Semi–analytical, piecewise temperature–time distributions in solid bodies of regular shape affected by uniform surface heat flux employing the Method Of Lines (MOL) and the eigenvalue method

The Transversal Method Of Lines (TMOL) or Rothe method is a general technique for solving parabolic partial differential equations that uses a two-point backward finite-difference formulation for the time derivative and differential spatial derivatives. This hybrid approach leads to transformed ordinary differential equations where the spatial coordinate is the independent variable and the time appears as an embedded parameter. The transformed ordinary differential equations may have constant or variable coefficients depending on the coordinate system and are first-order accurate. In this work, TMOL is applied to the 1-D heat equation for large plates, long cylinders and spheres with constant thermophysical properties, uniform initial temperature and prescribed surface heat flux. The analytic solutions of the adjoint heat equations are performed with the symbolic Maple software. It is demonstrated that the approximate semi-analytic TMOL temperature distributions for the three simple bodies are much better than first-order accurate. This signifies that TMOL temperature distributions are not only valid for short times, but they are valid for the entire heating period involving short, moderate and long times.

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Transversal Method of Lines for Unsteady Heat Conduction With Uniform Surface Heat Flux

Journal of Heat Transfer ◽

10.1115/1.4028082 ◽

2014 ◽

Vol 136 (11) ◽

Cited By ~ 3

Author(s):

Antonio Campo ◽

José Garza

Keyword(s):

Heat Flux ◽

Differential Equations ◽

Heat Equation ◽

Ordinary Differential Equations ◽

Surface Heat Flux ◽

Method Of Lines ◽

Surface Heat ◽

Second Order ◽

Partial Differential ◽

Stationary Heat

The transversal method of lines (TMOL) is a general hybrid technique for determining approximate, semi-analytic solutions of parabolic partial differential equations. When applied to a one-dimensional (1D) parabolic partial differential equation, TMOL engenders a sequence of adjoint second-order ordinary differential equations, where in the space coordinate is the independent variable and the time appears as an embedded parameter. Essentially, the adjoint second-order ordinary differential equations that result are of quasi-stationary nature, and depending on the coordinate system may have constant or variable coefficients. In this work, TMOL is applied to the unsteady 1D heat equation in simple bodies (large plate, long cylinder, and sphere) with temperature-invariant thermophysical properties, constant initial temperature and uniform heat flux at the surface. In engineering applications, the surface heat flux is customarily provided by electrical heating or radiative heating. Using the first adjoint quasi-stationary heat equation for each simple body with one time jump, it is demonstrated that approximate, semi-analytic TMOL temperature solutions with good quality are easily obtainable, regardless of time. As a consequence, usage of the more involved second adjoint quasi-stationary heat equation accounting for two consecutive time jumps come to be unnecessary.

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Effortless Application of the Method of Lines for the Inverse Estimation of Temperatures in a Large Slab With Two Different Surface Heating Waveforms

Journal of Heat Transfer ◽

10.1115/1.2993141 ◽

2008 ◽

Vol 131 (2) ◽

Cited By ~ 3

Author(s):

Antonio Campo ◽

John Ho

Keyword(s):

Heat Flux ◽

Heat Conduction ◽

Heat Conduction Problem ◽

Method Of Lines ◽

Time History ◽

Numerical Differentiation ◽

Heat Fluxes ◽

Exposed Surface ◽

The Method Of Lines ◽

Temperature Time

The boundary inverse heat conduction problem (BIHCP) deals with the determination of the surface heat flux or the surface temperature from measured transient temperatures inside a conducting body where the initial temperature is known. This work addresses a BIHCP related to the spatiotemporal heat conduction in a large slab when a time-variable heat flux is prescribed at an exposed surface and the other surface is thermally insulated. Two different heating waveforms are studied: a constant heat flux and a time-dependent triangular heat flux. The numerical temperature-time history at the insulated surface of the large slab provides the “temperature-time measurement” with one temperature sensor. Framed in the theory of the method of lines (MOL) first and employing rudimentary concepts of numerical differentiation later, the main objective of this paper is to develop a simple computational methodology to estimate the temporal evolution of temperature at the exposed surface of the large slab receiving the two distinct heat fluxes. In the end, it is confirmed that excellent predictions of the surface temperatures versus time are achievable for the two cases tested while employing the smallest possible system of two heat conduction differential equations of first-order.

