Mathematical Model Of Temperature Field Research Articles

Mathematical model of temperature field and its analytical solution for Freeze-Sealing Pipe-Roof method induced by radial offset of adjacent jacking pipes

In order to achieve the purpose of sealing water between adjacent jacked pipes in construction of long-distance and large-section tunnel, an innovative method called Freeze-Sealing Pipe Roof (FSPR) was applied to the Gongbei tunnel, which is a key part of Hong Kong-Zhuhai-Macau Bridge in China. Since the pre-constructed jacking pipes provides great convenience for flexible placement of freezing tubes inside, it is crucial to study the temperature field of different freezing states. This paper first established a new mathematical model considering radial offset of jacking pipe according to the practical engineering, and then the analytical solution to steady-state temperature field was also derived. Moreover, temperature distribution of four typical cross-sections were analyzed to evaluate the freezing effect on different paths. Results show that the analytical results are in good agreement with the numerical results, and the quasi-double-circle freezing model can be well applied to the practical problem of radial offset. Soil between steel pipes will be lowered to a sufficiently low temperature and form a fairly reliable water-sealing curtain. By designing the radial offset between freezing tubes in solid and hollow pipes, the freezing effect can be better ensured. The conclusions have important guiding significance for design and construction of FSPR method in future projects.

Numerical Simulation of the Dynamic Change Law of Spontaneous Combustion of Coal Gangue Mountains.

To further study the problem of spontaneous combustion of coal gangue mountains, a multifield coupled simulation model was established based on the mathematical models of temperature field, gas concentration field, and seepage velocity field. In addition, the dynamic development law of these three physical fields in the process of spontaneous combustion is numerically simulated, and the relationship between gas concentration and temperature is studied and verified by experiments. The results show that in the initial stage of the thermal storage and heating process of the coal gangue mountain, the overall heating rate is small. With the passage of stacking time, a high-temperature area will gradually form inside, and the high-temperature area is concentrated near the windward side first and then spread to the leeward side. The oxygen on the windward side keeps a high concentration all of the time and gradually attenuates after entering the interior, and the overall concentration decreases with the extension of stacking time. The distribution law of carbon monoxide concentration is opposite to that of oxygen concentration. The variation law of carbon dioxide concentration is the same as that of temperature, that is, the concentration near the windward side is the highest, and then, it shows a distribution trend of first concentration and then divergence. The vortex phenomenon formed on the leeward side of the mountain outside and the chimney effect inside will aggravate the gas convection inside the gangue mountain, which makes the reaction continue so that the temperature of the gangue mountain keeps rising until it spontaneously ignites. There is a correlation between temperature and gas concentration. Among them, the concentration of carbon monoxide and carbon dioxide can well reflect the temperature change inside the coal gangue mountain, and the effect of carbon dioxide is better in the high-temperature area. The research results provide a theoretical reference for the prevention and control measures of spontaneous combustion of coal gangue mountains and have certain guiding significance.

Mathematical modeling of temperature fields in antenna reflectors of space vehicles

The main factor that determines the distortion of the shape of the reflectors in outer space is the temperature deformation due to the uneven distribution of heat fluxes in the structural elements. Therefore, it is important to develop models and methods for calculating temperature fields in reflectors with uneven distribution of heat fluxes on the surface. The use of such methods will reduce the number of expensive field experiments. The paper first constructs a mathematical model for calculating temperature fields in a parabolic reflex antenna, in the form of a paraboloid of rotation rotating at a constant angular velocity, taking into account the finite velocity of heat propagation as a boundary value problem of mathematical physics for the hyperbolic equation of thermal conductivity. that the thermophysical properties of the body are constant. At the initial moment of time, the body temperature is constant, and on the outer surface of the body are known values of heat flux which are continuous coordinate functions. To solve the obtained boundary value problem, a new integral transformation for a two-dimensional finite space was constructed. The formula of inverse transformation is given. Eigenvalues and eigenfunctions for the kernel of integral transformation are found using finite element methods and Galorkin. The division of the region into simplex elements was made. Thus the problem of finding eigenvalues and eigenfunctions was reduced to the algebraic problem of finding eigenvalues and eigenfunctions. After applying the constructed new integral transformation to the obtained boundary value problem, we obtained the Cauchy problem, the solution of which was found analytically. The obtained solution of the boundary value problem is twice continuously differentiated by spatial coordinates and once in time. The solution of the boundary value problem found can be used to modulate the temperature fields that occur in a parabolic reflex antenna. The paper first constructs a mathematical model for calculating temperature fields in a paraboloid rotating at a constant angular velocity, taking into account the finite velocity of heat propagation as a boundary value problem of mathematical physics for the hyperbolic equation of thermal conductivity with Neumann boundary conditions. Using the developed integral transformation, the temperature fields in the paraboloid in the form of convergent series by Fourier functions were found. The solution of the generalized boundary value problem of heat exchange of the paraboloid of rotation can be used to modulate the temperature fields that occur in the antenna reflectors of spacecraft.

