Equation For Temperature Distribution Research Articles

Modelling for Temperature Distribution Calculation using Surface Scattering of Free Electrons in Finite Metal Solid

This study generalizes the temperature distribution equation for finite metal solid, unifying previous separate models for thick and thin plates. Effects of surface scattering of free electrons on heat conduction is taken account. As plate thickness decreases, these scattering events increase, leading to elevated temperatures due to reduction in mean free paths. We propose a prediction model for temperature distribution incorporating these effects onto the existing Rosenthal solution. It exhibits excellent agreement with finite element analysis results across all thicknesses. Furthermore, from an engineering standpoint, two examples of how heat concentration occurs in material with geometry that promotes multiple surface scattering are presented.

HEAT TRANSFER ANALYSIS ON BASE OF HEAT SINK WITH UNIFORM HEAT GENERATION

The present work describes the important results of problem of combined conduction, convection and radiation from a rectangular heat sink base subjected to uniform internal heat generation in its base. The governing equations for temperature distribution within the base are obtained by a relevant energy balance between rate of heat generation, heat conducted, convected and radiated. The derived non-linear partial differential equations are converted into algebraic form using finite difference method with 2nd order accuracy. The obtained algebraic equations have been solved simultaneously by Gauss-siedel iterative method with stringent convergence criteria and a C code has been written for above purpose. The computational domain has been discretized uniformly in all directions. A grid convergence test has been carried out to optimize the size of element. A complete parametric study has been carried out to study the effect of material of heat sink, regime of convection and role of radiation on local temperature distribution and maximum heat sink base temperature. It has been observed that material with high conductivity decreases the local temperature gradient and thus the maximum temperature. The regime of convection has a great influence on maximum temperature of heat sink with 15K of drop in Tmax. for a change in flow regime from free convection to forced convection. A special study on exclusive role of convection on maximum temperature reveals that one should not ignore the role forced convection in heat dissipation. Key Words: Heat generation, Multimode heat transfer, Radiation

Stability analysis of nonlinear convective boundary layer flow on a shrinking surface in the presence of viscous dissipation

This study investigates analytically the stability analysis of a nonlinear convective boundary layer (BL) flow of the nanofluids GO-EG and GO-W on a shrinking surface in the presence of a magnetic field and viscous dissipation. We provide a second-order nonlinear ordinary differential equation (NODE) for temperature distribution (TD) and a third-order NODE for velocity profile (VP) coupling based on the thermophysical characteristics of the base fluid and nanofluid as well as similarity transformations (STs) in the fundamental governing equations of momentum and energy. The analytical method HAM is used to answer the equations that have been gathered. In many different industries, such as manufacturing, automotive, microelectronics, and defense, cooling of various types of equipment and devices has very important challenges. To fulfill the demands of the engineering and industrial industries, it is hoped that this issue would significantly improve the heat transformation ratio.

Knee synovial fluid flow and heat transfer, a power law model

For the purpose of understanding, the governing system of partial differential equations for synovial fluid flow velocity and temperature distribution in the knee joint has been successfully solved for the first time. Therefore, such an article is shedding light on the convective diffusion of the viscous flow along the articular surfaces of the joints through the introduction of power-law fluids with different features of permeability, and stagnation point flow along a magnetic field. Henceforth, the frictional energy causes the knee joint’s temperature to increase. By way of filtration, heated synovial fluid reaches the articular cartilage and provides heat to the bone and cartilage. The lubricant in the joint cavity is properly mixed with this cooled fluid. A rectangular region flow and diffusion model is used to define the issue, thermal diffusion and flow inside the intra-articular gap, as well as flow and thermal diffusion within the porous matrix covering the approaching bones at the joint. Using the similarity solution approach, the linked mixed boundary value problem is addressed. The fluid has been shown to resist moving into or out of the cartilage in certain sick and/or aging synovial joints, causing the temperature to increase. By changing the values of the parameters from their usual levels, it is observed that the temperature did increase in aged and sick joints which impact cartilage and/or synovial fluid degradation.

Modeling of the process of solar drying of grapes in indirect type installations with natural air convection

The paper models the process of natural convection of an indirect type dryer mathematically for the accumulation and absorption of heat which uses water. The data from the experiment conducted by the authors of the work is used as the initial data. The Reynolds equations and the temperature distribution equations are used for the mathematical model considering the Boussinesq hypothesis. The SIMPLE control volume method is used in the paper for the difference approximation of the initial equations. The temperature and velocity fields in the drying chamber are determined. It is revealed that the maximum speed in this mode is reached in the upper part of the dryer and it will be equal to 0.02-0.03 m/s.

