Analytical approach with Laplace transform to the inverse problem of one-dimensional heat conduction transfer: Application to second and third boundary conditions

2002 ◽  
Vol 32 (1) ◽  
pp. 29-41
Author(s):  
Masanori Monde ◽  
Hirofumi Arima ◽  
Yuhichi Mitsutake
Keyword(s):  
Inverse Problem ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Analytical Approach ◽  
One Dimensional

Microprocessor Silicon Die Temperature Prediction by Simplified Boundary Conditions

2015 ◽  
Author(s):  
Koji Nishi ◽  
Tomoyuki Hatakeyama ◽  
Shinji Nakagawa ◽  
Masaru Ishizuka
Keyword(s):  
Heat Transfer ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Thermal Resistance ◽  
Hot Spot ◽  
Three Dimensional ◽  
Thermal Design ◽  
Target Temperature ◽  
One Dimensional ◽  
Thermal Network

The thermal network method has a long history with thermal design of electronic equipment. In particular, a one-dimensional thermal network is useful to know the temperature and heat transfer rate along each heat transfer path. It also saves computation time and/or computation resources to obtain target temperature. However, unlike three-dimensional thermal simulation with fine pitch grids and a three-dimensional thermal network with sufficient numbers of nodes, a traditional one-dimensional thermal network cannot predict the temperature of a microprocessor silicon die hot spot with sufficient accuracy in a three-dimensional domain analysis. Therefore, this paper introduces a one-dimensional thermal network with average temperature nodes. Thermal resistance values need to be obtained to calculate target temperature in a thermal network. For this purpose, thermal resistance calculation methodology with simplified boundary conditions, which calculates thermal resistance values from an analytical solution, is also introduced in this paper. The effectiveness of the methodology is explored with a simple model of the microprocessor system. The calculated result by the methodology is compared to a three-dimensional heat conduction simulation result. It is found that the introduced technique matches the three-dimensional heat conduction simulation result well.

scholarly journals Simplified Grinding Temperature Model Study

2019 ◽  
Vol 6 (2) ◽  
pp. a1-a7
Author(s):  
N. V. Lishchenko ◽  
V. P. Larshin ◽  
H. Krachunov
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Heat Source ◽  
Boundary Conditions ◽  
Grinding Temperature ◽  
Temperature Model ◽  
One Dimensional ◽  
Heating And Cooling ◽  
Equation Of Heat Conduction ◽  
The One

A study of a simplified mathematical model for determining the grinding temperature is performed. According to the obtained results, the equations of this model differ slightly from the corresponding more exact solution of the one-dimensional differential equation of heat conduction under the boundary conditions of the second kind. The model under study is represented by a system of two equations that describe the grinding temperature at the heating and cooling stages without the use of forced cooling. The scope of the studied model corresponds to the modern technological operations of grinding on CNC machines for conditions where the numerical value of the Peclet number is more than 4. This, in turn, corresponds to the Jaeger criterion for the so-called fast-moving heat source, for which the operation parameter of the workpiece velocity may be equivalently (in temperature) replaced by the action time of the heat source. This makes it possible to use a simpler solution of the one-dimensional differential equation of heat conduction at the boundary conditions of the second kind (one-dimensional analytical model) instead of a similar solution of the two-dimensional one with a slight deviation of the grinding temperature calculation result. It is established that the proposed simplified mathematical expression for determining the grinding temperature differs from the more accurate one-dimensional analytical solution by no more than 11 % and 15 % at the stages of heating and cooling, respectively. Comparison of the data on the grinding temperature change according to the conventional and developed equations has shown that these equations are close and have two points of coincidence: on the surface and at the depth of approximately threefold decrease in temperature. It is also established that the nature of the ratio between the scales of change of the Peclet number 0.09 and 9 and the grinding temperature depth 1 and 10 is of 100 to 10. Additionally, another unusual mechanism is revealed for both compared equations: a higher temperature at the surface is accompanied by a lower temperature at the depth. Keywords: grinding temperature, heating stage, cooling stage, dimensionless temperature, temperature model.

A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab

2013 ◽  
Vol 135 (3) ◽  
Author(s):  
A. Caffagni ◽  
D. Angeli ◽  
G. S. Barozzi ◽  
S. Polidoro
Keyword(s):  
Heat Conduction ◽  
Convergence Rate ◽  
Boundary Conditions ◽  
One Dimensional ◽  
Integral Formulas ◽  
Alternative Approach ◽  
The Laplace Transform ◽  
Piecewise Continuous ◽  
Continuous Boundary ◽  
Duhamel’S Integral

Classical Green’s and Duhamel’s integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.

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