An analytical solution for a cracked piezoelectric plate according to the PKHS electric boundary condition

2005 ◽  
Vol 180 (1-4) ◽  
pp. 233-244 ◽  
Author(s):  
Y. Liu ◽  
Y.-H. Chen
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Piezoelectric Plate ◽  
Electric Boundary Condition

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

The Effect of Biot Number on a One-Dimensional General Conduction Solution

2021 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Boundary Conditions ◽  
Biot Number ◽  
Analytical Solutions ◽  
Heat Fluxes ◽  
Graphical Form ◽  
Large Temperature ◽  
Convection Coefficient ◽  
General Analytical Solution

Abstract Analytical solutions for thermal conduction problems are extremely important, particularly for verification of numerical codes. Temperatures and heat fluxes can be calculated very precisely, normally to eight or ten significant figures, even in situations involving large temperature gradients. It can be convenient to have a general analytical solution for a transient conduction problem in rectangular coordinates. The general solution is based on the principle that the three primary types of boundary conditions (prescribed temperature, prescribed heat flux, and convective) can all be handled using convective boundary conditions. A large convection coefficient closely approximates a prescribed temperature boundary condition and a very low convection coefficient closely approximates an insulated boundary condition. Since a dimensionless solution is used in this research, the effect of various values of dimensionless convection coefficients, or Biot number, are explored. An understandable concern with a general analytical solution is the effect of the choice of convection coefficients on the precision of the solution, since the primary motivation for using analytical solutions is the precision offered. An investigation is made in this study to determine the effects of the choices of large and small convection coefficients on the precision of the analytical solutions generated by the general convective formulation. Results are provided, in tablular and graphical form, to illustrate the effects of the choices of convection coefficients on the precision of the general analytical solution.

Analytical Solution for Solid Cylindrical Heat Source Model With Convective Boundary Condition

2019 ◽  
Vol 141 (12) ◽  
Author(s):  
Yang Zhou ◽  
Cheng Xu ◽  
David Sego ◽  
Dong-hai Zhang
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Analytical Solution ◽  
Heat Source ◽  
Groundwater Flow ◽  
Convective Boundary Condition ◽  
Convective Boundary ◽  
Energy Pile ◽  
Special Cases ◽  
Scs Model

Abstract The energy pile technology has been widely used, and the solid cylindrical heat source (SCS) model is usually adopted to describe the heat transfer process between the energy pile and the surrounding soil. This paper investigates the SCS model with a convective boundary condition (SCS-3 model), and realistic conditions such as transversely isotropic ground and groundwater flow are all included in the model. An analytical solution for the problem is established using Green's function method and the theory of moving heat sources. Solutions for the SCS model with a boundary condition of the first kind (SCS-1 model) and for the line source (LS) model with a convective boundary condition (LS-3 model) are recovered as special cases of the solution in this paper. Computational examples are presented, and comparisons between different models are made. First, the SCS-1 model is compared with the SCS-3 model, showing the error caused by neglecting the surface convective effect. Second, the LS-3 model is compared with the SCS-3 model, showing the error associated with neglecting the size of heat source. The effects of groundwater flow velocity and convective heat transfer coefficient on the temporal and spatial variations of these errors are also investigated.

Damped Vibration Response of an Axially Moving Wire Subject to an Oscillating Boundary Condition and the Application to Slurry Wiresaws

2020 ◽  
pp. 1-37
Author(s):  
Liming Li ◽  
Imin Kao
Keyword(s):  
Boundary Condition ◽  
Analytical Solution ◽  
Traveling Waves ◽  
Solution Space ◽  
Vibration Response ◽  
Temporal Variables ◽  
Rocking Motion ◽  
Vibration Responses ◽  
Oscillating Boundary ◽  
Axially Moving

Abstract This paper presents the analysis of a new class of differential continuum system with a solution of traveling waves containing coupled spatial and temporal variables. Herein, we derive the analytical solution of the damped vibration response of a longitudinally moving wire with damping, subject to an oscillating boundary condition. The vibration response is the outcome of combining four traveling waves, induced by a wave initiating from the oscillating boundary, and traveling between the two boundaries. The four different traveling waves are the independent bases of the vibration responses that span the solution space of vibration of such continuum system. The combination, or the interference, of these traveling waves in the undamped condition produces nodal points in the vibration response, which can be formulated through the analytical solution. The impacts of wire speed, oscillating frequency at the boundary and damping factors on the vibration response are investigated. Furthermore, the vibration induced by the oscillating motion of the boundary has a profound impact on the effectiveness of slicing ingots with rocking motion of oscillating wire guides in wiresaw manufacturing processes.

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.

scholarly journals Analytical Solution of Heat Conduction in a Symmetrical Cylinder Using the Solution Structure Theorem and Superposition Technique

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1522 ◽  
Author(s):  
Rasool Kalbasi ◽  
Seyed Mohammadhadi Alaeddin ◽  
Mohammad Akbari ◽  
Masoud Afrand
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Analytical Solution ◽  
Solution Structure ◽  
Structure Theorem ◽  
Initial Conditions ◽  
Hyperbolic Heat Conduction ◽  
Fourier Heat Conduction ◽  
Superposition Technique ◽  
Complex Origin

In this paper, non-Fourier heat conduction in a cylinder with non-homogeneous boundary conditions is analytically studied. A superposition approach combining with the solution structure theorems is used to get a solution for equation of hyperbolic heat conduction. In this solution, a complex origin problem is divided into, different, easier subproblems which can actually be integrated to take the solution of the first problem. The first problem is split into three sub-problems by setting the term of heat generation, the initial conditions, and the boundary condition with specified value in each sub-problem. This method provides a precise and convenient solution to the equation of non-Fourier heat conduction. The results show that at low times (t = 0.1) up to about r = 0.4, the contribution of T1 and T3 dominate compared to T2 contributing little to the overall temperature. But at r > 0.4, all three temperature components will have the same role and less impact on the overall temperature (T).

scholarly journals Two-Dimensional Analytical Solution of the Laminar Forced Convection in a Circular Duct with Periodic Boundary Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
M. R. Astaraki ◽  
N. Ghiasi Tabari
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Periodic Function ◽  
Forced Convection ◽  
Solid Wall ◽  
Convection Heat Transfer ◽  
Axial Direction ◽  
Energy Balance Equation ◽  
Circular Duct

In the present study analytical solution for forced convection heat transfer in a circular duct with a special boundary condition has been presented, because the external wall temperature is a periodic function of axial direction. Local energy balance equation is written with reference to the fully developed regime. Also governing equations are two-dimensionally solved, and the effect of duct wall thickness has been considered. The temperature distribution of fluid and solid phases is assumed as a periodic function of axial direction and finally temperature distribution in the flow field, solid wall, and local Nusselt number, is obtained analytically.

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