A Hybrid Model for Billet Tapping Temperature Prediction and Optimization in Reheating Furnace

To predict and optimize the billet heating process in reheating furnace for rolling mills, this paper proposes a hybrid model that combines data-driven model with traditional mechanism knowledge, abbreviated as HMDM. By examining the heat conduction mechanism, a billet temperature distribution equation is established. Then, the billet temperature distribution in each heating zone is calculated and spliced with the corresponding process parameters. The Stacked-AutoEncoder is utilized to extract the features of process parameters, and the Long Short Term Memory model is employed to predict the temperature. Finally, using the previous predictions, the parameters of the subsequent heating stage are optimized and adjusted during the heating process. The experimental results on the real steel plant verify the effectiveness of HMDM. For example, the temperature prediction error has been reduced to less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$4^{\circ }$</tex-math></inline-formula> C, and the number of billets with abnormal tapping temperature has been decreased by <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$42.9\%$</tex-math></inline-formula> .

Derivation of the unifying model and its discussion on the effective thermal conductivity of isotropic, porous and composite media

A detailed and primordial derivative process of the classical Maxwell problem applied to thermal conductivity was presented in this study, aiming to offer a foundation for further research on various correlations predicting the effective thermal conductivity (ETC) of porous-composite media, especially for the unifying equation with five fundamental structural models. This process of theoretical analysis is, in essence, solving the Laplace's Equation for temperature distribution in a spherical coordinate under certain initial and boundary as well as other assumed conditions. The particular solution processes also overcome various degrees of shortcomings existed in previous literatures in this field, e.g. problems of nondegenerate, insufficient boundary conditions and many tiny mistakes (typos or misprints). We hope this commentary letter can maintain the rigor of academic research in a rightful and suitable form, and prevent any potential confusion for future readers in this field.

Using the decomposition method to solve the fractional order temperature distribution equation: A new approach

Due to its importance in science, finding both exact and approximate solutions to fractional partial differential equations with boundary conditions is important for the research community. The natural decomposition method (NDM), which is based on the natural transformation method (NTM) and the Adomian decomposition method, is modified in this study to produce exact and approximate solutions for boundary value problems (BVPs) of partial differential equations (PDEs) with fractional coefficients. In addition, we present an exact solution to the temperature distribution in a slab constructed of materials with variable thermal conductivity's combined convection–radiation lumped system. We present these findings as numerical tables and graphs that show the convergence and stability rates. The study demonstrates that this approach is effective since it is simple to apply and produces reliable findings. We are the first to use this approach for such applications, as far as we are aware. Additionally, this method is applicable to a sizable class of BVPs for ordinary differential equations (ODEs) and PDEs.

Homotopy analysis method with application to thin-film flow of couple stress fluid through a vertical cylinder

Abstract This work solves the problem of thin-film withdrawal and drainage of a steady incompressible couple stress fluid on the outer surface of a vertical cylinder. The governing equations for velocity and temperature distributions are subjected to the boundary conditions and solved with the help of homotopy analysis method. The obtained expressions for flow profile, temperature profile, average velocity, volume flow rate, and shear stress confirmed that the thin-film flow of couple stress fluid highly depends on involved parameters say Stokes number St , vorticity parameter λ, couple stress parameter η, and Brinkman number Br presented in the graphical description as well.

FEA of friction stir welding of AA1100 and parametric optimization using genetic algorithm

3D uncoupled thermo-mechanical analysis of friction stir welding (FSW) of aluminum alloy AA1100 was carried out using FEM. The heat generated in the FSW process is applied via DFLUX subroutine of ABAQUS. Simulations with different process parameters such as welding speed and rotational speed were performed for the estimation of temperature generated during the FSW process. Thermal analysis’s output is applied as thermal loads during mechanical analysis. Subsequently, residual stress developed during the FSW process was found out. Residual stresses were measured along the selected path, which included node to node stress variations in longitudinal directions. Maximum temperature appears to occur at the center of the weld pool and decreases away from the weld line towards the base metal. The numerical and experimental results were successfully validated with error in the range of 0.5 - 6%. Then, a semi-analytical equation for temperature distribution considering moving point heat source is proposed to carry out single objective unconstrained optimization of process parameters (welding speed and rotational speed). Finally, optimization was carried out to estimate the maximum temperature generated during the FSW process using a genetic algorithm in MATLAB.

