Unsteady-state Heat Conduction Research Articles

A dual-reciprocity boundary element method based transient seepage model with nonlinear distributed point source dual-porosity gas reservoir

As a result of advancements in gas reservoir development technologies and a deeper understanding of seepage theory, numerous nonlinear seepage problems have emerged. Consequently, conventional analytical solution models are no longer applicable. Moreover, existing semi-analytical and numerical solution models suffer from issues like high computational complexity and limited flexibility, significantly constraining their practical utility. For this purpose, a gas reservoir transient seepage model is established based on the dual-reciprocity boundary element method (DRBEM). This model couples the wellbore storage coefficients with skin factor and introduces a novel dual-porosity transient interporosity seepage algorithm, enabling the resolution of transient seepage problems in dual-porosity gas reservoirs with nonlinear point source distributions. Additionally, it investigates the dynamic behavior of bottomhole pressure during the early production stage of gas wells, while capturing the dynamic pressure distribution within the seepage field. This approach offers the advantages of semi-analyticity, meshless, and eliminates the need for Laplace transforms, resulting in a significant reduction in computational complexity. Ultimately, the application of our model to well-pattern production with the production system transition issues, in conjunction with the log-log type curves of bottomhole pressure, and the dynamics of gas reservoir pressure, enables the analysis of interference characteristics between injection-production wells. This showcases the practical advantages of the model's application. The findings of this study provide valuable assistance for modeling and solving transient seepage problems in dual-porosity oil and gas reservoirs production. The application of DRBEM in unsteady-state diffusion, heat conduction, stress fields, and other potential field problems is relevant for reference.

Registration of melting temperature at phase boundaries in melting powder mixtures

An integral constituent of the induction surfacing process for parts hardening pertains to the thermal treatment of rigid alloy particulates and the flux contained within the surfacing amalgam. This scholarly exposition delineates the outcomes of a comprehensive investigation, oriented towards the quantification and simulation of thermal gradients at interphase boundaries within intricate amalgamations of melting and thermosetting powdery substrates. In order to monitor the thermal dynamics during induction surfacing, the application of the CA-microthermocouple methodology and the thermal indication technique utilizing self-propagating SHS-process compositions is posited. The study encompasses both computational simulations and the resolution of the unsteady-state heat conduction problem within the chosen model of composite contacting materials. The devised methodologies for the intricate tracking of thermal profiles during induction surfacing offer the capability to ascertain the temperatures at which the surfacing amalgam and its individual constituents undergo liquefaction, in conjunction with the temperature gradient exhibited by the surface of the targeted component. This facet holds marked significance in the context of optimizing the hardening regimen and ensuring the requisite attributes of the amalgamated materials post-fusion. The outcomes of this research bear practical relevance in industrial applications, wherein enhancements to the caliber and dependability of hardened components are sought, simultaneously facilitating the curtailment of production overheads.

Simulation of Heat Transfer in Laser Transmission Welding

Abstract A numerical simulation of the heat transfer during laser transmission welding is presented. A finite difference approach was used to solve the one-dimensional unsteady-state heat conduction problem and to investigate the effect of welding conditions on the time-dependent temperature profiles for PA 6. For the needs of the simulation, the process was divided into heating and heat redistribution periods. The absorption coefficient of the laser-transparent part was measured experimentally and that of the laser-absorbing part was fitted using experimental data. The predicted temperature profiles were combined with experimental meltdown data to estimate the heat-affected zone thickness in the welded specimens. Good agreement was found between the estimated and measured heat-affected zone thickness values.

New operational relations for mathematical models of local nonequilibrium heat transfer

