Temperature-dependent Heat Transfer Coefficient Research Articles

Exact Closed-Form Solution of the Nonlinear Fin Problem With Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient

In the current paper, the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient is revisited. In this problem, it has been assumed that the heat transfer coefficient is expressed in a power-law form and the thermal conductivity is a linear function of temperature. It is shown that its governing nonlinear differential equation is exactly solvable. A full discussion and exact analytical solution in the implicit form are given for further physical interpretation and it is proved that three possible cases may occur: there is no solution to the problem, the solution is unique, and the solutions are dual depending on the values of the parameters of the model. Furthermore, we give exact analytical expressions of fin efficiency as a function of thermogeometric fin parameter.

Effect of temperature-dependent surface heat transfer coefficient on the maximum surface stress in ceramics during quenching

We study the difference in the maximum stress on a cylinder surface σmax using the measured surface heat transfer coefficient hm instead of its average value ha during quenching. In the quenching temperatures of 200, 300, 400, 500, 600 and 800°C, the maximum surface stress σmmax calculated by hm is always smaller than σamax calculated by ha, except in the case of 800°C; while the time to reach σmax calculated by hm (fmmax) is always earlier than that by ha (famax). It is inconsistent with the traditional view that σmax increases with increasing Biot number and the time to reach σmax decreases with increasing Biot number. Other temperature-dependent properties also have a small effect on the trend of their mutual ratios with quenching temperatures. Such a difference between the two maximum surface stresses is caused by the dramatic variation of hm with temperature, which needs to be considered in engineering analysis.

Spectral collocation method for radiative–conductive porous fin with temperature dependent properties

In this work, spectral collocation method is presented to predict the thermal performance of convective–radiative porous fin with temperature dependent convective heat transfer coefficient, fin surface emissivity and internal heat generation. In this approach, the dimensionless fin temperature distribution is approximated by Lagrange interpolation polynomials at spectral collocation points. The differential form of the governing equation is formulated by the Darcy model, and is transformed to a matrix form of algebraic equation. The accuracy of the SCM is verified by compared with numerical results by the homotopy perturbation method and the finite volume method. The node convergence rate of the SCM approximately follows an exponential law, and the computational time of the SCM do not significantly increase with the increasing of collocation points. The effects of various geometric and thermo-physical parameters on the dimensionless fin temperature, fin efficiency and heat transfer rate are comprehensively analyzed. In addition, optimum design analysis is also carried out.

Estimation of temperature-dependent heat transfer coefficients in near-dry cutting

Cutting fluid is the most common method to control the cutting temperature. However, it is usually harmful to the environment and personal health. Near-dry cutting could be an effective substitute of cutting fluid. Most researchers focus on the lubrication during near-dry cutting processes. Actually, the cooling ability of near-dry cutting is very important for the control of cutting temperature and effectiveness of coolant. In this study, heat transfer coefficients (HTCs) of different cooling processes, including nature cooling, compressed air cooling, compressed cold air cooling, minimal quantity lubrication (MQL) cooling, and atomizing water cooling, have been estimated. Surface temperature, temperature of coolant carrier, and air velocity of coolant carrier have a great effect on the cooling ability of near-dry cutting system. The finite element method (FEM) is employed to simulate the cooling process and modify the estimated heat transfer coefficients. It is found that HTC is from 9.3 to 53.9 W/m2/°C with different surface temperatures in nature cooling. HTC increases with the increase of surface temperature. HTC is from 48.1 to 346.0 W/m2/°C with different air velocity and surface temperature in compressed air cooling. HTC has an increased trend with the increase of air velocity. HTC is from 52.6 to 405.6 W/m2/°C with different air velocities and surface temperatures in compressed cold air cooling. Lower air temperature increases HTC by 10∼20 %. HTC is from 96.5 to 718.7 W/m2/°C with different oil quantities and surface temperatures in MQL cooling. HTC increases with the increase of oil quantity of MQL. As a comparison, HTC is from 133.7 to 519.6 W/m2/°C with different water quantities and surface temperatures in atomizing water lubrication (AWL) cooling. HTC of AWL is much lower than that of MQL in the same condition, especially when the surface temperature is high. The boiling point of water is much lower than oil; subsequently, less water mist arrives at the cooled surface, so less heat is removed with AWL cooling than that with MQL cooling in the same condition. HTC is from 282.1 to 1890.3 W/m2/°C with different oil quantities and surface temperatures in cold-air MQL cooling. Lower carrier temperature increases the HTC greatly, even increases the HTC by 150 % when the surface temperature is up to 700 °C. The reason is that the coolant prevents high-temperature deterioration before the coolant arrives at a cooled surface.