Lattice Boltzmann method for simulation of solid–liquid conjugate boiling heat transfer surface with mixed wettability structures

Based on the one-component multiphase lattice Boltzmann method, a novel solid–liquid conjugate boiling heat transfer pseudo-potential lattice Boltzmann (LB) model is tentatively proposed in this paper. By respectively introducing the physical property parameters of solids and liquids into the relaxation time τT of the temperature distribution equation, different energy transfer rates in solid, liquid, and vapor regions can be successfully predicted. After verifying the accuracy, stability, and reasonability of this model, the bubble detaching behavior and boiling heat transfer performance on the rectangular cavity structure are analyzed through setting different contact angles of the cavity surface and plane heating surface. The results show that the hydrophobic cavity surface can initialize bubble nucleation earlier and obviously increase the bubble detaching frequency because of its gas-bounding character, while the hydrophilic plane heating surface can restrict the expansion of bubbles and delay the appearance of film boiling. Moreover, for uniform wettability surfaces, the bubble detaching period varies in the quadratic equation with the surface contact angle due to the interaction of surface tension and buoyancy, and there is a minimum detaching period. While for the mixed wettability surfaces, the bubble detaching period also has a minimum value with the decrease in the contact angle the cavity surface, but the average bubble detaching diameter basically does not change with the cavity surface contact angle; moreover, the cavity surface contact angle corresponding to the minimum detaching period also increases with the increase in the plane heating surface contact angle. In addition, for the boiling heat transfer surface with cavity structure, the maximum heat flux and temperature gradient occur on the cavity surface, and the local heat flux of the hydrophobic cavity surface is higher than that of the hydrophilic cavity surface. This work will provide useful help for the further development of the conjugate boiling heat transfer LB model and clarify the mechanism of enhanced boiling heat transfer on a mixed wettability surface.

Local Thermal Nonequilibrium (LTNE) Modeling of a Partially Porous Channel With Spatial Variation in Biot Number

Abstract The local thermal nonequilibrium (LTNE) model has been used widely for analyzing heat transfer during internal flow through porous media, including when a channel is only partially filled with a porous medium. In such problems, the Biot number describes the rate of convective heat transfer between solid and fluid phases. While uniform Biot number models are commonly available, recent advances in functionally graded materials necessitate the analysis of spatially varying Biot number in such geometries. This paper presents LTNE-based heat transfer analysis for fully developed flow in a channel partially filled with porous medium and with spatially varying Biot number to describe solid-fluid interactions in the porous medium. Fully uncoupled ordinary differential equations for solid and fluid temperature distributions are derived under three different boundary condition models. Solid and fluid temperature fields are presented for a variety of Biot number distributions, including quadratically and periodically varying functions. An explanation of the nature of temperature distribution predictions for such problems is provided. For special cases, the results presented here are shown to reduce to past work on constant Biot number. This work improves the theoretical understanding of porous media heat transfer and facilitates the use of such theoretical models for functionally graded materials.

Theoretical Research on Flow and Heat Transfer Characteristics of Hydrostatic Oil Film in Flat Microfluidic Boundary Layer

The hydrostatic bearing is the core component of ultra-precision computer numerical control (CNC) machine tools. Because the temperature rise in the oil film of hydrostatic bearings seriously affects the working accuracy of the bearings, it is important to study the flow and heat transfer characteristics of the oil film. Based on the physical model of an incompressible viscous fluid flowing in a flat microfluidic boundary layer, velocity, temperature and heat flux distribution equations of oil film are constructed by theories of heat transfer and hydrodynamics. Then, the effects of several parameters on velocity distribution, temperature distribution and heat flux distribution are analyzed, such as the upper plate velocity, the channel length, and so on. The results show that the dimensionless velocity of the oil film decreases with the increase in the upper plate velocity and the channel length. The oil film temperature distribution can be divided into three zones: the increasing zone, stabilizing zone and decreasing zone. The heat flux decreases linearly with the increase in the plate thickness, and increases linearly with the increase in the temperature difference.

Development and validation of evaporation model for a multi-component fuel considering volume-average internal mass and enthalpy

We developed a droplet evaporation model for a multi-component fuel, considering the nonuniform distributions of both mass fraction and temperature inside multi-component fuel droplet. The distribution equations satisfy the conservation of droplet mass and enthalpy. We proposed mass fraction and temperature distribution equations inside a droplet to accurately determine variations between the numerical assumptions of a liquid phase and a gas phase. We found that an obvious gradient appears in the mass fraction and temperature distribution profiles at various ambient temperatures. To verify the accuracy of the developed model, we further compared the predicted droplet lifetimes with the experimental results at various ambient temperatures. The consideration of the pre-evaporation process led to more accurate droplet lifetime prediction using the present model. We also investigated the differences between the present model and the uniform distribution model. The difference in the predicted droplet lifetime becomes more than 10% at practical spray combustion conditions. Thus, internal distribution of mass fraction ad temperature must be considered in practical spray combustion simulations.

