Analytical Temperature Distribution Research Articles

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.

Analytical and numerical temperature distribution in a 3-D triple-layer skin tissue subjected to a multi-point laser beam

Analytical and numerical temperature distribution in a 3-D triple-layer skin tissue subjected to a multi-point laser beam

Thermal stress analysis of disc brake using analytical and numerical methods

This article presents study of the thermal stress development in brake disc and the associated life cycle of the disc. The thermal stress analysis of disc brake under the first brake application and the influences of thermal loads on stress development of the disc have been investigated. The temperature distribution was conducted as a function of disc thickness and braking time. The study was done on the disc brake of Sports Utility Vehicle with a model of DD6470C. Partial solution approach was used to solve analytical temperature distribution through the thickness. The model was done using representative areas of the disc exposed to high temperature whose distribution result was obtained as a function of disc thickness and braking time. The solutions of coupled thermal transient fields and stress fields were obtained based on thermal-structural coupled analysis. Based on the model developed for the study, the positions of high and low stress formations were investigated, and it has been observed that thermal stress and temperature gradient show similar behavior through the thickness of disc. Generally, high temperature and stress components were found on the rubbing surfaces of the disc.

Investigation of transient heat transfer in multi-scale PCM composites using a semi-analytical model

This paper presents a semi-analytical model to describe the transient heat transfer in phase change material (PCM)-based composites. One such example is micro-encapsulated paraffin in gypsum plaster walls for building applications. Without the availability of an analytical solution for the problem at hand, and given the computational intensity of full-scale numerical solution of the problem, there is a need for alternate approaches that can be used in the design of PCM walls. The key assumption underlying the proposed model is that the spherical paraffin particles are small enough relative to the thickness of the wall, and therefore at each time instant each particle is surrounded by a spatially uniform, albeit time-dependent matrix temperature. This evolving matrix temperature is used as a boundary condition in order to solve for the analytical temperature distribution at the particle-scale, from which the heat flow into the particle can be determined. This procedure avoids spatial discretization of the micro-scale and results in a macro-scale model in which the paraffin particles appear as sinks/sources at each nodal point. The heat equation is then solved using the Method of Lines, which reduces a parabolic partial differential equation into a set of ordinary differential equations. The proposed model is used to simulate the constant flux wall conditions often seen in thermal characterization experiments of PCM walls and structures. Simulation results elucidate the impact of the particle radius and interfacial resistance on the transition at the end of the thermal management phase. Simulations of cyclic environmental temperatures more relevant to building applications show that PCM volume loadings as low as 5% can reduce the energy demands of an HVAC system by 15 to 20%. Moreover, the model is also shown to provide excellent agreement with the work of Šavija and Schlangen, who simulated the transient thermal response of hardening concrete using the commercial finite element package FEMMASSE.

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”.

Assessment of a double-MRT pseudopotential lattice Boltzmann model for multiphase flow and heat transfer simulations

In this work, a double-MRT lattice Boltzmann (LB) model for thermal multiphase flow is presented and analyzed. The model is based on a pseudopotential scheme, and introduces a second equation with a non-diagonal relaxation matrix and an explicit definition of the equilibrium distribution in moment space. This alternative approach formally reproduces the target energy equation with no additional terms up to the relevant expansion scale, avoiding traditional explicit corrections in the source term or post-collision distributions. It also preserves the parallelization properties of the algorithm and proves that it can accurately simulate numerical tests with known analytical solutions, namely density and temperature distribution in a stratified van der Waals fluid, and interface evolution in a Stefan flow problem. Furthermore, a two-step grid independency test with fixed dimensionless numbers was analyzed and successfully applied in the simulation of bubble generation on a heated horizontal plate, showing that this methodology can be performed to prove consistency and order of convergence of the resulting LB scheme. For this particular problem, the influence of boundary conditions and numerical treatment of non-local terms in the energy equation are discussed in detail.

