Conduction Heat Transfer Equation Research Articles

Building-scale experimental validation of a new model for walls with phase change materials

The current article presents the building-scale experimental validation of a new model for walls equipped with phase change materials. The one-dimensional conduction heat transfer equation is solved using an explicit finite-difference method coupled with an enthalpy method to consider the variable phase change materials thermal capacity. The model is implemented in the transient system simulation tool simulation tool and referred to as Type 3258, and the method used to couple this component to the multi-zone building (known as Type 56) is discussed. The experimental investigation is performed in two identical full-scale test-cells (only one with phase change materials) exposed to the ambient environment. Experimental data are first used to benchmark a transient system simulation tool simulation model of the test-cell without phase change materials. After calibration, the phase change materials model is experimentally validated with data from the phase change materials-equipped test-cell. Compared to the benchmarking procedure (without phase change materials), the root mean square deviation values between the experiments and the simulations (with phase change materials) are lower or in the same order of magnitude. Simulation accurately reproduces the impact of adding phase change materials to the test-cell as observed in the experiments, such as mitigation and time-shift effects. A discussion about computation time concludes the present study.

Development and numerical validation of a new model for walls with phase change materials implemented in TRNSYS

This paper presents a model of a wall with variable properties dedicated to modelling phase change materials (PCMs) in building envelopes. The model is implemented in the TRNSYS simulation tool and referred to as Type 3258. The 1-D conduction heat transfer equation is solved using an explicit finite-difference method coupled with an enthalpy method to consider the variable PCM thermal capacity. This model includes temperature-dependent thermal conductivity and PCM-specific effects like hysteresis and supercooling. The stability conditions are discussed and the algorithm implemented in TRNSYS is described. A numerical validation performed on wall test cases proposed by the International Energy Agency is presented, showing that the developed model is in agreement with reference models. The paper also discusses the impact of temporal and spatial discretization on the model performance. Modelling problems encountered when using an effective heat capacity method (compared to an enthalpy method) and when representing supercooling are also discussed.

Comparison of the earth's crust geothermal field in Northern Tien Shan region with seismic characteristics

The deep crustal temperatures of the Northern Tien Shan region are being calculated on the basis of direct measurements of the heat flow and empirical relation of geothermal parameters and seismic velocities. For this purpose the applied method of numerical solution of conductive heat transfer equation takes into account additional sources of heat release due to radioactive decay. Thermoelastic deformations, reaching 20-40% of the total lithostatic pressure at depths of 35-60 km, are determined. The criteria controlling the distribution of hypocenters of earthquakes have been determined in accordance with the morphology of the regions of excess temperatures and thermal stresses.

Investigation of reservoir temperature in a gas reservoir in Middle East: case study

Having a good estimation of geothermal gradient and the reservoir temperature has a great impact on the methodological reservoir management in the entire reservoir life, from natural depletion phase up to draining the last portion of hydrocarbon by applying an appropriate EOR method. This issue could become crucial in a gas condensate reservoir management as the reservoir temperature has great influence on the time and amount of precipitated condensate in the reservoir and on the surface. It is usual to consider a constant reservoir temperature throughout the field. Also, encountering a constant geothermal gradient, with a similar fluid in a reservoir, is expected. During drilling campaign of two appraisal wells (one in the crest and other in the flank) of one of Middle East gas reservoir, it is aproximately found one geothermal gas gradient in both well; however, with some displacement in the wells. To find out a scientific justification for this phenomenon, a study was set out and results were presented in this paper. By using the analytical solution to unsteady-state conduction heat transfer for a cylinder with a radius of rm and infinite length and the available temperature data from FBDSTs, the exact reservoir temperature at different depths is calculated. It is also shown that by using the analytical solution ‘affected thermal well bore radius’ can be estimated. The affected thermal well bore radius is defined as the radius that beyond it, formation rock is not affected by any down-hole temperature variation. Based on the elaborated work, in addition to confirmation of having one geothermal gradient in the reservoir, it is found out that the ‘affected thermal well bore radius’ in wells is about 20–60 cm. Also by using the basic steady-state conduction heat transfer equation and assuming that the amount of transferred heat via layers of earth is constant, it is qualitatively shown that having different reservoir temperatures at constant depth throughout a giant deep-reservoir is normal and it should be accounted for the reservoir simulation.

