The Transient Temperature Distribution in One-Dimensional Heat-Conduction Problems With Nonlinear Boundary Conditions

1969 ◽  
Vol 91 (1) ◽  
pp. 77-82 ◽  
Author(s):  
W. Z˙yszkowski
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Nonlinear Boundary ◽  
Internal Heat ◽  
Nonlinear Boundary Conditions ◽  
Transient Temperature ◽  
One Dimensional ◽  
Parabolic Profile ◽  
Transient Temperature Distribution ◽  
Approximate Analytical Solutions

The transient, one-dimensional temperature distribution is determined for bodies with internal heat generation and nonlinear boundary condition in the form: k·e·gradθ+ε(θn−T0n)+ε1(θ−T0)=0 Approximate analytical solutions are derived with the aid of Biot’s variational method. The additional boundary condition introduced by Lardner is modified, and this modification makes it possible to solve the problem. The solution has been obtained assuming a parabolic profile of temperature distribution. Formulas are given for plates, cylinders, and spheres. Some results are illustrated with the graphs, and compared with the exact solution for the case of convective heat transfer.

The Transient Temperature Distribution in a Slab Subject to Thermal Radiation

1965 ◽  
Vol 87 (1) ◽  
pp. 117-130 ◽  
Author(s):  
R. D. Zerkle ◽  
J. Edward Sunderland
Keyword(s):  
Temperature Distribution ◽  
Thermal Radiation ◽  
Radiant Heat ◽  
Analog Computer ◽  
Graphical Form ◽  
Transient Temperature ◽  
Transient Temperature Distribution ◽  
Temperature Distributions ◽  
Approximate Analytical Solutions ◽  
Wide Range

The transient, one-dimensional temperature distribution is determined for a slab, insulated on one face, and subjected to thermal radiation at the other face. The slab is initially at a uniform temperature and is assumed to be homogeneous, isotropic, and opaque; the physical properties are assumed to be independent of temperature. Transient temperature distributions for both heating and cooling situations are obtained by means of a thermal-electrical analog computer. A diode limiter circuit is used to simulate the nonlinear radiant heat flux. The transient temperature distributions are presented in a dimensionless, graphical form for a wide range of variables. Approximate analytical solutions are also given which complement and extend the solution charts over ranges of parameters not covered in the charts.

scholarly journals Determination of temperature and thermal stresses distribution in power boiler elements with use inverse heat conduction method

2011 ◽  
Vol 32 (3) ◽  
pp. 191-200 ◽  
Author(s):  
sławomir Grądziel
Keyword(s):  
Boundary Condition ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Thermal Stresses ◽  
Thermal Boundary Condition ◽  
Transient Temperature ◽  
Inverse Heat Conduction ◽  
Transient Temperature Distribution ◽  
Power Boiler

Determination of temperature and thermal stresses distribution in power boiler elements with use inverse heat conduction method The following paper presents the method for solving one-dimensional inverse boundary heat conduction problems. The method is used to estimate the unknown thermal boundary condition on inner surface of a thick-walled Y-branch. Solution is based on measured temperature transients at two points inside the element's wall thickness. Y-branch is installed in a fresh steam pipeline in a power plant in Poland. Determination of an unknown boundary condition allows for the calculation of transient temperature distribution in the whole element. Next, stresses caused by non-uniform transient temperature distribution and by steam pressure inside a Y-branch are calculated using the finite element method. The proposed algorithm can be used for thermal-strength state monitoring in similar elements, when it is not possible to determine a 3-D thermal boundary condition. The calculated temperature and stress transients can be used for the calculation of element durability. More accurate temperature and stress monitoring will contribute to a substantial decrease of maximal stresses that occur during transient start-up and shut-down processes.

Post Fire Transient Temperature Distribution in Drum Type Packages in the Absence of Heat Generation

2003 ◽  
Author(s):  
Allen C. Smith
Keyword(s):  
Thermal Conductivity ◽  
Temperature Distribution ◽  
Heat Generation ◽  
Heat Storage ◽  
Internal Heat ◽  
Thermal Response ◽  
Transient Temperature ◽  
Test Cases ◽  
Transient Temperature Distribution ◽  
Parametric Studies

This study investigates the temperature distribution in an idealized cylindrical package subjected to the HAC Fire transient, with no internal heat generation. Cases for overpack materials with thermal conductivity spanning two orders of magnitude are considered. The results show that the peak internal temperature is determined by the thermal conductivity of the overpack material, for this case. The thermal wave effect, where the interior temperature continues to rise after the end of the fire exposure, is present in all three of the test cases. For contents with no heat generation, the most desirable overpack materials would have low thermal conductivity and low heat storage capability. The study complements the parametric studies of effects of thermal properties on thermal response of packages which were previously reported.

