Incompatible Boundary Conditions in Heat Equation Coupled With Air System Models

2020 ◽  
Author(s):  
Jose M. Chaquet ◽  
Roque Corral
Keyword(s):  
Boundary Conditions ◽  
Transfer Problem ◽  
Flow Direction ◽  
Heat Diffusion ◽  
Working Fluid ◽  
Good Prediction ◽  
Heat Diffusion Equation ◽  
Fluid Network ◽  
Ill Posed ◽  
Physical Boundary

Abstract Heat transfer problem is one of the main challenges in the design process of turbomachinery components for aeronautic applications. Good prediction capabilities are required to estimate metal temperatures, specially in those regions where the working fluid reaches temperatures near to the material melting point. In this context, it is common to perform multi-physics simulations involving different solvers. Special care must be taken at the interfaces between the several domains to avoid non-physical solutions. Concretely speaking, the coupling process between a thermal code (discretized heat diffusion equation solver) and a fluid network (low fidelity models representing air flows) is studied. Several non-physical boundary conditions examples are provided. The models are solved using an in-house thermal code called Saturn. The effects both in the results and in the convergence process are described. Non-physical boundary conditions provoke instabilities in the flow direction at some parts of the fluid network. A method to analyze the compatibility and convergence of the coupled problem is described and used in the examples. Also, some heuristics to achieve converge in the ill-posed models are commented.

A 1+1D Thermal Dynamic Model of a Li-Ion Battery Cell

2010 ◽  
Author(s):  
Matteo Muratori ◽  
Ning Ma ◽  
Marcello Canova ◽  
Yann Guezennec
Keyword(s):  
Temperature Distribution ◽  
Boundary Conditions ◽  
Thermal Management ◽  
Cooling System ◽  
Heat Diffusion ◽  
Li Ion Battery ◽  
Battery Cell ◽  
Heat Diffusion Equation ◽  
One Dimensional ◽  
Li Ion

Li-ion batteries are today considered the prime solution as energy storage system for EV/PHEV/HEV, due to their high specific energy and power. Since their performance, life and reliability are influenced by the operating temperature, great interest has been devoted to study different cooling solutions and control algorithms for thermal management. In this context, this paper presents a computationally efficient modeling approach to characterize the internal temperature distribution of a Li-ion battery cell, conceived to serve as a tool to aid the design of cooling systems and the development of thermal management systems for automotive battery packs. The model is developed starting from the unsteady heat diffusion equation, for which an analytical solution is obtained through the integral transform method. First, a general one-dimensional thermal model is developed to predict the temperature distribution inside a prismatic Li-ion battery cell under different boundary conditions. Then, a specific case with convective boundary conditions is studied with the objective of characterizing a cell cooled by a forced air flow. To characterize the effects of the cooling system on the temperature distribution within the cell, the one-dimensional solution is then extended to a 1+1D model that accounts for the variability of the boundary conditions in the flow direction. The calibration and validation of the specific model presented will be presented, adopting a detailed 2D FEM simulator as a benchmark.

scholarly journals Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽  
2021 ◽  
Vol 11 (3) ◽  
pp. 573
Author(s):  
Marzia Sara Vaccaro ◽  
Francesco Paolo Pinnola ◽  
Francesco Marotti de Sciarra ◽  
Raffaele Barretta
Keyword(s):  
Boundary Conditions ◽  
Structural Engineering ◽  
Beam Deflection ◽  
Soil Reaction ◽  
Differential Problem ◽  
Reaction Field ◽  
Ill Posed ◽  
Elastic Responses ◽  
Benchmark Solutions ◽  
Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

scholarly journals Short-time heat diffusion in compact domains with discontinuous transmission boundary conditions

2015 ◽  
Vol 26 (01) ◽  
pp. 59-110 ◽  
Author(s):  
Claude Bardos ◽  
Denis Grebenkov ◽  
Anna Rozanova-Pierrat
Keyword(s):  
Asymptotic Expansion ◽  
Boundary Conditions ◽  
Order Term ◽  
Heat Content ◽  
Resistance Coefficient ◽  
Small Time ◽  
Heat Diffusion ◽  
Heat Problem ◽  
Short Time ◽  
Mathematical Justification

We consider a heat problem with discontinuous diffusion coefficients and discontinuous transmission boundary conditions with a resistance coefficient. For all bounded (ϵ, δ)-domains Ω ⊂ ℝn with a d-set boundary (for instance, a self-similar fractal), we find the first term of the small-time asymptotic expansion of the heat content in the complement of Ω, and also the second-order term in the case of a regular boundary. The asymptotic expansion is different for the cases of finite and infinite resistance of the boundary. The derived formulas relate the heat content to the volume of the interior Minkowski sausage and present a mathematical justification to the de Gennes' approach. The accuracy of the analytical results is illustrated by solving the heat problem on prefractal domains by a finite elements method.

