Uniform Surface Heat Flux Research Articles

Nu–Gr correlation for laminar natural convection heat transfer from a sphere submitted to a constant heat flux surface

The work numerically investigated laminar natural convection heat transfer from the single sphere with a constant heat flux surface in air over the wide range of Grashof number (10≤Gr≤107\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$10 \\le Gr \\le 10^{7}$$\\end{document}). The more efficient and precise numerical method based on Bejan et al. was employed here, the accuracy of which has been confirmed through validation against a single sphere case. The heat transfer characteristics were systemically analyzed in terms of isothermal contours and streamlines around the sphere, dimensionless temperature and velocity profiles. Additionally, local Nusselt number as well as local pressure and friction drag coefficients were studied with different Grashof number. In comparison to the sphere with uniform heat flux surface, the heat transfer from the isothermal sphere was found to be enhanced attributable to a more robust buoyancy force and a steeper temperature gradient. Moreover, the average Nusselt number for the sphere with a constant heat flux between 60.4 and 98.6% of the isothermal sphere’s value, this range being contingent upon the specific Grashof number. What’s more, the proposed correlation addresses a notable void in the predictive understanding of heat transfer from the sphere with uniform heat flux, which is scenario prevalent in various engineering applications, particularly for the cooling of electrical and nuclear systems, and offer values for academic research.

On the efficacy of surface-attached air bubbles as thermal insulators for pressure-driven internal flow

There exists much research examining the role of surface-attached air bubbles in drag reduction. Most of this literature considers isothermal flows and so ignores temperature differences, e.g. with the solid boundary. Here, we relax this assumption and ask whether surface-attached air bubbles may prove useful as thermal insulators, e.g. when the solid temperature differs from that of the cargo liquid (water). Theoretical and numerical solutions, e.g. for the variation of the Nusselt number with bubble thickness, are presented for cases characterized by a uniform surface heat flux (USF). We examine channel and pipe flow geometries, and consider instances where the net mass flow rate within the (continuous) air bubble is zero or non-zero. When the thermal boundary condition is changed to uniform surface temperature (UST), our analysis limits attention to numerical solutions. We identify and discuss a remarkable connection between the drag reduction problem and the USF thermal insulation problem: the proportional change of water temperature with bubble thickness is identical to the proportional change of drag. Also, and although our analysis is conducted in the ‘perfect plastron limit’, we can, e.g. by evaluating hydrodynamic and thermal slip lengths, contrast our work against related studies where heat transfer occurs through the ridges or pillars that affix the air layer in place. This comparison indicates that the oft-applied adiabatic interface assumption may prove restrictive in some regions of the parameter space. We conclude by examining the implications of our work in the context of UST micro-channels, which are relevant to various lab-on-a-chip technologies.

The role of thermal effusivity on the incipient growth of the surface temperature in a semi-infinite region absorbing heat flux at the surface

Heat conduction theory stipulates that two thermo-physical properties of materials: the thermal conductivity “k” and the thermal diffusivity “ ” influence the temperature evolution in regular and irregular bodies as a response to various cooling/heating conditions. The traditional statement involving the two thermo-physical properties is examined at length in the present study for the case of a semi-infinite region. The primary objective of the present study is to investigate the influence of the less known thermo-physical property called the thermal effusivity “e” on the incipient surface temperature raise in a semi-infinite body affected by uniform surface heat flux. The secondary objective of the study is to identify a key figure-of-merit named the dimensionless threshold time that separates the incipient temperature elevation in a semi-infinite region from the incipient temperature elevation in a large wall of finite thickness under the same uniform surface heat flux. The outcome of the methodical analysis suggests that the accurate estimate for the dimensionless threshold time in the semi-infinite region should be 0.10.

