Two-dimensional Heat Transfer Equation Research Articles

NUMERICAL SIMULATION OF THE PROCESS OF SELF-HEATING OF PLANT RAW MATERIALS FOR THE PURPOSE OF DETERMINING THE TIME OF THE FIRE INITIATION

Problem statement. The problem of forecasting the dynamics of heating the embankment of vegetable raw materials due to the action of microorganisms is considered. The activity of microorganisms leads to a local increase in temperature in the embankment of vegetable raw materials, which leads, over time, to a fire in the elevator. Estimating the period of time during which an elevator fire may occur is a particularly important task. The purpose of the article. Development of a numerical multi-parametric model for calculating the process of heating the embankment of vegetable raw materials in the elevator in order to determine the time when a fire may occur in the storage. Methodology. A two-dimensional heat transfer equation (energy equation) is used for mathematical modeling of the heating process of an embankment of plant raw materials. This equation takes into account the possibility of calculating the temperature field in the presence of layers with different thermal properties in the plant material embankment. For the numerical integration of the two-dimensional heat transfer equation, two finite-difference schemes are used. To construct the first finite-difference scheme, the analytical splitting of the modeling heat transfer equation into two steps is carried out. At each step of the splitting, the unknown temperature value is determined according to an explicit “running calculation” scheme. The second finite difference scheme is an explicit scheme for the numerical integration of the two-dimensional heat transfer equation. Scientific novelty. An effective numerical model has been developed that allows to quickly determine how the temperature field inside the embankment of plant raw materials changes with time by the method of a computational experiment in order to determine the time of a possible fire at the elevator. The model takes into account the most significant physical factors affecting the heating process of the plant material embankment. Practical significance. On the basis of the developed model, a computer program was created that allows in real time to determine the temperature field inside the embankment of plant raw materials on the elevator. The numerical model will be useful for analyzing the risk of fire in elevators and developing measures to reduce the occurrence of this extreme situation. Conclusions. A numerical model and a computer program that implements it on a computer have been created, which allows the method of computational experiment to study the dynamics of heating the embankment of vegetable raw materials and predict the time of possible fire. The developed computer program can be implemented on low and medium power computers. The results of a computational experiment are presented.

The effect of axially variable thermal and mechanical loads on the 2D thermoelastic response of FG cylindrical shell

ABSTRACTThis article presents two-dimensional thermoelastic analysis of a functionally graded cylindrical pressure vessel subjected to axially variable thermal and mechanical loads. Thermal and mechanical properties are assumed to be variable along the radius based on the power law variation. Two-dimensional heat transfer equation along the axial and radial directions is analytically solved before thermoelastic analysis of the problem. First-order shear deformation theory is used for description of displacement field. Energy method as well as Euler equation is used to obtain final constitutive differential equations of the system. Four differential equations of the system are solved using the eigenvalue and eigenvector method for various patterns of axially variable loadings. Axial and radial distributions of displacement, strain, and stress components are calculated in terms of different parameters such as variable thermal and mechanical loadings and nonhomogeneous index. The obtained results indicate that the boundary conditions of the cylinder have significant effect on thermoelastic response of the vessel. The axial distribution of displacements, strains, and stresses are validated with the existing literature.

Non-equilibrium two-dimensional thermal evolutions in target materials irradiated by femtosecond laser pulse

AbstractNon-equilibrium heat transfer described by a two-step temperature model was developed to study the thermal evolution through target materials irradiated by femtosecond laser pulse. Two-dimensional heat transfer equations were solved numerically. The temperature dependent thermo-physical properties of the electron and the lattice are considered in the model. The Gaussian spatial and temporal distribution of the heat and temperature of the electron and the lattice is presented. The effect of reflectivity, electron — lattice coupling factor, and the spot size was studied using copper targets.

Compensation Control Model of Superheat and Cooling Water Temperature for Secondary Cooling of Continuous Casting

AbstractIn this paper, a compensation control model of secondary cooling process of billet continuous casting for quality steel has been presented. The effects on the spray control of the various parameters such as steel superheat, casting speed, cooling water temperature and chemical component of steel were considered. The parameters of control model were determined to associate with the two‐dimensional heat transfer equation and solved by finite‐difference method. Effects of steel superheat and cooling water temperature on surface temperature, solidification structure and solidifying end point were discussed. Results indicate that steel superheat significantly affects solidification structure and solidifying end point but has a little effect on slab surface temperature. Moreover, secondary cooling water temperature affects surface temperature and solidifying end point but has a little effect on solidification structure. The surface temperature and solidifying end point can be maintain stabilized through applying the compensation control model when steel superheat and cooling water temperature vary. The models have been validated by industrial measurements. The results show that the simulations are in very good agreement with the real casting situation.

