Development of Analytical Solution for a Two-Phase Stefan Problem in Artificial Ground Freezing Using Singular Perturbation Theory

2020 ◽  
Vol 142 (12) ◽  
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Ahmad F. Zueter ◽  
Victor Auger ◽  
Mahmoud A. Alzoubi ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Singular Perturbation ◽  
Stefan Problem ◽  
Underground Structure ◽  
Computational Cost ◽  
Singular Perturbation Theory ◽  
Singular Perturbation Method ◽  
Two Phase ◽  
Ground Freezing ◽  
Artificial Ground Freezing

Abstract Artificial ground freezing (AGF) has historically been used to stabilize underground structure. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of interest to develop accurate and reliable analytical frameworks for minimizing computational cost. This paper proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in AGF. Specifically, the singular perturbation method separates two distinct temporal scales to capture the subcooling and freezing stages in the ground. The ground was considered as a porous medium with volume-averaged thermophysical properties. Further, Stefan number was assumed to be small, and effects of a few site-dependent parameters were investigated. The analytical solution was verified by numerical results and found to have similar conclusions yet with much lesser computational cost. Keywords: artificial ground freezing, Stefan-like problems, singular perturbation, porous media, outward solidification.

Singular Perturbation Solution for a Two-Phase Stefan Problem in Outward Solidification

2019 ◽  
Author(s):  
Minghan Xu ◽  
Saad Akhtar ◽  
Mahmoud A. Alzoubi ◽  
Agus P. Sasmito
Keyword(s):  
Boundary Condition ◽  
Porous Media ◽  
Analytical Solution ◽  
Singular Perturbation ◽  
Stefan Problem ◽  
Moving Boundary ◽  
Computational Cost ◽  
Perturbation Solution ◽  
Two Phase ◽  
Moving Boundary Condition

Abstract Mathematical modeling of phase change process in porous media can help ensure the efficient design and operation of thermal energy storage and pipe freezing. Numerical methods generally require high computational power to be applicable in practice. Therefore, it is of great interest to develop accurate and reliable analytical frameworks. This study proposes a singular perturbation solution for a two-phase Stefan problem that describes outward solidification in a finite annular space. The problem solves cylindrical heat conduction equations for both solid and liquid phases, with consideration of a moving boundary condition. Perturbation method takes the advantages of small Stefan number as the perturbation parameter, which intrinsically occurs in porous media. Furthermore, a boundary-fixing technique is used to remove nonlinearity in the moving boundary condition. Two different time scales are separately expanded and evaluated to facilitate the construction of a composite asymptotic solution. The analytical solution is verified against a general numerical model using enthalpy method and local volume-averaged thermal properties. The results indicate that the temperature profile of both phases can be well modeled by singular perturbation theory. The analytical solution is found to have similar conclusions to the numerical analysis with much lesser computational cost.

Effect of Freeze Pipe Eccentricity in Artificial Ground Freezing Applications

2020 ◽  
Author(s):  
Ahmad F. Zueter ◽  
Minghan Xu ◽  
Mahmoud A. Alzoubi ◽  
Agus P. Sasmito
Keyword(s):  
Energy Consumption ◽  
Radiation Heat Transfer ◽  
Conservation Equation ◽  
Underground Mines ◽  
Phase Front ◽  
Two Phase ◽  
Ground Freezing ◽  
Deep Underground ◽  
Artificial Ground Freezing ◽  
Conjugate Model

Abstract Building concentric tubes is one of biggest practical challenges in the construction of freeze-pipes of artificial ground freezing (AGF) applications for deep underground mines. In this study, the influence of tubes eccentricity on phase-front expansion (i.e., expansion of the frozen body) and energy consumption of AGF systems is analyzed. A 1+1D semi-conjugate model that solves two-phase transient energy conservation equation is derived. The model is firstly validated against experimental data and then verified with a fully-conjugate model from the literature. After that, the model is extended to a field scale of typical deep underground mines to study freeze-pipe eccentricity. The results show that an eccentric freeze pipe can reduce the phase-front expansion by around 25%, as compared with a concentric one. Also, the geometrical profile of the phase-front is significantly influenced by the freeze-pipe eccentricity. Furthermore, in the passive zone, where AGF coolants are isolated from the ground to reduce energy consumption, freeze pipe eccentricity can increase the coolant heat gain by 10%. This percentage can increase up to 200% if radiation heat transfer is minimized.

