Quadratic Temperature Distribution Research Articles

Effect of Inhomogeneous Temperature in Chip‐Scale Infrared Thermal Sources: A Revisited Blackbody Radiation Formula with Experimental Validation

AbstractMiniature and low‐cost light sources are highly desirable for numerous optical microsystems. Among these, devices based on blackbody radiation of a filament heated at a few hundred degrees, perfectly fit with the requirements of producing a broad spectral range falling in the infrared range, owing to Planck's law. These light sources are of primary interest for Fourier transform infrared (FTIR) spectroscopy. Although thermal light production is simple, achieving precise light intensity is not a trivial task. Herein, the impact of the inhomogeneous temperature on the emitted radiation is studied. Blackbody radiation formulae are revisited for miniature sources, taking into account the temperature distribution and using the principle of superposition of non‐coherent sources. A theoretical model is formulated by dividing the source into multiple annular elementary sources of different temperature. This results in effective, corrected blackbody emission. Analytical formulae are derived in the case of a quadratic temperature distribution. For the experimental validation, a silicon‐based source, made of a platinum resistive micro‐heater on top of heavily doped silicon, is fabricated and experimentally characterized at temperatures ranging from 300 to 520 K. The experimental results show good agreement with the model predictions in the explored wavelength range of (λ = 2.5–4.8 µm).

Analytical solution for unsteady Walters-B fluid flow by a deforming surface with acceleration using OHAM based package BVPh2.0

Current study aims at simulating fluid flow due to a deformable heated surface in an otherwise static viscoelastic fluid obeying Walters-B model. Velocity of the surface is supposed to grow as time from its initiation of motion progress. Simulations in this work are based on the assumption of quadratic surface temperature distribution. Temperature rise attributed to the frictional heating effect is accounted for in the analysis. By choosing appropriate base functions, homotopy solutions are developed for reasonably large values of material fluid parameter. Reliability of the analytical results is established by computing averaged squared residual of the system. The contributions of the surface acceleration and elasticity on the boundary layer formation are enlightened through the plots of velocity components and temperature. Skin friction measuring the stress experienced by the surface is evaluated and examined under different controlling parameters. The paper also presents a numerical solution using NDSolve of MATHEMATICA in a special case of steady flow, and such solution agrees very well with the corresponding homotopy solution.

A study of heat transfer and entropy generation in von Kármán flow of Reiner-Rivlin fluid due to a stretchable disk

Present paper concentrates on viscous heating effects in fluid flow along a stretchable rotating surface in a Reiner-Rivlin fluid. Heating process of the disk is based on a quadratic surface temperature distribution. After invoking boundary layer assumptions, a self-similarity solution is assumed resulting in a coupled non-linear system. This system is evaluated numerically by a collocation method based MATLAB package bvp4c. Computational results reveal a marked formation of boundary layer provided that cross-viscosity coefficient is sufficiently large. Asymptotic solutions (for large axial distance) are derived by computing eigenvalues of the system at an equilibrium point. Some associated concepts such as moment coefficient (measuring the resisting torque), radial wall shear, disk pumping efficiency and heat transfer rate are critically scrutinized in the light of new physical attributes namely cross-viscosity coefficient and radial stretching in this research. Furthermore, second law of thermodynamics is utilized to investigate entropy production rate in the boundary layer. A noteworthy finding is that moment coefficient in von Kármán flow is substantially lowered when fluid exhibits non-Newtonian effects. Entrained flow is predicted to accelerate whenever radial stretching is considered.

Vibration analysis of dynamic mode scanning thermal microscope nanomachining probe

Abstract Here dynamic mode Scanning Thermal Microscope (SThM) nanomachining technique is presented. The hypothesis states that “Excitation on the base can control the depth of nanomachining process by SThM probe”. The mathematical model is determined based on the Euler-Bernoulli beam. Partial differential equation of motion is solved by separating variables for linear equations and assumed modes method for nonlinear equations. Thermal vibrations is associated with a temperature dependent axial force. Constant, linear and quadratic temperature distribution over the probe are investigated while the probe is harmonically excited on the base at its approximated fundamental resonance frequency. The shifted resonance frequencies and amplitude of response in the excited frequency and shifted resonance frequencies are determined. The results indicate that temperature difference shifts the resonance frequencies, and its influence on vibration amplitudes are significant. It is explained that the effect of temperature distribution function is significant on resonance frequencies shift but negligible on amplitudes. It is showed, the amplitude is decreased in all frequencies with increasing the temperature difference. Finally it is demonstrated that with adding a known excitation on the base the depth of nanomachining process can be controlled. It is declared increasing the temperature improves the quality of final nanomachined surface.