Study of a two-dimension transient heat propagation in cylindrical coordinates by means of two finite difference methods

The analytical approach of unsteady conduction heat transfer under actual conditions represent a very difficult (if not insurmountable) problem due to the issues related to finding analytical solutions for the conduction heat transfer equation. Various techniques have been developed in order to overcome these difficulties, among which the alternate directions method and the decomposition method. Both of them are particularly suited for two-dimension heat propagation. The paper deals with both techniques in order to verify whether the results provided are in good accordance. The studied case consists of a long hollow cylinder, and considers that the time-dependent temperature field varies both in the radial and the axial directions. The implicit technique is used in both methods and involves the simultaneous solving of a set of equations for all of the nodes for each time step successively for each of the two directions. Gauss elimination is used to obtain the solution of the set, representing the nodal temperatures. After using the two techniques the results show a very good agreement, and since the decomposition is easier to use in terms of computer code and running time, this technique seems to be more recommendable.

Aerothermoelastic Analysis of Functionally Graded Plates Using Generalized Differential Quadrature Method

In the present paper, the aerothermoelastic behavior of Functionally Graded (FG) plates under supersonic airflow is investigated using Generalized Differential Quadrature Method (GDQM). The structural model is considered based on the classical plate theory and the von Karman strain-displacement relations are utilized to involve the nonlinear behavior of the plate. To consider the supersonic aerodynamic loads on the plate, the first order piston theory is applied. The material properties of the FG panel are assumed to be temperature independent and alter in the thickness direction according to a power law distribution. The temperature distribution on the surface of the plate is assumed to be constant and in the thickness direction is obtained by one-dimensional steady conductive heat transfer equation. The discretized governing equations via GDQM are solved by the fourth order Runge-Kutta method. Comparison of the obtained results with those available in literature confirms the accuracy and ability of the GDQM to perform the aerothermoelastic analysis of FG plates. Also, the effect of some important parameters such as Mach number, in-plane thermal load, plate aspect ratio and volume fraction index on the plate aerothermoelastic behavior is examined.

A new heat balance equation for sap flow calculation during continuous linear heating in tree sapwood

This article introduces a new “linear heat balance (LHB) method” that is based on exact (physically correct) equation analytically derived from fundamental conduction–convection heat transfer equation. Therefore, the aim of this paper is to derive and analyze a new heat balance equation for sap flow calculation from measured temperature differences around continuously heated needle. The formula is derived from the steady-state conduction and convection heat transfer equation with linear source of heat power. This paper deals with the theoretical (analytical and numerical) and experimental analysis of heat and water transfer in the conductive xylem (sapwood) of trees. The analysis and verification of the new formula is based on numerical simulation (using finite element method) and the new LHB method is demonstrated on experimentally measured data. This new approach is important not only because of more exact sap flow calculation but also for information about thermal conductivity that can be used for water content estimation.

Subchannel analysis of Al2O3 nanofluid as a coolant in VMHWR

AbstractThe main objective of this study is to predict the thermal hydraulic behavior of nanofluids as the coolant in the fuel assembly of variable moderation high performance light water reactor (VMHWR). VMHWR is the new version of high performance light water reactor (HPLWR) conceptual design. Light water reactors at supercritical pressure (VMHWR, HPLWR), being currently under design, are the new generation of nuclear reactors. Water-based nanofluids containing various volume fractions of Al2O3 nanoparticles are analyzed. The conservation equations and conduction heat transfer equation for fuel and clad have been derived and discretized by the finite volume method. The transfer of mass, momentum and energy between adjacent subchannels are split into diversion crossflow and turbulent mixing components. The governed non linear algebraic equations are solved by using analytical iteration methods. Finally the nanofluid analysis results are compared with the pure water results.

Geometry optimization of a nanofluid-based direct absorption solar collector using response surface methodology

A nanofluid filled direct absorption solar collector (DASC) system in which incident sunlight is absorbed directly by a working fluid, provides a promising alternative to conventional solar collectors. Most of the previous numerical and experimental studies evaluated the effect of various nanofluids on the thermal performance of a pre-designed collector, and did not consider the effect of varying collector dimensions on its overall performance. In this study, a numerical model of nanofluid flow and temperature distribution in a DASC is proposed by solving the radiative transfer equations of particulate media and combining conduction and convection heat transfer equations. Response surface methodology (RSM) was then applied to understand the effect of varying dimensions on thermal efficiency and entropy generation of the DASC collector design. Based on the produced response surfaces, multi response optimization was performed to find the collector optimized geometry within the studied range of dimensions.