Mapping Thickness Dependent Thermal Conductivity of GaN

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
Elbara Ziade ◽  
Jia Yang ◽  
Gordie Brummer ◽  
Denis Nothern ◽  
Theodore Moustaks ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Pump Beam ◽  
Continuous Wave ◽  
High Electron Mobility Transistors ◽  
Heat Diffusion ◽  
Phase Lag ◽  
High Electron ◽  
Heat Diffusion Equation ◽  
Phase Images ◽  
Gan Film

Frequency domain thermoreflectance (FDTR) is used to create quantitative maps of thermal conductivity and thickness for a thinning gallium nitride (GaN) film on silicon carbide (SiC). GaN was grown by molecular beam epitaxy on a 4H-SiC substrate with a gradient in the film thickness found near the edge of the chip. The sample was then coated with a 5 nm nickel adhesion layer and a 85 nm gold transducer layer for the FDTR measurement. A piezo stage raster scans the sample to create phase images at different frequencies. For each pixel, a periodically modulated continuous-wave laser (the red pump beam) is focused to a Gaussian spot, less than 2 um in diameter, to locally heat the sample, while a second beam (the green probe beam) monitors the surface temperature through a proportional change in the reflectivity of gold. The pump beam is modulated simultaneously at six frequencies and the thermal conductivity and thickness of the GaN film are extracted by minimizing the error between the measured probe phase lag at each frequency and an analytical solution to the heat diffusion equation in a multilayer stack of materials. A scanning electron microscope image verifies the thinning GaN. We mark the imaged area with a red box. A schematic of the GaN sample in our measurement system is shown in the top right corner, along with the two fitting properties highlighted with a red box. We show the six phase images and the two obtained property maps: thickness and thermal conductivity of the GaN. Our results indicate a thickness dependent thermal conductivity of GaN, which has implications of thermal management in GaN-based high electron mobility transistors.

Fast Fluid Thermodynamics Simulation by Solving Heat Diffusion Equation

2021 ◽  
Vol 11 (04) ◽  
pp. 1-11
Author(s):  
Wanwan Li
Keyword(s):  
Diffusion Equation ◽  
Real Time ◽  
Stokes Equations ◽  
Fluid Simulation ◽  
Heat Diffusion ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Heat Diffusion Equation ◽  
Acceleration Techniques ◽  
Speed Up

In mechanical engineering educations, simulating fluid thermodynamics is rather helpful for students to understand the fluid’s natural behaviors. However, rendering both high-quality and realtime simulations for fluid dynamics are rather challenging tasks due to their intensive computations. So, in order to speed up the simulations, we have taken advantage of GPU acceleration techniques to simulate interactive fluid thermodynamics in real-time. In this paper, we present an elegant, basic, but practical OpenGL/SL framework for fluid simulation with a heat map rendering. By solving Navier-Stokes equations coupled with the heat diffusion equation, we validate our framework through some real-case studies of the smoke-like fluid rendering such as their interactions with moving obstacles and their heat diffusion effects. As shown in Fig. 1, a group of experimental results demonstrates that our GPU-accelerated solver of Navier-Stokes equations with heat transfer could give the observers impressive real-time and realistic rendering results.

Meshless Local Petrov-Galerkin Method for Heat Transfer Analysis

2013 ◽  
Author(s):  
Singiresu S. Rao
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Boundary Conditions ◽  
Transfer Coefficient ◽  
Transfer Problem ◽  
Radiation Heat Transfer ◽  
Heat Transfer Problem ◽  
Radiation Heat ◽  
Mlpg Method ◽  
Transfer Problems

A meshless local Petrov-Galerkin (MLPG) method is proposed to obtain the numerical solution of nonlinear heat transfer problems. The moving least squares scheme is generalized, to construct the field variable and its derivative continuously over the entire domain. The essential boundary conditions are enforced by the direct scheme. The radiation heat transfer coefficient is defined, and the nonlinear boundary value problem is solved as a sequence of linear problems each time updating the radiation heat transfer coefficient. The matrix formulation is used to drive the equations for a 3 dimensional nonlinear coupled radiation heat transfer problem. By using the MPLG method, along with the linearization of the nonlinear radiation problem, a new numerical approach is proposed to find the solution of the coupled heat transfer problem. A numerical study of the dimensionless size parameters for the quadrature and support domains is conducted to find the most appropriate values to ensure convergence of the nodal temperatures to the correct values quickly. Numerical examples are presented to illustrate the applicability and effectiveness of the proposed methodology for the solution of heat transfer problems involving radiation with different types of boundary conditions. In each case, the results obtained using the MLPG method are compared with those given by the FEM method for validation of the results.

Computer simulation of target link explosion in laser programmable redundancy for silicon memory

1986 ◽  
Vol 1 (2) ◽  
pp. 368-381 ◽  
Author(s):  
L.M. Scarfone ◽  
J.D. Chlipala
Keyword(s):  
Thermal Evolution ◽  
Three Dimensional ◽  
Heat Diffusion ◽  
Dielectric Films ◽  
Threshold Temperature ◽  
Heat Diffusion Equation ◽  
Heating Effects ◽  
Transparent Dielectric ◽  
Difference Form ◽  
Pulse Energies

Pulses of Q-switched Nd-YAG radiation have been used to remove polysilicon target links during the implementation of laser programmable redundancy in the fabrication of silicon memory. The link is encapsulated by transparent dielectric films that give rise to important optical interference effects modifying the laser flux absorbed by the link and the silicon substrate. Estimates of these effects are made on the basis of classical plane-wave procedures. Thermal evolution of the composite structure is described in terms of a finite-difference form of the three-dimensional heat diffusion equation with a heat generation rate having a Gaussian spatial distribution of intensity and temporal shapes characteristic of commercial lasers. Temperature-dependent thermal diffusivity and melting of the polysilicon link are included in the computer modeling. The calculations account for the discontinuous change in the link absorption coefficient at the transition temperature. A threshold temperature and corresponding pressure, sufficiently high to rupture the dielectric above the link and initiate the removal process, are estimated by treating the molten link as a hard-sphere fluid. Numerical results are presented in the form of three-dimensional temperature distributions for 1.06 and 0.53 μm radiation with pulse energies 3.5 and 0.15μJ, respectively. Similarities and differences between heating effects produced by long (190 ns FWHM/740 ns duration) and short (35 ns FWHM/220 ns duration) pulses are pointed out.

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