Unsteady Heat Conduction Equation Research Articles

Thermocapillary migration of a spherical drop in an arbitrary transient Stokes flow

The present problem is concerned with an arbitrary transient Stokes flow past a viscous drop. The interfacial tension gradient is assumed to be dependent on temperature which is unsteady and hence governed by unsteady heat conduction equation. Solenoidal decomposition method is used to solve the hydrodynamic problem. The unknown coefficients have been computed by using non-isothermal boundary conditions. The drag force and torque exerted on the surface of the drop are computed in the form of Faxen’s laws. Some special cases like flow due to an unsteady Stokeslet and thermal field due to a heat source have been discussed. Asymptotic expansions for drag and torque have been calculated. Further, we have computed migration velocity.

Nonlinear transient response of functionally graded shallow spherical shells subjected to mechanical load and unsteady temperature field

This paper is concerned with the transient deformation of functionally graded (FG) shallow spherical shells subjected to time-dependent thermomechanical load. Based on Timoshenko-Mindlin hypothesis and von Karman nonlinear theory, a set of nonlinear governing equations of motion for FG shallow spherical shells in regard to transverse shear deformation and all the inertia terms are established using Hamilton's principle. The collocation point method and Newmark-beta scheme in conjunction with the finite difference method are adopted to solve the governing equations of motion and the unsteady heat conduction equation numerically. In the numerical examples, the transient deflection and stresses of FG shallow spherical shells with various material properties under different loading conditions are presented.

A mechanistic model for baking of leavened aerated food

A mechanistic model for baking of leavened aerated food is proposed. The model accounts for heat conduction, moisture diffusion, diffusion of CO2 that is produced by fermentation, and the resistance to bubble expansion by the viscoelasticity of the dough which is described by Oldroyd B constitutive equation. Unsteady state heat conduction and moisture diffusion equations are solved accounting for bubble expansion due to heating, evaporation of moisture as well as diffusion of CO2 that is produced by fermentation to obtain the evolution of temperature, moisture and air volume fraction profiles as well as dough rise. The bubble expansion due to CO2 diffusion is coupled to temperature and moisture profiles. The model predicts that the growth of bubble exhibits a lag time followed by an exponential growth phase consistent with experimental observations. Eventhough the surface region is more expanded initially, at longer times, it is found to be more dense. The calculated air volume fraction profiles at longer times indicated an expanded inner region of more or less uniform density and a denser surface crust region of decreasing air volume fraction (or increasing density). The thickness of the crust region is found to increase with baking time. The average air volume fraction is found to increase with time because of the combined effects of air expansion as well as CO2 diffusion. The model predictions of dough rise as well as force exerted by the rising dough compare well with the experimental data (Singh and Bhattacharya, 2005; Romano et al., 2007).

A Framework for the Analysis of Thermal Losses in Reciprocating Compressors and Expanders

This article presents a framework that describes formally the underlying unsteady and conjugate heat transfer processes that are undergone in thermodynamic systems, along with results from its application to the characterization of thermodynamic losses due to irreversible heat transfer during reciprocating compression and expansion processes in a gas spring. Specifically, a heat transfer model is proposed that solves the one-dimensional unsteady heat conduction equation in the solid simultaneously with the first law in the gas phase, with an imposed heat transfer coefficient taken from suitable experiments in gas springs. Even at low volumetric compression ratios (of 2.5), notable effects of unsteady heat transfer to the solid walls are revealed, with thermally induced thermodynamic cycle (work) losses of up to 14% (relative to the work input/output in equivalent adiabatic and reversible compression/expansion processes) at intermediate Péclet numbers (i.e., normalized frequencies) when unfavorable solid and gas materials are selected, and closer to 10–12% for more common material choices. The contribution of the solid toward these values, through the conjugate variations attributed to the thickness of the cylinder wall, is about 8% and 2% points, respectively, showing a maximum at intermediate thicknesses. At higher compression ratios (of 6) a 19% worst-case loss is reported for common materials. These results suggest strongly that in designing high-efficiency reciprocating machines the full conjugate and unsteady problem must be considered and that the role of the solid in determining performance cannot, in general, be neglected.