Impacts of transient heat transfer modeling on prediction of advanced cladding fracture during LWR LBLOCA

This study presents the importance of coherency in modeling thermal-hydraulics and mechanical behavior of a solid for an advanced prediction of cladding thermal shock fracture. In water quenching, a solid experiences dynamic heat transfer rate evolutions with phase changes of the fluid over a short quenching period. Yet, such a dynamic change of heat transfer rates has been overlooked in the analysis of thermal shock fracture. In this study, we are presenting quantitative evidence against the prevailing use of a constant heat transfer coefficient for thermal shock fracture analysis in water. We conclude that no single constant heat transfer could suffice to depict the actual stress evolution subject to dynamic fluid phase changes. Use of the surface temperature dependent heat transfer coefficient will remarkably increase predictability of thermal shock fracture of brittle materials. The presented results show a remarkable stress prediction improvement up to 80–90% of the actual stress with the use of the surface temperature dependent heat transfer coefficient. For thermal shock fracture analysis of brittle fuel cladding such as oxidized zirconium-based alloy or silicon carbide during LWR reflood, transient subchannel heat transfer coefficients obtained from a thermal-hydraulics code should be used as input for stress analysis. Such efforts will lead to a fundamental improvement in thermal shock fracture predictability over the current experimental empiricism for cladding fracture analysis during reflood.

Estimation of critical dimensions for a trapezoidal-shaped steel fin using hybrid differential evolution algorithm

This paper deals with the inverse prediction of parameters in a trapezoidal fin with temperature-dependent thermal conductivity and heat transfer coefficient. Three critical dimensions along with the relevant heat transfer coefficient at the fin base have been simultaneously predicted for satisfying a given temperature distribution on the surface of the trapezoidal fin. The inverse problem is solved by a hybrid differential evolution-nonlinear programming (DE-NLP) optimization method. For a given fin material which is considered to be stainless steel, it is found from the present study that many feasible dimensions exist which satisfy a given temperature distribution, thereby providing flexibility in selecting any dimensions from the available alternatives by appropriately regulating the base heat transfer coefficient. A very good estimation of the unknown parameters has been obtained even for temperature distribution involving random measurement errors which is confirmed by the comparisons of the reconstructed distributions. It is concluded that for a given fin material, the hybrid DE-NLP algorithm satisfactorily estimates feasible dimensions of a trapezoidal fin even with random measurement error of 11 %.

Online Thermal Monitoring Models for Induction Machines

This paper presents a new transfer function approach for calculating both the average temperature and hot spot temperature of an induction machine with the consideration of temperature-dependent material properties and speed-dependent heat transfer coefficient. Green's functions are introduced and taken as the new transfer functions. Online thermal monitoring models for induction machines are obtained based on Green's function method. Using the presented method, the hot spot temperature and average temperature can be calculated accurately and rapidly. The proposed method is validated via finite-element method and experimental measurement.

Application of Collocation Spectral Method to Solve a Convective-Radiative Longitudinal Fin with Temperature Dependent Internal Heat Generation, Thermal Conductivity and Heat Transfer Coefficient

Application of Collocation Spectral Method to Solve a Convective-Radiative Longitudinal Fin with Temperature Dependent Internal Heat Generation, Thermal Conductivity and Heat Transfer Coefficient

Solution of boundary heat transfer coefficients between hot stamping die and cooling water based on FEM and optimization method

The thermal physical parameters have significant effects on the calculation accuracy of physical fields, and the boundary heat transfer coefficient between the die and water is one of the most important thermal physical parameters in the hot stamping. In order to attain the boundary heat transfer coefficient, the testing devices and test procedures are designed according to the characteristic of heat transfer in the hot stamping die. A method of estimating the temperature-dependent boundary heat transfer coefficient is presented, and an inverse heat conduction software is developed based on finite element method, advance-retreat method and golden section method. The software is used to calculate the boundary heat transfer coefficient according to the temperatures measured by NiCr–NiSi thermocouples in the experiment. The research results show that, the convergence of the method given in the paper is well, the surface temperature of sample has a significant effect on the boundary heat transfer coefficient between the die and water. The boundary heat transfer coefficient increases as the surface temperature of sample reduces, and the variation is nonlinear.

Wavelet collocation solution of non-linear Fin problem with temperature dependent thermal conductivity and heat transfer coefficient

In this paper, Wavelet Collocation Method has been used to solve nonlinear n problem with temperature dependent thermal conductivity and heat transfer coecient. Thermal conductivity of n materials varies any type so that we consider thermal conductivity as the general function of temperature. Here we consider three particular cases, where we assume that thermal conductivity is constant, linear and exponential function of temperature. In each case eciency of n is evaluated. The whole analysis is presented in dimensionless form and the eect of variability of n parameter, exponent and thermal conductivity parameter on temperature distribution and n eciency is shown graphically and discussed in detail.

