Equation For Temperature Distribution Research Articles

Theoretical Calculation and Numerical Simulation Based on Radial Fin of Triangular Profile

In most heat exchange equipment, in order to increase heat dissipation between solid surface and external fluid, the heat transfer surface is often made into finned form. This is because the method of increasing the surface area of the fin wall is used to reduce the heat resistance of the convective heat transfer and enhance the heat transfer. There are various types of fin wall, including straight fin, radial fin and needle fin. The radial fin of triangular profile belongs to the radial fin with variable thickness profile. Based on the theoretical calculation and numerical simulation of the excess temperature distribution and fin efficiency of the radial fin with triangular profile, the excess temperature distribution equation and the calculation formula of fin efficiency on the wall of the radial fin with triangular profile are derived, and the curve of excess temperature distribution and fin efficiency is fitted and output by MATLAB, the theoretical basis for the design and application of this kind of radial fin is provided.

A New Mixed Nonpolynomial Spline Method for the Numerical Solutions of Time Fractional Bioheat Equation

In this paper, a numerical approximation for a time fractional one-dimensional bioheat equation (transfer paradigm) of temperature distribution in tissues is introduced. It deals with the Caputo fractional derivative with order for time fractional derivative and new mixed nonpolynomial spline for second order of space derivative. We also analyzed the convergence and stability by employing Von Neumann method for the present scheme.

Mathematical modeling of heat distribution on carbon fiber Poly(ether-ether-ketone) (PEEK) composite during laser ablation

Since large amounts of energy are transferred precisely to the material in a very short time, laser and material parameters strongly affect the laser process quality. Mathematical modeling of heat distribution during cavity formation on carbon fiber reinforced PEEK composite material was performed. The temperature distribution equation has been obtained by the Fourier method.At the first stage, 1 J energy laser beam was sent onto the material and the cavity on the composite material was obtained. The constants in the temperature distribution equation obtained by making measurements over the cavity were found. Then the cavities were created with 2, 3 and 4 J laser beams to prove its reliability of the model. The results obtained from the measurement on the cavities and calculated from the temperature distribution equation were compared.Since the unidirectional carbon fibers were used, the obtained cavities have an elliptical shape. Verification experiments were carried out using two different heat conduction constants for in direction along fibers and perpendicular to carbon fibers. Experimental results and the mathematical model are in good agreement.

Modelling of the Temperature Changes Induced by Transcutaneous Spinal Direct Current Stimulation (tsDCS)

Transcutaneous spinal direct current stimulation is a neuromodulation technique recently exploited to regulate the activity and enhance plasticity of spinal neural structure. Despite the few side-effects reported by clinical studies, the assessment of potential localized temperature increases in tissues is still not solved. In this study, the temperature increase induced by a 3 mA stimulation in the tissue target (i.e., the spinal cord), in its surrounding tissues and in other tissues possibly more vulnerable to temperature increase were assessed through a computational approach. That solves Laplace equation and BioHeat equation for electric field and temperature distribution, respectively, on six whole body high resolution anatomical models, including three models of pregnant women at different gestational ages. Results show that temperature increase distribution in those targets is guided by a complex interaction between different mechanism in which electric stimulation plays a secondary role, in particular when blood perfusion is active. The very low heating assessed (below 1.5 m °C) is not consistent with the hypothesis that the induced temperature increase would critically activate metabolic changes in the target tissues here considered or would contribute to the few side effect that the applications of spinal direct current stimulation protocols have shown.

A simplified analytical model to predict heating performance of single U-tube ground heat exchangers

In underground U-tube heat exchangers (boreholes) it is important to predict its heating performance to design and select the proper parameters such as length, diameter, material etc. to have an optimized borehole from the point of view of heat capacity and economical aspects. For this reason, having trusty equations is vital to foresee borehole heating performance and applying it in design issues. In this study a single vertical U-tube borehole with constant wall temperature is considered and analytical equations for temperature distribution in the surrounding ground around the borehole is evaluated based on one and two dimensional heat conduction respectively. The analytical equation is compared to experimental data for a borehole with 50 m depth in which warm water of 40 C is pumped into it a time period of 120 hours and the heat transfer rate per unit length is recorded. The comparison between analytical expression and experimental data shows a good agreement between them. Also the borehole entropy generation number is studied and the optimized parameters are evaluated to minimize it. It is concluded that for the considered borehole, entropy generation number is decreased by increasing its length and by decreasing the borehole radius and pipe outer radius.