Variational principle for unsteady heat conduction equation

The semi-inverse method is used to establish a variational principle for an unsteady heat conduction equation.

FEM-Based Modeling to Simulate Thermal Deformation Process for High-Speed Ball Screw Drive Systems

Along with increasing speed and acceleration of numerically controlled machine tools, the influence of thermal dynamics characteristics on operating accuracy becomes more and more important. Improvement of thermal dynamics characteristics has turned out to be one of the crucial problems to develop machine tool of high performance. The positioning error of a feed drive system, mostly caused by the thermal deformation of a ball screw shaft, can directly affect the working accuracy of the machine tool. In this study, we applied the computational approach using the finite element method (FEM) to simulate the thermal expansion process for estimating the deformation of the ball screw system. In the numerical analysis, the deformation of the ball screw shaft and nut was modeled via a linear elasticity approach along with the assumption that the material was elastic, homogeneous, and isotropic. To model the reciprocating motions of the nut at a speed of 60m/min respecting to the screw shaft, we utilized the three-dimensional unsteady heat conduction equation with the frictional heat from the sources of the ball screw shaft, nut and bearings to calculate the temperature distributions for determining the temperature rises and axial thermal deformations in a ball screw shaft under operating situations. Simulations were conducted to explore the connection between the temperature increase of nut and the thermal deformation of the ball screw drive system, revealing the need of a compensation scheme for thermal error to improve the operating accuracy of machine tools.

Modelling of temperature field with physical characteristics dependence on temperature

The model of a metal plate heating by laser beam, taking into account the dependence of density, heat capacity and heat conduction of material on the temperature, is considered in the paper. For describing the temperature distribution, three-dimensional nonlinear unsteady heat conduction equation is used. For its solution the coordinate wise splitting method was used, which reduces the three dimensional problem to a sequence of one-dimensional. At each time step, a system of nonlinear algebraic equations is formed, for its solving the Newton's method was used. The system of linear algebraic equations with a tridiagonal matrix is formed at each iтteraсtion. For reducing the dimensionality of generating systems, the method for constructing adaptive difference grids was used, in which nodes are concentrated in the areas of large solution gradients. The model can be applied to compare the results obtained with regard to the dependence of physical characteristics of material on the temperature and without it.

A mechanistic model for baking of unleavened aerated food

A mechanistic model for baking of unleavened aerated food is proposed. The aerated food was assumed to consist initially of uniformly distributed same sized bubbles. The liquid medium was assumed to be Newtonian. Bubble coalescence, nucleation of bubbles and formation of interconnected channels were neglected. Unsteady state heat conduction and moisture diffusion equations were solved to obtain the evolution of temperature, moisture and air volume fraction profiles as well as cake rise. Consistent with experimental observations reported in the literature, the height of aerated food was found to increase with time, reach a maximum and decreased at longer times. The cake rise was found to be faster and larger at higher oven temperatures and for aerated food of higher initial air fraction and larger sugar content. The model predictions of cake rise agreed fairly well with the experimental data of Pernell, Luck, Foegeding, & Daubert, 2002 for fitted initial air volume fractions with the agreement being better at longer times for decreasing air volume fractions. The calculated air volume fraction profiles indicated an expanded inner region of more or less uniform density and a denser surface crust region whose thickness increased with baking time.