Objectives . Recently, interest in studying local nonequilibrium processes has increased in the context of the development of laser technologies, the possibility of reaching ultrahigh temperatures and pressures, and the need for a mathematical description of various physical processes under extreme conditions. In simulating local nonequilibrium processes, it becomes necessary to take into account the internal structure of investigation subjects, which significantly complicates the classical transport models. An important stage here is to construct mathematical models of various physical fields in which their spatiotemporal nonlocality should be taken into account. For these purposes, hyperbolic equations are used for a wide class of phenomena and, first of all, for unsteady-state heat conduction processes based on the generalized Maxwell–Cattaneo–Luikov–Vernotte phenomenology. Mathematical models in the form of boundary value problems for hyperbolic equations are called generalized boundary value problems. These problems differ significantly in solving difficulty from the classical ones based on Fourier phenomenology. The specificity of these problems is the relative simplicity of the initial mathematical models, together with the difficulty of solving them in an analytically closed form. Hence, very little success has been achieved in finding exact analytical solutions to problems of this kind. The most acceptable approach to solving them is operational calculus. However, it gives analytical solutions in the Laplace transform space as complex functional structures, the inverse transforms of which are not available in well-known reference books on operational calculus. On this path, serious computational difficulties arise. The study aimed to analyze a set of nonstandard transforms arising from the operational solution of mathematical models of local nonequilibrium heat transfer and to obtain their inverse transforms. Methods . Methods and theorems of operational calculus, methods of contour integration of complex transforms, and the theory of special functions were used. Results . Operational calculus was developed for mathematical models of local nonequilibrium heat transfer in terms of the theory of unsteady-state heat conduction for hyperbolic equations (wave equations). Nonstandard operational transforms, the inverse transforms of which are unavailable in the literature, were considered. It was shown that the presented transforms are common to operational solutions of a wide class of generalized boundary value problems for hyperbolic equations in the theory of heat conduction, diffusion, hydrodynamics, vibrations, propagation of electricity, thermomechanics, and other areas of science and technology. Partially bounded and finite domains were explored. Illustrative examples were given, namely, the results of numerical experimental studies of a local nonequilibrium heat transfer process that took into account the finiteness of the heat transfer rate, which had a wave character. The latter was expressed by the presence of the Heaviside step function in the analytical solution of the problem. The physical meaning of the finiteness of the heat transfer rate was substantiated. The isochron was constructed for the temperature function in a partially bounded domain. It was shown that the temperature profile has a discontinuity on the surface of the propagating wave front. This leads to the retention of heat outflow beyond the discontinuity boundary. This is a characteristic feature of the analytical solutions of the wave equations, along with the possibility to describe the analytical solution of the problem as equivalent integral relations, which noticeably simplify numerical calculations. Conclusions . The inverse transforms of nonstandard operational (Laplace) transforms were presented, which are contained in the operational solutions of a wide class of problems of local nonequilibrium (heat, mass, momentum) transfer processes, electrical circuits, hydrodynamics, oscillation theory, thermomechanics, and others. Illustrative examples were given, and the possibility of transition from one form of an analytical solution to another equivalent form was shown. The presented analytical solutions of hyperbolic heat transfer models in canonical domains are new in classical thermal physics.

An analytical approach of heat transfer modelling with thermal stresses in circular plate by means of Gaussian heat source and stress function

In the proposed research work we have used the Gaussian circular heat source. This heat source is applied with the heat flux boundary condition along the thickness of a circular plate with a finite radius. The research work also deals with the formulation of unsteady-state heat conduction problems along with homogeneous initial and non-homogeneous boundary condition around the temperature distribution in the circular plate. The mathematical model of thermoelasticity with the determination of thermal stresses and displacement has been studied in the present work. The new analytical method, Reduced Differential Transform has been used to obtain the solution. The numerical results are shown graphically with the help of mathematical software SCILAB and results are carried out for the material copper.

Kansa method for solving initial-value problem of hyperbolic heat conduction in nonhomogeneous medium

The Kansa method is a well-established method for solving partial differential equation-based problems. Nevertheless, it is still a relatively young, 30-year-old method, especially compared to the finite difference method. Consequently, there are still many problems that it is not sure that the Kansa method is suitable for solving them. An example of such a situation is the unsteady-state hyperbolic heat conduction problem described by Cattaneo-Vernotte model in a medium with spatially-variable thermal parameters alongside with time-dependent Neumann boundary conditions or irregularly shaped computational domain. In this paper, such problems were considered in the one- and two-dimensional variants using the Kansa method in the so-called coefficient formulation. The study conducted in the article showed that the Kansa method is highly likely suitable for solving problems of this type. A good agreement of the Kansa method solution with the finite difference method, the finite element method and the analytical solution was obtained. Also, characteristic features of hyperbolic heat flow were observed in the solution: delay in the heat flux transmission, the introduction of strong damping into the system, the intensification of the amplitude of periodic temperature and heat flux changes, the phase shift in the temperature oscillation. The article also addresses the problem of selecting the p exponent. It has been shown that the values recommended in the literature −1/2 and 1/2 may not be appropriate for the considered class of the problem. It has also been shown that for this class of problem, much better results are obtained when using very small positive values of the p exponent.

Heat and Mass Transfer Modeling to Predict Temperature Distribution during Potato Frying after Pre-Treatment with Pulsed Electric Field.