Identification of nonlinear heat transfer laws from boundary observations

We consider the problem of identifying a nonlinear heat transfer law at the boundary, or of the temperature-dependent heat transfer coefficient in a parabolic equation from boundary observations. As a practical example, this model applies to the heat transfer coefficient that describes the intensity of heat exchange between a hot wire and the cooling water in which it is placed. We reformulate the inverse problem as a variational one which aims to minimize a misfit functional and prove that it has a solution. We provide a gradient formula for the misfit functional and then use some iterative methods for solving the variational problem. Thorough investigations are made with respect to several initial guesses and amounts of noise in the input data. Numerical results show that the methods are robust, stable and accurate.

Thermal Analysis of Semicircular Fins Considering Temperature-dependent Thermal Conductivity and Heat Transfer Coefficient

This paper investigates the semicircular fin efficiency considering the fact that both thermal conductivity of the fin material and the heat transfer coefficient for dissipation of heat from the fin surface are temperature dependent and can render changes in fin efficiency for large temperature differences. Based on a steady-state two-dimensional thermal analysis, the non-linear differential equation of semicircular fin model has been solved using finite-difference method. Two-dimensional temperature distribution of the fin surface has been successfully correlated to a third-degree polynomial. Results show that for a cooling fin with increase in tube size fin efficiency remains nearly the same or decreases for a given number of fin depending on whether fin number is large or small and whether tube size is slightly large or much larger. With increase in fin number for a given tube size, efficiency remains nearly the same or increases depending on whether fin numbers are a few more or many more and whether tube size is small or large. The efficiency range is 0.2875–0.311 for aluminium and 0.295–0.308 for copper corresponding to tube size from 0.01 m to 0.09 m and fin number ranging from 4 to 24. As expected, fin efficiency has been found to increase with enhancement in base temperature and reduction in ambient temperature. Results mentioned above are consistent for both copper and aluminium as fin material.

Heat Transfer in Hot Stamping of High-Strength Boron Steel Sheets

An optimization-based numerical procedure was developed to determine the temperature-dependent interface heat transfer coefficient (IHTC) between blank and tools during the hot stamping of boron steel. During the quenching period, IHTC changed with the temperature difference between blank and lower tool. The maximum value of 4300 W/m2 K was achieved at ΔT = 798 K (525 °C). The IHTC decreased with temperature difference and reached the lowest value (1400 W/m2 K) at about ΔT = 573 K (300 °C).

Approximate solution of the nonlinear heat transfer equation of a fin with the power-law temperature-dependent thermal conductivity and heat transfer coefficient

In this paper, differential transform method (DTM) is used to solve the nonlinear heat transfer equation of a fin with the power-law temperature-dependent both thermal conductivity and heat transfer coefficient. Using DTM, the differential equation and the related boundary conditions transformed into a recurrence set of equations and finally, the coefficients of power series are obtained based on the solution of this set of equations. DTM overcame on nonlinearity without using restrictive assumptions or linearization. Results are presented for the dimensionless temperature distribution and fin efficiency for different values of the problem parameters. DTM results are compared with special case of the problem that has an exact closed-form solution, and an excellent accuracy is observed.

Effect of oxide scale on temperature-dependent interfacial heat transfer in hot stamping process

An optimization-based numerical procedure was developed to determine the temperature-dependent interfacial heat transfer coefficient (IHTC). The effects of temperature, pressure and oxide scale thickness were analyzed, for oxide thickness between 9μm and 156μm and pressure from 8MPa to 42MPa. Oxide scales and contact pressure both show distinctive effects on IHTC in the cooling process. The average IHTC decreases about 2461W/(m2°C) with the increase of oxide scale thickness and increases 2620W/(m2°C) with the increase of pressure. Based on the two-way ANOVA, the effect of contact pressure influences the IHTC most. Their mutual interaction is negligible. The IHTC decreases when the average temperature between the blank and die surface is above 250°C and increases when the latent heat release.

Application of the two-dimensional differential transform method to heat conduction problem for heat transfer in longitudinal rectangular and convex parabolic fins

In this article, approximate analytical (series) solutions for the temperature distribution in a longitudinal rectangular and convex parabolic fins with temperature dependent thermal conductivity and heat transfer coefficient are derived. The transient heat conduction problem is solved for the first time using the two-dimensional differential transform method (2D DTM). The effects of some physical parameters such as the thermo-geometric parameter, exponent and thermal conductivity gradient on temperature distribution are studied. Furthermore, we study the temperature profile at the fin tip.