Bubble growth model in uniformly superheated binary liquid mixture

A novel model was developed to investigate the fundamental heat transfer and mass diffusion mechanisms of bubble growth in uniformly superheated ethanol–water mixture. In the proposed model, the energy equation was applied coupling with the quadratic temperature distribution within the thermal boundary layer. The mass diffusion effect was accounted by the introduction of species conservation equation in combination with the quadratic concentration distribution within the concentration boundary layer. Peng-Robinson equation of state and activity coefficient calculation were also adopted for the estimation of vapor-liquid equilibrium. In the present study, the maximum mass diffusion limited growth rate was proposed to quantify and illustrate the effect of mass diffusion on bubble growth. The results show that the bubble growth process in a binary mixture can be divided into three distinctive stages. The later stage of bubble growth is mainly subject to mass diffusion and partly to heat transfer at low ethanol concentrations.

Bubble growth characteristics in bi-component liquid: Influence of component concentration and liquid temperature

A thermodynamic model for bubble growth in superheated liquid mixture was developed to investigate the heat transfer and mass diffusion mechanisms. The influence of component concentration and liquid temperature on bubble growth characteristics is the main focus of the present study. In the proposed model, the energy equation and the component diffusion equation for the liquid were coupled with the assumed quadratic temperature and concentration distribution within the liquid boundary layer. The vapor-liquid equilibrium of the binary mixture was estimated by the non-random two-liquid (NRTL) equation. The comparison between the current calculated results with the available experimental data demonstrates the accuracy of the proposed bubble growth model. The results show that the later stage of bubble growth in a binary mixture is primarily controlled by mass diffusion and also influenced by heat transfer. The bubble growth characteristics strongly depend on the mass fraction of the more volatile component and the liquid temperature. Both higher liquid temperature and concentration of the light component are beneficial to enhance the bubble growth. The effect of mass diffusion on bubble growth becomes weaker with increasing liquid temperature and decreasing initial mass fraction of the more volatile component.

A novel model for the bubble growth in the cavitation region of an injector nozzle

A novel thermodynamic model was developed to calculate the bubble growth in the cavitation region of an injector nozzle. The influence of liquid pressure on bubble growth process was evaluated for both homogenous nucleus and heterogeneous nucleus. In this model, the energy equation coupling with the quadratic temperature distribution within the thermal boundary layer was applied with consideration of temporal variation of bubble pressure, radius, velocity and the thickness of the thermal boundary layer. The results show that the evolution of the bubble growth can be divided into three stages, i.e., surface tension controlled stage, transitory stage, and heat transfer controlled stage. The evolution processes of bubble growth from a homogeneous nucleus or a heterogeneous nucleus are much alike other than in the surface tension controlled stage.

A new class of exact solutions for three-dimensional thermal diffusion equations

A new class of exact solutions has been obtained for three-dimensional equations of themal diffusion in a viscous incompressible liquid. This class enables the description of the temperature and concentration distribution at the boundaries of a liquid layer by a quadratic law. It has been shown that the solutions of the linearized set of thermal diffusion equations can describe the motion of a liquid at extreme points of hydrodynamic fields. A generalization of the classic Couette flow with a quadratic temperature and concentration distribution at the lower boundary has been considered as an example. The application of the presented class of solutions enables the modeling of liquid counterflows and the construction of exact solutions describing the flows of dissipative media.

Ultra-high temperature chirped fiber Bragg gratings produced by gradient stretching of viscoelastic silica

By applying a suitable quadratic temperature distribution at a temperature within the viscoelastic softening region for silica, a regenerated chirped grating with bandwidth of 9.8 nm is produced from a uniform grating using post strain-tuning under load. Simulated and experimental results are in good agreement.

