Transient Conduction Equation Research Articles

Energy harvesting from thermally induced vibrations of antenna panels

• A novel piezoelectric energy harvester excited by thermally induced vibration is proposed. • A fully coupled model considering thermal-mechanical-electrical interactions is established. • A new computational framework with high accuracy is developed for the multi-physics problem. • The feasibility of the designed energy harvester on the antenna panel has been proven. Though vibration energy harvesting technology has been extensively explored in the past decades, harnessing energy from thermally induced vibration has been rarely investigated. This study, for the first time, proposes a piezoelectric energy harvester (PEH) excited by time-varying thermal loading in outer space to power wireless electronics in antenna panels of satellites. A novel thermal-mechanical-electrical coupling model is developed to accurately predict the dynamic response of the system. Firstly, based on the comprehensive analysis of spatial heat fluxes, the transient thermal conduction equations are derived via the variational principle. Subsequently, different from conventional incremental finite elements, the thermoelasticity of the panel is characterized by the absolute nodal formulation. Taking advantage of invariant matrices, an enhanced mathematical model is constructed to improve the computational efficiency of the thermoelastic forces and their Jacobian matrices. Furthermore, an electromechanically-coupled analytical model is put forward for the PEH installed on the antenna panel. Finally, an integrated computational framework is established to iteratively solve the multi-physics coupled problem with second-order accuracy. A corresponding finite element model is also built for verification. The effectiveness and efficiency of the developed multi-physics model are demonstrated through a comparison with the simulation results. In particular, the proposed analytical model not only considers the bidirectional interaction between the elastic deformation and heat absorption, but also incorporates the coupling relationship between the piezoelectric effect and structural vibration. Moreover, the investigation results provide pivotal insights into the design of the energy harvesting system excited by thermally induced vibration.

Heat transfer modelling of an isolated bubble in sodium pool boiling

Sodium tubular boiler receivers in concentrated solar thermal power plants are a viable option to provide near-isothermal heat for industrial applications. Accurately predicting the boiling behaviour in the receivers is imperative to improve the performance of these plants. A physics-based reduced-order bubble growth model is developed in this work to gain a fundamental understanding of the boiling process in the receiver. The model predicts the growth rate of an isolated bubble in a sodium pool based on heat transferred from the microlayer, the macrolayer, the thermal boundary layer and the bulk liquid surrounding the bubble. A transient 2D conduction equation is solved to model the cooling of the wall below the bubble due to evaporation of the microlayer and the macrolayer. The model is used to study the growth of a sodium bubble in a liquid pool and analyse the relative contribution of different heat transfer mechanisms to the growth process. It is found that the microlayer heat transfer is the dominant mechanism controlling bubble growth in sodium. Furthermore, a parametric study of the effect of wall superheat, contact angle and temperature of the bulk liquid on the bubble growth process in sodium shows that the bubble size increases with increasing superheat, contact angle and bulk liquid temperature.

Evaluation of the Effect of Cooling Conditions and Heat Transfer Coefficients on Solidification of Al-Cu Binary Alloy in Static Casting Process

Source-based method for modelling solidification problems have been modified and presented in the current work. It coupled the effect of thermal radiation to macro-transport codes and was solved using finite volume method. The problem was formulated based on the classic continuum energy conservation equation for transient conduction controlled solidification system. Radiation heat transfer and latent heat evolution were added as source terms and solved with appropriate numerical treatments to obtain a system of linearized source terms. This circumvented the need for the application of any analytical solution to the intricate heat transfer regimes included in the model. The effect of cooling was carried out under various cooling conditions imposed on different surfaces of the mould for the solidifying metal. The resultant influence of cooling on the solid fraction evolution during static casting was then evaluated. The simulated cooling curves show that thermal radiation have no influence on the rate of heat extraction and the results show that the predicted cooling curves and solid fraction updates are similar to the results of previous models. The predicted curves at the top section of the open mould however show a little deviation due to effect of surface tension gradient forces. It was further revealed that heat transfer coefficients has more effect cooling curves and temperature contours at the lateral mould surfaces than the interior of the casting which is in agreement with theory of Newtonian cooling.