Analytical model for estimating bending angle in laser bending of 304 stainless steel/Q235 carbon steel laminated plate

Compared with the single-component metal plate, the stainless steel-carbon steel laminated plate (SCLP) combines the stainless steel layer and the carbon steel layer with a special preparation technique. In order to estimate the bending angle of laminated plates accurately, it is of great significance to establish an analytical model. Based on the temperature gradient mechanism, the temperature distribution equation of the SCLP is established by the piecewise function. Then, the depth of the plastic zone is calculated by the recrystallization temperature according to the temperature distribution along the thickness direction of the SCLP. Moreover, by fitting the yield strength curves of stainless steel and carbon steel, the average compressive stress of the plastic zone is calculated through the integral method. By optimizing the calculation of the plastic depth and the average compressive stress in the plastic zone, the analytical model is established based on mechanical equilibrium equations. The experimental verification shows that the average error of bending angle using the proposed model is 9.95%, while Liu's model is 38.02%. The proposed model provides a calculation method for estimating the bending angle, which contributes to improving the accuracy of the analytical model in laser bending of SCLP.

Multiphysics simulation and experiment of a thrust bearing in scroll compressors

Abstract In this study, a quasi-dynamic lubrication model of a thrust bearing is developed to identify the behaviors of scroll compressors. The present model is comprehensive and accurate with fewer assumptions. It is composed of the Reynolds equation for a lubrication film, an energy equation for temperature distribution, and a stress–strain equation for deformation owing to fluid pressure, forces, moments, and thermal stress. For model validation, the film thickness of the thrust bearing was compared with the results of numerical simulation and experiment at two operating conditions. The numerical simulation is consistent with the experiment, showing an error within 5%.

Non-Linear Model for a Spiral Porous Fin Subjected to Darcy Flow

In this study, the temperature distribution equation for a spiral porous fin is presented. Based on Darcy’s model, a mathematical equation of the energy is derived and a suitable dimensionless form is outlined to highlight some characteristic parameters, namely, the spiral fin pitch, the porosity, and the modified Rayleigh number. The behavior of the solution is analyzed for two cases of interest, taking into account the temperature-dependent thermal conductivity of the fin encountered in a hostile environment. A Numerical method is applied to solve this non-linear problem. It is found that the thermal transfer is not affected by the change of the spiral fin pitch, whereas increasing the porosity or the parameter β* makes higher fin temperature and improve the fin efficiency.

Sine–cosine wavelets operational matrix method for fractional nonlinear differential equation

In this paper, we present a solution method for fractional nonlinear ordinary differential equations. We propose a method by utilizing the sine–cosine wavelets (SCWs) in conjunction with quasilinearization technique. The fractional nonlinear differential equations are transformed into a system of discrete fractional differential equations by quasilinearization technique. The operational matrices of fractional order integration for SCW are derived and utilized to transform the obtained discrete system into systems of algebraic equations and the solutions of algebraic systems lead to the solution of fractional nonlinear differential equations. Convergence analysis and procedure of implementation for the proposed method are also considered. To illustrate the reliability and accuracy of the method, we tested the method on fractional nonlinear Lane–Emden type equation and temperature distribution equation.

Mathematical modelling of laser ablation of random oriented short glass fiber reinforced Polyphenylene sulphide (PPS) polymer composite

Many sectors using high technology need to use materials with more than one property. Reinforced plastics (FRP) have been used frequently in the transportation and energy sectors because of its superior specific mechanical properties especially high strength-to-weight ratio. Lasers are often used in cutting, drilling and surface treatments since they have the ability to perform high energy transfer in various forms. Because of the quality degradation in the laser process, especially in the efficient processing of FRPs, high energy and other laser parameters compatible with these energy levels are required. Thermal damage is one of the most important parameters affecting the quality of FRPs.Mathematical Modelling of laser ablation of random oriented short glass fiber reinforced plastic were performed. Temperature distribution equation has been obtained by Fourier method. Finite difference method was applied to simulate temperature distribution and Heat affected Zone (HAZ) size on FRP surface. Temperatures at any distance from laser spot were obtained by solving one-dimensional heat conduction equation. The melting and vaporization temperatures of PPS were used as boundary conditions to find Fourier coefficients in the temperature distribution equation. In order to test the reliability of the proposed mathematical model, laser pulses of different energies were sent to the material. As a result of the measurements made from the obtained cavities, the reliability of the proposed model has been proven.