Mathematical modeling heat and mass transfer in a vertical hydraulic fracture crack during inflation and cleaning*

When designing hydraulic fracturing for high-temperature formations, it is important to know the temperature change in the fracture during the injection of fracturing fluid. The temperature profile in the hydraulic fracture is necessary to calculate the optimal composition of the fracturing fluid, which necessarily includes a crosslinker (crosslinker) and a breaker (breaker), the concentration of which is calculated by the temperature at the end of the crack. Currently, this concentration is calculated based on the maximum temperature of the formation, which can lead to a decrease in the efficiency of hydraulic fracturing, since a breaker will not completely destroy the crosslinked gel. Therefore, when a well is brought into operation after the stimulation, proppant removal may occur, reducing the effectiveness of stimulation to zero. In this regard, the optimization of the decision-making process in the design of hydraulic fracturing in terrigenous and carbonate reservoirs by calculating the optimal parameters of process fluids based on predicting heat and mass transfer processes occurring during processing is a very urgent task. A tool has been developed to improve the design efficiency of hydraulic fracturing based on mathematical modeling of temperature fields in a hydraulic fracture during its development and during the period of technological sludge. A mathematical model that describes the temperature dynamics in a hydraulic fracture taking into account fluid leakage into the formation represents the evolutionary equation of convective heat transfer with a source, which is defined as the density of the heat flux from the formation. To check the adequacy of the model of temperature dynamics in a hydraulic fracture, a model of temperature recovery in a fracture is presented with the subsequent adaptation of simulation results to actual data. Developed mathematical models can be used in hydraulic fracturing simulators.

Application of smooth particle hydrodynamics (SPH) method in gravity casting shrinkage cavity prediction

Gravity casting has been applied extensively. Cavity will occur in the casting and form shrinkage cavity in solidification due to volume reduction and lowering of liquid metal level, which will affect casting quality. The formation process of shrinkage cavity cannot be directly observed and needs to be displayed by using the numerical simulation method. The smoothed particle hydrodynamics (SPH) method is a pure Lagrange method with no grids and has now been applied in casting numerical simulation, but the prediction model of the shrinkage cavity in gravity casting solidification based on the SPH method has not been reported. In this paper, the mathematical model for temperature field (including latent heat treatment) and shrinkage cavity prediction in solidification is established based on the SPH method. The correctness for the temperature field (including latent heat treatment) is validated by computing temperature field in the solidification of an L-shaped aluminum alloy casting, and correctness for the prediction of shrinkage cavity is validated by computing the shrinkage cavity in the solidification of a cylindrical steel casting.

Numerical Simulation of Transient Temperature Field of Single Fissure Flow in Mining Rock Mass

In this paper, the transient temperature field distribution of single fissure water flow is studied in order to study the fluid-solid coupling heat transfer mechanism of the rock mass. By simulating the flow state of underground water flow along single fissure of rock mass, it establishes the seepage model of single fissure. In addition, the mathematical model of temperature field under the influence of seepage field is established by simplifying the equations of single fissure and temperature control. The results show that the heat balance is closely related to temperature difference and time under fluid-solid coupling, and the greater the change of permeability velocity, the greater the change of temperature distribution. With the increase of time and heat transfer, the temperature difference between the water flow temperature and the wall surface of the rock mass gradually decreases, and the temperature isograde change rate along the flow direction of the water flow decreases gradually.

Mechanism of Bubble Formation in a Combined In-Mold Decoration and Microcellular Foaming Injection Molding Process

A combined in-mold decoration and microcellular foaming injection molding process (IMD/Mucell) is proposed to improve the surface quality of foamed parts. Through theoretical inference, the mathematical models of temperature field, velocity profile and forces on bubble were established with combining traditional mathematical models and the different heat transfer coefficient of the mold inner surface. The effects of the non-isothermal mold on the bubble structure and forming process of inter bubble were investigated with experiments and numerical simulation. Results showed that the critical pressure of the cavity existed in the melt filling and bubble expanding stage, and it affected the cells structure obviously at the filling stage. The lift force took the bubbles to the film side based on the velocity profile of viscous fluid, and maximal bubble was closer to the transition layer with film due to higher temperature. In the combined process, the evolution rule of superficial bubbles was analyzed, and results showed that the surface quality was improved in the film side.

