Numerical nonlinear inverse problem of determining wall heat flux

The inverse problem of determining time-variable surface heat flux in a plane wall, with constant or temperature dependent thermal properties, is numerically studied. Different kinds of incident heat flux, including rectangular waveform, are assumed. The solution is numerically solved as a function estimation problem, so that no a priori information for the functional waveforms of the unknown heat flux is needed. In all cases, a solution in the form of a piece-wise function is used to approach the incident flux. Transient temperature measurements at the boundary, from the solution of the direct problem, served as the simulated experimental data needed as input for the inverse analysis. Both direct and inverse heat conduction problems are solved using the network simulation method. The solution is obtained step-by-step by minimising the classical functional that compares the above input data with those obtained from the solution of the inverse problem. A straight line of variable slope and length is used for each one of the stretches of the desired solution. The influence of random error, number of functional terms and the effect of sensor location are studied. In all cases, the results closely agree with the solution.     

Heat flux sensor with minimal impact on boundary conditions

A technique for determining the heat transfer on the far surface of a wall based on measuring the heat flux and temperature on the near wall is presented. Although heat transfer measurements have previously been used to augment temperature measurements in inverse heat conduction methods, the sensors used alter the heat flow through the surface, disturbing the very quantity that is desired to be measured. The ideal sensor would not alter the boundary condition that would exist were the sensor not present. The innovation of this technique in that it has minimal impact on the wall boundary condition. Since the sensor is placed on the surface of the wall, no alteration of the wall is needed. The theoretical basis for the experimental technique as well as experimental results showing the heat flux sensor performance is presented.     

Thermofluidynamic analysis of the flow in a sharp 180° turn channel

The aim of the present study is to obtain surface flow visualisation, as well as local and spanwise averaged heat transfer measurements near a 180° sharp turn in a rectangular channel. The channel aspect ratio (width to height ratio) varies from 1 to 5 and the ratio between the width of the channel and that of the partition wall is always equal to 5. Heat transfer measurements are performed by means of the heated-thin-foil technique, which practically corresponds to a constant heat flux boundary condition, and by using infrared (IR) thermography. Two different heating conditions, in particular heating from one side (asymmetrical), or from two sides (symmetrical), are implemented. The convective heat transfer coefficient is evaluated from the measured temperature maps and the local bulk temperature of the flow which is obtained by making a one-dimensional balance along the channel. Results are presented in terms of local, or averaged, Nusselt number which is normalised with the classical Dittus and Boelter correlation. The fluid used during the test is air and the Reynolds number, based on the flow average velocity and channel hydraulic diameter, is varied between 16,000 and 60,000.     

Solution of non-linear inverse heat conduction problems using the method of lines

 Two space marching methods for solving the one-dimensional nonlinear inverse heat conduction problems are presented. The temperature-dependent thermal properties and the boundary condition on the accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The methods have the advantage that time derivatives are not replaced by finite differences and the good accuracy of the method results from an appropriate approximation of the first time derivative using smoothing polynomials. The extension of the first method presented in this study to higher dimensions inverse heat conduction problems is straightforward. Received on 3 May 1999     

Use of experiment and an inverse method to study interface heat transfer during solidification in the investment casting process

A technique to determine the thermal boundary conditions existing during the solidification of metallic alloys in the investment casting process is presented. Quantitative information about these conditions is needed so that numerical models of heat transfer in this process produce accurate results. In particular, the variation of the boundary conditions both spatially and temporally must be known. The method used involves the application of a new inverse heat conduction method to thermal data recorded during laboratory experiments of aluminium alloy solidification in investment casting shell moulds. The resultant heat transfer coefficient for the alloy/mould interface is calculated. An experimental programme to determine requisite mould thermal properties was also undertaken. It was observed that there is significant variation of the alloy/mould heat transfer coefficient during solidification. It is found to be highly dependent on the alloy type and on the vertical position below the initial free surface of the liquid metal. The aluminium casting alloys used in this study were 413, A356, 319 (Aluminum Association designations), and commercially pure aluminium. These alloys have significantly different freezing ranges. In particular, it was found that alloys with a high freezing range solidify with rates of heat transfer to the mould which are very sensitive to metallostatic head.     

A study of internal heat transfer in nonuniform porous structures

The results of theoretical and experimental studies of heat transfer and pressure drop in nonuniform porous materials and systems are presented. In experiments, measurements were made of the air flow rate, inlet and outlet air pressures, and air and porous sample temperatures. Experimental determination of the heat transfer coefficient in porous structures is associated with certain difficulties. The problem of determining a temperature difference between coolant and porous skeleton is the most complex. As a rule, under laboratory conditions this difference is small and cannot be found with sufficient accuracy. In the present work, the method of determination of the internal heat transfer coefficient is based on solving the inverse unsteady heat transfer problem for porous structures. Using this approach, the heat transfer coefficient is calculated indirectly or on the basis of the porous material temperature variation over time.     

