Steady-State Two- and Three-Dimensional Heat Conduction: Solutions with Separation of Variables

2018 ◽  
pp. 147-204
Author(s):  
Sadık Kakaç ◽  
Yaman Yener ◽  
Carolina P. Naveira-Cotta
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Separation Of Variables ◽  
Three Dimensional

Steady-State Behavior of a Three-Dimensional Pool-Boiling System

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Michel Speetjens
Keyword(s):  
Steady State ◽  
Model Problem ◽  
Separation Of Variables ◽  
Pool Boiling ◽  
Three Dimensional ◽  
Electronics Cooling ◽  
Nucleate Boiling ◽  
Cooling Capacity ◽  
State Behavior ◽  
Steady State Behavior

Pool-boiling serves as the physical model problem for electronics cooling by means of phase-change heat-transfer. The key for optimal and reliable cooling capacity is better understanding of the conditions that determine the critical heat-flux (CHF). Exceeding CHF results in the transition from efficient nucleate-boiling to inefficient film-boiling. This transition is intimately related to the formation and stability of multiple (steady) states on the fluid-heater interface. To this end, the steady-state behavior of a three-dimensional pool-boiling system has been studied in terms of a representative mathematical model problem. This model problem involves only the temperature field within the heater and models the heat exchange with the boiling medium via a nonlinear boundary condition imposed on the fluid-heater interface. The steady-state behavior is investigated via a bifurcation analysis with a continuation algorithm based on the treatment of the model with the method of separation of variables and a Fourier-collocation method. This revealed that steady-state solutions with homogeneous interface temperatures may undergo bifurcations that result in multiple solutions with essentially heterogeneous interface temperatures. These heterogeneous states phenomenologically correspond with vapor patches (“dry spots”) on the interface that characterize transition conditions. The findings on the model problem are consistent with laboratory experiments.

Three-Dimensional, Steady-State Heat Conduction in Cylinders of General Anisotropic-Media

1979 ◽  
Vol 101 (3) ◽  
pp. 548-553 ◽  
Author(s):  
Y. P. Chang ◽  
K. C. Poon
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Fourier Transforms ◽  
Coupling Effect ◽  
Three Dimensional ◽  
Anisotropic Media ◽  
State Heat ◽  
Change Of Variables ◽  
Steady State Heat ◽  
Kummer's Equation

This paper provides the analytical solution of three-dimensional steady-state heat conduction in solid and hollow cylinders of general anisotropic-media. By the use of Fourier transforms and a change of variables the partial differential equation is reduced to Kummer’s equation. Some calculated results for a solid cylinder are shown and discussed. A parameter γ is found to represent the coupling effect of three-dimensional anisotropy. For small values of γ, an approximate solution is recommended. The inequality σ > 0 which was found in an earlier paper is further discussed.

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.

Generalized Solution for Two-Dimensional Transient Heat Conduction Problems With Partial Heating Near a Corner

2019 ◽  
Vol 141 (7) ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck
Keyword(s):  
Boundary Condition ◽  
Heat Flux ◽  
Heat Conduction ◽  
Steady State ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Generalized Solution ◽  
Convective Heating ◽  
Two Dimensional ◽  
Partial Heating

A generalized solution for a two-dimensional (2D) transient heat conduction problem with a partial-heating boundary condition in rectangular coordinates is developed. The solution accommodates three kinds of boundary conditions: prescribed temperature, prescribed heat flux and convective. Also, the possibility of combining prescribed heat flux and convective heating/cooling on the same boundary is addressed. The means of dealing with these conditions involves adjusting the convection coefficient. Large convective coefficients such as 1010 effectively produce a prescribed-temperature boundary condition and small ones such as 10−10 produce an insulated boundary condition. This paper also presents three different methods to develop the computationally difficult steady-state component of the solution, as separation of variables (SOV) can be less efficient at the heated surface and another method (non-SOV) is more efficient there. Then, the use of the complementary transient part of the solution at early times is presented as a unique way to compute the steady-state solution. The solution method builds upon previous work done in generating analytical solutions in 2D problems with partial heating. But the generalized solution proposed here contains the possibility of hundreds or even thousands of individual solutions. An indexed numbering system is used in order to highlight these individual solutions. Heating along a variable length on the nonhomogeneous boundary is featured as part of the geometry and examples of the solution output are included in the results.

Calculation of three-dimensional nearly singular boundary element integrals for steady-state heat conduction

2015 ◽  
Vol 60 ◽  
pp. 137-143 ◽  
Author(s):  
Guizhong Xie ◽  
Liangwen Wang ◽  
Jianming Zhang ◽  
Dehai Zhang ◽  
Hao Li ◽  
...  
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Boundary Element ◽  
Three Dimensional ◽  
Singular Boundary ◽  
State Heat ◽  
Steady State Heat
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