Free vibration analysis of functionally graded rotating conical shells in thermal environment

Due to the wide applications of functionally graded shells in high temperature conditions, the study of dynamic behavior of these structures in thermal conditions is of the key importance. The vibration of rotating functionally graded conical shells with temperature-dependency of material properties is studied in this paper. The Donnel shell theory is employed to extract governing equations including initial thermomechanical stresses as well as centrifugal forces, Coriolis forces, and initial hoop tensions due to rotation. The initial thermomechanical stresses are obtained from the nonlinear heat conduction equation with temperature-dependent thermal conductivity for temperature distribution and linear bending equations for initial deflections. The method of generalized differential quadrature is employed to solve the problem. The obtained results are compared with the available literature and finally, the effects of angular velocities, thermal conditions, volume fraction index, semi-vertex angles and lengths of the cone on frequency are investigated.

Heat transfer on simplified pre-period stage of tunnel fire

Tunnel fire is a part of applied thermal problems. With increase of transient temperature of the tunnel fire on the structure surface (i.e. tunnel lining), the heat transfer from the surface is possibly varying transient temperature distribution within the structure. The transient temperature distribution is also possibly damaging the composition of structure (micro-crack) because of critical damage temperature. Therefore, the transient temperature distribution has a significantly important role on defining mechanical and physical properties of structure and determining thermal-induced damaged region. The damage at pre-period stage of tunnel fire is perhaps more significant than that at the other period stages because of thermal gradient. Consequently, a theoretical model was developed for simplifying complicated thermal engineering during pre-period stage of tunnel fire. A hollow solid model (HSM) in a combination of dimensional analysis and heat transfer theory with Bessel?s Function and Duhamel?s Theorem were employed to verify a theoretical equation for dimensionless transient temperature distribution (DTTD) under linear transient thermal loading (LTTL). Experimental and numerical methods were also adopted to approve the results from this theoretical equation. The heating rate (M) is a primary variable for discussing DTTD on three means. The heating rate of 10.191, 10 and 240?C/min were applied to experimental and numerical studies. The experimental and numerical results are consistent with the theoretical solution, successfully verifying that the theoretical solution can predict the DTTD well in field. This equation can be used for thermal/tunnel engineers to evaluate the damaged region and to obtain the parameters related to DTTD.

Study on nucleation position and wetting state for dropwise condensation on rough structures with different wettability using multiphase lattice Boltzmann method

Dropwise condensation on rough structures with different wettability are numerically simulated using Lattice Boltzmann method. The latent heat during condensation process together with the energy equation is taken into consideration in this paper by solving the temperature distribution equation with a source term for the phase change. Typical droplet nucleation positions and wetting states are found out at different surface wettability through numerical results. It is concluded that with the increase of strength coefficient of the fluid-solid interaction or surface hydrophobicity, the nucleation position rises from the bottom to top of the pillar, and the wetting state of droplet changes from the wetting Wenzel state to the nonwetting Cassie one.

The Rotating Flow of Magneto Hydrodynamic Carbon Nanotubes over a Stretching Sheet with the Impact of Non-Linear Thermal Radiation and Heat Generation/Absorption

The aim of this research work is to investigate the innovative concept of magnetohydrodynamic (MHD) three-dimensional rotational flow of nanoparticles (single-walled carbon nanotubes and multi-walled carbon nanotubes). This flow occurs in the presence of non-linear thermal radiation along with heat generation or absorption based on the Casson fluid model over a stretching sheet. Three common types of liquids (water, engine oil, and kerosene oil) are proposed as a base liquid for these carbon nanotubes (CNTs). The formulation of the problem is based upon the basic equation of the Casson fluid model to describe the non-Newtonian behavior. By implementing the suitable non-dimensional conditions, the model system of equations is altered to provide an appropriate non-dimensional nature. The extremely productive Homotopy Asymptotic Method (HAM) is developed to solve the model equations for velocity and temperature distributions, and a graphical presentation is provided. The influences of conspicuous physical variables on the velocity and temperature distributions are described and discussed using graphs. Moreover, skin fraction coefficient and heat transfer rate (Nusselt number) are tabulated for several values of relevant variables. For ease of comprehension, physical representations of embedded parameters such as radiation parameter ( R d ) , magnetic parameter ( M ) , rotation parameter ( K ), Prandtl number ( P r ), Biot number ( λ ) , and heat generation or absorption parameter ( Q h ) are plotted and deliberated graphically.

Fractional blood flow in oscillatory arteries with thermal radiation and magnetic field effects

A fractional model is proposed to study the effect of heat transfer and magnetic field on the blood flowing inside oscillatory arteries. The flow is due to periodic pressure gradient and the fractional model equations include body acceleration. The proposed velocity and temperature distribution equations are solved using the Laplace and Hankel transforms. The effect of the fluid parameters such as the Reynolds number (Re), the magnetic parameter (M) and the radiation parameter (N) is studied graphically with changing the fractional-order parameter. It is found that the fractional derivative is a valuable tool to control both the temperature and velocity of blood when flow parameters change under treatment, for example. Besides, this work highlights the fact that in the presence of strong magnetic field, blood velocity and temperature reduce. A reversed effect is observed where the applied thermal radiation increase; the velocity and temperature of blood increase. However, the temperature remains high around the artery centerline, which is appropriate during treatment to avoid tissues damage.

