Research on the Variation of Average Temperature of Heat Conduction Process under the Second Boundary Condition

2012 ◽  
Vol 562-564 ◽  
pp. 1951-1954
Author(s):  
Yong Yan Wang ◽  
Chuan Qi Su ◽  
Hong Cai Zheng ◽  
Nan Qin ◽  
Jia Bin Shi
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Heat Conduction ◽  
Integral Form ◽  
Energy Principle ◽  
Conservation Of Energy ◽  
Average Temperature ◽  
Equation Of Heat Conduction ◽  
Heat Conduction Process ◽  
Adiabatic Boundary Condition

The variation law of the average temperature with time in general case is derived by the differential equation of heat conduction which it is the reflection of the conservation of energy principle. The expression of the average temperature under the second boundary condition is given by the integral form of initial and boundary conditions. And what can be also derived are that the average temperature has a linear relationship with time when the boundary heat flux is constant, and it does not change with time under the adiabatic boundary condition.

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.

The Temperature Field Caused by Sphere Inclusion in Dielectric Irradiated by Multi-Pulse Laser

2013 ◽  
Vol 423-426 ◽  
pp. 452-455
Author(s):  
Cai Hua Huang ◽  
Xiao Hua Sun ◽  
Yi Hua Sun
Keyword(s):  
Differential Equation ◽  
Temperature Field ◽  
Heat Conduction ◽  
Finite Difference Method ◽  
Pulse Duration ◽  
Pulse Laser ◽  
Single Pulse ◽  
Inclusion Size ◽  
Equation Of Heat Conduction ◽  
Repetition Interval

The thermal effect caused by absorbing inclusions irradiated by multi-pulse laser is different from that of single pulse laser. The temperature field induced by multi-pulse laser depends markedly on both inclusion size and pulse duration, and repetition interval of pulse. Based on the differential equation of heat conduction, the temperature field caused by single absorbing inclusion is solved by use of finite difference method. The effect of inclusion size, pulse duration and repetition interval of pulse on the evolution of temperature field at the center of inclusion and interface between inclusion and dielectric are discussed qualitatively.

scholarly journals Integral method of solving heat-conduction problems with boundary condition of the second-kind. 2. Analysis of accuracy

2018 ◽  
Vol 63 (3) ◽  
pp. 318-332 ◽  
Author(s):  
V. A. Kot
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Heat Conduction ◽  
Integral Method ◽  
Boundary Value ◽  
Approximation Error ◽  
The Body ◽  
Temperature Front ◽  
Boundary Characteristics ◽  
Very High

An algorithm of finding polynomial solutions of boundary-value problems on nonstationary heat conduction with a time-dependent boundary condition of the secondary kind for bodies having a plane geometry, a cylindrical symmetry, or a spherical symmetry is presented. Thе algorithm is based on the introduction into consideration of the boundary characteristics in the form of a definite set of k-fold derivatives and n-fold integrals with respect to the time function of the heat flow on the surface of a body representing a boundary condition. Two stages of the heat-conduction process were considered separately: 1) the temperature front does not reach the center of a body and 2) the temperature front reaches the center of the body, and it is heated throughout its thickness. By the example of symmetric heating of a lengthy plate with a constant and variable heat flows, a very high accuracy of the proposed approach based on the integral method of boundary characteristics (BChIM) was demonstrated. As compared to the method of additional boundary characteristics, the BChIM makes it possible to de crease the relative approximation error (at one and the same polynomial degrees N) by  three to five orders of magnitude and by larger values and brings it to a negligibly low level (0.00028 %  at N = 11 and 0.000025 % at N = 14). It was established that, with each next approximation (with addition of three degrees into the polynomial), the approximation error decreases by an order of magnitude for the first stage of the process. An efficient algorithm of finding the eigenvalues of a boundary-value problem on heat conduction, based on the introduction into consideration of an additional function corresponding to the largest, in sequence order, boundary integral characteristic, is prtsented for the second stage of the process. Thе algorithm makes it possible to transform the integro-differential equation obtained on the basis of the BChIM into the ordinary differential equation with zero initial conditions. The calculations of the temperature at the center of the plate have shown that the approximation accuracy of the approach proposed is very high.

scholarly journals Application of Jacobi Polynomial and Multivariable Aleph- Function in Heat Conduction in Non-Homogeneous Moving Rectangular Parallelepiped

2021 ◽  
Vol 45 (03) ◽  
pp. 439-448
Author(s):  
DINESH KUMAR ◽  
FRÉDÉRIC AYANT
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Temperature Distribution ◽  
Jacobi Polynomial ◽  
Rectangular Parallelepiped ◽  
Equation Of Heat Conduction ◽  
Function Of Two Variables

The present paper deals with an application of Jacobi polynomial and multivariable Aleph-function to solve the differential equation of heat conduction in non-homogeneous moving rectangular parallelepiped. The temperature distribution in the parallelepiped, moving in a direction of the length (x-axis) between the limits x = −1 and x = 1 has been considered. The conductivity and the velocity have been assumed to be variables. We shall see two particular cases and the cases concerning Aleph-function of two variables and the I-function of two variables.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Enhancing Accuracy of Runge–Kutta-Type Collocation Methods for Solving ODEs

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 174
Author(s):  
Janez Urevc ◽  
Miroslav Halilovič
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Integration ◽  
Integral Form ◽  
Collocation Methods ◽  
Nonlinear Odes ◽  
New Class ◽  
New Methods ◽  
Runge Kutta

In this paper, a new class of Runge–Kutta-type collocation methods for the numerical integration of ordinary differential equations (ODEs) is presented. Its derivation is based on the integral form of the differential equation. The approach enables enhancing the accuracy of the established collocation Runge–Kutta methods while retaining the same number of stages. We demonstrate that, with the proposed approach, the Gauss–Legendre and Lobatto IIIA methods can be derived and that their accuracy can be improved for the same number of method coefficients. We expressed the methods in the form of tables similar to Butcher tableaus. The performance of the new methods is investigated on some well-known stiff, oscillatory, and nonlinear ODEs from the literature.

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