Combined natural convection heat and mass transfer in an enclosure having finite thickness walls

Unsteady three-dimensional conjugate heat and mass transfer in an enclosure having finite thickness heat-conducting walls has been analyzed numerically. The governing unsteady, three-dimensional flow, energy and contaminant transport equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless terms of the vector potential functions, the vorticity vector, the temperature and the concentration, have been solved using an iterative implicit finite-difference method. Main attention was paid to the effects of the Rayleigh number, buoyancy ratio and the dimensionless time on the flow structure and heat and mass transfer regimes. It should be noted that the dominant cause of the oscillations in the dimensionless time dependences of the average Nusselt number on the heat source surface and the average Sherwood number on the contaminant source surface at Ra>5⋅105 is the mutual influence of the analyzed object geometry and the thermo-diffusivity impact on the flow. The change in the buoyancy ratio can lead to the essential modifications of the flow, temperature and concentration fields owing to the significant influence of the concentration gradient.

Optimizations of a GPU accelerated heat conduction equation by a programming of CUDA Fortran from an analysis of a PTX file

The Fortran language has been commonly used for many kinds of scientific computation. In this paper, we focus on the solution of an unsteady heat conduction equation, which is one of the simplest problems for thermal dynamics. Recently, a GPU (graphics processing unit) has been enhanced with a Fortran programming language capability employing CUDA (compute unified device architecture), known as CUDA Fortran. We find that the speed performance of a system using an ordinary program coding of CUDA Fortran is lower than that of systems using a program coding of CUDA C. We also find that intermediate assembly files PTX (parallel thread execution) of the two languages are not coincident. Therefore, by comparing the PTX files from the two coding programs we could detect the bottleneck that causes the speed reduction. We propose three optimization techniques that can enable the calculated speeds using CUDA Fortran and CUDA C to be coincident. The optimizations can be performed by the Fortran language when improved by an analyzed PTX file. It is thus possible to improve the performance of CUDA Fortran by adding a correction to it, which happens to be at a programming language level.

Study of coupled fluid and solid for a hypersonic lending edge

Coupled simulation of hypersonic flow and heat exchange is investigated. Moreover, structural stress based on pseudo static assumption is also studied. The unsteady Navier-Stokes equations are solved for the flow field, and SST k-ω model is used as the turbulent model. The unsteady heat conduction equation is solved for the structure. At the fluid-structure interface the flow field acquires temperature boundary condition from the structure and the structure temperature field obtains heat flux boundary condition from flow field. The method proposed here is proved by the slipstream experiment of the infinite circular pipe. The transformation of the temperature and stress of the two-dimensional circular pipe is analyzed in detail during aerodynamic heating. The results show that the high temperature field is gradually enlarged with the increase of aerodynamic heating time. At the same time the structural thermal strain is also gradually enhanced. The minimal deformed area will appear when θ is equal to 60°. The thermal deformation of circular pipe has very little effect on the flow field.

Vacuum Paddle Fast Pyrolysis Reactor Design and Internal Heat Transfer Investigation

Fast pyrolysis is cracking process in which macromolecule of biomass is broken into small molecular under moderate temperature and at high heating speed. Bio-oil produced by the process can be used as potential alternatives of diesel oil and other chemicals. By investigate different kinds of reactor, a new type reactor called vacuum paddle fast pryoysis reactor is developed with low manufacture cost and operating fee and more stable performance. Two stage condensing system composed of 1st quench stage and 2nd cooling heat exchanger is designed, which is proven effective in avoiding secondary cracking of biomass oil and eliminating block of equipment by tar. At same time, a model describing heat transfer in new reactor is established and solved by unsteady heat conduction equation, which give conclusion on minimum agitating speed and therefore, provide basic theoretical foundation. Keywords: fast pyrolysis, vacuum paddle reactor, condenser, heat transfer

Solution to the 1-D unsteady heat conduction equation with internal Joule heat generation for thermoelectric devices