Computational Model Development for Hybrid Tilting Pad Journal Bearings Lubricated with Supercritical Carbon Dioxide

Fluid film bearings lubricated with supercritical carbon dioxide (sCO2) eliminate the infrastructural requirement for oil lubricant supply and sealing in turbomachinery for sCO2 power systems. However, sCO2’s thermohydrodynamic properties, which depend on pressure and temperature, pose a challenge, particularly with computational model development for such bearings. This study develops a computational model for analyzing sCO2-lubricated tilting pad journal bearings (TPJBs) with external pressurization. Treating sCO2 as a real gas, the Reynolds equation for compressible turbulent flows solves the pressure distribution using the finite element method, and the Newton−Raphson method determines the static equilibrium position by simultaneously calculating forces, moments, flow rates of externally pressurized sCO2, and pressure drop due to flow inertia. The finite difference method solves the energy equation for temperature distribution. The density and viscosity of sCO2 are converged using the successive substitution method. The obtained predictions agree with the previous and authors’ computational fluid dynamics predictions, thus validating the developed model. Hybrid lubrication increases the minimum film thickness and stiffness up to 80% and 65%, respectively, and decreases the eccentricity ratio by up to 65% compared to those of pure hydrodynamic TPJB, indicating significant improvement in the load capacity. The bearing performance is further improved with increasing sCO2 supply pressure.

Design of cold-formed steel wall studs subject to non-uniform elevated temperature distributions

The current design rules for cold-formed steel studs used in wall systems subject to non-uniform elevated temperature distributions are complicated with iterations and different elevated temperature section properties. This paper first investigates the accuracy of the current effective width method (EWM) and direct strength method (DSM) based design equations, and then simplifies the calculation process without compromising the accuracy in calculating the load ratios (elevated to ambient temperature capacity ratio). It also improves the accuracy of the current DSM based design equations. It then investigates the possibilities of using uniform temperature based DSM design equations for non-uniform temperature distributions and identifies the possibilities of using hot flange mechanical properties and uniform temperature based DSM design equations. Hence, it proposes a simplified uniform temperature based DSM design method with hot flange mechanical properties and a correction factor based on hot and cold flange temperatures, yield strength, length and depth of studs.

Boundary Layer Flow and Heat Transfer Analysis by Using the Correlation Coefficient and Regression Model Over a Stretching Bullet-Shaped Object

The boundary layer theory is important when fluid flows over a solid surface that is moving or stationary. In presence of the boundary layer, the effective shape of the body may change leading to changes in pressure distribution, as a result, the overall lift and drag forces change. Therefore, the Boundary layer theory helps in designing aerofoil’s, to compute the lift and drag forces for the aerospace and automobile designers, to control the heat transfer rate from the device, etc. So, the present problem will help design the various types of bullet-shaped objects in the field of automobile engineering. Therefore, the current problem has focused on the two-dimensional axisymmetric BL flow over a stretching bullet-shaped object for the effect of magnetic field strength (M), linear stretching parameter (M), and surface thickness parameter (s). Therefore, the main goal of this work is to determine the relation by applying the correlation coefficient among the physical parameters and velocity field, temperature field, shear stress (τw), Nusselt number (Nux). Hence, the novelty of the current paper is to develop the relationship among the dependent and independent parameters by the correlation coefficient and also developed the numerical method to solve these highly nonlinear equations. The numerical results are discussed for the three different values of the stretching ratio parameter and two values of the surface thickness parameter. The velocity and temperature distribution equations are compressed into a system of ODEs with similarity transformations. These ODEs are then solved using a spectral quasi-linearization method (SQLM) by applying Taylor series expansions that can be used to linearize the non-linear terms in the equations. These resulting linearized systems of equations are determined by the spectral collocation method. The convergence of the numerical solutions was performed by using the residual error of the PDEs. The error analysis is established for the validity of the present model. This error norm is applied to establish the validity and convergence of the numerical solution. The outcome of the mentioned dimensionless parameters over the fluid velocity field, temperature field, skin friction coefficient (Cf), and Nusselt number (Nux) are displayed graphically. It is observed that the parameters M and M are positively correlated with fluid velocity distribution within the BL but the surface thickness parameter(s) are negatively correlated. The rate of temperature increases for the parameter M and Pr but decreases for M and s. Therefore, the boundary layer thickness reduces for increasing the values of M and M but increases for increasing the values of s. The velocity of the fluid is about 80% higher in the case of a thinner surface (s = 0.2) than the thicker surface (s = 2.0) and the heat transfer rate is also igher in the case of a thinner surface comparatively thicker surface. The innovation of this present problem lies in the unification of more physical parameters into the governing equations and an attempt to give a thorough analysis of how the flow properties are affected by these parameters.