Cloaking: Controlling Thermal and Hydrodynamic Fields Simultaneously

Hydrodynamic and thermal cloaking effects have attracted interest among researchers using metamaterials. This paper explores an analytical solution using only isotropic materials to achieve cloaking under forced convective heat transfer. It is shown that, under a background of uniform flow and linearly varying temperature, a circular bilayer cloak with an inner layer made of impenetrable and insulated material and an outer layer made of a porous medium having a constant permeability and effective conductivity does not disturb the external flow and temperature fields. Moreover, the cloaked region is shielded from the external flow and temperature fields. Analytical temperature and velocity distributions within the porous outer layer are presented to demonstrate both the hydrodynamic and thermal cloaking effects of such a bilayer. Finally, numerical results, based on practically attainable material properties, are given to confirm the analytical model.

The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

Thermal Gradients with Sintered Solid State Electrolytes in Lithium-Ion Batteries

The electrolyte is one of the three essential constituents of a Lithium-Ion battery (LiB) in addition to the anode and cathode. During increasingly high power and high current charging and discharging, the requirement for the electrolyte becomes more strict. Solid State Electrolyte (SSE) sees its niche for high power applications due to its ability to suppress concentration polarization and otherwise stable properties also related to safety. During high power and high current cycling, heat management becomes more important and thermal conductivity measurements are needed. In this work, thermal conductivity was measured for three types of solid state electrolytes: Li 7 La 3 Zr 2 O 12 (LLZO), Li 1.5 Al 0.5 Ge 1.5 (PO 4 ) 3 (LAGP), and Li 1.3 Al 0.3 Ti 1.7 (PO 4 ) 3 (LATP) at different compaction pressures. LAGP and LATP were measured after sintering, and LLZO was measured before and after sintering the sample material. Thermal conductivity for the sintered electrolytes was measured to 0.470 ± 0.009 WK − 1 m − 1 , 0.5 ± 0.2 WK − 1 m − 1 and 0.49 ± 0.02 WK − 1 m − 1 for LLZO, LAGP, and LATP respectively. Before sintering, LLZO showed a thermal conductivity of 0.22 ± 0.02 WK − 1 m − 1 . An analytical temperature distribution model for a battery stack of 24 cells shows temperature differences between battery center and edge of 1–2 K for standard liquid electrolytes and 7–9 K for solid state electrolytes, both at the same C-rate of four.

Size effect of thermal shock crack patterns in ceramics: Insights from a nonlocal numerical approach

A new weakly coupled thermoelastic ordinary state-based peridynamic (SB-PD) model considering strain-softening characteristics is developed to investigate the size effect on thermal shock crack patterns of ceramic slabs in water quenching tests. In the new weakly coupled thermoelastic ordinary (SB-PD) model, the general SB-PD model is derived by considering thermal effects. The new microscopic thermal conductivity that establishes the relationship between the microscopic geometries and macroscopic geometric conditions is derived from the analytical temperature distributions. Moreover, the new general form of the critical stretch takes the peridynamic state into consideration and is presented based on an equivalent energy relationship in which the softening tensile parts are taken into consideration. The numerically predicted thermal shock crack patterns are in good agreement with previous experimental observations, which demonstrates that the new nonlocal model is effective in capturing thermal shock crack patterns in cold quenching tests. Moreover, differences from the effects of the brittle ceramic specimen size and the water quenching temperature that affect the energy evolution and crack propagation speed during thermal shock fracturing processes are investigated.

Plain correlation asymptotes to predict the center and mean temperatures and total heat transfer in simple solid objects with uniform surface temperature at limiting small-time conditions

Within the platform of unsteady, one-dimensional heat conduction in simple solid objects (large plate, long cylinder and sphere) cooled/heated with uniform surface temperature, the three important thermal quantities that need to be quantified are: the center temperature, the mean temperature and the total heat transfer. One objective of the study is the accurate determination of new dimensionless critical times for the mean and center temperatures that describe the large time sub-domain using a symbolic algebra software. Another objective of the study is to use regression analysis to construct plain correlation asymptotes for the center temperature, the mean temperature and the total heat transfer in the small time sub-domain using as input data the exact, analytical temperature distributions for the plate, cylinder and sphere expressed by infinite series for all time. Excellent agreement for the center temperature, mean temperature and total heat transfer are achieved in the small time sub-domain using the two different computational procedures.