An Optical Fiber-Based Data-Driven Method for Human Skin Temperature 3-D Mapping.

Human skin temperature mapping provides abundant information of physiological conditions of human body, which provides supplementary or alternative indicators for disease monitoring or diagnosis. The existing models of temperature mapping or temperature field distribution of human skin are generally established by finite element method. Due to the complexity of biological systems, it is challenging to achieve high accuracy mathematical models of temperature field of human skin. The goal of this study is to establish human skin temperature three-dimensional (3-D) mapping platform by integrating optical fibers and improved genetic algorithm-back propagation (GA-BP) neural network. The proposed data-driven method is capable of acquiring entire human skin temperature 3-D mapping by simply measuring a few points on human skin. Multiple experiments were conducted to validate the proposed method on different areas of human skin in different ambient environments. In each experiment setting, the measured data and the model output data were compared. The mean absolute error in all the validation experiments is 0.11°C, which is lower than that in the state of the art using physical modeling for skin temperature prediction and more close to clinical accuracy. The results show that the proposed approach is accurate and reliable, which may provide a platform technology for human skin temperature mapping that can be used in both medical and scientific studies as well as home monitoring.

Mathematical modeling of temperature fields and concrete strength during winter concreting of building structures

In this paper we present a mathematical model of a three-dimensional non-stationary temperature field for concrete that hardens in winter conditions. The developed mathematical model describes heat transfer processes basing on the differential heat equation. We also consider the classical boundary conditions of the I, II, III and IV kinds. We compiled a computer interpolation of the family of nomograms in the second order regression equation in order to calculate the relative theoretical strength of concrete after heat treatment.

Mathematical modeling of temperature fields with consideration for curing kinetics of thick-walled fiberglass plate

The kinetic parameters of curing of the binder VSK-14-2m are determined. The heat release power in the specified hardening regime in the temperature range from 100 up to 200°C is estimated on the basis of the obtained data. A mathematical model of heat transfer when forming the thick-walled fiberglass plate is proposed. Temperature differences of the outer and middle layers of the plate and temperature gradients over the plate thickness are revealed—significant temperature differences of the outer and middle layers of the plate (more than 30°C) and temperature gradients of layers (more than 7 K/mm) are observed, resulting in decrease in mechanical characteristics. Comparison of theoretical and experimental data on the temperature distribution over the plate thickness is performed. The level of correlation with each other is 99.5%.

Modeling of temperature fields in the working chamber of the process furnace for REE synthesis

The results of mathematical modeling of temperature fields in the working chamber of the process furnace for special purposes are shown. Studied laboratory furnace is test equipment, which is used for practicing the stages of the technological process of continuous solid-phase synthesis of nanopowders of various purpose, such as obtaining of luminophore powders with rare earth elements oxides in its composition. Mathematical model adequacy is tested empirically

Study of Convective Heat Transfer Coefficient of High Pressure Water Descaling

In order to improving the product quality of hot rolled plate, the iron scale was removed by high pressure water descaling before hot rolling. The billet temperature dropped when a large amount of high pressure water injected on the billet surface. Establishing reasonably mathematical model of temperature field was very important, because it was related to formulate correctly rolling technology. High pressure water descaling convection heat transfer coefficient was an important parameter in the mathematical model of the temperature field. This paper calculated the high pressure water convection heat transfer coefficient by the method of numerical simulation, and regressed the mathematical model of the high pressure water coefficient of convective heat transfer by nonlinear regression method. The author used this mathematical model for finite element analysis in a steel mill, the results showed that the simulation results agreed with the experimental results, the mathematical model of high pressure water descaling convective heat transfer coefficient was reasonable.

Research of Temperature Field in Oil-Immersed Transformer Considering the Oil Speed

In this paper, we have Elaborated the mathematical model of temperature field and flow field of the oil-immersed transformer, and analysis its structure of thermal .We established a temperature finite element model of an oil-immersed transformer using the method of flow-solid-thermal coupling. Using the software of ANSYS, simulating on a 250MVA oil-immersed transformer, we obtain the steady-state temperature distribution and the winding hottest locations. Analyze the effect of oil-speed to the temperature field and location of the hot spot temperature of oil-immersed transformer. The results show that when oil flow rate is increases in the normal range, Transformer temperature rise corresponding slowly, and its location hottest temperature slightly pulled accordingly. The fiber measure different speeds Oil immersed transformer winding hot spot temperature to provide a basis for positioning.