A manoeuvre control strategy for flexible-joint manipulators with joint dry friction

Robotica ◽  
2009 ◽  
Vol 28 (4) ◽  
pp. 621-635 ◽  
Author(s):  
H. Salmasi ◽  
R. Fotouhi ◽  
P. N. Nikiforuk
Keyword(s):  
Singular Perturbation ◽  
Trajectory Tracking ◽  
Control Strategy ◽  
Integral Manifold ◽  
Tracking Error ◽  
Singular Perturbation Theory ◽  
Singular Perturbation Method ◽  
Flexible Joint ◽  
Perturbation Theorem ◽  
Joint Friction

SUMMARYA new control strategy based on the singular perturbation method and integral manifold concept is introduced for flexible-joint manipulators with joint friction. In controllers so far developed based on the singular perturbation theory, the dynamics of actuators of flexible-joint manipulators are partially modelled, and the coupling between actuators and links is ignored. This assumption leads to inaccuracy in control performance and error in trajectory tracking which is crucial in high-precision manipulation tasks. In this paper, a comprehensive dynamic model which takes into account the coupling between actuators and links is developed and a composite controller is then designed based on the singular perturbation theorem and integral manifold concept. To overcome the joint friction, a novel method is introduced in which a linear feed-forward torque is designed using the principle of work and energy. Finally, the experimental set-up of a single rigid-link flexible-joint manipulator in the Robotics Laboratory at the University of Saskatchewan is used to verify the proposed controller. Experimental results employing the new controller show that the trajectory tracking error during and at the end of the motion of the robot manipulator is significantly reduced.

Impedance control of a two degree-of-freedom planar flexible link manipulator using singular perturbation theory

Robotica ◽  
2005 ◽  
Vol 24 (2) ◽  
pp. 221-228 ◽  
Author(s):  
G. R. Vossoughi ◽  
A. Karimzadeh
Keyword(s):  
Singular Perturbation ◽  
Sliding Mode ◽  
Impedance Control ◽  
Singular Perturbation Theory ◽  
Flexible Link ◽  
Singular Perturbation Method ◽  
Flexible Links ◽  
Assumed Mode Method ◽  
Mode Method ◽  
Flexible Link Robots

In this article, impedance control of a two link flexible link manipulators is addressed. The concept of impedance control of flexible link robots is rather new and is being addressed for the first time by the authors. Impedance Control provides a universal approach to the control of flexible robots, in both constrained and unconstrained maneuvers. The initial part of the paper concerns the use of Hamilton's principle to derive the mathematical equations governing the dynamics of joint angles, vibration of the flexible links and the constraining forces. The approximate elastic deformations are then derived by means of the Assumed-Mode-Method (AMM). Using the singular perturbation method, the dynamic of the manipulator is decomposed into fast and slow subsystems. The slow dynamic corresponds to the rigid manipulator and the fast dynamic is due to vibrations of flexible links. The sliding mode control (SMC) theory has been used as the means to achieve the 2nd order target impedance for the slow dynamics. A controller based on state feedback is also designed to stabilize the fast dynamics. The composite controller is constructed by using the slow and fast controllers. Simulation results for a 2-DOF robot in which only the 2nd link is flexible confirm that the controller performs remarkably well under various simulation conditions.

scholarly journals A Numerical Method for the Solution of the Two-Phase Fractional Lamé–Clapeyron–Stefan Problem

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2157
Author(s):  
Marek Błasik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Numerical Method ◽  
Stefan Problem ◽  
Moving Boundary ◽  
Second Phase ◽  
Time Step ◽  
Partial Differential ◽  
Two Phase

In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.