Thermal management of flexible wearable electronic devices integrated with human skin considering clothing effect

The flexible wearable electronic devices can be directly integrated with human skin to monitor the vital signs of human body, which will be covered with clothes. An analytical model is developed in this paper to study the flexible wearable electronic devices integrated with human skin considering clothing effect. The coupling model of Pennes bio-heat transfer equation and Fourier heat conduction equation is adopted to consider the thermal behavior of the whole system. The clothes are modeled as a kind of composite material consisting of clothing fibers and air under the complicated boundary condition including convection, conduction and radiation between the environment. The predicted analytical temperature distribution of the system is validated by finite element analysis (FEA). And the influences of systematic parameters on characteristic temperatures, i.e., the temperature of electronic component and the maximum temperature at the device/skin interface are investigated comprehensively to discuss the design of heat dissipation and the clothing effect. The results presented are able to provide guidelines for the thermal management of flexible wearable electronic devices under real working conditions.

Approximate analytical treatment of annular fins of rectangular profile for teaching fin heat transfer: Utilization of the mean value theorem for integrals

In the teaching of annular fins of rectangular profile, the main obstacle lies in solving the quasi one-dimensional heat conduction equation, the modified Bessel equation of first kind. In the modified Bessel equation, the variable coefficient [Formula: see text] multiplying the first order derivative of temperature [Formula: see text] is problematic. The principal objective of the present paper on engineering education is to solve the modified Bessel equation of first kind in approximate analytical form. Specifically, we seek to apply the mean value theorem for integrals to the variable coefficient [Formula: see text], viewed as an auxiliary function in the domain of the annular fin extending from the inner radius [Formula: see text] to the outer radius [Formula: see text]. It is demonstrated in a convincing manner that approximate analytical temperature distributions having exponential functions are easy to obtain without resorting to the exact analytical temperature distribution embodying four modified Bessel functions of first kind. Furthermore, the easiness in calculating heat transfer rates in annular fins of rectangular profile for realistic combinations of the two controlling parameters: the normalized radius ratio and the dimensionless thermo-geometrical parameter is verifiable.

Approximate, analytical procedure for rectangular annular fins by accommodating the Cauchy–Euler equation

An approximate, analytical treatment is presented for the rectangular annular fin by transforming the complicated modified Bessel equation of zero order into a rudimentary Cauchy-Euler equation. The essential step in the computational procedure revolves around a simple manipulation of the radial coordinate that sets up a variable coefficient in the third term of the modified Bessel equation of zero order. In the third term, the radial variable will be replaced by the mean radius of the inner and outer radius, whereas the radial variable prevails in the first and second terms. This action paves the way to the easier Cauchy-Euler equation. For a collection of rectangular annular fins of interest in engineering applications, approximate, analytical temperature distributions and heat transfer rates (via the fin efficiency) written in terms of two binomials demonstrate excellent quality levels in all cases. Additionally, relative error distributions are presented in detailed manner using as the baseline cases the classical exact, analytical temperature distributions and heat transfer rates expressed in terms of the complicated modified Bessel functions of first and second kind.

Exact, analytic temperature distributions of pin fins with constant thermal conductivity and power‐law type heat transfer coefficient

AbstractIn this Technical Note, the problem of determining the temperature distribution in a pin fin with power‐law heat transfer coefficients is addressed. It is demonstrated that the governing fin equation, a nonlinear second‐order differential equation, is exactly solvable for the entire range of the exponent n in the power‐law heat transfer coefficients. The exact, closed‐form analytical solutions in implicit form are convenient for physical interpretation and optimization for maximum heat transfer. Furthermore, it is proved that the exact solutions have three different structures: (1) dual in the range of , (2) unique or dual in the range of , and (3) unique in the range of . Additionally, exact analytical expressions for the fin efficiency and the fin effectiveness are provided, both as a function of the dimensionless fin parameter for the gamma of n under study.