Analytical study on liquid-solid particles interaction in the presence of heat and mass transfer through a wavy channel

A comprehensive study is presented on the liquid and solid particles interaction propagating through a finite symmetric wavy channel. A non-Newtonian power law model is used to determine the flow behavior with heat and mass transfer. Furthermore, a mathematical model for electro-osmotic flow with Magnetohydrodynamics effects has been presented. The mathematical model comprises on two-dimensional conservation equations for momentum, energy, concentration, continuity with Ohm's law, chemical reaction, electrokinetic body force and viscous dissipation are formulated with the help of Cartesian coordinate system. The electric field terms are formulated into electrical potential terms using Debye length approximation, Nernst–Planck equation, and Poisson-Boltzmann equation. Moreover, an approximation of long wavelength and small Reynolds number have been used to obtain the final equations for velocity, concentration, and temperature distribution. Series solutions are presented for velocity, concentration, and temperature distribution with the help of Homotopy perturbation method up to second order. The graphical results are presented against all the involved parameters. Applications for the present study involves hemodynamics flow in biomedical engineering as well as thermal electro-osmotic micro-pumps. Furthermore, the present study is also relevant to improve thermal process in microfluidics, electrophoretic process in physiology and chromatography.

Design of Steam Preheating and Ignition Program of In Situ Combustion

The performance of ignition directly affects the success of in situ combustion. Because of the low temperature and heat loss in the formation, it takes a longer time to ignite and the success rate of ignition is relatively low. The steam injection preheating can shorten the ignition time and reduce the ignition cost greatly. However, thus far, there is relatively no complete design method for steam injection. On the basis of the heat conduction equation of formation, the paper evaluates the effect of the preheating process by laboratory experiments and derives a mathematical equation of temperature distribution of steam injection preheating. In addition, considering the ignition threshold value and oxidation characteristics of crude oil, the paper also designs preheating parameters of steam injection. With the mathematical equation and preheating parameters, the performance of ignition can be optimized to establish table combustion rapidly. Finally, on the basis of the field data, the paper presents a pra...

Optimal Perturbation Iteration Method for Solving Nonlinear Heat Transfer Equations

In this paper, the new optimal perturbation iteration method (OPIM) is introduced and applied for solving nonlinear differential equations arising in heat transfer. The effectiveness of the proposed method will be tested by considering two specific applications: the temperature distribution equation in a thick rectangular fin radiation to free space and cooling of a lumped system with variable specific heat. Comparing different methods shows that the results obtained by optimal perturbation iteration method are very good agreement with the numerical solutions and perform better than the most existing analytic methods.

Influence of viscous dissipation on unsteady MHD natural convective flow of Casson fluid over an oscillating vertical plate via FEM

Abstract In this paper, an unsteady magnetohydrodynamic natural convection, heat transfer, electrically conductive non-Newtonian Casson fluid over an oscillating vertical porous plate taken in to the account with an influence of viscous dissipation. An efficient computational Finite Element Method (FEM) employed to solve non-dimensional boundary layer partial differential equations for velocity and temperature distribution with the influence of emerging dimensionless parameters. Also, the local Skin-friction and local Nusselt number coefficients are obtained and analyzed through graphical and tabular forms. The results shows that increasing of Eckert number leads to increase in the velocity and temperature distribution while decrease in the local Skin-friction and Nusselt number coefficients. An increasing of Casson parameter leads to decrease in the velocity distribution and the local Skin-friction coefficient. Compared the present results with earlier reported studies of analytical solutions of Newtonian and non-Newtonian fluid flow problems. The current results are in excellent agreement with earlier existed results. The velocity and temperature distributions are closure for various mesh (grid) size. Therefore, these computational results are stable and converge. FEM is flexible and simplest method to solve this type of problems in various cases like Newtonian and non-Newtonian fluid flow problems.

Exact and approximate analytic solutions of the nonlinear convective fin problem with temperature-dependent thermal conductivity

ABSTRACTThis article shows that the well-known nonlinear boundary value problem of the differential equation for temperature distribution of convective straight fins with temperature-dependent thermal conductivity is exactly solvable in an implicit form. Furthermore, an exact solution in an explicit form is derived. Also, an accurate analytic solution (series solution) is obtained by a new variation of the Adomian decomposition method.