Method of Lines to Solve the Linear Temperature Field of LENS

2015 ◽  
Vol 1120-1121 ◽  
pp. 1441-1445
Author(s):  
Chun Fang Xue
Keyword(s):  
Thermal Conductivity ◽  
Temperature Field ◽  
Transfer Problem ◽  
Method Of Lines ◽  
Main Idea ◽  
Functionally Graded ◽  
Temperature Fields ◽  
Linear Temperature ◽  
Steady Temperature Field ◽  
Net Shaping

This article introduces a semi-analytical numerical method ——method of lines(MOLs) to solve steady temperature field of Laser Engineered Net shaping (LENS). The main idea of MOLs is to semi-discretized the governing equation of thermal transfer problem into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. The steady linear temperature fields of functionally graded materials were obtained using MOLs and the regularities of different temperature functions were also found. The effects of thermal conductivity coefficient under different formal functions on thermal temperature fields were analys. Numerical results showed that different material thermal conductivity function had obvious different effect on the temperature field.

Method of Lines to Solve 2-D Steady Temperature Field of FGM

2007 ◽  
Vol 353-358 ◽  
pp. 2003-2006 ◽  
Author(s):  
Wei Tan ◽  
Chang Qing Sun ◽  
Chun Fang Xue ◽  
Yao Dai
Keyword(s):  
Thermal Conductivity ◽  
Temperature Field ◽  
Transfer Problem ◽  
Method Of Lines ◽  
Main Idea ◽  
Functionally Graded ◽  
Temperature Fields ◽  
Steady Temperature ◽  
Thermal Transfer ◽  
Steady Temperature Field

Method of Lines (MOLs) is introduced to solve 2-Dimension steady temperature field of functionally graded materials (FGMs). The main idea of the method is to semi–discretized the governing equation of thermal transfer problem into a system of ordinary differential equations (ODEs) defined on discrete lines by means of the finite difference method. The temperature field of FGM can be obtained by solving the ODEs with functions of thermal properties. As numerical examples, six kinds of material thermal conductivity functions, i.e. three kinds of polynomial functions, an exponent function, a logarithmic function, and a sine function are selected to simulate spatial thermal conductivity profile in FGMs respectively. The steady-state temperature fields of 2-D thermal transfer problem are analyzed by the MOLs. Numerical results show that different material thermal conductivity function has obvious different effect on the temperature field.

scholarly journals Impact of Tunnel Temperature Variations on Surrounding Rocks in Cold Regions

2019 ◽  
Author(s):  
Qingyang Yu ◽  
Chao Zhang ◽  
Zhenxue Dai ◽  
Chao Du ◽  
Mohamad Reza Soltanian ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Temperature Field ◽  
Temperature Distribution ◽  
Three Dimensional ◽  
Temperature Fields ◽  
Insulation Layer ◽  
Cold Regions ◽  
Ambient Temperatures ◽  
Temperature Conditions ◽  
Layer Control

Temperature is an important factor in designing and maintaining tunnels, especially in cold regions. We present three-dimensional numerical simulations of tunnel temperature fields at different temperature conditions. We study the tunnel temperature field in two different conditions with relatively low and high ambient temperatures representing winter and summer of northeast China. We specifically study how these temperature conditions affect tunnel temperature and its migration to surrounding rocks. We show how placing an insulation layer could affect the temperature distribution within and around tunnels. Our results show that the temperature field without using an insulation layer is closer to the air temperature in the tunnel, and that the insulation layer has shielding effects and could plays an important role in preventing temperature migration to surrounding rocks. We further analyzed how thermal conductivity and thickness of insulation layer control the temperature distribution. The thermal conductivity and thickness of insulation layer only affect the temperature of the surrounding rocks which are located at distances below ~20 m from the lining.