Periodic Heat Transfer in Convective Fins Based on Dual-Phase-Lag Theory

Heat transfer enhancement through extended surfaces is crucial in modern engineering applications such as microelectromechanical systems and electronic components. Conduction heat transfer is the only way to achieve this objective when the mixing augmentation is not possible. This paper investigates the effects of non-Fourier thermal conduction in convective straight fins with arbitrary constant cross section under periodic boundary conditions by introducing the exact analytical solution for the dual-phase-lag heat conduction. The corresponding analytical approach is developed through the Laplace transform method and inversion theorem. To help advance the understanding of fin thermophysical behavior, a generalized model of conduction heat transfer equation is used for studying all of the interpretations to Fourier-based model, namely, parabolic thermal diffusion, hyperbolic thermal wave, and dual-phase-lag models. Heat flux and temperature gradient relaxation times are the characteristics of the dual-phase-lag model, and the simulation results are strictly a function of these two parameters. Therefore, fin temperature distributions are presented for flux precedence and gradient precedence heat flow regimes. Calculations are performed to investigate the influence of temperature gradient relaxation time on the hyperbolic heat conduction characteristics of non-Fourier fins.

Numerical analysis of heat transfer characteristics in energy piles

AbstractSingle energy pile is studied numerically by both commercial code (CFX 13.0) and open‐source software (OpenFOAM). Both fluid and solid domains are simulated. The temperature distribution in the soil and fluid shows the consistency of the two software. Because of the high computational cost for the simulation of multiple piles, a simplification approach is proposed. The fluid temperatures are updated by iterative analytical calculation while the soil temperatures are simulated by conductive heat transfer equation using OpenFOAM. By this effort, the computation of the whole energy pile system is feasible. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Performance improvement of a nanofluid solar collector based on direct absorption collection (DAC) concepts

Nanofluids are engineered colloidal suspensions of nanoparticles in base fluids, which have good properties of radiation absorption and heat transfer and are a kind of potential working fluids for solar collector based on direct absorption collection (DAC) concepts. A simulation model of nanofluid solar collector was built based on DAC concepts by solving the radiative transfer equations of particulate media and combining conduction and convection heat transfer equations. The system efficiency and temperature distributions are analyzed by considering the absorption and scattering of nanoparticles and the absorption of the matrix. The simulation results were in accordance with the experiments’. The nanofluids improved the outlet temperature and the efficiency by 30–100K and by 2–25% than the base fluid. The photothermal efficiency of a 0.01% graphite nanofluid is 122.7% of that of a coating absorbing collector. The study indicated that nanofluids, even of low-content, have good absorption of solar radiation, and can improve the outlet temperatures and system efficiencies.

RETRACTED: Subchannel analysis of TiO2 nanofluid as the coolant in VVER-1000 reactor

The main objective of this study is to predict the thermal-hydraulic behavior of TiO2 nanofluid as the coolant in the hot fuel assembly of VVER-1000 reactor core. The subchannel's continuity conservation, momentum conservation and energy conservation were established. The conservation equations and conduction heat transfer equation for fuel and clad have been derived and discretized by Finite volume method. The turbulent and diversion cross flow were considered between adjacent subchannels. All subchannels were correlated and numerical analysis was made from the inlet of the core to the outlet. The velocity and enthalpy of the coolant was obtained in each subchannel along core's axial direction, therefore the surface temperature of the fuel rods were obtained as well. Finally the analysis results for TiO2 are compared with the pure water results. The achieved results show that as the concentration of TiO2 nanoparticles increases, the temperature of the coolants grows up and the central fuel temperature can be reduced.

Subchannel analysis of nanofluids application to VVER-1000 reactor

In this study a thermal-hydraulic analysis is performed of nanofluids as a coolants in subchannels in hot fuel assembly of VVER-1000 reactor by subchannel method. Water-based nanofluids containing various volume fractions of Al2O3 nanoparticles are analyzed. The conservation equations and conduction heat transfer equation for fuel and clad have been derived and discretized by finite volume method. The transfer of mass, momentum and energy between adjacent subchannels are split into diversion crossflow and turbulent mixing components. The governing non linear algebraic equations are solved by numerical iteration methods. Finally the nanofluids analysis results are compared with the pure water results. To validate the applied approach, the model and COBRA-EN code results are compared for the case of pure water. Results show that the heat transfer increases with nanoparticles volume concentration in the subchannel geometry.

Estimation of residual life of boiler tubes using steamside oxide scale thickness

In thermal power plants, remaining-life-estimation of boiler tubes is required at regular intervals for a safer and a better functionality of boilers. In this paper, a new method is proposed for the residual life estimation of service exposed boiler tubes using Non-Destructive Ultrasonic Oxide scale thickness measurements, average metal temperature and creep master curve. While steady state conduction heat transfer equations are solved to calculate the average metal temperature, creep master curve is generated from short term stress rupture data of rupture life less than 5000 h on a virgin material. In the present study, the residual life of T22 (2.25Cr-1Mo) service exposed Platen Superheater tube is estimated using two master creep curves, i.e. Larson-Miller Parametric (LMP) method of standard ASME T22 creep data and Wilshire approach of short term stress rupture data of T22. As the residual life is calculated from fundamental conduction heat transfer theory and creep rupture data, the proposed method can be applied for different grades of boiler materials.