Boundary Condition Design to Heat a Moving Object at Uniform Transient Temperature Using Inverse Formulation

2004 ◽  
Vol 126 (3) ◽  
pp. 619-626 ◽  
Author(s):  
Hakan Ertu¨rk ◽  
Ofodike A. Ezekoye ◽  
John R. Howell
Keyword(s):  
Boundary Condition ◽  
Temperature Distribution ◽  
Design Problem ◽  
Manufacturing Industry ◽  
Three Dimensional ◽  
Chemical Deposition ◽  
Iterative Solution ◽  
Conveyor Belt ◽  
Temperature History ◽  
Transient Temperature

The boundary condition design of a three-dimensional furnace that heats an object moving along a conveyor belt of an assembly line is considered. A furnace of this type can be used by the manufacturing industry for applications such as industrial baking, curing of paint, annealing or manufacturing through chemical deposition. The object that is to be heated moves along the furnace as it is heated following a specified temperature history. The spatial temperature distribution on the object is kept isothermal through the whole process. The temperature distribution of the heaters of the furnace should be changed as the object moves so that the specified temperature history can be satisfied. The design problem is transient where a series of inverse problems are solved. The process furnace considered is in the shape of a rectangular tunnel where the heaters are located on the top and the design object moves along the bottom. The inverse design approach is used for the solution, which is advantageous over a traditional trial-and-error solution where an iterative solution is required for every position as the object moves. The inverse formulation of the design problem is ill-posed and involves a set of Fredholm equations of the first kind. The use of advanced solvers that are able to regularize the resulting system is essential. These include the conjugate gradient method, the truncated singular value decomposition or Tikhonov regularization, rather than an ordinary solver, like Gauss-Seidel or Gauss elimination.

Chaotic Vibration of a Two-dimensional Non-strictly Hyperbolic Equation

2018 ◽  
Vol 61 (4) ◽  
pp. 768-786 ◽  
Author(s):  
Liangliang Li ◽  
Jing Tian ◽  
Goong Chen
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Hyperbolic Equation ◽  
Method Of Characteristics ◽  
Nonlinear Boundary ◽  
Chaotic Oscillation ◽  
Nonlinear Boundary Conditions ◽  
Rigorous Proof ◽  
Two Dimensional ◽  
Chaotic Vibration

AbstractThe study of chaotic vibration for multidimensional PDEs due to nonlinear boundary conditions is challenging. In this paper, we mainly investigate the chaotic oscillation of a two-dimensional non-strictly hyperbolic equation due to an energy-injecting boundary condition and a distributed self-regulating boundary condition. By using the method of characteristics, we give a rigorous proof of the onset of the chaotic vibration phenomenon of the zD non-strictly hyperbolic equation. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs. Numerical simulations are also provided.

Direct observation of three-dimensional transient temperature distribution in SiC Schottky barrier diode under operation by optical-interference contactless thermometry (OICT) imaging

2022 ◽  
Author(s):  
Keiya Fujimoto ◽  
Hiroaki Hanafusa ◽  
Takuma Sato ◽  
Seiichiro HIGASHI
Keyword(s):  
Temperature Distribution ◽  
Schottky Barrier ◽  
Hot Spot ◽  
Local Heating ◽  
Three Dimensional ◽  
Schottky Barrier Diode ◽  
Transient Temperature ◽  
Optical Interference ◽  
Transient Temperature Distribution ◽  
Heating And Cooling

Abstract We have developed optical-interference contactless thermometry (OICT) imaging technique to visualize three-dimensional transient temperature distribution in 4H-SiC Schottky barrier diode (SBD) under operation. When a 1 ms forward pulse bias was applied, clear variation of optical interference fringes induced by self-heating and cooling were observed. Thermal diffusion and optical analysis revealed three-dimensional temperature distribution with high spatial (≤ 10 μm) and temporal (≤ 100 μs) resolutions. A hot spot that signals breakdown of the SBD was successfully captured as an anormal interference, which indicated a local heating to a temperature as high as 805 K at the time of failure.

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