Based on unsteady state heat conduction, a mathematical model has been developed to describe the simultaneous heat and moisture transfer during potato frying. For the first time, the equation was solved using both enthalpy and Variable Space Network (VSN) methods, based on a moving interface defined by the boiling temperature of water in a potato disc during frying. Two separate regions of the potato disc namely fried (crust) and unfried (core), were considered as heat transfer domains. A variable boiling temperature of the water in potato discs was required as an input parameter for the model as the water is evaporated during frying, resulting in an increase in the soluble solid concentration of the potato sample. Pulsed electric field (PEF) pretreatment prior to frying had no significant effect on the measured moisture content, thermal conductivity or frying time compared to potatoes that did not receive a PEF pretreatment. However, a PEF pretreatment at 1.1 kV/cm and 56 kJ/kg reduced the temperature variation in the experimentally measured potato center by up to 30%. The proposed heat and moisture transfer model based on unsteady state heat conduction successfully predicted the experimental measurements, especially when the equation was solved using the enthalpy method.

Modeling temperature profile in mass concrete at early ages of cement hydration

Thermally induced cracks due to temperature gradient in mass concrete have adverse effects on its durability and service life. Heat released during the hydration of Portland cement in early age mass concrete can be quite excessive depending on the ambient temperature, cement content of the concrete mix and the size. Finite difference model using Crank Nicholson implicit method was developed based on the two dimensional unsteady state heat conduction. Optimized MATLAB based software was developed for simulation and data visualization. A mass concrete block cast with standard mix ratio and water cement ratio was used to verify the efficacy of the model. Type-K thermocouple and digital thermometer were used to monitor the temperature at time intervals. The temperature profile showed a hotter core and cooler surface except for the initial placement temperature, which exhibited a uniform temperature for all thermocouple locations. Peak temperature values were recorded within the first day of concrete placement. The model successfully predicted the temperature profile of the mass concrete at early ages of cement hydration. With the knowledge of the ambient temperature and the configuration of the mass concrete, the model can reliably predict the temperature profile from which potential for thermal cracks occurrence can be determined to enable suitable proactive preventive and control measures.

Analytical Solution vs. Numerical Solution of Heat Equation Flow Through Rod of Length 8 Units in One Dimension

The paper presented the basic treatment of the solution of heat equation in one dimension. Heat is a form of energy in transaction and it flows from one system to another if there is a temperature difference between the systems. Heat flow is the main concern of sciences which seeks to predict the energy transfer which may take place between material bodies as result of temperature difference. Thus, there are three modes of heat transfer, i.e., conduction, radiation and convection. Conduction can be steady state heat conduction, or unsteady state heat conduction. If the system is in steady state, temperature doesn’t vary with time, but if the system in unsteady state temperature may varies with time. However, if the temperature of material is changing with time or if there are heat sources or sinks within the material the situation is more complex. So, rather than to escape all problem, we are targeted to solve one problem of heat equation in one dimension. The treatment was from both the analytical and the numerical view point, so that the reader is afforded the insight that is gained from analytical solution as well as the numerical solution that must often be used in practice. Analytical we used the techniques of separation of variables. It is worthwhile to mention here that, analytical solution is not always possible to obtain; indeed, in many instants they are very cumber some and difficult to use. In that case a numerical technique is more appropriate. Among numerical techniques finite difference schema is used. In both approach we found a solution which agrees up to one decimal place.

Mathematical modeling of electrothermal processes using the example of high-frequency welding of a batch of symmetric polymer workpieces

The article substantiates the necessity and practical relevance of the mathematical modeling of processes of the high-frequency electrothermy of polymers. A mathematical model of high-frequency welding of polymeric and composite products has been developed. A mathematical model of high-frequency welding of a batch of polymeric and composite products has been developed, which is a system of differential equations of unsteady-state heat conduction with internal heat sources, which allows studying the heat distribution in the entire volume of the processed product of any geometric shape. The paper presents the testing results for the developed mathematical model at high-frequency welding of a batch of polymer parts. This made it possible to identify the effect of a non-insulated electrode heating up during sequential processing of a batch of parts on the position of the coordinate of the maximum heating point and to develop a technique for its search and maintenance of the required position. This, in turn, is necessary for organizing control of the high-frequency processing operation.