Inclusion of unsteady heat conduction in regular bodies subject to uniform surface heat flux in a heat transfer course

The present study is concerned with the analysis of unsteady heat conduction in regular bodies (large plane wall, long cylinder, and sphere) with constant initial temperature, prescribed surface heat flux and thermal properties of the solid that are invariant with temperature. Surprisingly, this important topic is absent in textbooks on heat transfer. The exact evaluation of the dimensionless surface temperatures varying with the dimensionless time in the regular bodies over the entire dimensionless time domain [Formula: see text] is carried out with a symbolic algebra code. Thereafter, regression analysis is applied to the data gathered for the dimensionless surface temperatures varying with the dimensionless time of the regular bodies inside the dimensionless “small time” time sub-domains [Formula: see text]. The dimensionless threshold time [Formula: see text] is a decisive parameter that establishes the borderline between the “small time” sub-domain [Formula: see text] and the “large time” sub-domain [Formula: see text] comprising the entire dimensionless time domain [Formula: see text]. Based on regression analysis, compact asymptotes are constructed for the dimensionless surface temperatures varying with the dimensionless time inside the dimensionless “small time” sub-domain [Formula: see text]. At the end, agreements with the dimensionless exact, analytical surface temperature distributions (the baseline solutions) valid for the dimensionless time sub-domain [Formula: see text]. are considered of excellent quality.

Heat transfer enhancement in laminar water flow through a square channel by streamwise ultrasound irradiation

This research focuses on the heat transfer of laminar water flow in a square channel under 28–120 kHz and 60–120 W ultrasonic waves, irradiated along the streamwise direction. The experiment was conducted at Reynolds numbers of 400–2000, and the uniform surface heat flux of the pipe was set as 3.1–12.9 kW/m2. In this study, ultrasound-induced streaming was investigated using a particle image velocimetry (PIV) technique, and cavitation bubbles were detected using a digital camera. From the results, the maximum heat transfer enhancement was 155% with 28 kHz and 120 W ultrasound at q = 12.9 kW/m2, Re = 400, and x/Dh = 2.58. The Nu ratio, which represents the capability of ultrasound-induced heat transfer enhancement, was also determined. A maximum Nu ratio of 2.017 was achieved with 28 kHz and 120 W ultrasound at q = 7.2 kW/m2 and Re = 400 or St = 28.16 × 104. With a relatively low Re, the poorest heat transfer was observed at St values ranging from 40 × 104 to 120 × 104. This study provides a predictive formula for the Nu ratio, with deviations ranging from 6.89 to 13.2%. Furthermore, the PIV analysis suggested that acoustic streaming of Eckart type appears under the ultrasound irradiation and its velocity enhancement effect becomes maximal under the conditions for the maximum Nu ratio. It is also confirmed that the ultrasound irradiation can enhance the nucleation of bubbles at the pipe wall; ultrasound-induced dynamics of these nucleated bubbles are expected to further disturb near-wall liquid motion and contribute to heat transfer enhancement.

Group Method of Uniform Surface Heat Flux from a Vertical Cone Using Laminar Free Convection

Group Method of Uniform Surface Heat Flux from a Vertical Cone Using Laminar Free Convection

A sensitivity analysis of the effect of curvature ratio on the thermal-hydraulic performance in helically coiled tubes with corrugated walls

To enhance thermal-hydraulic performance by rising the heat transfer rate with reasonable pressure loss, heat transfer augmentation methods should be considered in the heat exchanger designs. The use of two distinct passive heat transfer augmentation methods, helically coiled tube structure and helically corrugated tube wall, is considered in this study to investigate numerically the effect of coil diameter on the heat transfer and flow characteristics of single-phase liquid water flowing in the corrugated coiled tubes. Curvature ratio, the ratio of tube-to-coil diameter, and the use of corrugation are one of the most influential factors on heat transfer in coiled tubes due to their contributions to the formation of secondary and swirl flows. The curvature ratios between 0.0958 and 0.2875 are considered for smooth and equivalent corrugated coiled tubes with the same tube diameter of 5.75 mm and length of 390 mm. In all simulations, the water flows through the investigated tubes at Dean numbers between 150 and 3270 under boundary conditions of uniform surface heat flux. The influence of coil diameter on Nusselt number, friction factor, and thermal-hydraulic performance is examined by a commercial software. As a result, the enhancement in the thermal-hydraulic performance of corrugated coiled tubes compared to smooth straight ones is much higher, up to 270%, at Reynolds numbers below 2300 than those at higher Reynolds numbers. Additionally, the Genetic Aggregation Response Surface Methodology for the estimation of output parameters and their sensitivity analysis is employed by evaluating the Reynolds number and curvature ratio as inputs, and Nusselt number, friction factor, and performance evaluation criteria as outputs. Finally, the output parameters are estimated with an error range of ±5%, and the Reynolds number on Nusselt number and performance evaluation criteria, and curvature ratio on friction factor are realized to be more effective.

