scholarly journals MATHEMATIC MODEL OF AND METHOD FOR SOLVING THE DIRICHLET HEAT-EXCHANGE PROBLEM FOR ONE-SHEET ROTARY HYPERBOLOID

2020 ◽  
Vol 1 (126) ◽  
pp. 23-36
Author(s):  
Mykhailo Berdnyk
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Boundary Problem ◽  
Heat Conductivity ◽  
Integral Transformation ◽  
Mathematic Model ◽  
The Body ◽  
Initial Moment ◽  
Thin Wall ◽  
Conductivity Equation

It is the first generalized 3D mathematical model developed for calculating temperature fields in the thin-wall one-sheet rotary hyperboloid, which rotates with constant angular velocity around the axis OZ, ; the model was created with the help of known equations of generating lines in cylindrical coordinate system with taking into account finite velocity of heat conductivity and in the form of the Dirichlet boundary problem for the hyperbolic equation of heat conduction under condition that heat-conduction properties of the body were constant, and no internal sources of the heat were available. At initial moment of time, the body temperature was constant; values of temperature on outside surfaces of the body were known and presented continuous function of coordinate.The hyperbolic heat-conductivity equation was derived from the generalized energy transfer equation for the moving element of continuous medium with taking into account finiteness of the heat conductivity velocity.In order to solve the boundary problem, the desired temperature field was represented as a complex Fourier series. The obtained boundary problems for the Fourier coefficients were found with the help of Laplace integral transformations and the new integral transformation for two-dimensional finite space. Intrinsic values and intrinsic functions for the integral transformation kernel were found by method of finite element and the Galerkin methods. Besides, the domain was divided into simplex element.As a result, the temperature field in the thin-wall one-sheet rotary hyperboloid was found in the form of convergent series in Fourier functions.

The Mathematic Model and Method for Solving the Dirichlet Heat-Exchange Problem for Empty Isotropic Rotary Body

2018 ◽  
Vol 277 ◽  
pp. 168-177
Author(s):  
Mykhailo Berdnyk
Keyword(s):  
Temperature Field ◽  
Dirichlet Problem ◽  
Heat Exchange ◽  
Angular Velocity ◽  
Integral Transformation ◽  
Mathematic Model ◽  
Constant Angular Velocity ◽  
Temperature Fields ◽  
Finite Velocity ◽  
Finite Space

It is the first generalized 3D mathematic model, which is created for calculating temperature fields in the empty isotropic rotary body, which is restricted by end surfaces and lateral surface of rotation and rotates with constant angular velocity around the axis OZ, with taking into account finite velocity of the heat conductivity in the form of the Dirichlet problem. In this work, an integral transformation was formulated for the 2D finite space, with the help of which a temperature field in the empty isotropic rotary body was determined in the form of convergence series by the Fourier functions.

scholarly journals The nonstationary thermoelectric elasticity problem for a long piezoceramic cylinder

2021 ◽  
pp. 181-190
Author(s):  
D. A Shlyakhin ◽  
M. A Kalmova
Keyword(s):  
Thermal Conductivity ◽  
Temperature Field ◽  
Heat Conductivity ◽  
The Body ◽  
Body Volume ◽  
Outer Face ◽  
External Temperature ◽  
Measuring Device ◽  
Wall Boundary Conditions ◽  
Temperature Load

