On the Calculation of Coated Fins

2002 ◽  
Author(s):  
C. Cortes ◽  
L. I. Diez ◽  
A. Campo
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Formal Model ◽  
The Finite Element Method ◽  
Dimensional Model ◽  
Transfer Enhancement ◽  
One Dimensional ◽  
Conjugate System ◽  
Primary Material ◽  
Surrounding Fluid

Coated fins constitute a new concept in heat transfer enhancement. This type of fin is made from a primary material (the substrate) that usually possesses a low-to-moderate thermal conductivity. To augment the transfer of heat from the primary material to a surrounding fluid, a viable avenue is to coat the substrate with a thin layer of a high conductivity material (the coating). Undoubtedly, the formal model for a two-material fin is complicated because it involves a conjugate system of two heat conduction equations in two space variables. As a simpler alternative, Campo (2001) proposed a simplified quasi one-dimensional model that engages an ordinary differential equation with embedded spatial means of the thermal conductivities of the substrate and the coating. The objective of the present study is to extend the statistically-based ideas for a one material fin to two-material fins of variable thickness. To this end, a system of two heat conduction equations, coupled with the applicable boundary conditions, is solved with the Finite Element Method (FEM). The adequacy of the approximate algebraic route for the estimation of fin efficiencies is tested against the numerically-determined fin efficiencies supplied by the FEM.

BATCH DOUGH VISCOSITY INFLUENCE ON FLOW-PRESSURE CHARACTERISTICS OF THE KNEADING MACHINE

2019 ◽  
Author(s):  
В.С. РУБАН ◽  
В.И. АЛЕШИН ◽  
Д.С. БЕЗУГЛЫЙ
Keyword(s):  
Batch Process ◽  
System Of Equations ◽  
Systems Of Equations ◽  
The Finite Element Method ◽  
Dimensional Model ◽  
Symmetric Form ◽  
Pressure Function ◽  
One Dimensional ◽  
Flow Pressure ◽  
Transport Problems

Рассмотрены уравнения баланса и концентрационных потоков, базирующихся на моделях, позволяющих анализировать одноименные модели реологии течения в канале шнека блока замеса тестомесильной машины. Анализ процесса транспортировки и замеса на основе одномерной модели выявил необходимость использования сигмоидальной функции коэффициента напоропроводности от давления. Переход от одномерных задач к многомерным задачам переноса связан с преобразованием систем уравнений к симметричному виду. Полученные системы уравнений после использования теоремы Грина могут быть решены методом конечных элементов. The balance equation and concentration flows based on the models which make it possible to analyze the eponymous models of flow rheology in the block screw channel in a dough mixing machine has been considered. The analysis of the transportation and batch process based on one-dimensional model proved the necessity to apply sigmoidal coefficient of pressure function. The transition from one-dimensional problems to multidimensional transport problems is associated with the transformation of systems of equations to a symmetric form. The resulting system of equations after using Green’s theorem can be solved by the finite element method.

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.

scholarly journals Simple One-Dimensional Model of Heat Conduction which Obeys Fourier’s Law

2001 ◽  
Vol 86 (24) ◽  
pp. 5486-5489 ◽  
Author(s):  
P. L. Garrido ◽  
P. I. Hurtado ◽  
B. Nadrowski
Keyword(s):  
Heat Conduction ◽  
Dimensional Model ◽  
Fourier’S Law ◽  
One Dimensional ◽  
Fourier's Law

scholarly journals The Application of the Collocation Method Using Hermite Cubic Splines to Nonlinear Transient One-Dimensional Heat Conduction Problems

1975 ◽  
Vol 97 (4) ◽  
pp. 562-569 ◽  
Author(s):  
T. C. Chawla ◽  
G. Leaf ◽  
W. L. Chen ◽  
M. A. Grolmes
Keyword(s):  
Differential Equation ◽  
Heat Conduction ◽  
Finite Difference ◽  
Differential Equations ◽  
Collocation Method ◽  
Gaussian Quadrature ◽  
Time Step ◽  
One Dimensional ◽  
Hermite Splines ◽  
Nonlinear Transient

A collocation method for the solution of one-dimensional parabolic partial differential equations using Hermite splines as approximating functions and Gaussian quadrature points as collocation points is described. The method consists of expanding dependent variables in terms of piece-wise cubic Hermite splines in the space variable at each time step. The unknown coefficients in the expansion are obtained at every time step by requiring that the resultant differential equation be satisfied at a number of points (in particular at the Gaussian quadrature points) in the field equal to the number of unknown coefficients. This collocation procedure reduces the partial differential equation to a system of ordinary differential equations which is solved as an initial value problem using the steady-state solution as the initial condition. The method thus developed is applied to a two-region nonlinear transient heat conduction problem and compared with a finite-difference method. It is demonstrated that because of high-order accuracy only a small number of equations are needed to produce desirable accuracy. The method has the desirable characteristic of an analytical method in that it produces point values as against nodal values in the finite-difference scheme.

