The Quaternion Fourier Number Transform

Abstract We numerically investigate the melting and solidi?cation behavior of phase change materials encapsulated in a small-radii cylinder subjected to a cyclic convective boundary condition (square wave). Initially, we explore the effect of the Stefan and Biot numbers on the non-dimensionalized time required (i.e. reference Fourier number Tref ) for a PCM initially held at Tcold to melt and reach the cross?ow temperature Thot. The increase in either Stefan or Biot number decreases Tref and can be predicted accurately using a correlation developed in this work. The variations of the PCM melt fraction, surface temperature, and heat transfer rate as a function of Fourier number are reported and analyzed for the above process. We further study the effect of the cyclic Fourier number on the periodic melting and freezing process. The melting or freezing front initiates at the outer periphery of the PCM and propagates towards the center. At higher frequencies, multiple two-phase interfaces are generated (propagating inward), and higher overall heat transfer is achieved as the surface temperature oscillates in the vicinity of the melting temperature, which increases the effective temperature difference driving the convective heat transfer.

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Coaxial Thermocouples for Heat Transfer Measurements in Long-Duration High Enthalpy Flows

Sensors ◽

10.3390/s20185254 ◽

2020 ◽

Vol 20 (18) ◽

pp. 5254

Author(s):

Shizhong Zhang ◽

Qiu Wang ◽

Jinping Li ◽

Xiaoyuan Zhang ◽

Hong Chen

Keyword(s):

Heat Transfer ◽

Heat Flux ◽

Surface Temperature ◽

Thermal Effusivity ◽

Fast Response ◽

Test Time ◽

Physical Parameters ◽

Fourier Number ◽

High Enthalpy ◽

Long Duration

Coaxial thermocouples have the advantages of fast response and good durability. They are widely used for heat transfer measurements in transient facilities, and researchers have also considered their use for long-duration heat transfer measurements. However, the model thickness, transverse heat transfer, and changes in the physical parameters of the materials with increasing temperature influence the accuracy of heat transfer measurements. A numerical analysis of coaxial thermocouples is conducted to determine the above influences on the measurement deviation. The minimum deviation is obtained if the thermal effusivity of chromel that changes with the surface temperature is used to derive the heat flux from the surface temperature. The deviation of the heat flux is less than 5.5% when the Fourier number is smaller than 0.255 and 10% when the Fourier number is smaller than 0.520. The results provide guidance for the design of test models and coaxial thermocouples in long-duration heat transfer measurements. The numerical calculation results are verified by a laser radiation heating experiment, and heat transfer measurements using coaxial thermocouples in an arc tunnel with a test time of several seconds are performed.

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An Experimental and Theoretical Study of Heat Conduction and Vulcanization of Rubber Compounds in Molds

Rubber Chemistry and Technology ◽

10.5254/1.3536114 ◽

1987 ◽

Vol 60 (1) ◽

pp. 140-158 ◽

Cited By ~ 14

Author(s):

Dancheng Kong ◽

James L. White ◽

Frederick C. Weissert ◽

Nobuyuki Nakajima

Keyword(s):

Heat Conduction ◽

Curing Kinetics ◽

Damkohler Number ◽

Fourier Number ◽

Fundamental Study ◽

Damköhler Number ◽

Rubber Compounds ◽

Cure Temperature ◽

Black Other ◽

Kinetics Of

Abstract A fundamental study on curing of rubber compounds in molds is presented. We have measured the thermal conductivity of a range of rubber compounds determining the influence of carbon black, other fillers, and oil. The heats of reaction associated with the curing kinetics of model compounds were measured. A mathematical model is proposed to predict the temperature profiles for curing a reactive slab. This involves inclusion of an energy generation rate, which depends on time and temperature. This is expressed through a Damkohler number. Solutions of the heat conduction equation are interpreted in terms of the Fourier number and the Damkohler number. Calculations are carried out using experimentally determined thermal conductivities and curing kinetics. Thick parts are shown to heat up more slowly (associated with the Fourier number) and to show greater overshoots of cure temperature (associated with the Damkohler number).

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Some Analytical and Numerical Solutions to Inverse Problems Applied to Optimizing Phase-Transformation Tracking in Gas Quenching

Journal of Heat Transfer ◽

10.1115/1.1517271 ◽

2003 ◽

Vol 125 (1) ◽

pp. 1-10 ◽

Cited By ~ 11

Author(s):

Michael Vynnycky ◽

Je´ro^me Ferrari ◽

Noam Lior

Keyword(s):

Phase Transformation ◽

Numerical Solutions ◽

Asymptotic Methods ◽

Convective Heat Transfer Coefficient ◽

Asymptotic Series ◽

Cooling Curve ◽

Nonlinear Heat Equation ◽

Attractive Alternative ◽

Fourier Number ◽

Gas Quenching

A transient inverse heat conduction problem focused on gas quenching of steel plates and rings is posed and solved, both analytically and numerically. The quenching objective is to calculate the transient convective heat transfer coefficient which would produce an optimized phase transformation cooling curve. The governing nonlinear heat equation is nondimensionalised, and a small parameter, the reciprocal of the Fourier number, is identified. This allows the construction of an analytic solution in the form of an asymptotic series. For higher values of the reciprocal Fourier number, a numerical scheme incorporating the function specification and Keller Box methods is used to generate solutions. Comparison of the results proves favorable, and suggests that for this inverse problem asymptotic methods provide an attractive alternative to solely numerical ones.