А new closed-loop solution for the coupled nonstationary problem of thermoelectric elasticity is designed for a long piezoceramic radially polarized cylinder. The case of the nonstationary load acting on its inner cylindrical surface is considered as a function of temperature change at a given law of the convection heat exchange on the outer face wall (boundary conditions of heat conductivity of the 1st and 3rd types). Electrodynamic cylinder surfaces are connected to a measuring device with a high input resistance (electric idling). We investigate the problem where the rate of the temperature load changes does not affect the inertial characteristics of the elastic system. It makes it possible to expand the initial linear computational relations with the equilibrium, electrostatics and heat conductivity equations with respect to the radial component of the displacement vector, electric potential as well as the function of temperature field changes. Hyperbolic LS-theory of the thermal conductivity is used in the computations. The problem is solved with a generalized method of biorthogonal finite integral transformation based on a multicomponent ratio of eigen functions of two homogeneous boundary value problems. The structural algorithm of this approach allows identifying a conjugated operator, without which it is impossible to solve non-self-conjugated linear problems in mathematical physics. The resulted computational relations make it possible to determine the stress-strain state, temperature and electric fields induced in the piezoceramic element under an arbitrary external temperature effect. By connecting the electroelastic system to the measuring tool, we can find voltage. Firstly, the analysis of the numerical results allows identifying the rate of the temperature load changes, at which it is necessary to use the hyperbolic theory of thermal conductivity. Secondly, it allows determining the physical characteristics of the piezoceramic material for the case when the rate of changing the body volume leads to a redistribution of the temperature field. The developed computational algorithm can be used to design non-resonant piezoelectric temperature sensors.

Integral Transformation of Multidimensional Heat Conduction Problems in Heterogeneous Media

2017 ◽  
Author(s):  
Carolina Palma Naveira Cotta ◽  
Renato Machado Cotta ◽  
Anderson Pereira de Almeida
Keyword(s):  
Heat Conduction ◽  
Heterogeneous Media ◽  
Integral Transformation

scholarly journals To the Solution of Geometric Inverse Heat Conduction Problems

2021 ◽  
Vol 24 (1) ◽  
pp. 6-12
Author(s):  
Yurii M. Matsevytyi ◽  
◽  
Valerii V. Hanchyn ◽  
Keyword(s):  
Heat Conduction ◽  
Iterative Process ◽  
Regularization Parameter ◽  
Outer Boundary ◽  
Linear Equations ◽  
The Body ◽  
Two Dimensional ◽  
Inverse Heat Conduction ◽  
System Of Linear Equations ◽  
Inverse Heat Conduction Problems

On the basis of A. N. Tikhonov’s regularization theory, a method is developed for solving inverse heat conduction problems of identifying a smooth outer boundary of a two-dimensional region with a known boundary condition. For this, the smooth boundary to be identified is approximated by Schoenberg’s cubic splines, as a result of which its identification is reduced to determining the unknown approximation coefficients. With known boundary and initial conditions, the body temperature will depend only on these coefficients. With the temperature expressed using the Taylor formula for two series terms and substituted into the Tikhonov functional, the problem of determining the increments of the coefficients can be reduced to solving a system of linear equations with respect to these increments. Having chosen a certain regularization parameter and a certain function describing the shape of the outer boundary as an initial approximation, one can implement an iterative process. In this process, the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of coefficients in the previous iteration and the vector of the increments of these coefficients, obtained as a result of solving a system of linear equations. Having obtained a vector of coefficients as a result of a converging iterative process, it is possible to determine the root-mean-square discrepancy between the temperature obtained and the temperature measured as a result of the experiment. It remains to select the regularization parameter in such a way that this discrepancy is within the measurement error. The method itself and the ways of its implementation are the novelty of the material presented in this paper in comparison with other authors’ approaches to the solution of geometric inverse heat conduction problems. When checking the effectiveness of using the method proposed, a number of two-dimensional test problems for bodies with a known location of the outer boundary were solved. An analysis of the influence of random measurement errors on the error in identifying the outer boundary shape is carried out.

AN ANALYTICAL MODEL OF A NON-STATIONARY TEMPERATURE FIELD IN A RESERVOIR WITH A HYDRAULIC FRACTURING

2021 ◽  
Vol 7 (2) ◽  
pp. 75-94
Author(s):  
Ramil F. SHARAFUTDINOV ◽  
Filyus F. Davletshin
Keyword(s):  
Temperature Field ◽  
Hydraulic Fracturing ◽  
Analytical Model ◽  
Oil Recovery ◽  
Temperature Anomaly ◽  
Nonstationary Temperature ◽  
Initial Moment ◽  
Positive Temperature ◽  
Fluid Filtration ◽  
Stage Of Development