An Analytical Solution to the One‐Dimensional Heat Conduction–Convection Equation in Soil

Soil heat transfer occurs by conduction and convection. Soil temperatures below infiltrating water can provide a signal for water flux. In earlier work, analysis of field measurements with a sine wave model indicated that convection heat transfer made significant contributions to the subsurface temperature oscillations. In this work, we used a Fourier series to describe soil surface temperature variations with time. The conduction and convection heat transfer equation with a multi‐sinusoidal wave boundary condition was solved analytically using a Fourier transformation. Soil temperature values calculated by the single sine wave model and by the Fourier series model were compared with field soil temperature values measured at depths of 0.1 and 0.3 m below an infiltrating ponded surface. The Fourier series model provided better estimates of observed field temperatures than the sine wave model. The new model provides a general way to describe soil temperature under an infiltrating water source.

Numerical estimation of time lags and decrement factors for wall complexes including Multilayer Thermal Insulation, in two different climatic zones

In the current contribution, time lags and decrement factors for a building wall including different Multilayer Thermal Insulation (MTI) configurations have been calculated numerically. For this investigation purpose, a numerical model which can take care of composite walls was developed. The combined conduction, radiation and convection heat transfer equation was solved explicitly via a control volume discretization. In order to investigate the thermal behavior of a wall under certain outdoor climates, meteorological data that served as boundary conditions were applied to the outer surface of this wall. The outdoor air temperatures were averaged over a number of years after been obtained from hourly measurements in both locations under study. Regarding the solar radiation, it was calculated using a clear-sky model during the summer period. The influence of geographic coordinates’ location on the insulating performance of MTI applications was investigated while the wall configurations were supposed to be located in two different climatic zones. The selection of a particular wall configuration involved not only the outdoor climate, but also the whole building characteristics, orientation, etc. The results of the present study are useful for further development in order to conceive and design the optimum MTI configuration, adjusted in specific geographic coordinates and orientations.

Application of the variational iteration method to nonlinear non‐Fourier conduction heat transfer equation with variable coefficient

AbstractIn this paper, a variational iteration method (VIM) has been applied to nonlinear non‐Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results of VIM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems, and it is predicted that VIM can find a wide application in new engineering problems. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.20362

Heat transfer modeling of multilayer reactive foil

Nanostructured reactive foils have attracted a great deal of interest recently due to their unique mechanical and thermal properties. Reactive thin foil can be used as a controllable, localized heating source for joining applications, which enable soldering and brazing of materials at room temperature. In this paper, multiphysics-based numerical simulations were performed to simulate reactive thin foils. The conductive heat transfer equation and mass diffusion equations were solved for 1D, 2D, and 3D cases. 1D simulation examined the effect of the chemical reaction rate on the flame speed; 2D simulation gave the detailed temperature and composition distribution in the bi-layers; and 3D simulation demonstrated the localized heating effect of reactive thin foils on a silicon wafer substrate.

A dimensionless analysis of heat and mass transport in an adsorber with thin fins; uniform pressure approach

A numerical study on heat and mass transfer in an annular adsorbent bed assisted with radial fins for an isobaric adsorption process is performed. A uniform pressure approach is employed to determine the changes of temperature and adsorbate concentration profiles in the adsorbent bed. The governing equations which are heat transfer equation for the adsorbent bed, mass balance equation for the adsorbent particle, and conduction heat transfer equation for the thin fin are non-dimensionalized in order to reduce number of governing parameters. The number of governing parameters is reduced to four as Kutateladze number, thermal diffusivity ratio, dimensionless fin coefficient and dimensionless parameter of Γ which compares mass diffusion in the adsorbent particle to heat transfer through the adsorbent bed. Temperature and adsorbate concentration contours are plotted for different values of defined dimensionless parameters to discuss heat and mass transfer rate in the bed. The average dimensionless temperature and average adsorbate concentration throughout the adsorption process are also presented to compare heat and mass transfer rate of different cases. The values of dimensionless fin coefficient, Γ number and thermal diffusivity ratio are changed from 0.01 to 100, 1 to 10 − 5 and 0.01 to 100, respectively; while the values of Kutateladze number are 1 and 100. The obtained results revealed that heat transfer rate in an adsorbent bed can be enhanced by the fin when the values of thermal diffusivity ratio and fin coefficient are low (i.e., α ⁎ = 0.01, Λ = 0.01). Furthermore, the use of fin in an adsorbent bed with low values of Γ number (i.e. Γ = 10 − 5 ) does not increase heat transfer rate, significantly.