Three-Dimensional Transient Heat Conduction Equation Solution for Accurate Determination of Heat Transfer Coefficient

Abstract Accurate quantification of local heat transfer coefficient (HTC) is imperative for design and development of heat exchangers for high heat flux dissipation applications. Liquid crystal and infrared thermography (IRT) are typically employed to measure detailed surface temperatures, where local HTC values are calculated by employing suitable conduction models, e.g., one-dimensional (1D) semi-infinite conduction model on a material with the low thermal conductivity and low thermal diffusivity. Often times, this assumption of 1D heat diffusion and ignoring its associated lateral conduction effects leads to significant errors in HTC determination. Prior studies have identified this problem and quantified the associated errors in HTC determination for some representative cooling concepts, by accounting for lateral heat diffusion. In this paper, we have presented a procedure for solution of three-dimensional (3D) transient conduction equation using alternating direction implicit (ADI) method and an error minimization routine to find accurate HTCs at relatively lower computational cost. Representative cases of a single jet and an array jet impingement under maximum crossflow condition have been considered here, for IRT and liquid crystal thermography, respectively. Results indicate that the globally averaged HTC obtained using the 3D model was consistently higher than the conventional 1D model by 7–14%, with deviation levels reaching as high as 20% near the stagnation region. Proposed methodology was computationally efficient and is recommended for studies aimed toward local HTC determination.

Computational Techniques for The Evaluation of Inhomogeneity Parameters on Transient Conduction in Functionally Graded Layers

Transient thermal response of a functionally graded material (FGM) layer is considered and individual effects of inhomogeneity parameters on temperature distribution are examined. Transient conduction equation has variable coefficients controlling conductivity, mass density and specific heat capacitance due to the material property variation along the thickness of the graded layer. In order to solve the time dependent conduction equation for the unknown interior temperatures, computational methods are employed based on finite difference and finite element methods. Governing partial differential equation is discretized in space and time grids and computer codes are developed to implement explicit and implicit schemes. Results of explicit and implicit schemes are compared with those found by finite element method. A very good agreement is achieved for the applied boundary and initial conditions. Parametric study reveals the individual influences of various inhomogeneity parameters of FGM upon time dependent temperature distribution of a functionally graded layer. The results of the direct comparison study indicate that inhomogeneity parameters for specific heat and mass density have greater influence on temperature distribution than that for thermal conductivity.

Simultaneous estimation of thermal properties of orthotropic material with non-intrusive measurement

This article presents a simple in-house laboratory experiment method for the simultaneous estimation of principal thermal conductivity and specific heat capacity of an orthotropic composite with non-intrusive temperature measurement. The principle of method involves in symmetrical heating of two identical samples with a planar heater sandwiched in-between them and measuring the temperature of non-heated surface non-intrusively using IR camera. The forward problem that mimics the heat conduction from heater to sample and then to ambient is a 3-D transient conduction equation which is solved using finite volume method whereas inverse problem is solved using Levenberg-Marquardt algorithm. At first numerical estimations are carried out with synthetic temperature data to decide upon the parameters to be identified/fixed as well as to find optimal experimental variables. Then, using measured temperature response of the non-heated surface, the thermal properties of unidirectional fiber reinforced carbon composite are estimated. Induced bias errors in the estimated parameters because of error in the fixed parameters are also approximately quantified. The estimated transverse thermal conductivity is found to be 0.72 W/m K whereas the in-plane conductivities parallel and perpendicular to fiber are 5.43 W/m K, 0.64 W/m K respectively. Also, the estimated specific heat capacity is found to be 1044 J/kg K. A maximum deviation of 10% from the absolute Guarded Hot Plate measurement is found in the estimated thermal conductivity parallel to fiber.

10.31: Simplified analytical determination of the temperature distribution and the load bearing resistance of slim‐floor beams

ABSTRACTThis contribution presents an analytical method to determine the distribution of temperature within a steel section integrated into a concrete slab, which is subjected to a time‐dependent heat flux. The method developed is based on an analytical solution of the general heat equation for transient conduction simplified to be usable for the calculation of the load bearing resistance of the slim‐floor beam. This simplified analytical solution is validated by comparison against results obtained from numerical simulations. A possible extension of the developed method to fire protected sections is given.