Heat transfer experiments in rotating boundary layer flow

The influence of Coriolis force on heat transfer in a rotating transitional boundary layer has been experimentally investigated. The experiments have been conducted for local Görtler numbers up to 150. Heat transfer measurements have been performed for a flat plate with nearly uniform heat flux applied to the surface, where the temperature was measured by the thermochromic liquid crystal method. The results indicate that heat transfer is enhanced when Coriolis force acts towards the wall, i.e., on the pressure surface. The velocity measurements under equivalent conditions show that Coriolis instability induces counter-rotating longitudinal vortices which augment the lateral transport of the fluid on the pressure surface. On the other hand, the heat transfer on the suction surface remains at the same level as compared to the case without system rotation. As a consequence, the heat transfer coefficient on the pressure surface is 1.8 times higher than that measured on the suction surface when averaged over the measured surface.     

Heat transfer of spray cooling using alumina/water nanofluids with full cone nozzles

The transient temperatures of a metal testing plate during spray cooling using alumina/water nanofluids were measured. The heat transfer coefficient (HTC) was calculated by an inverse heat-conduction technique using the measured temperatures. The results show a decrease of approximately 20?% of the HTC of spray cooling with the nanoparticle suspension changing from 0 to 16.45?%. The nature and the reason of the HTC deduction were investigated and the HTC correlations with the mass fluxes and nanoparticle fraction were specifically reported.     

Flow boiling heat transfer and pressure drop of R-134a in a mini tube: an experimental investigation

This paper presents the results of an experimental study carried out with R-134a during flow boiling in a horizontal tube of 2.6 mm ID. The experimental tests included (i) heat fluxes in the range from 10 to 100 kW/m², (ii) the refrigerant mass velocities set to the discrete values in the range of 240-930 kg/(m² s) and (iii) saturation temperature of 12 and 22 °C. The study analyzed the heat transfer, through the local heat transfer coefficient along of flow, and pressure drop, under the variation of these different parameters. It was possible to observe the significant influence of heat flux in the heat transfer coefficient and mass velocity in the pressure drop, besides the effects of saturation temperature. In the low quality region, it was possible to observe a significant influence of heat flux on the heat transfer coefficient. In the high vapor quality region, for high mass velocities, this influence tended to vanish, and the coefficient decreased. The influence of mass velocity in the heat transfer coefficient was detected in most tests for a threshold value of vapor quality, which was higher as the heat flux increased. For higher heat flux the heat transfer coefficient was nearly independent of mass velocity. The frictional pressure drop increased with the increase in vapor quality and mass velocity. Predictive models for heat transfer coefficient in mini channels were evaluated and the calculated coefficient agreed well with measured data within a range 35% for saturation temperature of 22 °C. These results extend the ranges of heat fluxes and mass velocities beyond values available in literature, and add a substantial contribution to the comprehension of boiling heat transfer phenomena inside mini channels.     

A simultaneous PIV and heat transfer study of bubble interaction with free convection flow

Whole field velocity and point temperature and surface heat flux measurements were performed to characterise the interaction of a single rising ellipsoidal air bubble with the free convection flow from a heated flat surface immersed in water at different angles of inclination. Two thermocouples and a hot film sensor were used to characterise heat transfer from the surface, while a time-resolved digital particle image velocimetry technique was used to map the bubble induced flow in a plane parallel to the surface. Heat flux fluctuations, preceding and following the bubble passage, were shown to correlate with the variation in both local flow velocities and fluid temperatures. The largest increases in heat transfer were recorded when both flow and temperature effects combined to enhance the convective cooling simultaneously. Such conditions were shown to be most likely met when the block was inclined at 45°, thus forcing the bubble to slide closer to the heated surface and hence to the thermal boundary layer.     

Two-phase flow of refrigerants in 85 μm-wide multi-microchannels: Part II – Heat transfer with 35 local heaters

This article is the second part of a study on flow boiling of R236fa and R245fa. This part presents the heat transfer coefficients obtained in a 12.7 mm silicon evaporator composed of 135 microchannels with 85 μm wide and 560 μm high channels separated by 46 μm wide fins. There were 35 local heaters and temperature measurements arranged in a 5 × 7 array. The heat transfer results were uniform in the lateral direction to the flow (attributable to the inlet restriction) and a function of the heat flux, vapor quality and mass flux. The steady-state standard deviation of the local base temperature was less than 0.2 °C, inferring that the boiling process was very stable. For wall heat fluxes over 45 kW/m², the heat transfer coefficient curves were V-shaped, decreasing for intermittent flow regimes and increasing for annular flow. The three-zone model of Thome et al. (2004) was the best heat transfer prediction method when setting the dryout thickness equal to the channel roughness.     