Effect of Freeze Pipe Eccentricity in Selective Artificial Ground Freezing Applications

2021 ◽  
pp. 1-33
Author(s):  
Ahmad Zueter ◽  
Ali Madiseh ◽  
Ferri Hassani ◽  
Agus Pulung Sasmito
Keyword(s):  
Energy Consumption ◽  
Radiation Heat Transfer ◽  
Conservation Equation ◽  
Underground Mines ◽  
Frozen Ground ◽  
Two Phase ◽  
Ground Freezing ◽  
1D Model ◽  
Artificial Ground Freezing ◽  
Conjugate Model

Abstract Building concentric tubes is one of biggest practical challenges in the construction of freeze-pipes of selective artificial ground freezing (S-AGF) applications for underground mines. In this study, the influence of tubes eccentricity on phase-front expansion (i.e., expansion of the frozen body) and energy consumption of S-AGF systems is analyzed. A 1+1D semi-conjugate model that solves two-phase transient energy conservation equation is derived based on the enthalpy method. The 1+1D model is firstly validated against experimental data and then verified with a fully-conjugate model from our previous work. After that, the 1+1D model is extended to a field-scale of typical underground mines to examine the effect of freeze-pipe eccentricity. The results show that concentric freeze-pipes form the desired frozen ground volume 15% faster than eccentric freeze-pipes. Also, the geometrical profile of the phase-transition-front of the frozen ground is found to be significantly influenced by the freeze-pipe eccentricity. Furthermore, in the passive zone, where S-AGF coolants are isolated from the ground to reduce energy consumption, freeze pipe eccentricity can increase the coolant heat gain by 20%. This percentage can increase up to 200 % if radiation heat transfer is minimized.

Simulation of Two Phase Flow Dynamics in Flexible Riser Exit Geometries for Oil and Gas Applications

2018 ◽  
Vol 39 ◽  
pp. 1-13
Author(s):  
Ikpe E. Aniekan ◽  
Owunna Ikechukwu ◽  
Satope Paul
Keyword(s):  
Oil And Gas ◽  
Two Phase Flow ◽  
Flow Analysis ◽  
Computational Cost ◽  
Maximum Velocity ◽  
Oil Well ◽  
Phase Flow ◽  
Two Phase ◽  
The Third ◽  
Riser Pipe

Four different riser pipe exit configurations were modelled and the flow across them analysed using STAR CCM+ CFD codes. The analysis was limited to exit configurations because of the length to diameter ratio of riser pipes and the limitations of CFD codes available. Two phase flow analysis of the flow through each of the exit configurations was attempted. The various parameters required for detailed study of the flow were computed. The maximum velocity within the pipe in a two phase flow were determined to 3.42 m/s for an 8 (eight) inch riser pipe. After thorough analysis of the two phase flow regime in each of the individual exit configurations, the third and the fourth exit configurations were seen to have flow properties that ensures easy flow within the production system as well as ensure lower computational cost. Convergence (Iterations), total pressure, static pressure, velocity and pressure drop were used as criteria matrix for selecting ideal riser exit geometry, and the third exit geometry was adjudged the ideal exit geometry of all the geometries. The flow in the third riser exit configuration was modelled as a two phase flow. From the results of the two phase flow analysis, it was concluded that the third riser configuration be used in industrial applications to ensure free flow of crude oil and gas from the oil well during oil production.

scholarly journals Renormalization group and fractional calculus methods in a complex world: A review

2021 ◽  
Vol 24 (1) ◽  
pp. 5-53
Author(s):  
Lihong Guo ◽  
YangQuan Chen ◽  
Shaoyun Shi ◽  
Bruce J. West
Keyword(s):  
Perturbation Theory ◽  
Renormalization Group ◽  
Fractional Calculus ◽  
Singular Perturbation ◽  
World View ◽  
Singular Perturbation Theory ◽  
Asymptotic Behavior Of Solutions ◽  
Quantum Field ◽  
Self Similarity ◽  
Way Of Thinking

Abstract The concept of the renormalization group (RG) emerged from the renormalization of quantum field variables, which is typically used to deal with the issue of divergences to infinity in quantum field theory. Meanwhile, in the study of phase transitions and critical phenomena, it was found that the self–similarity of systems near critical points can be described using RG methods. Furthermore, since self–similarity is often a defining feature of a complex system, the RG method is also devoted to characterizing complexity. In addition, the RG approach has also proven to be a useful tool to analyze the asymptotic behavior of solutions in the singular perturbation theory. In this review paper, we discuss the origin, development, and application of the RG method in a variety of fields from the physical, social and life sciences, in singular perturbation theory, and reveal the need to connect the RG and the fractional calculus (FC). The FC is another basic mathematical approach for describing complexity. RG and FC entail a potentially new world view, which we present as a way of thinking that differs from the classical Newtonian view. In this new framework, we discuss the essential properties of complex systems from different points of view, as well as, presenting recommendations for future research based on this new way of thinking.

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