Temperature response factors at different boundary conditions for modelling the single borehole heat exchanger

Design and simulation of borehole heat exchangers rely on the solution of the transient conduction equation. The typical approach for predicting the ground temperature variations in the short and long term is to recursively apply basic thermal response factors available as analytical functions or as pre-estimated tabulated values. In this paper a review of the existing response factor models for borehole heat exchangers (BHE) analysis is presented and a numerical model, built in Comsol environment is employed for calculating the temperature distribution in time and space around a single, finite length, vertical cylindrical heat source also taking into account its position with reference to the ground surface (effects of the adiabatic length or “buried depth” D). The temperature values are recast as dimensionless response factors in order to compare them with analytical solutions where available. Furthermore new temperature response factors suitable for describing the single Finite Cylindrical Source (FCS) under different operating modes (i.e. boundary conditions) are generated. Boundary conditions include imposed heat transfer rate, imposed temperature and a combination of both conditions, where spatially uniform temperature at the BHE interface is attained while also keeping constant the applied heat transfer rate.

Hourly Calculation Method of Air Source Heat Pump Behavior

The paper describes an hourly simplified model for the evaluation of the energy performance of heat pumps in cooling mode maintaining a high accuracy and low computational cost. This approach differs from the methods used for the assessment of the overall energy consumption of the building, normally placed in the so-called white or black box models, where the transient conduction equation is deterministically and stochastically solved, respectively. The present method wants to be the expression of the grey box model, taking place between the previous approaches. The building envelope is defined using a building thermal model realized with a 3 Resistance 1 Capacitance (3R1C) thermal network based on the solution of the lumped capacitance method. The simplified model evaluates the energy efficiency ratio (EER) of a heat pump through the determination of the hourly second law efficiency of a reversed Carnot cycle. The results of the simplified method were finally compared with those provided by EnergyPlus, a dynamic building energy simulation program, and those collected from an outdoor test cell in real working conditions. The results are presented in temperatures and energy consumptions profiles and are validated using the Bland-Altman test.

Investigation of single-droplet/wall collision heat transfer characteristics using infrared thermometry

This study experimentally investigated the heat transfer characteristics during the collision of a single droplet with a plate heated to temperatures ranging from 176 to 375°C. The heated plate was made of infrared (IR)-transparent sapphire with a very thin IR-opaque platinum film on top, which allowed the top-surface temperature to be measured from below using an IR camera. The dynamics of the droplets and the corresponding top-surface temperature distribution of the heated plate during collision were acquired using synchronized high-speed video and IR cameras. The three-dimensional transient conduction equation for the heated plate was solved numerically using the measured surface temperatures as boundary conditions, and the time- and space-resolved surface heat flux was obtained. Various physical characteristics associated with the heat transfer during the collision of a single droplet with a heated plate were analyzed, including the local heat flux distribution, effective heat transfer area, instantaneous heat transfer rate, total heat transfer, and vapor film thickness. In addition, the crucial surface temperature at which the total heat transfer from a single-droplet collision is significantly degraded, known as the dynamic Leidenfrost point, was detected.

Determination of the time-dependent perfusion coefficient in the bio-heat equation

Identification of the time-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation from Cauchy data is investigated. Necessary and sufficient conditions for the unique solvability of this inverse problem are provided.

Optimum residence time for steel productivity and energy saving in a hot rolled reheating furnace

A mathematical model is developed to calculate temperature distribution of a slab in a hot rolled reheating furnace by considering thermal radiation in the furnace and transient conduction in the slab. The furnace is modeled as a radiating medium with spatially varying temperature. Radiative heat flux within the furnace, including the effects of the furnace walls, combustion gases, skid beams, and buttons, is calculated through finite volume method with weighted sum of gray gases model and is applied as the boundary condition in the transient conduction equation of the slab. After validating the predictions of the present models with experimental data, the variations in the heating characteristics of the slab are investigated with residence time in the furnace at two different charging and furnace gas temperatures. The optimum residence time for the slab in the furnace is examined with regard to slab exit temperature, temperature difference between the upper surface and centerline of the slab, and skid mark.

Reconstruction of the Space- and Time-Dependent Blood Perfusion Coefficient in Bio-Heat Transfer

The identification of the space- and time-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. While boundary and initial conditions are prescribed, additional temperature measurements are considered inside the solution domain. The problem is approached both from a global and a local perspective. In the global approach a Crank–Nicolson-type scheme is combined with the Tikhonov regularization method. In the local approach, we compute both the time first-order and space second-order derivatives by means of first kind integral equations. A comparison between the numerical results obtained using the two methods shows that the local approach is more accurate and stable than the global one.