At the present stage of development of the oil and gas industry, considerable attention is paid to methods of increasing oil recovery of productive reservoirs. One of the most popular methods of intensifying oil production today is hydraulic fracturing. The efficiency and success of hydraulic fracturing largely depends on the parameters of the formed fracture; in this regard, the development of methods for evaluating the parameters of hydraulic fracturing fractures is an urgent task. Non-stationary thermometry is a promising area for monitoring the quality of hydraulic fracturing. To date, thermometry is used to localize the locations of multiple fractures in horizontal wells. In this paper, we study the application of non-stationary thermometry for estimating the parameters of a vertical hydraulic fracturing fracture. An analytical model of non-isothermal single-phase fluid filtration in a reservoir with a vertical fracture is developed. To calculate the temperature field in the formation and the fracture, the convective heat transfer equation is used, taking into account the thermodynamic effects (Joule — Thomson and adibatic), for the fracture, the heat and mass transfer between the fracture and the formation area is also taken into account. To assess the correctness of the model, the analytical solution is compared with the results of numerical modeling in the Ansys Fluent software package. The nonstationary temperature field is calculated for the constant sampling mode. It is established that at the initial moment of time after the well start-up, a negative temperature anomaly is formed due to the adiabatic effect, the value of which increases with a decrease in the fracture width. Over time, the temperature of the fluid flowing into the well increases due to the Joule — Thomson effect, and the value of the positive temperature anomaly increases as the width and permeability of the fracture decreases due to an increase in the pressure gradient in it. The developed analytical model can be used to solve inverse problems for estimating hydraulic fracturing parameters based on non-stationary temperature measurements in the wellbore of producing wells.

Temperature Field Analysis of Laser Medium with Rough Surface Heat Conduction

2012 ◽  
Vol 32 (6) ◽  
pp. 0614001
Author(s):  
连天虹 Lian Tianhong ◽  
王石语 Wang Shiyu ◽  
过振 Guo Zhen ◽  
李兵斌 Li Bingbin ◽  
林林 Lin Lin ◽  
...  
Keyword(s):  
Temperature Field ◽  
Heat Conduction ◽  
Rough Surface ◽  
Surface Heat ◽  
Field Analysis ◽  
Laser Medium

Mathematic Model and Method for Solving the Heat-Exchange Problem in Electron-Beam Welding of Arbitrary Areas

2019 ◽  
Vol 291 ◽  
pp. 173-182
Author(s):  
Mykhailo Berdnyk
Keyword(s):  
Electron Beam ◽  
Electron Beam Welding ◽  
Integral Transformation ◽  
Mathematic Model ◽  
Convergent Series ◽  
Dirichlet Boundary Conditions ◽  
Temperature Fields ◽  
The Finite Element Method ◽  
Finite Space ◽  
First Time

For the first time in this article, a mathematical model has been developed for calculating the temperature fields in arbitrary areas in electron-beam welding; this model was created in the form of a boundary value problem of mathematical physics for a parabolic equation of heat conductivity with Dirichlet boundary conditions. A new integral transformation was constructed for a two-dimensional finite space, with the use of which, as well as the finite element method and Galerkin's method, a temperature field has been determined in the form of a convergent series.

On Heat and Cells and Proteins

Physiology ◽  
2004 ◽  
Vol 19 (1) ◽  
pp. 11-15 ◽  
Author(s):  
Dörthe M. Katschinski
Keyword(s):  
Temperature Field ◽  
Heat Shock ◽  
Temperature Control ◽  
Cold Shock ◽  
Control Strategies ◽  
Cellular Function ◽  
The Body ◽  
Cold Shock Proteins ◽  
Shock Proteins

Two principal forms of temperature-control strategies have evolved, i.e., poikilothermic and homeothermic life. Even in homeothermic animals, the temperature field of the body is not homogenous. These observed temperature differences can affect cellular function directly or via the expression of heat shock or cold shock proteins.

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