Prediction of the temperature–time history in ordinary bodies induced by surface heat flux utilizing the enhanced method of discretization in time and the finite difference method

Purpose The purpose of this paper is to address unsteady heat conduction in two subsets of ordinary bodies. One subset consists of a large plane wall, a long cylinder and a sphere in one dimension. The other subset consists of a short cylinder and a large rectangular bar in two dimensions. The prevalent assumptions in the two subsets are: constant initial temperature, uniform surface heat flux and thermo-physical properties invariant with temperature. The engineering applications of the unsteady heat conduction deal with the determination of temperature–time histories in the two subsets using electric resistance heating, radiative heating and fire pool heating. Design/methodology/approach To this end, a novel numerical procedure named the enhanced method of discretization in time (EMDT) transforms the linear one-dimensional unsteady, heat conduction equations with non-homogeneous boundary conditions into equivalent nonlinear “quasi–steady” heat conduction equations having the time variable embedded as a time parameter. The equivalent nonlinear “quasi–steady” heat conduction equations are solved with a finite difference method. Findings Based on the numerical computations, it is demonstrated that the approximate temperature–time histories in the simple subset of ordinary bodies (large plane wall, long cylinder and sphere) exhibit a perfect matching over the entire time domain 0 < t < ∞ when compared against the rigorous exact temperature–time histories expressed by classical infinite series. Furthermore, using the method of superposition of solutions in the convoluted subset (short cylinder and large rectangular crossbar), the same level of agreement in the approximate temperature–time histories in the simple subset of ordinary bodies is evident. Originality/value The performance of the proposed EMDT coupled with a finite difference method is exhaustively assessed in the solution of the unsteady, one-dimensional heat conduction equations with prescribed surface heat flux for: a subset of one-dimensional bodies (plane wall, long cylinder and spheres) and a subset of two-dimensional bodies (short cylinder and large rectangular bar).

BORU YÜZEYİNDEKİ KAPSÜL TİPİ KABARTMANIN VE AL2O3 SU NANOAKIŞKANIN ISI TRANSFERİNE ETKİSİNİN SAYISAL ANALİZİ

This study aims to numerically investigate and evaluate the enhancement of heat transfer by new capsule dimples on tube surfaces for flow of water and Al2O3-water nanofluid with different concentrations, under uniform surface heat flux. The originality of this work lies in combining two passive heat transfer enhancement methods such as geometrical improvements and nanofluids together. Capsule dimples with different depths were considered. Al2O3-water nanofluid was modeled as a single-phase flow based on the mixture properties. The effects of dimple depth and nanoparticle concentrations on Nusselt number, friction factor and performance evaluation criteria (PEC) were studied. Numerical computations were performed using ANSYS Fluent commercial software for 2000 14000 Reynolds number range. It was found that when laminar, transient and fully developed turbulent flow cases are considered, increase in the dimple depth increases the Nusselt number and friction factor for both pure water and Al2O3-water nanofluids cases. Also, the friction factor increases as dimple depth increases. Results show that increase in PEC is more pronounced in the laminar region than in the transition region, it starts to decrease for turbulent flows. For nanofluid, PEC values are considerably higher than pure water cases. The variation of PEC for capsule dimpled tubes are dependent on flow regimes and dimple depths. Increasing the nano particle volume concentration and dimple depth in laminar flows increase the PEC significantly.

