization). In this invited article we review new control designs for moving-boundary PDEs of both parabolic and hyperbolic types and illustrate them by applications, respectively, in additive manufacturing (3D printing) and freeway traffic.

Sign in / Sign up

Export Citation Format

Share Document