Time Estimate for the Incipient Surface Melting of Regular-Shaped Metals Receiving Uniform Heat Flux Inside a Metal Melting Furnace

This article addresses the continuous heating of regular-shaped metals (large plate, long cylinder, and sphere) at ambient temperature placed in a metal melting furnace. Under the assumption of temperature-independent thermophysical properties of the metal, the heat conduction problem entails to unsteady one-dimensional (1D) heat conduction with a boundary condition of uniform heat flux. Based on the exact, analytic spatiotemporal temperature distributions for the regular-shaped metals, the objective of this study is to construct simple predictive formulas so that engineers can estimate the incipient melting of these metals when heated continually. The time at which melting at the metal surface is initiated, tmelt, corresponds to setting the surface temperature, Tsur, equal to the melting temperature, Tmelt. The analysis will be done under the premises of two asymptotic solutions: one a “large-time” solution and the other a “short-time” solution. A collection of six formulas of simple form for predicting the melting time, tmelt, will be developed for those regular-shaped metals (large plate, long cylinder, and sphere).

Explicit exact three-dimensional analytical temperature distribution in passively and actively cooled high-power fibre lasers

Efficiency and beam quality enhancement of high-power lasers require effective thermal management. In order to determine the temperature distribution in double-cladding high-power continuous wave fibre lasers, the heat transfer equation is analytically solved. Using the exact derived temperature distributions, the ultimate pump power before thermal damage, which is one of the key subjects of consideration in high-power laser design, is achieved.

Entropy generation analysis in a slab with non-uniform heat generation subjected to convection cooling

Entropy generation in a rectangular slab with a non-uniform internal heat generation is investigated. A non-dimensional analysis of the local and total entropy generation during steady state heat conduction through the slab is presented. The external surfaces of the slab are subjected to asymmetric convection cooling by the surrounding medium. Analytical temperature distribution within the slab is obtained. The study presents a parametric study of relevant dimensionless heat transfer parameters and their influence on entropy generation. The results show that the local and total entropy generation rates can be minimised for certain combinations of the heat transfer parameters. The results of calculations are presented graphically.

Experimental and Analytical Temperature Distributions during Oven‐Based Convection Heating

Mathematical models, combined with experimental evaluation, provide an approach to understand, design, and optimize food process operations. Magnetic resonance imaging (MRI), as an experimental technique, is used extensively in both medical and engineering applications to measure and quantify transport processes. Magnetic resonance (MR) was used in this study to assess a mathematical model based on Fourier's second law. The objective was to compare analytical solutions for the prediction of internal temperature distributions in foods during oven-based convective heating to experimental temperature measurements and determine at what point during the heating process a coupled heat and mass transport process should be considered. Cylindrical samples of a model food gel, Russet potato and rehydrated mashed potato were heated in a convection oven for specified times. Experimentally measured internal temperatures were compared to the internal temperatures predicted by the analytical model. Temperatures distributions in the axial direction compared favorably for the gel and acceptably for the Russet and mashed potato samples. The MR-acquired temperatures in the radial direction for the gel resulted in a shallower gradient than predicted but followed the expected trend. For the potato samples, the MR-acquired temperatures in the radial direction were not qualitatively similar to the analytical predictions due to moisture loss during heating. If temperature resolution is required in the radial direction, moisture losses merit the use of transport models that couple heat and mass transfer.

DUST EMISSION FROM A PARSEC-SCALE STRUCTURE IN THE SEYFERT 1 NUCLEUS OF NGC 4151

We report mid-infrared interferometric measurements (based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere, Chile, programme number 081.B-0092(A)) with ∼10 mas resolution, which resolve the warm (T = 285+25−50 K) thermal emission at the center of NGC 4151. Using pairs of Very Large Telescope 8.2 m telescopes with the Mid-infrared interferometric instrument and by comparing the data to a Gaussian model, we determined the diameter of the dust emission region, albeit only along one position angle, to be (2.0 ± 0.4) pc (FWHM). This is the first size and temperature estimate for the nuclear warm dust distribution in a Seyfert 1 galaxy. The parameters found are comparable to those in Seyfert 2 galaxies, thus providing direct support for the unified model. Using simple analytic temperature distributions, we find that the mid-infrared emission is probably not the smooth continuation of the hot nuclear source that is marginally resolved with K-band interferometry. We also detected weak excess emission around 10.5 μm in our shorter baseline observation, possibly indicating that silicate emission is extended to the parsec scale.