Modeling of the self-propagating reactions of nickel and aluminum multilayered foils

In this study, we performed simulations of self-propagating reactions of nanoscale nickel-aluminum multilayers using numerical methods. The model employs two-dimensional heat transfer equations coupled with heat generation terms from, (1) 1D parabolic growth of intermetallic phases Ni2Al3 and NiAl in the thickness direction and (2) phase transformations such as melting and peritectic reactions. The model uses temperature dependent physical and chemical data, such as interdiffusion coefficients, specific heat capacities, and enthalpy of reactions obtained from previous independent work. The equations are discretized using a lagged Crank–Nicolson method. The results show that initially, the reaction front velocity is determined by the rapid growth of Ni2Al3 and the front temperature is limited by the peritectic reaction at ∼1406 K. After the front completely traverses the foil and the temperature reaches the peritectic point, the reaction slows down and the temperature rises by the growth of NiAl which is the only stable phase at these temperatures. The reaction is completed when the initial constituents are consumed and the temperature reaches the melting point of NiAl. Subsequently, the foil cools and solidifies to the final phase dictated by the overall composition. The computational results show excellent fit to experimental velocity and temperature measurements.

Optimization of non-symmetric convective–radiative annular fins by differential quadrature method

The shape optimization of non-symmetric, convective–radiative annular fins is performed based on two-dimensional heat transfer analysis. The formulations are general so that the spatial and temperature dependent geometrical and thermal parameters can easily be implemented. The thermal conductivity of the fin is assumed to vary as a linear function of the temperature. The convective–radiative condition at the external surfaces of the fin and the effective convective condition at the fin base are considered. The differential quadrature method (DQM), as a simple, accurate and computationally efficient numerical tool, is used to solve the nonlinear two-dimensional heat transfer equation and the related nonlinear boundary conditions in the polar coordinate system. The accuracy of the method is demonstrated by comparing its results with those of the finite difference method. It is shown that by using fewer grid points, highly accurate results are obtained. Less computational effort of the method with respect to the finite difference method is shown.

A Note on Applying the Displacement Method in Assessing Fire Hazard of Glass Systems

Planar thermoelastieity problems have to be solved in assessing the fire hazard of glass systems. The displacement method was proposed to solve such problems. The two-dimensional heat transfer equation will be solved by the Green's theorem. A potential function was derived to give the displacement, strain, and stress. Similar to the Airy stress function, the displacement function used in this article will also satisfy the set of equations concerned. Behavior of thin glass panels in a fire is proposed to be studied by this displacement method in calculating the temperature distribution. Language: en

FINITE ELEMENT MODELING OF HEAT TRANSFERS IN VISCOELASTIC MATERIALS FOR LARGE DEFORMATIONS

The present work deals with the thermoviscoetastic elastomer behavior for large deformations. In the energy equation the source term describes the energy dissipated by internal damping. To determine this source term as a Junction of temperature, it is necessary to develop a Theological model for which the coefficients are temperature dependent. However, these coefficients cannot generally be quantified in relation to temperature. For a more accurate description of the real behavior of the material, the source term is determined from measurements through an inverse method. The two-dimensional heat transfer equation in Lagrangian form is then solved via a finite element method. A good agreement between the numerical and experimental results is obtained.

Modelling of a thermo-viscoelastic coupling for large deformations through finite element analysis

In this article we present the mathematical model leading to the equations of the thermoviscoelastic coupling for large deformations. We have chosen a Poynting-Thomson model to describe the rheological behavior of the material. In this model the source term generated by viscous frictions is independent of temperature. The two-dimensional heat transfer equation in Lagrangian form is solved via a finite element method for a two-layer elastomer-metal test piece under a shear force. The results of the numerical model are compared with experimental results which show that the source term cannot be temperature independent. For a more accurate description of the real behavior of the material the source term is then identified from experimental results via an inverse method. The numerical results agree well with the experimental ones.

The complex structure of a two-dimensional thermal wave absorbing laser radiation

A non-stationary two-dimensional heat transfer equation with non-linear coefficients and sources which model the process of the absorption of laser radiation is investigated using a numerical method. The intensity of the radiation is found from a one dimensional transfer equation.