Vibration of Orthotropic Circular Plate with Thermal Effect in Exponential Thickness and Quadratic Temperature Distribution

Vibration of Orthotropic Circular Plate with Thermal Effect in Exponential Thickness and Quadratic Temperature Distribution

Nonequilibrium molecular dynamics calculation of the thermal conductivity based on an improved relaxation scheme

A nonequilibrium molecular dynamics (NEMD) method using stochastic energy injection and removal as uniform heat sources and sinks is developed to calculate the thermal conductivity. The stochastic energy is generated by a Maxwell function generator and is imposed on only a few individual molecules each time step. The relaxation of the thermal perturbation is improved compared to other NEMD algorithms because there are no localized heat source and sink slab regions in the system. The heat sources are uniformly distributed in the right half of the system while the sinks are in the left half, which leads to a periodically quadratic temperature distribution that is almost sinusoidal. The thermal conductivity is then easily calculated from the mean temperatures of the right and left half systems rather than by fitting the temperature profiles. This improved relaxation NEMD scheme is used to calculate the thermal conductivities of liquid and solid argons. It shows that the present algorithm gives accurate results with fast convergence and small size effects. Other stochastic energy perturbation, e.g., thermal noise, can be used to replace the Maxwell-type perturbation used in this paper to make the improved relaxation scheme more effective.

Tunable dispersion slope compensator using a chirped fiber Bragg grating tuned by a fan-shaped thin metallic heat channel

An electrically tunable dispersion slope compensator using a chirped fiber Bragg grating tuned by quadratic temperature distribution is demonstrated. A fan-shaped thin metallic heat channel was used to generate a quadratic temperature gradient along the grating. By controlling the temperature of inner and outer boundaries of the heat channel, we achieved a dispersion slope from -60.82-65.89 ps/nm/sup 2/ for 1.2-nm bandwidth. The dispersion slope compensation in 160-Gb/s return-to-zero transmission is demonstrated by simulation.

On the boundary-layer structure of cavity flow in a porous medium driven by differential heating

This paper describes the boundary-layer structure of flow through a porous medium in a two-dimensional rectangular cavity driven by differential heating of the upper surface. The lower surface and sidewalls of the cavity are thermally insulated. In the limit of large Darcy–Rayleigh number, the solution involves a horizontal boundary layer near the upper surface where the main thermal gradients occur. For a monotonic temperature distribution at the upper surface, these drive fluid to the colder end of the cavity where it descends within a narrow vertical boundary layer before returning to the horizontal layer. The horizontal and vertical layers form an interactive system which is solved by a combination of asymptotic analysis and numerical computation. A complete solution is obtained for the case of a quadratic temperature distribution at the upper surface. The solution of the interactive boundary-layer system determines the almost constant temperature in the core region below the horizontal and vertical layers, which contains relatively weak variations in both the thermal and velocity fields.

Thermocapillary convection in a spherical drop partly attached to a spherical wall

For a liquid globule, partly captured in spherical caps of different meridian angles the thermo-capillary flow has been determined due to the change of temperature on the liquid surface. The variation of the applied temperature on the liquid surface causes a variation of the surface tension and therefore a tangential force moving from warm to cold. The streamlines are presented for linear and quadratic temperature distribution on the liquid surface. As the cap angle α increases the center of the ring vortex moves down and towards the free liquid surface. In addition the flow exhibits with increasing magnitude of cap coverage a reduced velocity distribution. For a quadratic temperature distribution the existence of two opposite vortex rings are shown for α<π/2. The upper vortex ring disappears as α becomes larger than π/2. In addition we notice the shift of the vortex core towards the free surface and bottom of the drop.

Boundary element analysis of interface cracks subjected to non-uniform thermal loading

This work applies the method of multi-region boundary element to analyze the thermal stress intensity factor (TSIF) of the bi-material interface cracks subjected to linear and quadratic temperature distribution. An attempt is also made to resolve the problem containing body force which is caused by the inhomogeneous thermal loading by, initially, separating the solution of the inhomogeneous problem of each material into homogeneous and particular solutions, as proposed by Sung. The particular solution can be obtained by expanding the body force into Fourier series and, then, solving each term of the Fourier series. Next, inserting the obtained particular solutions into the boundary conditions of the original problem allows us to reduce the inhomogeneous problem to a homogeneous one. Moreover, the program of thermal multi-region BEM (TMBEM), which neither requires a domain integral nor changes the kernel functions, is established by imposing the continuity conditions on the interfaces. Finally, the applications of TMBEM are illustrated by evaluating the TSIFs of the interface cracks of bi-material subjected to linear and quadratic temperature distributions.