Buckling and Free Vibration of Nonuniformly Heated Functionally Graded Carbon Nanotube Reinforced Polymer Composite Plate

2017 ◽  
Vol 17 (06) ◽  
pp. 1750064 ◽  
Author(s):  
Nivish George ◽  
P. Jeyaraj ◽  
S. M. Murigendrappa
Keyword(s):  
Temperature Field ◽  
Carbon Nanotube ◽  
Free Vibration ◽  
Composite Plate ◽  
Natural Frequencies ◽  
Vibration Mode ◽  
Functionally Graded ◽  
Temperature Fields ◽  
Mode Shapes ◽  
Reinforced Polymer

Buckling and free vibration behavior of functionally graded carbon nanotube reinforced polymer composite plate subjected to nonuniform temperature fields have been investigated using finite element approach. The effective material constants of the plate are obtained using the extended rule of mixture along with efficiency parameters of the carbon nanotube (to include geometry-dependent material properties). Influence of boundary conditions, aspect ratio, functional grading of the carbon nanotube, nonuniform thermal loading on thermal buckling and free vibration behavior of the heated plate are analyzed. It is observed that temperature fields and functional grading are influenced on the critical buckling temperature of the plates. Further, nature of functional grading showed significant change in buckling mode shapes irrespective of the boundary conditions. The first few natural frequencies of the plate under thermal load decreases as the temperature increases and they are influenced significantly by the nature of temperature field. Variations in free vibration mode shapes of the square plates found with not significant change as temperature increases. However, free vibration modes of the rectangular plates are sensitive to the nature of temperature field whenever there is a free edge associated with the boundary condition. Influence of functional grading on the free vibration mode shapes is not significant in contrast with the free vibration natural frequencies. The magnitude of free vibration natural frequencies of functional grade-X type carbon nanotube reinforcement showed higher in comparison with other two types of reinforcements considered here.

Analysis on Steady Temperature of Honeycomb-Core Panel Based on Coupled Aerothermal Heating and Honeycomb Conduction

2011 ◽  
Vol 217-218 ◽  
pp. 1197-1201
Author(s):  
Wei Liang ◽  
Yu Fu ◽  
Zhen Qi Liu ◽  
Lu Lu Yang ◽  
Han Chao Mai
Keyword(s):  
Thermal Conductivity ◽  
Temperature Field ◽  
Numerical Method ◽  
Physical Model ◽  
Thermal Conduction ◽  
Honeycomb Structure ◽  
Honeycomb Core ◽  
Steady Temperature ◽  
Equivalent Thermal Conductivity ◽  
Steady Temperature Field

A physical model is formulated to evaluate the steady temperature field of honeycomb-core panel. The model takes into account the coupled effect of aerothermal heating and radiate energy from front and rear plate and honeycomb thermal conduction. The equations that are established based on the model are solved in numerical method and the equivalent thermal conductivity is obtained. The model is also used to investigate the effect of coming fluid and the geometric parameters of honeycomb structure on the TPS capacity.

Nonlinear Steady Temperature Fields in Non—Homogeneous Materials

2007 ◽  
Vol 561-565 ◽  
pp. 1957-1960
Author(s):  
Yao Dai ◽  
Qi Sun ◽  
Wei Tan ◽  
Chang Qing Sun
Keyword(s):  
Thermal Conductivity ◽  
Differential Equations ◽  
Method Of Lines ◽  
Variable Coefficients ◽  
Temperature Fields ◽  
Gradient Material ◽  
Original Form ◽  
Steady Temperature ◽  
Heat Shielding ◽  
Parameter Values

Functionally gradient material (FGM) developed for heat-shielding structure is often used in the very high temperature environment. Therefore, the material property parameters are not only functions of spatial coordinates but also ones of temperature. The former leads to partial differential equations with variable coefficients, the latter to nonlinear governing equations. It is usually very difficult to obtain the analytical solution to such thermal conduction problems of FGMs. If the finite element method is adopted, it is very inconvenient because material parameter values must be imputed for each element. Hence, a semi-analytic numerical method, i.e., method of lines (MOLs) is introduced. The thermal conductivity functions do not need to be discretized and remain original form in ordinary differential equations. As a numerical example, the nonlinear steady temperature fields are computed for a rectangular non-homogeneous region with the first, the second and the third kinds of boundary conditions, where three kinds of functions, i.e. power, exponential and logarithmic ones are adopted for the thermal conductivity. Results display the important influence of non-linearity on temperature fields. Moreover, the results given here provide the better basis for thermal stress analysis of non-homogenous and non-linear materials.