Experimental study on heat transfer of dense phase flow through a horizontal tube under constant wall flux conditions

A novel application of particles is utilization as dense phase flow for heat capture, transportation and storage. The use of horizontal dense phase flow as heat transfer fluid in solar systems is discussed with respect to current technologies. The investigation aims to analyze the heat transfer behavior of a horizontal dense phase flow under the condition of uniform surface heat flux. The proposed technologies transport particles as semi-continuous plug flow with a bottom layer, particles at the bottom layer will regularly renew once plugs pass by. Tests were conducted in a self-pilot scale-up platform. The measured wall-to-mixture heat transfer coefficient varies from ∼303 to 813 W/m2K in the particle mass flux range from 125 to 777 kg/m2s. Three heat transfer equations will be provided with the experimental data. A heat transfer equation describes the wall-to-mixture heat transfer coefficient linearly increases with particle mass flux. Then, a general empirical correlation in a dimensionless form predicted results well with experimental data of the present work and other data reported in the literature. Furthermore, a mechanistic heat transfer model is proposed according to a partial penetration mechanism and multi-layer flow model. The predicted results agree with experimental results within 12% of the average error. Considering the findings in this paper, researchers may have ideas to improve the heat transfer of horizontal dense phase flow to utilize this technology to the parabolic concentrator for the solar thermal industry.

Natural Convection Flow over a Vertical Permeable Circular Cone with Uniform Surface Heat Flux in Temperature-Dependent Viscosity with Three-Fold Solutions within the Boundary Layer

The aim of this study is to investigate the effects of temperature-dependent viscosity on the natural convection flow from a vertical permeable circular cone with uniform heat flux. As part of numerical computation, the governing boundary layer equations are transformed into a non-dimensional form. The resulting nonlinear system of partial differential equations is then reduced to local non-similarity equations which are solved computationally by three different solution methodologies, namely, (i) perturbation solution for small transpiration parameter (ξ), (ii) asymptotic solution for large ξ, and (iii) the implicit finite difference method together with a Keller box scheme for all ξ. The numerical results of the velocity and viscosity profiles of the fluid are displayed graphically with heat transfer characteristics. The shearing stress in terms of the local skin-friction coefficient and the rate of heat transfer in terms of the local Nusselt number (Nu) are given in tabular form for the viscosity parameter (ε) and the Prandtl number (Pr). The viscosity is a linear function of temperature which is valid for small Prandtl numbers (Pr). The three-fold solutions were compared as part of the validations with various ranges of Pr numbers. Overall, good agreements were established. The major finding of the research provides a better demonstration of how temperature-dependent viscosity affects the natural convective flow. It was found that increasing Pr, ξ, and ε decrease the local skin-friction coefficient, but ξ has more influence on increasing the rate of heat transfer, as the effect of ε was erratic at small and large ξ. Furthermore, at the variable Pr, a large ξ increased the local maxima of viscosity at large extents, particularly at low Pr, but the effect on temperature distribution was found to be less significant under the same condition. However, at variable ε and fixed Pr, the temperature distribution was observed to be more influenced by ε at small ξ, whereas large ξ dominated this scheme significantly regardless of the variation in ε. The validations through three-fold solutions act as evidence of the accuracy and versatility of the current approach.