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Analysis of significant measures for thermal conductivity

Izmeritel`naya Tekhnika ◽

10.32446/0368-1025it.2020-3-35-42 ◽

2020 ◽

pp. 35-42

Author(s):

Yuri P. Zarichnyak ◽

Vyacheslav P. Khodunkov

Keyword(s):

Thermal Conductivity ◽

Heat Flux ◽

Surface Heat Flux ◽

Heat Flux Density ◽

Measurement Method ◽

Conductivity Measurement ◽

Surface Heat ◽

Measuring Instrument ◽

New Class ◽

Achievable Accuracy

The analysis of a new class of measuring instrument for heat quantities based on the use of multi-valued measures of heat conductivity of solids. For example, measuring thermal conductivity of solids shown the fallacy of the proposed approach and the illegality of the use of the principle of ambiguity to intensive thermal quantities. As a proof of the error of the approach, the relations for the thermal conductivities of the component elements of a heat pump that implements a multi-valued measure of thermal conductivity are given, and the limiting cases are considered. In two ways, it is established that the thermal conductivity of the specified measure does not depend on the value of the supplied heat flow. It is shown that the declared accuracy of the thermal conductivity measurement method does not correspond to the actual achievable accuracy values and the standard for the unit of surface heat flux density GET 172-2016. The estimation of the currently achievable accuracy of measuring the thermal conductivity of solids is given. The directions of further research and possible solutions to the problem are given.

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COMBINED CONVECTION BOUNDARY-LAYER FLOW ALONG A VERTICAL CYLINDER WITH A POROUS WALL AND SURFACE HEAT FLUX

Heat Transfer Research ◽

10.1615/heattransres.2012003599 ◽

2012 ◽

Vol 43 (6) ◽

pp. 483-507

Author(s):

Abbasali Abouei Mehrizi ◽

Ganji Domairry ◽

V. Khamseh

Keyword(s):

Boundary Layer ◽

Heat Flux ◽

Boundary Layer Flow ◽

Surface Heat Flux ◽

Vertical Cylinder ◽

Porous Wall ◽

Surface Heat ◽

Convection Boundary Layer ◽

Convection Boundary ◽

Layer Flow

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Transient Response of a Steady Vertical Flow which is Subjected to a Change in Surface Heat Flux in Porous Media

International Symposium on Transient Convective Heat Transfer ◽

10.1615/ichmt.1996.transientconvheattransf.500 ◽

1996 ◽

Author(s):

Simon D. Harris ◽

Derek B. Ingham ◽

Ioan Pop

Keyword(s):

Heat Flux ◽

Porous Media ◽

Transient Response ◽

Surface Heat Flux ◽

Surface Heat ◽

Vertical Flow

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Two Successive Thermal Inverse Problems Solved for Plasma Facing Components Inside JET Tokamak: Estimation of Surface Heat Flux and Thermal Resistance of a Surface Carbon Layer

Proceedings of the 15th International Heat Transfer Conference ◽

10.1615/ihtc15.pls.009415 ◽

2014 ◽

Author(s):

jonathan gaspar ◽

Fabrice Rigollet ◽

Jean-Laurent Gardarein ◽

Christophe Le Niliot ◽

Yann Corre

Keyword(s):

Heat Flux ◽

Inverse Problems ◽

Thermal Resistance ◽

Surface Heat Flux ◽

Carbon Layer ◽

Surface Heat ◽

Surface Carbon ◽

Plasma Facing Components

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Heat Transfer Characteristics of Downward Facing Hot Horizontal Surfaces Using Mist Jet Impingement

Volume 6: Thermal-Hydraulics ◽

10.1115/icone25-67537 ◽

2017 ◽

Cited By ~ 2

Author(s):

Ashutosh Kumar Yadav ◽

Parantak Sharma ◽

Avadhesh Kumar Sharma ◽

Mayank Modak ◽

Vishal Nirgude ◽

...