Heat transfer to air–water annular flow in a horizontal pipe

Two-phase air–water flow and heat transfer in a 25 mm internal diameter horizontal pipe were investigated experimentally. The water superficial velocity varied from 24.2 m/s to 41.5 m/s and the air superficial velocity varied from 0.02 m/s to 0.09 m/s. The aim of the study was to determine the heat transfer coefficient and its connection to flow pattern and liquid film thickness. The flow patterns were visualized using a high speed video camera, and the film thickness was measured by the conductive tomography technique. The heat transfer coefficient was calculated from the temperature measurements using the infrared thermography method. It was found that the heat transfer coefficient at the bottom of the pipe is up to three times higher than that at the top, and becomes more uniform around the pipe for higher air flow-rates. Correlations on local and average Nusselt number were obtained and compared to results reported in the literature. The behavior of local heat transfer coefficient was analyzed and the role of film thickness and flow pattern was clarified.     

Flow of a micropolar fluid due to a porous stretching sheet and heat transfer

The present work investigates the micropolar fluid flow due to a permeable stretching sheet and the resulting heat transfer. Unlike the existing numerical works on the flow phenomenon in the literature, the prime interest here is to analytically work out shape of the solutions and identify whether they are unique. Indeed, unique solutions are detected and presented in the exact formulas for the associated boundary layer equations. Temperature field influenced by the microrotation is also mathematically resolved in the cases of constant wall temperature, constant heat flux and Newtonian heating. To discover the salient physical features of many mechanisms acting on the considered problem, it is adequate to have the analytical velocity and temperature fields and also closed-form skin friction/couple stress/heat transfer coefficients, all as given in the current paper. For instance, the practically significant rate of heat transfer is represented by a single formula valid for all three temperature cases.     

Nonsimilar Solutions for Mixed Convection in Non-Newtonian Fluids Along Horizontal Surfaces in Porous Media

A nonsimilar boundary layer analysis is presented for the problem of mixed convection in power-law type non-Newtonian fluids along horizontal surfaces with variable heat flux distribution. The mixed convection regime is divided into two regions, namely, the forced convection dominated regime and the free convection dominated regime. The two solutions are matched. Numerical results are presented for the details of the velocity and temperature fields. A discussion is provided for the effect of viscosity index on the surface heat transfer rate.     

Momentum and heat transfer of a special case of the unsteady stagnation-point flow

This paper investigates the unsteady stagnation-point flow and heat transfer over a moving plate with mass transfer,which is also an exact solution to the unsteady Navier-Stokes(NS)equations.The boundary layer energy equation is solved with the closed form solutions for prescribed wall temperature and prescribed wall heat flux conditions.The wall temperature and heat flux have power dependence on both time and spatial distance.The solution domain,the velocity distribution,the flow field,and the temperature distribution in the fluids are studied for different controlling parameters.These parameters include the Prandtl number,the mass transfer parameter at the wall,the wall moving parameter,the time power index,and the spatial power index.It is found that two solution branches exist for certain combinations of the controlling parameters for the flow and heat transfer problems.The heat transfer solutions are given by the confluent hypergeometric function of the first kind,which can be simplified into the incomplete gamma functions for special conditions.The wall heat flux and temperature profiles show very complicated variation behaviors.The wall heat flux can have multiple poles under certain given controlling parameters,and the temperature can have significant oscillations with overshoot and negative values in the boundary layers.The relationship between the number of poles in the wall heat flux and the number of zero-crossing points is identified.The difference in the results of the prescribed wall temperature case and the prescribed wall heat flux case is analyzed.Results given in this paper provide a rare closed form analytical solution to the entire unsteady NS equations,which can be used as a benchmark problem for numerical code validation.     