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Uncoupled Wave Systems and Dispersion in an Infinite Solid Cylinder

Journal of Applied Mechanics ◽

10.1115/1.3176089 ◽

1989 ◽

Vol 56 (2) ◽

pp. 347-355 ◽

Cited By ~ 2

Author(s):

Yoon Young Kim

Keyword(s):

Cylindrical Surface ◽

Wave Motion ◽

Fourier Number ◽

Large Wave ◽

Surface Conditions ◽

Asymptotic Nature ◽

Wave Numbers ◽

Complex Wave ◽

Insight Into ◽

Free Cylindrical Surface

In this study, it is shown that there exist uncoupled wave systems for general non-axisymmetric wave propagation in an infinite isotropic cylinder. Two cylindrical surface conditions corresponding to the uncoupled wave systems are discussed. The solutions of the uncoupled wave systems are shown to provide proper bounds of Pochhammer’s equation for a free cylindrical surface. The bounds, which are easy to construct for any Fourier number in the circumferential direction, can be used to trace the branches of Pochhammer’s equation. They also give insight into the modal composition of the branches of Pochhammer’s equation at and between the intersections of the bounds. More refined dispersion relations of Pochhammer’s equation are possible through an asymptotic analysis of the itersections of the branches of Pochhammer’s equation with one family of the bounds. The asymptotic nature of wave motion corresponding to large wave numbers, imaginary or complex, for Pochhammer’s equation is studied. The wave motion is asymptotically equivoluminal for large imaginary wave numbers, and is characterized by coupled dilatation and shear for large complex wave numbers.

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Generalized Correlations for Predicting Optimal Spacing of Decaying Heat Sources in a Conducting Medium

Journal of Heat Transfer ◽

10.1115/1.4033461 ◽

2016 ◽

Vol 138 (9) ◽

Author(s):

Young-Jin Baik ◽

Ich-Long Ngo ◽

Jang Min Park ◽

Chan Byon

Keyword(s):

Thermal Resistance ◽

Numerical Study ◽

Surrounding Medium ◽

Conducting Medium ◽

Heat Sources ◽

Two Dimensional ◽

Fourier Number ◽

Array Type ◽

Interpolation Technique ◽

Optimal Spacing

This paper presents a numerical study for predicting the optimal spacing (OS) of decaying heat sources/sinks in a conducting medium. The optimal configuration that minimizes the overall thermal resistance between the cylinder array and surrounding medium is tracked using interpolation technique. Consequently, the dimensionless OS obtained is of the order of 0.442th power of the Fourier number (Fo) defined as a function of the decaying time constant, which differs from the 0.5th value reported in previous study. In addition, the overall thermal resistance is shown to be highly dependent on the dimensionless spacing and Fo, while the OS also depends on the array type of the cylinders. Based on the extensive numerical study, closed-form correlations are proposed for predicting the OS of decaying heat sources/sinks in both quadratic and hexagonal arrangements. These results can be widely utilized for optimally positioning heat sources/sinks with two dimensional configurations.

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Determination of thermal diffusivity of the material by numerical-analytical model of a semi-bounded body

Izvestiya Ferrous Metallurgy ◽

10.17073/0368-0797-2020-6-474-480 ◽

2020 ◽

Vol 63 (6) ◽

pp. 474-480

Author(s):

A. K. Sokolov

Keyword(s):

Thermal Diffusivity ◽

Heated Layer ◽

Algebraic Expression ◽

Temperature Perturbation ◽

Convective Heating ◽

Time Interval ◽

Algebraic Equations ◽

Fourier Number ◽

Absolute Deviation ◽

Bounded Body

A mathematical description of the material thermal diffusivity aт in a semi-bounded body is proposed with a relatively simple algorithm for its numerical and analytical by solving the inverse problem of thermal conductivity. To solve the problem, it is necessary to obtain the temperature values of the unbounded plate as a result of a thermophysical experiment. A plate can be conditionally considered as a semi-bounded body as long as the Fourier number Fo ≤ Foк (Foк ≈ 0.04–0.06). It is assumed that the temperature distribution over cross-section of the heated layer of the plate R is sufficiently described by a power function whose exponent depends linearly on the Fourier number. A simple algebraic expression is obtained for calculating aт in the time interval ∆τ from the dynamics of temperature change T(Rп , τ) of a plate surface with thickness Rп heated under boundary conditions of the second kind. Temperature of the second surface of the plate T(0, τ) is used only to determine the time of the end of experiment τк. The moment of time τк, in which the temperature perturbation reaches the adiabatic surface x = 0, can be set by the condition T(Rп , τк) – T(0, τ = 0) = 0,1 K. The method of approximate calculation of dynamics of changes in depth of the heated layer R by the values of Rп , τк , and τ is proposed. Calculation of a т for the time interval ∆τ is reduced to an iterative solution of a system of three algebraic equations by matching the Fourier number, for example, using a standard Microsoft Excel procedure. Estimation of the accuracy of a т calculation was made by the test (initial) temperature field of the refractory plate with the thickness Rп = 0.05 m, calculated by the finite difference method under the initial condition T(x, τ = 0) = 300 (0 ≤ x ≤ Rп) at radiation-convective heating. The heating time was 260 s. Calculation of aт, i was performed for 10 time moments τi + 1 = τi + Δτ, τ = 26 s. Average mass temperature of the heated layer for the whole time was T = 302 K. The arithmetic-mean absolute deviation of aт(T = 302 K) from the initial value at the same temperature was 2.8 %. Application of the method will simplify the conduct and processing of experiments to determine the thermal diffusivity of materials.

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