Keyword(s):

Heat Transfer ◽

Heat Flux ◽

Surface Heat Flux ◽

Cooling System ◽

Water Pressure ◽

Jet Impingement ◽

Surface Heat ◽

Heat Transfer Characteristics ◽

Impingement Cooling ◽

Transfer Characteristics

Impinging jet cooling technique has been widely used extensively in various industrial processes, namely, cooling and drying of films and papers, processing of metals and glasses, cooling of gas turbine blades and most recently cooling of various components of electronic devices. Due to high heat removal rate the jet impingement cooling of the hot surfaces is being used in nuclear industries. During the loss of coolant accidents (LOCA) in nuclear power plant, an emergency core cooling system (ECCS) cool the cluster of clad tubes using consisting of fuel rods. Controlled cooling, as an important procedure of thermal-mechanical control processing technology, is helpful to improve the microstructure and mechanical properties of steel. In industries for heat transfer efficiency and homogeneous cooling performance which usually requires a jet impingement with improved heat transfer capacity and controllability. It provides better cooling in comparison to air. Rapid quenching by water jet, sometimes, may lead to formation of cracks and poor ductility to the quenched surface. Spray and mist jet impingement offers an alternative method to uncontrolled rapid cooling, particularly in steel and electronics industries. Mist jet impingement cooling of downward facing hot surface has not been extensively studied in the literature. The present experimental study analyzes the heat transfer characteristics a 0.15mm thick hot horizontal stainless steel (SS-304) foil using Internal mixing full cone (spray angle 20 deg) mist nozzle from the bottom side. Experiments have been performed for the varied range of water pressure (0.7–4.0 bar) and air pressure (0.4–5.8 bar). The effect of water and air inlet pressures, on the surface heat flux has been examined in this study. The maximum surface heat flux is achieved at stagnation point and is not affected by the change in nozzle to plate distance, Air and Water flow rates.

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Nanofluid Flow on a Shrinking Cylinder with Al2O3 Nanoparticles

Mathematics ◽

10.3390/math9141612 ◽

2021 ◽

Vol 9 (14) ◽

pp. 1612

Author(s):

Iskandar Waini ◽

Anuar Ishak ◽

Ioan Pop

Keyword(s):

Heat Transfer ◽

Heat Flux ◽

Friction Factor ◽

Surface Heat Flux ◽

Transfer Rate ◽

Surface Heat ◽

Al2o3 Nanoparticles ◽

Nanofluid Flow ◽

Curvature Parameter ◽

Over Time

This study investigates the nanofluid flow towards a shrinking cylinder consisting of Al2O3 nanoparticles. Here, the flow is subjected to prescribed surface heat flux. The similarity variables are employed to gain the similarity equations. These equations are solved via the bvp4c solver. From the findings, a unique solution is found for the shrinking strength λ≥−1. Meanwhile, the dual solutions are observed when λc<λ<−1. Furthermore, the friction factor Rex1/2Cf and the heat transfer rate Rex−1/2Nux increase with the rise of Al2O3 nanoparticles φ and the curvature parameter γ. Quantitatively, the rates of heat transfer Rex−1/2Nux increase up to 3.87% when φ increases from 0 to 0.04, and 6.69% when γ increases from 0.05 to 0.2. Besides, the profiles of the temperature θ(η) and the velocity f’(η) on the first solution incline for larger γ, but their second solutions decline. Moreover, it is noticed that the streamlines are separated into two regions. Finally, it is found that the first solution is stable over time.

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