ИНЖЕНЕРНЫЕ ФОРМУЛЫ ДЛЯ ОЦЕНКИ ВЛИЯНИЯ ЕСТЕСТВЕННОЙ КОНВЕКЦИИ У ПОВЕРХНОСТИ ТВС ВВЭР-1000 НА РЕЗУЛЬТАТЫ ИЗМЕРЕНИЯ ЕЕ РАЗМЕРОВ УЛЬТРАЗВУКОВЫМ МЕТОДОМ В БАССЕЙНЕ ВЫДЕРЖКИ АЭС

Engineering formulas are presented for calculating the value of methodological error in measuring the distances from the ultrasonic sensor to the surface of the irradiated fuel assembly of WWER-1000 caused by decay heat from the fuel assembly. The methodical error arises due to the water temperature gradient along ultrasonic wave path. This is due to the presence of natural convection at the surface of the fuel assembly discharged from the reactor into cooling pool at nuclear power plant. The paper presents a methodology for calculating the methodical error. It is assumed that the water temperature between the sensor and the surface of the fuel assembly is determined by convective heat transfer between the fuel assembly and the water in cooling pool at nuclear power plant. The surface of VVER-1000 fuel assemblies is modeled by a flat vertical plate with a uniform surface heat flux. The propagation of an ultrasonic wave in a medium between the surface of a fuel assembly and an ultrasonic sensor is described in the approximation of geometric acoustics. The results of numerical calculations according to the presented method are executed in the form of nomograms. Engineering formulas for calculating the value of methodological error are obtained by processing an array of calculated data. The obtained formulas are used in the development of measuring systems for monitoring the deformation of fuel assemblies of WWER-1000 in cooling pond of nuclear power plant.

Lucid characterization of unsteady heat conduction in large, square crossbars and cubes subject to uniform surface heat flux by way of “quasi–steady” Helmholtz equations

The technical paper addresses unsteady heat conduction in large, square crossbars and cubes heated continually with uniformly surface heat flux. The uniform surface heat flux is customarily provided by electrical, radiative and pool fire heating in engineering applications. This type of heating fall under the category of non-homogeneous Neumann boundary conditions. The existing traditional method for solving the unsteady, heat conduction equation in two and three dimensions with constant thermo-physical properties and subject to non-homogeneous uniform heat flux boundary conditions is the principle of superposition of partial solutions of simpler unsteady, heat conduction equations. For the latter, the exact, analytical temperature distributions are customarily expressed by infinite series regardless of the solution method. For the numerical evaluation of spatio-temporal temperatures, the main inconvenience of the infinite series lies in the poor convergence when the time is small. To prevent the impeding obstacle, the Transversal Method Of Lines (TMOL) is implemented in this work to transform the unsteady, two- and three-dimensional heat conduction equations into degraded “quasi steady”, two- and three-dimensional heat conduction equations, namely the homogeneous Helmhotz equations. Thereby, the exact, analytical solutions of the converted partial differential equations of elliptic type and homogeneous were determined by the standard method of separation of variables. The resulting approximate, semi-analytical two- and three-dimensional temperature distributions for large, square crossbars and cubes do not contain elaborate infinite series. Instead, the temperature disdtributions contain hyperbolic functions elucidating worthy compactness and superb quality, regardless of the time, either “small time”, “moderate time” or “large time”.

Thermal convection studies in liquid metal flow inside a horizontal duct under the influence of transverse magnetic field

An experiment is conducted to study the effect of magnetic field on heat transfer in a magnetohydrodynamic flow of molten lead–lithium, in a stainless steel thin-wall duct, with a uniform surface heat flux at its bottom wall. A novel technique of measuring fluid temperature inside the duct is applied to map the temperature profile in a flow cross section, at both ends of the heated section, using a 4 × 4 array of 16 equidistantly placed thermocouples. Surface temperature, as well as electric potential, is measured at seven different locations along the heated section. Based on the temperature profiles obtained at the outlet of the heated section, various flow regimes have been identified over the experimentally investigated flow parameters. Distinctively, three flow regimes have been recognized depending on the dominance of buoyancy force, electromagnetic force, and inertial force. In order to better understand the experimental data, numerical simulations are performed using COMSOL. In the buoyancy force dominated flow regime, a quasi-two-dimensional turbulence flow is predominant, determining the overall heat transfer mechanism. In the electromagnetic force dominated regime, the perturbation due to buoyancy force is suppressed by the magnetic field. Finally, in the inertial force dominated regime, the electromagnetic force and buoyancy force do not play a significant role in determining the heat transfer mechanism. The transition between observed flow regimes has been identified in terms of Grashof, Reynolds, and Hartmann numbers, and the Nusselt number has been calculated for quantitative comparison of heat transfer in these flow regimes.