Abstract Thermoelectric devices are semiconductor devices which are capable of either generating a voltage when placed in between a temperature gradient, exploiting the Seebeck effect, or producing a temperature gradient when powered by electricity, exploiting the Peltier effect. The devices are usually employed in environments with time-varying temperature differences and input/output powers. Therefore it becomes important to understand the behaviour of thermoelectric devices during thermal and electrical transients in order to properly simulate and design complex thermoelectric systems which also include power electronics and control systems. The purpose of this paper is to provide the transient solution to the one-dimensional heat conduction equation with internal heat generation that describes the transfer and generation of heat throughout a thermoelectric device. The solution proposed can be included in a model in which the Peltier effect, the thermal masses and the electrical behaviour of the system are considered too; this would be of great benefit because it would allow accurate simulations of thermoelectric systems. While the previous literature does not focus on the study of thermal transients in thermoelectric applications and usually considers constant the temperatures at the hot and cold sides, this paper proposes a dynamic exchange of heat through the hot and cold side, both in steady-state and transients. This paper also presents an analytical solution which is then computed by Matlab to simulate a physical experiment. Simulation results show excellent correlation with experimentally determined values, thus validating the solution.

Flow and coupled heat transfer in the cavity between the rotor and stator

The turbulent flow and coupled heat transfer in the cavity between the rotor and stator is numerically simulated. Reynolds-averaged Navier-Stokes equations closed with equations of the k-ɛ turbulence model are used to calculate the viscous compressible gas flow characteristics and heat transfer; the unsteady heat conduction equation is used to calculate the temperature field in the metal. The influence of the mass flow rate of the coolant on the flow structure and efficiency of cooling of the rotor and stator walls is studied. The calculated results are compared with experimental data.

Flow and conjugate heat exchange in a rotating cavity with an axially supplied working medium

Numerical modeling of the turbulent flow and conjugate heat exchange in a cavity bounded by a rotor and a stator is carried out. Coupled thermal calculations are based on the unsteady heat-conduction equation describing the temperature distribution within a solid and Reynolds-averaged Navier–Stokes equations closed using the k–e turbulence model, which allow calculation of the velocity, pressure, and temperature distributions in the fluid-filled region. The space-time distributions of the temperature and the heat flux on the metal–fluid interface are obtained in two- and three-dimensional formulations of the problem on a structured and an unstructured grid.

Transient Turbulent Heat Transfer for Heating of Water in a Short Vertical Tube

The transient turbulent heat transfer coefficients in a short vertical Platinum test tube were systematically measured for the flow velocities (u=4.0 to 13.6 m/s), the inlet liquid temperatures (Tin=296.93 to 304.81 K), the inlet pressures (Pin=794.39 to 858.27 kPa) and the increasing heat inputs (Q0 exp(t/τ), exponential periods, τ, of 18.6 ms to 25.7 s) by an experimental water loop comprised of a multistage canned-type circulation pump with high pump head. The Platinum test tubes of test tube inner diameters (d=3 and 6 mm), heated lengths (L=66.5 and 69.6 mm), effective lengths (Leff=56.7 and 59.2 mm), ratios of heated length to inner diameter (L/d=22.16 and 11.6), ratios of effective length to inner diameter (Leff/d=18.9 and 9.87) and wall thickness (δ=0.5 and 0.4 mm) with average surface roughness (Ra=0.40 and 0.45 µm) were used in this work. The surface heat fluxes between the two potential taps were given the difference between the heat generation rate per unit surface area and the rate of change of energy storage in the test tube obtained from the faired average temperature versus time curve. The heater inner surface temperature between the two potential taps was also obtained by solving the unsteady heat conduction equation in the test tube under the conditions of measured average temperature and heat generation rate per unit surface area of the test tube. The transient turbulent heat transfer data for Platinum test tubes were compared with the values calculated by authors’ correlation for the steady state turbulent heat transfer. The influence of inner diameter (d), ratio of effective length to inner diameter (Leff/d), flow velocity (u) and exponential period (τ) on the transient turbulent heat transfer is investigated into details and the widely and precisely predictable correlation of the transient turbulent heat transfer for heating of water in a short vertical tube is given based on the experimental data and authors’ studies for the transient critical heat fluxes (CHFs) of subcooled water flow boiling in a short vertical tube caused by exponentially, ramp-wise and stepwise increasing heat inputs. The correlation can describe the transient turbulent heat transfer coefficients obtained in this work for the wide range of the temperature difference between heater inner surface temperature and average bulk liquid temperature, ΔTL, of 10 to 160 K with d=3 to 6 mm, Leff/d=9.87 to 18.9, u=4.0 to 13.6 m/s and τ=18.6 ms to 25.7 s within ±15 % difference.