A computational technique for thermal analysis in coaxial cylinder of one-dimensional flow of fractional Oldroyd-B nanofluid

It is a focused fact in thermal analysis that if a particular alloy changes from liquid to solid from higher temperature to lower temperature, then reactions between fluid particles occur at equal rates of heat transfer. In this manuscript, mathematical synchronisation in coaxial cylinders of one-dimensional flow of fractional Oldroyd-B nanofluid is developed for heat transfer in the absence of pressure gradient. The modern fractional approach-based non-singular and local kernel is particularised on the boundary layered cylindrical governing equations for temperature distribution and velocity profile. To get rid of the complexity of computation from governing equations based on cylindrical domain, the powerful techniques of integral transforms are employed. Moreover, the fractionalised Oldroyd-B nanofluid, containing a mixture of small nanoparticles, namely, (molybdenum disuphide), (copper), (aluminium oxide), (silver) with ethylene glycol as a base fluid, is also discussed for velocity profile and temperature distribution. In the light of thermal relaxation time and Prandtl number, our results suggest that smaller values of thermal relaxation time generate oscillation between temperature distribution and Prandtl number produced thermal fluctuations with an intrinsic structure.

A Thermal Effect Model for the Impact of Vertical Groundwater Migration on Temperature Distribution of Layered Rock Mass and Its Application

On the basis of the one-dimensional heat conduction–convection equation, a thermal effect model for vertical groundwater migration in the stratified rock mass was established, the equations for temperature distribution in layered strata were deduced, and the impacts of the vertical seepage velocity of groundwater and the thermal conductivity of surrounding rocks on the temperature field distribution in layered strata were analyzed. The proposed model was employed to identify the thermal convection and conduction regions at two temperature-measuring boreholes in coal mines, and the vertical migration velocity of groundwater was obtained through reverse calculation. The results show that the vertical temperature distribution of the layered rock mass is subject to the migration of the geothermal water; the temperature curve of the layered formation is convex when the geothermal water travels upward, but concave when the water moves downward. The temperature distribution in the stratified rock mass is also subject to the thermal conductivity of the rock mass; greater thermal conductivity of the rock mass leads to a larger temperature difference among regions of the rock mass, while weaker thermal conductivity results in a smaller temperature difference. A greater velocity of the vertical migration of geothermal water within the surrounding rock leads to a larger curvature of the temperature curve. The model was applied to a study case, which showed that the model could appropriately describe the variation pattern of the ground temperature in the stratified rock mass, and a comparison between the modeling result and the measured ground temperature distribution revealed a high goodness of fit of the model with the actual situation.

Simulation of Diffusion Processes in Chemical and Thermal Processing of Machine Parts

To solve a number of technological issues, it is advisable to use mathematical modeling, which will allow us to obtain the dependences of the influence of the technological parameters of chemical and thermal treatment processes on forming the depth of the diffusion layers of steels and alloys. The paper presents mathematical modeling of diffusion processes based on the existing chemical and thermal treatment of steel parts. Mathematical modeling is considered on the example of 38Cr2MoAl steel after gas nitriding. The gas nitriding technology was carried out at different temperatures for a duration of 20, 50, and 80 h in the SSHAM-12.12/7 electric furnace. When modeling the diffusion processes of surface hardening of parts in general, providing a specifically given distribution of nitrogen concentration over the diffusion layer’s depth from the product’s surface was solved. The model of the diffusion stage is used under the following assumptions: The diffusion coefficient of the saturating element primarily depends on temperature changes; the metal surface is instantly saturated to equilibrium concentrations with the saturating atmosphere; the surface layer and the entire product are heated unevenly, that is, the product temperature is a function of time and coordinates. Having satisfied the limit, initial, and boundary conditions, the temperature distribution equations over the diffusion layer’s depth were obtained. The final determination of the temperature was solved by an iterative method. Mathematical modeling allowed us to get functional dependencies for calculating the temperature distribution over the depth of the layer and studying the influence of various factors on the body’s temperature state of the body.