Experimental investigation on varying flame characteristics of benzoic resin solid fuel pellets

This paper discusses about the varying flame characteristics of a solid fuel known as benzoic resin as a function of surface temperature and time. Benzoic resin is a naturally occurring biomass and the solid fuel is synthesized using different species of tree bark from the genus Styrax. Hence, an induction heating steel plate fitted with liquefied petroleum gas (LPG) flame ignitor was employed to evaluate these characteristics. Furthermore, a mathematical model is developed to estimate and predict the changing flame behavior of benzoic resin solid fuel. Experiments were executed for different heating surface temperatures such as 70 °C, 110 °C, 130 °C, 160 °C, 180 °C, 210 °C, 240 °C, and 270 °C. The experimental results revealed that there was a change in the flame behavior at 70 °C and 110 °C surface temperatures. The nature of the flames were transition and turbulent, but the laminar flame was detected only at the induction heating surface temperatures of 160 °C and 270 °C. Also, the laminar flame angle is 90° under uniform surface heat flux. Moreover, increase in heat release rate was observed for increasing combustion time.

Semi–analytical, piecewise temperature–time distributions in solid bodies of regular shape affected by uniform surface heat flux employing the Method Of Lines (MOL) and the eigenvalue method

The central objective of the present study is to generate an approximate, semi–analytical piecewise solution of the one-dimensional, unsteady heat conduction equation in regular solid bodies (large wall, long cylinder and sphere) with uniform heat flux at the exposed surfaces. The main obstacle in these heat conduction problems lies in the boundary condition of uniform heat flux that is non-homogeneous, and of course is not conducive to the direct utilization of the method of separation of variables. To circumvent this restrictive obstacle, the traditional solution procedure relies on the principle of superposition of solutions composed of three degraded sub-solutions: one sub-solution in terms of the space coordinate and time, another sub-solution in terms of the space coordinate and another sub-solution in terms of time. In the present study, we exploit a radical different approach applying the Method Of Lines (MOL) forthwith to the one-dimensional, unsteady, heat conduction equation with prescribed uniform heat flux. Essentially, MOL discretizes the spatial partial derivatives in the one-dimensional, unsteady, heat conduction equation, while leaving the time partial derivative continuous. Correspondingly, the MOL transformation gives way to an adjoint system of linear, ordinary differential equations of first order in time, which is solved analytically (not numerically) with the potent eigenvalue method in the platform of a symbolic algebra code. The outcome of the two-part computational procedure provides a sequence of semi-analytical piecewise temperature distributions at the pre-selected lines in the large wall, long cylinder and sphere, which are expressed in terms of linear combinations of exponential functions of time comprising the required set of eigenvalues and eigenvectors. At the end, it was determined that the semi-analytical piecewise surface, center and mean temperature-time distributions based on seven lines exhibit excellent quality in the entire time domain.

Study on Laminar Natural Convection Heat Transfer from a Hemisphere with Uniform Heat Flux Surface