Longitudinal-Fin Heat Sink Optimization Capturing Conjugate Effects Under Fully Developed Conditions

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Georgios Karamanis ◽  
Marc Hodes
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Thermal Resistance ◽  
Heat Sink ◽  
Transfer Problem ◽  
Optimization Method ◽  
Conjugate Problem ◽  
Temperature Fields ◽  
Microchannel Cooling ◽  
Fin Thickness

We develop a method requiring minimal computations to optimize the fin thickness and spacing in a fully shrouded longitudinal-fin heat sink (LFHS) to minimize its thermal resistance under conditions of hydrodynamically and thermally developed laminar flow. Prescribed quantities are the density, viscosity, thermal conductivity and specific heat capacity of the fluid, the thermal conductivity and height of the fins, the width and length of the heat sink, and the pressure drop across it. Alternatively, the length of the heat sink may be optimized as well. The shroud of the heat sink is assumed to be adiabatic and its base isothermal. Our results are relevant to, e.g., microchannel cooling applications where base isothermality can be achieved by using a heat spreader or a vapor chamber. The present study is distinct from the previous work because it does not assume a uniform heat transfer coefficient, but fully captures the velocity and temperature fields by numerically solving the conjugate heat transfer problem in dimensionless form using an existing approach. We develop a dimensionless formulation and compute a dense tabulation of the relevant parameters that allows the thermal resistance to be calculated algebraically over a relevant range of dimensionless parameters. Hence, the optimization method does not require the time-consuming solution of the conjugate problem. Once the optimal dimensionless fin thickness and spacing are obtained, their dimensional counterparts are computed algebraically. The optimization method is illustrated in the context of direct liquid cooling.

scholarly journals Modified method of direct in problems of thermal conductivity of annular plates

2020 ◽  
pp. 302-309
Author(s):  
Yuliia Sovych
Keyword(s):  
Thermal Conductivity ◽  
Boundary Conditions ◽  
Projection Method ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Temperature Fields ◽  
System Of Equations ◽  
Annular Plates ◽  
Modified Method ◽  
Initial Boundary Value

In this paper, to solve the initial boundary value problem of thermal conductivity using a numerical-analytical method - a modified method of lines is proposed. The initial equations of thermal conductivity defined in the cylindrical coordinate system are considered in the spatial formulation, which greatly complicates them. As an object on which they are defined, an annular plate is considered, the overall dimensions of which are commensurate. In the problems of calculating of thermal effects in load-bearing elements the first step is to determine the temperature fields, especially if the overall dimensions of the structures are proportional. Such elements include non-thin annular plates. The boundary conditions are considered in a general form too - these are the conditions for convective heat transfer, which using the passage to the limit, turn into boundary conditions of the first and second types. The application of the modified method of lines to reduce the dimensionality of the initial system of equations of nonstationary thermal conductivity used to determine the temperature fields of the load-bearing elements is shown in this paper. The application of the modified method of lines involves solving these initial boundary value problems in two stages. At the first stage, the dimensionality of the initial equations with respect to variable z is reduced. The Bubnov-Galerkin-Petrov projection method is used to reduce the dimensionality. The so-called functions-"caps" are accepted as basic functions, which are related to the lines plotted on the definition domain of the problem. The projection method is also used to reduce the dimension of the initial and boundary conditions, that allows to formulate a reduced initial-limit problem, which is convenient to solve using the numerical finite-difference method, using explicit or implicit difference schemes. The most successful form of writing the original equations was found, which ensures ease of application of dimensionality reduction of the initial system of equations using a modified method of lines. The calculation took into account the impact of the environment. Reduced equations, boundary and initial conditions are obtained. As a result, the reduced problem has a form convenient to its solution by modern numerical methods.

Method of lines for temperature field of functionally graded materials

2005 ◽  
Vol 12 (2) ◽  
pp. 230-232
Author(s):  
Yao Dai ◽  
Qi Sun ◽  
Gui-xiang Hao ◽  
Xiu-fa Yan ◽  
Yong-dong Li
Keyword(s):  
Temperature Field ◽  
Functionally Graded Materials ◽  
Method Of Lines ◽  
Functionally Graded ◽  
Graded Materials
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