A Numerical Well-Balanced Scheme for One-Dimensional Heat Transfer in Longitudinal Triangular Fins

The temperature profile for fins with temperature-dependent thermal conductivity and heat transfer coefficients will be considered. Assuming such forms for these coefficients leads to a highly nonlinear partial differential equation (PDE) which cannot easily be solved analytically. We establish a numerical balance rule which can assist in getting a well-balanced numerical scheme. When coupled with the zero-flux condition, this scheme can be used to solve this nonlinear partial differential equation (PDE) modelling the temperature distribution in a one-dimensional longitudinal triangular fin without requiring any additional assumptions or simplifications of the fin profile.

Estimation of thermo-physical and transport properties with Bayesian inference using transient liquid crystal thermography experiments

An inverse methodology is proposed to estimate thermo-physical and transport properties individually and simultaneously from in-house experimental data obtained using the transient Liquid Crystal Thermography (LCT) technique. A vertical rectangular fin made of mild steel and size of 75 × 250 × 3 (L × W × t) (all in mm) has been used. Thermochromic Liquid Crystals (TLCs) are used to obtain transient temperature distribution along the fin surface to determine the temperature dependent heat transfer coefficient, hθ and the thermal diffusivity, α of the fin. The variation of heat transfer coefficient is considered as a power law function of temperature excess (hθ = a''θb(x,t)) and is derived from the basic Nusselt number equation, Nuθ = aRabθ used for laminar natural convection for a vertical plate in ambient air. Using this functional form, the 1–D transient fin equation solved using the finite difference technique for assumed values of 'a' and 'α'. Treating the inverse problem as a one parameter estimation in 'a' or 'α' or a two parameter estimation problem in 'a' and 'α', the sum of the squares of the difference between the TLC measured and simulated temperatures are minimized with the Bayesian frame work in the inverse model to determine the point estimates for 'a' and 'α'. Two point estimates namely the (i) mean and (ii) maximum a posterior (MAP) are used to report the retrieved quantities together with the associated standard deviation.

A novel analysis for calculating the smallest envelope shape of wet fins with a nonlinear mode of surface transport

A solely analytical method is investigated to determine the minimum fin shape of wet fins subject to variable heat transfer coefficients. The analysis is applicable to any surface condition, including dry, fully wet and partially wet. An optimization procedure based on the variational principle is derived to develop an analytical formulation. The effects of various design parameters, including base temperature, relative humidity, surrounding temperature and surrounding pressure, are studied to identify the dependent variables affecting the minimum envelope shape to transfer energy efficiently. The analysis also shows the influence of the variable heat transfer coefficient on the temperature distribution and profile shape with the intention of optimizing the envelope shape criterion. A convergent optimum profile shape is established for the constant heat transfer coefficient or the temperature-dependent heat transfer coefficient with a decreasing function along the length of the fin. On the other hand, divergent profiles under wet conditions may transfer heat optimally for an increasing heat transfer coefficient from the fin base to the fin tip. A simple method is highlighted to produce a new profile by choosing a constraint tip temperature to improve the shape near the tip in order to have an ease in manufacturing process.

Estimation of temperature dependent heat transfer coefficient in a vertical rectangular fin using liquid crystal thermography

This paper presents a methodology for the estimation of temperature dependent heat transfer coefficient for a vertical rectangular fin by using the inverse heat transfer method with Liquid crystal thermography (LCT) data. Steady state, laminar natural convection experiments have been done on a vertical rectangular fin of size 150×250×4, (L×w×t, all dimensions are in mm). The variation of heat transfer coefficient is considered as a power law function of temperature excess (h=aoθb) and is derived from the basic Nusselt number equation used for laminar natural convection, Nu=aRab. With this functional form, the one dimensional fin equation in finite difference form is repeatedly solved using the Gauss–Seidel iterative method. Treating this as a one parameter estimation in ‘a’ the sum of the squares of the difference between the simulated and Thermochromic Liquid Crystal (TLC) measured temperatures is minimized with the Golden section search algorithm to retrieve ‘a’. Estimate of ‘a’ and the accompanying uncertainties are first reported for synthetically generated temperature distribution for assumed values of ‘a’. The effect of noise on the estimate of ‘a’ is discussed. This is followed by retrievals with experimentally obtained TLC temperature distribution for a range of heat inputs to the fin base. The required temperature distributions for accomplishing the retrievals over the surface are obtained using calibrated R40C5W Thermochromic Liquid Crystal (TLC) sheets. As an additional proof of the accuracy of the method, the retrieved value of ‘a’ is used to simulate the temperature distribution in the fin which is then compared with the actual TLC measured temperature distribution.