ICONE19-43190 TRANSIENT TURBULENT HEAT TRANSFER FOR HEATING OF WATER IN A SHORT VERTICAL TUBE

The transient turbulent heat transfer coefficients in a short vertical Platinum test tube were systematically measured for the flow velocities (u=4.0 to 13.6 m/s), the inlet liquid temperatures (T_<in>=296.93 to 304.81 K), the inlet pressures (Pin=794.39 to 858.27 kPa) and the increasing heat inputs (Q_0 exp(t/τ), exponential periods, τ, of 18.6 ms to 25.7 s) by an experimental water loop comprised of a multistage canned-type circulation pump with high pump head. The Platinum test tubes of test tube inner diameters (d=3 and 6 mm), heated lengths (L=66.5 and 69.6 mm), effective lengths (L_<eff>=56.7 and 59.2 mm), ratios of heated length to inner diameter (L/d=22.16 and 11.6), ratios of effective length to inner diameter (L_<eff>/d=18.9 and 9.87) and wall thickness (δ=0.5 and 0.4 mm) with average surface roughness (Ra=0.40 and 0.45 μm) were used in this work. The surface heat fluxes between the two potential taps were given the difference between the heat generation rate per unit surface area and the rate of change of energy storage in the test tube obtained from the faired average temperature versus time curve. The heater inner surface temperature between the two potential taps was also obtained by solving the unsteady heat conduction equation in the test tube under the conditions of measured average temperature and heat generation rate per unit surface area of the test tube. The transient turbulent heat transfer data for Platinum test tubes were compared with the values calculated by authors' correlation for the steady state turbulent heat transfer. The influence of inner diameter (d), ratio of effective length to inner diameter (L_<eff>/d), flow velocity (u) and exponential period (τ) on the transient turbulent heat transfer is investigated into details and the widely and precisely predictable correlation of the transient turbulent heat transfer for heating of water in a short vertical tube is given based on the experimental data and authors' studies for the transient critical heat fluxes (CHFs) of subcooled water flow boiling in a short vertical tube caused by exponentially, ramp-wise and stepwise increasing heat inputs. The correlation can describe the transient turbulent heat transfer coefficients obtained in this work for the wide range of the temperature difference between heater inner surface temperature and average bulk liquid temperature (ΔT_L=10 to 160 K) with the inner diameters (d=3 to 6 mm), L_<eff>/d (=9.87 to 18.9), the flow velocities (u=4.0 to 13.6 m/s) and the exponential periods (τ=18.6 ms to 25.7 s) within ±15 % difference.

Numerical Simulation of Convective Heat Transfer Modes in a Rectangular Area With a Heat Source and Conducting Walls

