A Method for Computing the Analytical Solution of the Steady-State Heat Equation in Multilayered Media

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Ivor Dülk ◽  
Tamás Kovácsházy
Keyword(s):  
Steady State ◽  
Analytical Solution ◽  
Separation Of Variables ◽  
Layered Media ◽  
Steady State Solution ◽  
Multilayered Media ◽  
Number Of Layers ◽  
Layer Medium ◽  
Steady State Heat ◽  
Weighting Coefficients

The computation of the analytical solution of the steady temperature distribution in multilayered media can become numerically unstable if there are different longitudinal (i.e., the directions parallel to the layers) boundary conditions for each layer. In this study, we develop a method to resolve these computational difficulties by approximating the temperatures at the junctions step-by-step and solving for the thermal field separately in only the single layers. First, we solve a two-layer medium problem and then show that multilayered media can be represented as a hierarchy of two-layered media; thus, the developed method is generalized to an arbitrary number of layers. To improve the computational efficiency and speed, we use varying weighting coefficients during the iterations, and we present a method to decompose the multilayered media into two-layered media. The developed method involves the steady-state solution of the diffusion equation, which is illustrated for 2D slabs using separation of variables (SOV). A numerical example of four layers is also included, and the results are compared to a numerical solution.

Analytical Solution to Steady-State Heat-Conduction Problems With Irregularly Shaped Boundaries

1971 ◽  
Vol 93 (4) ◽  
pp. 449-454 ◽  
Author(s):  
D. M. France
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Analytical Solution ◽  
Least Squares ◽  
Series Solution ◽  
Point Matching ◽  
State Heat ◽  
Temperature Heat ◽  
Steady State Heat ◽  
Least Squares Approach

A method of obtaining an analytical solution to two-dimensional steady-state heat-conduction problems with irregularly shaped boundaries is presented. The technique of obtaining the coefficients to the series solution via a direct least-squares approach is compared to the “point-matching” scheme. The two methods were applied to problems with known solutions involving the three heat-transfer boundary conditions, temperature, heat flux, and convection coefficient specified. Increased accuracy with substantially fewer terms in the series solution was obtained via the least-squares technique.

Experimental Measurements of Temperature Distribution in Three Dimensional Stacked Package

2007 ◽  
Author(s):  
Anand Desai ◽  
James Geer ◽  
Bahgat Sammakia
Keyword(s):  
Experimental Study ◽  
Heat Conduction ◽  
Steady State ◽  
Temperature Distribution ◽  
Numerical Model ◽  
Analytical Solution ◽  
Three Dimensional ◽  
Power Input ◽  
Experimental Measurements ◽  
Steady State Heat

This paper presents the results of an experimental study of steady state heat conduction in a three dimensional stack package. The temperatures are measured at different interfaces within the stacked package. Delphi devices are used in the experiment which enables controlled power input and surface temperature of the devices. The experiment is carried out for three different boundary conditions on the package. The power input in varied to study its effects. A numerical model is created to compare to the experimental results. The results are also compared with the analytical solution presented in Desai et al [5] and Geer et al [6]. The results indicate that the experimental, numerical and analytical solutions follow the same trend. The agreement between the experimental and numerical results improves when the lateral losses are taken into account.

scholarly journals Dynamic properties of piecewise linear systems with fractional time-delay feedback

2021 ◽  
pp. 146134842110076
Author(s):  
Jianchao Zhang ◽  
Jun Wang ◽  
Jiangchuan Niu ◽  
Yufei Hu
Keyword(s):  
Steady State ◽  
Time Delay ◽  
Analytical Solution ◽  
Goodness Of Fit ◽  
Numerical Solutions ◽  
Dynamic Properties ◽  
Piecewise Linear ◽  
Steady State Solution ◽  
State Solution ◽  
Comparison Results

The forced vibration of a single-degree-of-freedom piecewise linear system containing fractional time-delay feedback was investigated. The approximate analytical solution of the system was obtained by employing an averaging method. A frequency response equation containing time delay was obtained by studying a steady-state solution. The stability conditions of the steady-state solution, the amplitude–frequency results, and the numerical solutions of the system under different time-delay parameters were compared. Comparison results indicated a favorable goodness of fit between the two parameters and revealed the correctness of the analytical solution. The effects of the time-delay and fractional parameters, piecewise stiffness, and piecewise gap on the principal resonance and bifurcation of the system were emphasized. Results showed that fractional time delay occurring in the form of equivalent linear dampness and stiffness under periodic variations in the system and influenced the vibration characteristic of the system. Moreover, piecewise stiffness and gap induced the nonlinear characteristic of the system under certain parameters.