Research on Finite Element Simulation and Experiment of Temperature Field for Surface Cutting of Nanostructured WC/12Co Coatings

To investigate the distribution of cutting temperature for surface cutting of nanastructured WC/12Co coating used the PDC cutter, according to machining properties of WC/12Co coating and working conditions, on the basis of the theory of thermal equilibrium of moving thermal source, the mathematical model of temperature field was established through reasonable hypothesis. Simulations were performed for the temperature distribution on the surface and inside the WC/12Co coating specimen using finite element method. And experiment of surface temperature of WC/12Co coating used the PDC cutter was carried out. The finite element simulation results were in good agreement with the experiment data, which verified the validation and capability of the finite element model.

Method Research and Practice for Numerical Simulation Analysis of Temperature Field of Hot Runner of Injection Mould

In this article, it is studied how to establish and solve the mathematical model of temperature field of hot runner of injection mould, visual numerical simulation analysis for temperature field is realized with computer, and finally corresponding example is cited.

Research on Finite Element Simulation and Experiment of Temperature Field in Surface Cutting Used a New PDC Cutter

A new polycrystalline diamond compacts (PDC) cutter was designed and prepared. The new PDC cutter has fibers that are artificially ordered so as to have the desired geometric angles, and it has a larger number of the cutting edges, enough space for holding chips and interrupted cutting so that it is easier to release heat. According to machining properties of materials and working conditions, on the basis of the theory of thermal equilibrium of moving thermal source, the paper established the mathematical model of temperature field of interrupted cutting used the PDC cutter through reasonable hypothesis. Simulations were performed for the temperature distribution in the tool-work interface and inside the Al2O3 ceramic specimen using finite element method. And experiment of surface temperature of Al2O3 ceramic interrupted cutting used the PDC cutter was carried out. The validation and capability of the finite element model were demonstrated by a satisfying agreement between the finite element simulation results and experimental ones.

Thermal Deformation Analysis for the Guideway of Large-Type CNC Lathe Based on Finite Element Method

Friction between the guideway and the bench of large-type CNC lathe will cause thermal deformation of the guideway, which causes processing error of the lathe，thereby reduces machining precision of the workpiece. The authors establish the mathematical model of temperature field and thermal deformation of the guideway in the work process, numerically simulate the guideway thermal characteristics by ANSYS finite element analysis software, and obtain the distribution regularities of temperature field and thermal deformation and their major influencing factors, which provide a theoretical basis for optimizing design and thermal error compensation design of the lathe guideway.

The Numerical Simulation of Directional Solidification Process Temperature Field for Large-Scale Frustum of a Cone Ingot

By using Fluent software, the mathematical model of temperature field is established on directional solidification process for large-scale frustum of a cone ingot, and the result is analyzed by Origin software, Tecplot. The influences of different width/thickness ratio to directional solidification process of cone ingot are discussed in order to provide basis for design optimization and ingot quality improvement.

Study on Mathematical Model of Temperature Field in the Laser Welding Process

From the laser welding actual process, the welding heat source model of laser welding process was established, that is, superposition heat source. According to the knowledge of thermodynamics, the establishment of a welding process, the mathematical model of temperature distribution of laser welding process was obtained by laser welding heat source. Finally, the finite element simulation of welding temperature distribution was used. The simulated results were compared with the analytical results of mathematical model of temperature field, it was proved consistent between simulated results and analytical results, at the same time it can account for the correctness of the mathematical model of temperature field.

Numerical simulation of temperature distribution of deep field in high temperature mine

In order to study the temperature distribution of deep field, mathematical models of temperature field in field and surrounding rock were built based on heat transfer and seepage theory. Combined test data with mathematical model, the temperature distribution under heat-transfer and underground-water coupling was studied by using Golden Software Surfer and Matlab. The results show that distribution law of most isothermal lines is very similar in deep field, and temperature gradient is equal in general. At the same time, temperature distribution is influenced by underground-water and fault. In surrounding rock, seepage changes symmetrical distribution of temperature field and vector, and the temperature field may divide into inward-diffusion area and outward-diffusion area. Peripheral temperature of working will approach to the temperature of airflow. In inward diffusion area, the distribution of temperature and temperature vector is symmetric, and the direction of temperature vector point to the center of working. The action of airflow is stronger than seepage in inward diffusion area, however, the case opposite is true in outward diffusion area.