Effect of Pulsation and Acceleration of Liquid Metal Turbulent Flow Through a Horizontal Channel by Large Eddy Simulation

Abstract Pulsation and acceleration of liquid metal turbulent flow through a horizontal channel has been numerically studied using a large eddy simulation (LES) technique. The effect of inlet pulsation and acceleration on flow and heat transfer characteristics of low Prandtl number liquid metal flow have been investigated and reported here. Results have been presented for different Reynolds numbers, different amplitudes, and frequency with constant bottom wall thickness. The flow field is modeled as unsteady-state two-dimensional incompressible turbulent-forced convection flow. Turbulence is modeled using a LES technique. Two-dimensional unsteady-state heat conduction equation is solved to know the temperature distribution in the solid region. Finite difference method solver is developed for solving the governing equations using sixth-order accuracy of compact schemes. The average Nusselt number shows cyclic variation with respect to time in pulsation flows. The enhancement of heat transfer with pulsation at amplitude 0.4 and frequency 100 Hz is 6.51%. The rate of heat transfer increases in pulsation flow compared to quasi-steady flow. The inlet acceleration shows a significant effect on flow characteristics. The present results are compared with direct numerical simulation (DNS) results available in the literature and matching well with DNS data.

Analytical model for prediction of temperature distribution in early age mass concrete

Thermally induced cracks have far-reaching implications on the durability of concrete structures. When cement mixes with water, the reaction is exothermic implying the release of heat. In the case of mass concrete structures, quite a substantial increase in internal temperature may be experienced depending on the ambient temperature and cement content in the mix. The objective of the paper is to develop a mathematical model to predict the time dependent temperature profile in early age mass concrete. Mass concrete block was used to verify the model. Type-K thermocouples placed at various positions and digital thermometer was used to monitor the temperature distribution within the mass concrete block at intervals. The highest temperature values occurred within the core of the mass concrete after one day of concrete placement. Analytical model was developed by applying method of separation of variables and orthogonality relation to two dimensional unsteady state heat conduction equations. The model equation was evaluated and using MATLAB based computer programe. The model successfully predicted the temperature variation within the mass concrete with time. It is therefore suitable for use in the assessment of thermal cracks potential in mass concrete structures.Â

Study of Heat Conduction inside Rolling Calender Nip for Different Roll Temperatures

Calendering is a smoothening process at the final stage in textile industry, where fabric passes through the nips formed by two or more rolls in contact. The material used for making of rolls varies for different types of calenders. Depending on the quality of fabric required these rolls are hard or soft and can be heated to certain temperature, using induction process, hot water passage or heated oil passage inside the rolls. In rolling calendering process, fabric is pressed between two or more rolls, where nip is formed by the combination of alternate hard and soft rolls. In this paper effect of roll temperature and bonding time on fabric temperature has been discussed using single nip rolling calender, when fabric is inside the calender nip formed by hard and soft rolls having different temperatures using one dimensional unsteady state heat conduction equation.

A mechanistic model for swelling kinetics of waxy maize starch suspension

The evolution of 8% suspension of waxy maize starch granule size distribution when subjected to heating to 65, 70, 75, 80, 85 and 90 °C at a heating rate of 15 °C/min were measured. Granule swelling was more pronounced at higher temperatures with an increase in the average size from 13 μm to 25–28 μm, eventually approaching equilibrium. Cryo SEM images revealed porous structure of swollen granule. The proposed model for starch swelling accounts for entropy of mixing, enthalpy of water-starch interaction and elastic restoring force. The second virial coefficient of waxy maize in aqueous medium was characterized by static light scattering. Chemical potential profile as well as the temperature profile within the granule were predicted from the solution of unsteady state diffusion and heat conduction respectively. The granule size was then inferred from the chemical potential profile. The proposed model predicts the evolution of average granule size and granule size distribution accurately.

Exergy analysis for unsteady-state heat conduction

Exergy analysis has been used in various fields to quantify the efficiency of various energy transfer processes. So far, most exergy analyses have been performed by assuming steady-state conditions both for a system of focus and its environment. However, steady-state analyses may idealize a problem too much and thereby limit what we can know about the system, since the assumed system is only a snapshot of a certain condition or moment. In particular, a steady-state exergy analysis is not suitable for analyzing problems where the system has a large heat capacity and problems with varying boundary conditions. To understand the exergy transfer and consumption processes in a system whose transient characteristics are critical, an unsteady-state exergy analysis must be conducted.This study presents a methodology for unsteady-state exergy analysis that can be used to study heat conduction problems. Governing equations of entropy and exergy to be parallel to the governing equation of energy are derived in their partial differential forms, and their numerical methods are presented in detail. Based on the proposed methodology, an unsteady-state exergy analysis of a heat conduction problem for a concrete wall is carried out to demonstrate its transient exergy behavior.