scholarly journals Analysis of groundwater flow and stream depletion in L-shaped fluvial aquifers

2018 ◽  
Vol 22 (4) ◽  
pp. 2359-2375 ◽  
Author(s):  
Chao-Chih Lin ◽  
Ya-Chi Chang ◽  
Hund-Der Yeh
Keyword(s):  
Steady State ◽  
Groundwater Resources ◽  
Separation Of Variables ◽  
Measurement Data ◽  
Steady State Solution ◽  
State Solution ◽  
Stream Depletion ◽  
Numerical Studies ◽  
Present Solution ◽  
Depletion Rate

Abstract. Understanding the head distribution in aquifers is crucial for the evaluation of groundwater resources. This article develops a model for describing flow induced by pumping in an L-shaped fluvial aquifer bounded by impermeable bedrocks and two nearly fully penetrating streams. A similar scenario for numerical studies was reported in Kihm et al. (2007). The water level of the streams is assumed to be linearly varying with distance. The aquifer is divided into two subregions and the continuity conditions of the hydraulic head and flux are imposed at the interface of the subregions. The steady-state solution describing the head distribution for the model without pumping is first developed by the method of separation of variables. The transient solution for the head distribution induced by pumping is then derived based on the steady-state solution as initial condition and the methods of finite Fourier transform and Laplace transform. Moreover, the solution for stream depletion rate (SDR) from each of the two streams is also developed based on the head solution and Darcy's law. Both head and SDR solutions in the real time domain are obtained by a numerical inversion scheme called the Stehfest algorithm. The software MODFLOW is chosen to compare with the proposed head solution for the L-shaped aquifer. The steady-state and transient head distributions within the L-shaped aquifer predicted by the present solution are compared with the numerical simulations and measurement data presented in Kihm et al. (2007).

scholarly journals Analysis of groundwater flow and stream depletion in the L-shaped fluvial aquifer

2017 ◽  
Author(s):  
Chao-Chih Lin ◽  
Ya-Chi Chang ◽  
Hund-Der Yeh
Keyword(s):  
Steady State ◽  
Groundwater Resources ◽  
Separation Of Variables ◽  
Measurement Data ◽  
Steady State Solution ◽  
Design Tool ◽  
State Solution ◽  
Stream Depletion ◽  
Present Solution ◽  
Fluvial Aquifer

Abstract. Understanding the head distribution in aquifers is crucial for the evaluation of groundwater resources. This article develops an analytical model for describing flow induced by pumping in an L‐shaped fluvial aquifer bounded by impermeable bedrocks and two nearly fully penetrating streams. A similar scenario for numerical studies was reported in Kihm et al. (2007). The water level of the streams is assumed to be linearly varying with distance. The aquifer is divided into two sub-regions and the continuity conditions of hydraulic head and flux are imposed at the interface of the sub-regions. The steady-state solution describing the head distribution for the model without pumping is first developed by the method of separation of variables. The transient solution for the head distribution induced by pumping is then derived based on the steady-state solution as initial condition and the methods of finite Fourier transform and Laplace transform. Moreover, the solution for stream depletion rate (SDR) from each of the two streams is also developed based on the head solution and Darcy's law. Both head and SDR solutions in real time domain are obtained by a numerical inversion scheme called the Stehfest algorithm. The software MODFLOW-2005 is chosen to check the accuracy of the head solution for the L-shaped aquifer. The steady-state and transient head distributions within the L-shaped aquifer predicted by the present solution are compared with the numerical simulations and measurement data presented in Kihm et al. (2007). The SDR solution is employed to demonstrate its use as a design tool in determining well location for required amounts of SDR from nearby streams under a specific aquifer pumping rate.

Analytical Solution of Unsteady State Conduction Problem with Heat Source

2012 ◽  
Vol 204-208 ◽  
pp. 4364-4367
Author(s):  
Shun Yu Su ◽  
Qin Huang
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Infinite Plate ◽  
Unsteady State ◽  
Inhomogeneous Problem ◽  
Separation Of Variables Method

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.

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