By employing the modified model based on Bejan et al., laminar natural convection heat transfer from a hemisphere with uniform heat flux surface has been numerically investigated. Extensive results of two different surface boundary conditions are obtained for a wide range of Grashof numbers (10 ≤ Gr ≤ 107) and Prandlt number of 0.72. The characteristics of heat transfer and fluid flow are analyzed in terms of isotherm contours and streamline patterns, radial and tangential velocities, dimensionless temperature profiles, local friction and pressure drag coefficients, as well as local and average Nusselt numbers. Meanwhile, the effects of Grashof number and adiabatic surface on flow motion and heat transfer have been studied. No recirculation zone or flow separation generates over the top of the hemisphere compared to the isothermal sphere. Owing to the curvature effect, the maximum values of local friction and pressure drag coefficients appear at the corner point B. Comparisons with the previous results are also reported in detail. All the results are in good agreement with the numerical data. Moreover, both local and average Nusselt numbers show a positive dependence on Grashof number. The values of the non-adiabatic case are smaller than that of the adiabatic case due to the preheating effect. Finally, two precise and general correlations of average Nusselt number varying with Grashof numbers have been presented, which can provide an effective prediction for the heat transfer rate in engineering applications, and offer academic values for the future research.

CFD ANALYSIS OF LAMINAR FORCED CONVECTIVE HEAT TRANSFER FOR TiO2/WATER NANOFLUID IN A SEMI-CIRCULAR CROSS-SECTIONED MICRO-CHANNEL

In this study, forced convection flow and heat transfer characteristics of TiO2/water nanofluid flow with different nanoparticle volume fractions (1.0%, 2.0%, 3.0% and 4.0%) in semi – circular cross – sectioned micro – channel was numerically investigated. The three - dimensional study was conducted under steady state laminar flow condition where Reynolds number changing from 100 to 1000. CFD model has been generated by using ANSYS FLUENT 15.0 software based on finite volume method. The flow was under hydrodynamically and thermally developing flow condition. Uniform surface heat flux boundary condition was applied at the bottom surface of the micro – channel. The average and local Nusselt number and Darcy friction factor values were obtained using numerical results. Also, the effects of using nanofluid on local values of Nusselt number and Darcy friction factor were investigated. Numerical results indicate that the increasing of nanoparticle volume fraction of nanofluid, the average Nusselt number increases; however, there is no significant variation in average Darcy friction factor.

The Graetz Problem for the Ellis Fluid Model

Abstract The determination of temperature and auxiliary quantities such as local and average Nusselt numbers for thermally developing flow is referred as the Graetz problem. In the classical Graetz problem, the fluid entering the tube or channel is Newtonian in nature. Here, an extension of the classical Graetz problem is presented by assuming that the fluid entering the tube or channel obeys the Ellis constitutive equation. The energy equation for the considered problem is solved using the separation of variables technique supplemented with the MATLAB routine bvp4c for computation of the eigenvalues and numerical solution of the associated Sturm-Liouville boundary value problem. The problem is solved for two types of thermal boundary conditions, namely, uniform surface temperature and uniform surface heat flux for both flat and circular geometries. Expressions for bulk mean temperature and local and average Nusselt numbers are presented and discussed through tables and graphs.

Calibration of a plate sensor for determination of total heat transfer into a surface with a spatially varying heat flux

This study presents an investigation of the calibration of a plate sensor for determination of the total surface heat transfer into a stainless steel plate with a spatially variable heat flux. The total surface heat transfer is determined through use of the calibration integral equation method (CIEM) which is an elegant way of relating “input to output” for a conduction heat transfer process. The calibration data consists of a measured uniform heat transfer input to the surface of the plate along with the measured temperatures from five thermocouples installed at the same specified depth just below the back face of the plate. It is demonstrated that the calibration data set obtained from a uniform heat flux test can be used to predict the total surface heat transfer for a time and spatially varying heat flux input to the plate using the average of the measured thermocouple temperatures. The CIEM was modified by rescaling the time domain to account for the temperature dependent thermal diffusivity and rescaling the surface heat transfer to account for the temperature dependent thermal conductivity. Three calibration tests with uniform surface heat flux inputs and six reconstruction tests with time dependent spatially varying heat flux inputs were conducted. Pairs of calibration and reconstruction test data were used to investigate eighteen test cases. Accurate predictions of the total surface heat transfer into the calibrated plate sensor were obtained regardless of the spatial distribution of the surface heating, the calibration test data or the measurement noise magnitude in the reconstruction temperature.