The measurement of heat transfer from hot surfaces to non-wetting droplets

An experimental method to measure the heat transfer between a hot surface and a non-wetting droplet is reported in this paper. By means of transient, high resolution, infrared microscopy, surface temperature measurements with spatial and temporal resolutions of ∼100μm and ∼4ms, respectively, are obtained, by observing a thin metallic layer from beneath through an infrared-transparent substrate. Data from the infrared camera is generated at each time-step in the form of a set of temperatures, at closely-spaced locations on the surface of the infrared transparent plate. Subsequent solution of the transient thermal conduction equation within the substrate permits all thermal quantities (heat flux, energy, etc.) to be determined. As a typical result, the heat transferred by a 1.5mm droplet is measured to be 0.19J, with the heat flux peaking at 3.5MW/m2 during the 10ms it spends in the vicinity of the surface, and with a peak transient surface temperature reduction of 47°C. Error analysis indicates that the uncertainty in this measurement of heat transfer is about 15%.

Coupled simulation of heat transfer and temperature of the composite rocket nozzle wall

Simulation of the temperature distribution on the rocket nozzle wall in high temperature environment is most important for assessing safety and reliability of the nozzle. For that purpose, the coupling simulation of heat transfer and transient temperature of the rocket nozzle wall is carried out in this paper. Equations of radiative heat flux on surfaces in an enclosure with inhomogeneous, participating media is developed to compute the radiative heat flux. The convective heat transfer considering ablation of the C/C material has been studied. And radiative transfer between the plume and nozzle wall has been analyzed with the net radiation computing method in one enclosure. Also the differential equation for two-dimensional transient conduction in the cylindrical coordinate has been established, discretized and solved. The convection and radiation heat fluxes, also the transient temperature along the wall length and thickness of the composite nozzle wall after startup of the engine are investigated. Two benchmark problems have been taken, one is the nozzle of an experimental rocket engine fabricated with stainless substrate and a ZrO 2 coating, the other is constructed with multi-layer composite materials. The computing results show that: plume heating on the throat section is most serious and wall temperature in that place is much higher than the average temperature, and heat insulation of the throat is very important for rocket nozzle. The typical character of transient temperature is steady increase with time after startup of the rocket engine, both on the internal and outside walls of the rocket nozzle while the former increases rapidly. On the same section, the temperature decreases along the thickness from internal to outside of the wall, the difference is over 1000 K due to the heat insulating layer in the nozzle material. The coupled simulation method and related results in the present study can be best used to predict the safety performance of the rocket nozzle in startup stage. Results of coupling simulation method are compared with available results in references which show reasonable agreement.

Inverse temperature-dependent perfusion coefficient reconstruction

The identification of the temperature-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. If Neumann boundary conditions are prescribed, then the additional measurement sufficient to render a unique solution is a temperature measurement on a part of the boundary. A numerical approach based on a Crank–Nicolson-type finite-difference scheme combined with the first-order Tikhonov regularization method is developed. Numerical results are presented and discussed.

Space-dependent perfusion coefficient identification in the transient bio-heat equation

The identification of the space-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. While Dirichlet boundary conditions are assumed, the additional measurement necessary to render a unique solution is either a heat-flux measurement or a time-average temperature measurement inside the space region of interest. A numerical approach based on the Crank–Nicolson finite-difference scheme combined with the first-order Tikhonov’s regularization method is developed. Numerical results are presented and discussed.

An inverse coefficient identification problem for the bio-heat equation

In this article, we investigate both analytical and numerical techniques for the identification of the constant perfusion coefficient in the transient bio-heat conduction equation. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution may be a heat flux, an interior temperature or an average temperature measurement at a single instant. Numerical results obtained using the boundary element method combined with an ordinary non-linear minimization are presented and discussed.

Inverse space-dependent perfusion coefficient identification

The identification of the space-dependent perfusion coefficient in the one-dimensional transient bio-heat conduction equation is investigated. In this inverse coefficient identification problem, the additional measurement necessary to render a unique solution is a boundary temperature measurement. A numerical approach based on a Crank-Nicolson finite-difference scheme combined with Tikhonov's regularization methods is developed. Numerical results are presented and discussed.

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder

Damped-Wave Conduction and Relaxation in a Finite Sphere and Cylinder