Integrated thermal and mechanical analysis of composite plates and shells

Thermal and mechanical analyses of structures are usually performed with non-consistent tools leading to an excessive effort in data adaptation. The application of 2D finite elements for both tasks relaxes this deficiency. With thermal lamination theories assuming a linear or quadratic temperature distribution in the thickness direction, suitable elements are developed and experimentally verified for both steady-state and transient problems. On the basis of a 2D global analysis accurate transverse stresses can be determined from local equilibrium conditions. For laminated plates and cylindrical shells a procedure is presented which allows us to reduce the order of shape functions.

On the determination of mixed mode stress intensity factors of an angled crack in a disc using FEM

This paper is concerned with the rigorous determination of the stress intensity factors of an arbitrary located and oriented angled crack in discs using the finite element method. Three different loading conditions are examined: boundary loadings resulting from disc attachments and/or interference fit with a rotating shaft, body forces resulting from rotation at a constant angular velocity and thermal stresses associated with a quadratic radial temperature distribution. Three techniques are adopted in the evaluation of the resulting mixed-mode stress intensity factors: direct extrapolation methods, virtual crack extension and J-integral method. Verification with available referenced solutions for the simple case of a radial crack is provided and merits and limitations associated with the above three methods are discussed.

Use of the equivalent volumetric enthalpy variation in non-linear phase-change processes: freezing-zone progression and thawing-time determination

Unlike pure substances that melt and solidify at a fixed temperature, some materials, such as the Karlsruhe Test Substance (methyl cellulose gel) freeze and thaw out over a range of temperatures. In this paper, the phase change process is studied using an equivalent volumetric enthalpy variation corresponding to the thermal field in the freezing or thawing zone instead of the latent heat appearing in the Stefan solution for the phase change of a pure substance. In order to determine the equivalent volumetric enthalpy variation, two different temperature distributions are assumed in the freezing zone of the substance undergoing a phase change. Results obtained from calculations applying this approach are compared with experimental measurements. Measurements of freezing-zone progression were made in the course of three freezing experiments where the first kind of boundary conditions were imposed on the samples. In addition, 35 values for thawing time have been taken from the literature presenting experiments made with the third kind of boundary conditions. Comparisons showed the best agreement when a quadratic temperature distribution was adopted. The proposed method seems to be useful for many practical applications, particularly because it allows calculations of the freezing or thawing time without necessitating the use of empirical parameters in Plank's original equation

Multi‐domain two‐ and three‐dimensional thermoelasticity by BEM

AbstractA computationally advantageous multi‐domain boundary element formulation is presented for uncoupled thermoelastic analysis of two‐ and three‐dimensional isotropic media. Particular integrals capable of representing exactly a quadratic temperature distribution in both two and three dimensions are employed. These particular integrals are derived using an orderly approach. When temperature data are provided at discrete points, the best fit quadratic distribution is generated in the least square error sense which, together with a multi‐domain approach, makes it possible to represent an arbitrary temperature distribution within acceptable bounds. Numerical examples are given to show the efficiency and effectiveness of the present formulation, which is now incorporated in a general purpose boundary element code named GPBEST.

Alternate BEM formulations for 2- and 3-D anisotropic thermoelasticity

Direct boundary element formulations are presented for the elastostatic analysis of two- and three-dimensional general anisotropic media subject to known temperature distributions. The thermal effects are incorporated in the analysis through a volume integral or alternatively, using particular integrals. A multi-region boundary element approach is adopted, in which particular integrals are derived assuming a quadratic temperature distribution in each sub-region. A multiple regression scheme is used for the best fit quadratic temperature distribution in each sub-region for temperature data given at discrete points. The multi-region analysis capability of the numerical implementation enhances its capacity to model discontinuous temperature fields as well as piecewise homogeneous media, such as laminated composites. The formulations presented are incorporated in a general purpose system and are illustrated with examples.