Laminar conjugate heat transfer in a rectangular area having finite thickness heat-conducting walls at local heating has been analyzed numerically. The heat source located on the left wall is kept at constant temperature during the whole process. Conjugate heat transfer is complicated by the forced flow. The governing unsteady, two-dimensional flow and energy equations for the gas cavity and unsteady heat conduction equation for solid walls, written in dimensionless form, have been solved using implicit finite-difference method. The solution has been obtained in terms of the stream function and the vorticity vector. The effects of the Grashof number Gr, the Reynolds number Re, and the dimensionless time on the flow structure and heat transfer characteristics have been investigated in detail. Results have been obtained for the following parameters: 103≤Gr≤107, 100≤Re≤1000, and Pr=0.7. Typical distributions of thermohydrodynamic parameters describing features of investigated process have been received. Interference of convective flows (forced, natural, and mixed modes) in the presence of conducting solid walls has been analyzed. The increase in Gr is determined to lead to both the intensification of the convective flow caused by the presence of the heat source and the blocking of the forced flow nearby the upper wall. The nonmonotomic variations in the average Nusselt number with Gr for solid-fluid interfaces have been obtained. The increase in Re is shown to lead to cooling of the gas cavity caused by the forced flow. Evolution of analyzed process at time variation has been displayed. The diagram of the heat convection modes depending on the Grashof and Reynolds numbers has been obtained. The analysis of heat convection modes in a typical subsystem of the electronic equipment is oriented not only toward applied development in microelectronics, but also it can be considered as test database at creation of numerical codes of convective heat transfer simulation in complicated energy systems. Comparison of the obtained results can be made by means of both streamlines and temperature fields at different values of the Grashof number and Reynolds number, and the average Nusselt numbers at solid-fluid interfaces.

One- and Three-Dimensional Wall Heat Flux Calculations in a O2/H2 System

OVER the past several decades, considerable effort has been dedicated to model combustion in rocket chambers to understand and predict the conjugate heat transfer to the chamberwalls [1]. Validation of computational fluid dynamics (CFD) design tools requires, in turn, reliable experimental data assessment. This involves the need to acquire a comprehensive set of data in the same facility, including wall heat fluxes along with inflow measurements over a broad range of pressures [2]. The experimental data for CFD validation acquired to date were obtained from a range of facilities of different sizes, various internal geometry or fuel composition, and injection configurations [3–9]; therefore, wall heat fluxes were considerably different among these studies. The strength, the life cycle, and the cooling system effectiveness are highly dependent on heat transfer into and out of the system [10], and the number of studies addressing the heat transfer into the chamber walls are still inadequate. In a previous study, which was the first of its kind, Marshall et al. [11] obtained the heat flux measurements using Gordon heat flux gauges from coaxial temperature measurements that were placed along the chamber wall. The heat fluxes were, then, calculated by solving the transient axisymmetric heat flux equation. Conley et al. [12] calculated the heat fluxes from the temperature measurements at several longitudinal locations. In these studies, the heatfluxeswere calculated by solving the steady-state one-dimensional (1-D) heat conduction equation and adding a correction term to compensate the heat absorption by the chamberwalls. Thevery nature of heat transfer in the combustion chamber is, in fact, three-dimensional (3-D). Thus, calculation of heat fluxes based on a 1-D assumption may lead to errors. Vaidyanathan et al. [13] calculated wall heat fluxes by numerically solving the two-dimensional (2-D) unsteady heat conduction equation. Although more accurate when compared with 1-D assumption, the 2-D heat conduction assumption still does not take into account the longitudinal heat transfer. The current study is aimed at calculating the heat fluxes in the combustion chamber by numerically solving the 3-D heat conduction equation to assess the validity of 1-D unsteady heat conduction assumption in wall heat flux calculation. Hence, the data presented next include wall heat fluxes calculated from 3-D heat transfer assumption for a GO2=GH2 single-element shear coaxial injector at 37 bar.

Analytical Method for Determining Heat Flux from Temperature-Sensitive-Paint Measurements in Hypersonic Tunnels

This paper describes the analytical method used for temperature-sensitive-paint measurements to obtain quantitative global heat flux diagnostics in hypersonic tunnels. An exact inverse solution of the one-dimensional unsteady heat conduction equation for a polymer layer (temperature-sensitive paint or temperature-sensitive paint and an insulator layer) on a base of any material is obtained to calculate heat flux into the polymer surface as an explicit function of a time-dependent surface temperature change measured by temperature-sensitive paint. In addition, an analytical solution of the three-dimensional unsteady heat conduction equation is given to consider the effect of the lateral heat conduction. Simulations and experiments are conducted to examine the analytical method and assess the relevant factors regarding the measurement uncertainty.