A novel heat transfer model of a phase change material using in solar power plant

Concentrating solar power (CSP) plant coupling with thermal energy storage is a popular technology in the recent years. The storage technology is the wave of the future in solar utilization field. A lot of scholars focus on studying of heat transfer performance by the experimental or simulating method. In this paper, a novel heat transfer model of three-dimensional unsteady state heat conduction in cylindrical coordinate has been proposed based on the theoretical analysis. The most difficulty problem during the solution is the materials, which is molten salt based on some real cases, that is a phase change material (PCM) with the melting point of 494 K. The computer code of this model is written by C language in MATLAB software. The results show that the charge time is 25200 s (about 7 h). Also, the characteristic of melting progress of the PCM has been obtained based on this program. The aim of this paper is to develop a new heat transfer model and provide a good reference to utilize solar energy.

Additional boundary conditions in unsteady-state heat conduction problems

Using some additional sought function and boundary conditions, a precise analytical solution of the heat conduction problem for an infinite plate was obtained using the integral heat balance method with symmetric first-order boundary conditions. The additional sought function represents the variation of temperature with time at the center of a plate and, due to an infinite heat propagation velocity described with a parabolic heat conduction equation, changes immediately after application of a first-order boundary condition. Hence, the range of its time and temperature variation completely incorporates the ranges of unsteadystate process times and temperature changes. The additional boundary conditions are such that their fulfilment is equivalent the fulfilment of a differential equation at boundary points. It has been shown that the fulfilment of an equation at boundary points leads to its fulfilment inside the region. The consideration of an additional sought function in the integral heat balance method provide a possibility to confine the solution of an equation in partial derivatives to the integration of an ordinary differential equation, so this method can be applied to the solution of equations, which do not admit the separation of variables (nonlinear, with variable physical properties of a medium, etc.).

A study on thermal calculation method for a plastic greenhouse with solar energy storage and heating

Plastic greenhouse has been widely used in agriculture and horticulture due to its prolonging period for crops growth. In this work, an approach that stores solar energy in the daytime and provides heat by earth-tube at night was proposed, and then applied to a plastic greenhouse to elevate the inside air temperature. A one-dimensional dynamic model was established to assist the design of the solar energy storage and heating system and to evaluate the system performance. Using the model developed in Matlab, the date-hour change patterns of characteristic temperatures in the plastic greenhouse were obtained, through calculating the heat gains of various surfaces and heat storage by hour from solar radiation and solving the unsteady-state heat conduction equation in the structure components of the greenhouse. The calculated results show good agreement with the measured data, indicating that the method is valid and can be applied to the design of solar energy storage and heating system as well as the thermal performance analysis of greenhouses.

Conjugate heat transfer analysis of turbulent forced convection of moving plate in a channel flow

The effect of wall thickness and heat conduction in the wall on turbulent forced convection flow characteristics in a vertical channel has been numerically studied. The effect of heated plate velocity related to fluid velocity in a vertical channel including heat conduction in the plate has been investigated numerically. The flow field is modeled as two dimensional unsteady state incompressible turbulent forced convection flow in a channel. Energy equation in the solid region is considered as two dimensional unsteady state heat conduction. Turbulence is modeled with a low Reynolds number k−ϵ model of Launder–Sharma. Conjugate heat transfer solver has been developed using high accuracy compact finite difference schemes. The effects of Reynolds number, wall thickness and thermal conductivity of the solid on the flow and heat transfer characteristics are reported. Results show that the interface temperature decreases and average convective heat transfer coefficient increases with increase in Reynolds number. The temperature decreases in the solid and fluid regions with increase in thickness of the solid region. The wall thickness and heat conduction in the wall significantly alter the convective heat transfer characteristics in channel flows. The heated plate velocity shows a significant effect on heat transfer characteristics in the solid and fluid regions. The average convective heat transfer coefficient increases with increase in speed of the moving plate. The present results are matching well with the experimental results available in the literature.

Solving Unsteady Boundary Value Problems Using Discrete-Analytic Method for Non-Iterative Simulation of Temperature Processes in Time

Method for solving a boundary value problem of inhomogeneous unsteady-state heat conduction transfer is considered. This physical process can be described by a boundary value problem for a partial differential equation of the 2nd order. Discrete-analytical method, which turns out the mathematical formulation of the initial problem to be normal system of differential equations, was used. There is the non-iterative solution of such system, which is the set of analytic functions. The theory of matrix functions, particularly the properties of matrix exponential, was applied to get the solution. This approach allows us to model the unsteady-state heat conduction processes with unstationary boundary conditions of different types, defined as time-dependent functions. Such modeling describes the real physical processes in structural materials more accurately.