Lie group analysis for thermophoretic and radiative augmentation of heat and mass transfer in a Brinkman-Darcy flow over a flat surface with heat generation

A boundary layer analysis is presented to investigate numerically the effects of radiation,thermophoresis and the dimensionless heat generation or absorption on hydromagnetic flow with heat and mass transfer over a flat surface in a porous medium.The boundary layer equations are transformed to non-linear ordinary differential equations using scaling group of transformations and they are solved numerically by using the fourth order Runge-Kutta method with shooting technique for some values of physical parameters.Comparisons with previously published work are performed and the results are found to be in very good agreement.Many results are obtained and a representative set is displayed graphically to illustrate the influence of the various parameters on the dimensionless velocity,temperature and concentration profiles as well as the local skin-friction coefficient,wall heat transfer,particle deposition rate and wall thermophoretic deposition velocity.The results show that the magnetic field induces acceleration of the flow,rather than deceleration(as in classical magnetohydrodynamics(MHD) boundary layer flow) but to reduce temperature and increase concentration of particles in boundary layer.Also,there is a strong dependency of the concentration in the boundary layer on both the Schmidt number and mass transfer parameter.     

An Enthalpy Control Volume Method for Transient Mass and Heat Transport with Solidification

A single domain enthalpy control volume method is developed for solving the coupled fluid flow and heat transfer with solidification problem arising from the continuous casting process. The governing equations consist of the continuity equation, the Navier–Stokes equations and the convection–diffusion equation. The formulation of the method is cast into the framework of the Petrov–Galerkin finite element method with a step test function across the control volume and locally constant approximation to the fluxes of heat and fluid. The use of the step test function and the constant flux approximation leads to the derivation of the exponential interpolating functions for the velocity and temperature fields within each control volume. The exponential fitting makes it possible to capture the sharp boundary layers around the solidification front. The method is then applied to investigate the effect of various casting parameters on the solidification profile and flow pattern of fluids in the casting process.     

Experimental investigation of forced convective heat transfer coefficient in nanofluids of Al2O3/EG and CuO/EG in a double pipe and plate heat exchangers under turbulent flow

Nanofluid is the term applied to a suspension of solid, nanometer-sized particles in conventional fluids; the most prominent features of such fluids include enhanced heat characteristics, such as convective heat transfer coefficient, in comparison to the base fluid without considerable alterations in physical and chemical properties. In this study, nanofluids of aluminum oxide and copper oxide were prepared in ethylene glycol separately. The effect of forced convective heat transfer coefficient in turbulent flow was calculated using a double pipe and plate heat exchangers. Furthermore, we calculated the forced convective heat transfer coefficient of the nanofluids using theoretical correlations in order to compare the results with the experimental data. We also evaluated the effects of particle concentration and operating temperature on the forced convective heat transfer coefficient of the nanofluids. The findings indicate considerable enhancement in convective heat transfer coefficient of the nanofluids as compared to the base fluid, ranging from 2% to 50%. Moreover, the results indicate that with increasing nanoparticles concentration and nanofluid temperature, the convective heat transfer coefficient of nanofluid increases. Our experiments revealed that in lower temperatures, the theoretical and experimental findings coincide; however, in higher temperatures and with increased concentrations of the nanoparticles in ethylene glycol, the two set of results tend to have growing discrepancies.     

粗糙表面之间接触热阻反问题研究

当两个固体表面相互接触时,由于接触面粗糙度的影响,界面间就形成了非一致接触,这种接触导致热流收缩,进而产生接触热阻. 目前的理论研究主要集中在正问题研究,对反问题的研究相对较少. 接触热阻反问题研究是通过研究部分边界温度、热流和部分测量点的温度来反演得到界面上的接触热阻. 反问题研究在很多工程领域都有应用,如航空航天、机械制造、微电子等,是工程中确定接触热阻一种快速有效的方法. 本文采用边界元法和共轭梯度法研究了二维空间随坐标变化的接触热阻反问题. 为了验证方法的准确性和可行性,假定在已知部分测量点温度和真实接触热阻的情况下,反演计算得到界面的温度和热流,进而得到接触热阻,并与真实接触热阻进行比较. 结果表明采用边界元法和共轭梯度法在无测量误差的情况下,可以准确反演获得界面的真实接触热阻. 若存在测量误差,反演计算结果对测量误差极其敏感,反演结果误差会由于测量误差的引入而被放大. 为处理这种不适定性, 采用最小二乘法对反演计算结果进行校正,结果表明采用最小二乘法能够避开反问题中一些偏离实际值较大的测量点,显著提高反演计算结果的准确性.      

An edge dislocation and a heat source at the center of a multicoated circular inhomogeneity

Closed-form solutions are derived to the problem of an edge dislocation or a steady line heat source at the center of a multicoated circular inhomogeneity by using the complex variable method and the transfer matrix method. The problem is reduced to a single linear algebraic equation which determines the single unknown real coefficient appearing in the complex stress functions defined in the surrounding matrix. The other unknown real coefficient in the complex stress functions in the inhomogeneity can then be conveniently determined.