Use of the Finite-Element Method in the Solution of Diffusion-Convection Equations

1971 ◽  
Vol 11 (02) ◽  
pp. 139-144 ◽  
Author(s):  
Y.M. Shum
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Heat Conduction ◽  
Variational Principle ◽  
Numerical Solutions ◽  
Transient Heat Conduction ◽  
The Finite Element Method ◽  
One Dimensional ◽  
Diffusion Convection ◽  
Element Method

Abstract A variational principle can be applied to the transient heat conduction equation with heat-flux boundary conditions. The finite-element method is employed to reduce the continuous spatial solution into a finite number of time-dependent unknowns. From previous work, it was demonstrated that the method can readily be applied to solve problems involving either linear or nonlinear boundary conditions, or both. In this paper, with a slight modification of the solution technique, the finite-element method is shown to be applicable to diffusion-convection equations. Consideration is given to a one-dimensional transport problem with dispersion in porous media. Results using the finite-element method are compared with several standard finite-difference numerical solutions. The finite-element method is shown to yield satisfactory solutions. Introduction The problem of finding suitable numerical approximations for equations describing the transport of heat (or mass) by conduction (or diffusion) and convection simultaneously has been of interest for some time. Equations of this type, which will be called diffusion-convection equations, arise in describing many diverse physical processes. Of particular interest to petroleum engineers is the classical equation describing the process by which one miscible fluid displaces another in a one-dimensional porous medium. Many authors have presented numerical solutions to this rather simple presented numerical solutions to this rather simple diffusion-convection problem using standard finite-difference methods, method of characteristics, and variational methods. In this paper another numerical method is employed. A finite-element method in conjunction with a variational principle for transient heat conduction analysis is briefly reviewed. It is appropriate here to mention the recent successful application of the finite-element method to solve transient heat conduction problems involving either linear, nonlinear, or both boundary conditions. The finite-element method was also applied to transient flow in porous media in a recent paper by Javandel and Witherspoon. Prime references for the method are the papers by Gurtin and Wilson and Nickell. With a slight modification of the solution procedure for treating the convective term as a source term in the transient heat conduction equation, the method can readily be used to obtain numerical solutions of the diffusion-convection equation. Consideration is given to a one-dimensional mass transport problem with dispersion in a porous medium. Results using the finite-element method yield satisfactory solutions comparable with those reported in the literature. A VARIATIONAL PRINCIPLE FOR TRANSIENT HEAT CONDUCTION AND THE FINITE-ELEMENT METHOD A variational principle can be generated for the transient conduction or diffusion equation. Wilson and Nickell, following Gurtin's discussion of variational principles for linear initial value problems, confirmed that the function of T(x, t) that problems, confirmed that the function of T(x, t) that leads to an extremum of the functional...........(1) is, at the same time, the solution to the transient heat conduction equation SPEJ P. 139

Three-dimensional kinetic theory approach for laser pulse heating

1999 ◽  
Vol 213 (5) ◽  
pp. 491-506 ◽  
Author(s):  
B. S. Yilbas ◽  
M Sami
Keyword(s):  
Heat Conduction ◽  
Kinetic Theory ◽  
Three Dimensional ◽  
Experimental Results ◽  
Heating Process ◽  
Theory Approach ◽  
Dimensional Model ◽  
Three Dimensional Model ◽  
One Dimensional ◽  
Heating Model

Lasers are widely used as a machine tool in the metal industry. One of the important areas of laser application is surface treatment of engineering metals. To improve the process parameters in the laser heating process, an exploration of the heating mechanism is fruitful. The present study is carried out to develop a three-dimensional model for a laser pulsed heating process using the electron kinetic theory approach. The heating model introduced relies on successive electronphonon collisions; therefore, it is this process that describes the heat conduction mechanism. This study is limited to heat conduction only. Consequently, the phase change process is not taken into account. To validate the theoretical predictions, an experiment is conducted to measure the surface temperature using an optical method. Moreover, a one-dimensional heating model developed previously is also considered and the predictions of three- and one-dimensional heating models as well as experimental results are compared. It is found that the three-dimensional model gives lower surface temperatures compared with the one-dimensional model considered. However, experimental results agree well with the results obtained from the three-dimensional model. In addition, an equilibrium time is introduced. In that case, energy gain of electrons via incident beam absorption balances the energy losses due to conduction through successive electron-phonon collisions.

scholarly journals Mathematical simulation of forest fire front influence on wood-based building using one-dimensional model of heat transfer

2020 ◽  
Vol 200 ◽  
pp. 03007
Author(s):  
Nikolay V Baranovskiy ◽  
Aleksey Malinin
Keyword(s):  
Heat Transfer ◽  
Mathematical Model ◽  
Heat Conduction ◽  
Mathematical Simulation ◽  
Forest Fire ◽  
Dimensional Model ◽  
One Dimensional ◽  
Building Enclosures ◽  
Fire Front ◽  
Fire Impact

The purpose of the present paper is to mathematical simulation of heat transfer in enclosures of wood-based building when exposed to thermal radiation from forest fire front. One-dimensional mathematical model is used. Mathematically, heat transfer in building enclosures is described by system of non-stationary equations of heat conduction with corresponding initial and boundary conditions. It is suggested to use several scenarios of forest fire impact. Temperature distribution on wall depth is obtained for different scenarios of forest fire impact on building enclosures.

A one-dimensional model for the heat conduction in ring-like body

2013 ◽  
Vol 60 (4) ◽  
pp. 495-508
Author(s):  
István Ecsedi ◽  
Attila Baksa
Keyword(s):  
Thermal Properties ◽  
Heat Conduction ◽  
Cross Section ◽  
Polar Angle ◽  
Arbitrary Cross Section ◽  
Dimensional Model ◽  
One Dimensional ◽  
Model Based

Abstract A one-dimensional model based on the Fourier’s theory of heat conduction is developed for ring-like bodies. The ring-like body is an incomplete or complete torus with arbitrary cross section. The thermal properties of considered rings are independent of the polar angle. Examples illustrate the application of model presented

Sign in / Sign up

Export Citation Format

Share Document