Steady-State Temperatures in an Anisotropic Strip

1990 ◽  
Vol 112 (1) ◽  
pp. 16-20 ◽  
Author(s):  
Zhang Xiangzhou
Keyword(s):  
Steady State ◽  
Solution Method ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Rigorous Solution ◽  
Steady State Temperature ◽  
Distribution Features ◽  
Matching Technique ◽  
Principal Directions ◽  
Strip Geometry

This article deals with the development of a rigorous solution to the steady-state temperature in an anisotropic strip. The solution is given with respect to a coordinate system (x, y), which conforms with the strip geometry but does not necessarily coincide with the principal directions of the anisotropic material. Using a partitioning–matching technique and the separation of variables method, exact expressions are obtained for temperatures in the strip under prescribed boundary temperature conditions. Numerical values of the temperatures and heat flux are provided in graphic form. Also, a discussion is presented regarding the solution method and the temperature distribution features in the heat conduction problem of an anisotropic medium.

Transient Damped Wave Conduction and Relaxation in Human Skin and Thermal Wear During Winter

2008 ◽  
Author(s):  
Kal Renganathan Sharma
Keyword(s):  
Steady State ◽  
Human Skin ◽  
Separation Of Variables ◽  
Relaxation Times ◽  
Thermal Relaxation ◽  
Transient Temperature ◽  
Heat Sources ◽  
Maximum Heat ◽  
Steady State Temperature ◽  
Fabric Layer

Damped wave conduction and relaxation in the human skin layer and thermal fabric layer are modeled with a temperature dependent heat source in the human tissue layer. Steady state temperature profiles are derived from the Fourier heat conduction equation. The general solution for the temperature is assumed to be a sum of the transient temperature and steady state temperature. This makes the boundary conditions in space for the skin and fabric layers homogeneous for the transient temperarature. The hyperbolic PDE is solved for by the method of separation of variables. The use of final condition in time in addition to the initial temperature condition leads to bounded infinite Fourier series solutions. These solutions are bounded and does not violate second law of thermodynamics. The model can be used to interpret experimental observations of maximum heat flux that is a parameter of the warm/cool feeling of human skin in winter. For large relaxation times of human skin tissue, τrs>(1+U*)2(b−a)216π2αs, the transient temperature can be expected to undergo oscillations. These oscillations will be supercritical and grow with time for strong heat sources, U* > 1 and may be subcritical damped oscillatory for weak heat sources, U* < 1. For large thermal relaxation times of thermal fabric material, τrf>a24π2αs, the transient temperature in the thermal fabric layer may be expected to be subcritical damped oscillatory.

Analytical Solution of Unsteady State Conduction Problem with Heat Source

2012 ◽  
Vol 204-208 ◽  
pp. 4364-4367
Author(s):  
Shun Yu Su ◽  
Qin Huang
Keyword(s):  
Heat Conduction ◽  
Steady State ◽  
Analytical Solution ◽  
Heat Source ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Infinite Plate ◽  
Unsteady State ◽  
Inhomogeneous Problem ◽  
Separation Of Variables Method

Separation-of-variables method is one of the analytical solution methods to solve unsteady state heat conduction problems. But unsteady state conduction with heat source is an inhomogeneous problem. It can not be solved by separation-of-variables method directly. The combination of variables division method and separation-of-variables method was applied in this paper to deal with heat conduction with heat source in an infinite plate wall. The problem was divided to a steady state and inhomogeneous heat conduction problem, and an unsteady state and homogeneous heat conduction problem by variables division method. The steady state and inhomogeneous problem can be integrated directly. The unsteady state and homogeneous problem can be transformed to the problem that can be solved by separation-of-variables method directly through variable substitution. The unsteady state temperature field in the infinite plate wall was then obtained.

scholarly journals A Head Formulation for the Steady-State Analysis of Water Distribution Systems Using an Explicit and Exact Expression of the Colebrook–White Equation

Water ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1163
Author(s):  
Mengning Qiu ◽  
Avi Ostfeld
Keyword(s):  
Steady State ◽  
Distribution System ◽  
Distribution Systems ◽  
Water Distribution ◽  
Solution Method ◽  
Water Distribution System ◽  
Academic Research ◽  
Exact Expression ◽  
Contamination Source

Steady-state demand-driven water distribution system (WDS) solution is the bedrock for much research conducted in the field related to WDSs. WDSs are modeled using the Darcy–Weisbach equation with the Swamee–Jain equation. However, the Swamee–Jain equation approximates the Colebrook–White equation, errors of which are within 1% for ϵ/D∈[10−6,10−2] and Re∈[5000,108]. A formulation is presented for the solution of WDSs using the Colebrook–White equation. The correctness and efficacy of the head formulation have been demonstrated by applying it to six WDSs with the number of pipes ranges from 454 to 157,044 and the number of nodes ranges from 443 to 150,630. The addition of a physically and fundamentally more accurate WDS solution method can improve the quality of the results achieved in both academic research and industrial application, such as contamination source identification, water hammer analysis, WDS network calibration, sensor placement, and least-cost design and operation of WDSs.

scholarly journals Identification of DC Thermal Steady-State Differential Inductance of Ferrite Power Inductors

Energies ◽  
2021 ◽  
Vol 14 (13) ◽  
pp. 3854
Author(s):  
Salvatore Musumeci ◽  
Luigi Solimene ◽  
Carlo Stefano Ragusa
Keyword(s):  
Steady State ◽  
Input Voltage ◽  
Switching Frequency ◽  
Ohmic Losses ◽  
Steady State Temperature ◽  
Wide Range ◽  
Average Current ◽  
Power Inductors ◽  
Saturable Inductor ◽  
Thermal Steady State

In this paper, we propose a method for the identification of the differential inductance of saturable ferrite inductors adopted in DC–DC converters, considering the influence of the operating temperature. The inductor temperature rise is caused mainly by its losses, neglecting the heating contribution by the other components forming the converter layout. When the ohmic losses caused by the average current represent the principal portion of the inductor power losses, the steady-state temperature of the component can be related to the average current value. Under this assumption, usual for saturable inductors in DC–DC converters, the presented experimental setup and characterization method allow identifying a DC thermal steady-state differential inductance profile of a ferrite inductor. The curve is obtained from experimental measurements of the inductor voltage and current waveforms, at different average current values, that lead the component to operate from the linear region of the magnetization curve up to the saturation. The obtained inductance profile can be adopted to simulate the current waveform of a saturable inductor in a DC–DC converter, providing accurate results under a wide range of switching frequency, input voltage, duty cycle, and output current values.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

Steady-State Thermal Stresses in Wedge-Shaped Cutting Tools

1975 ◽  
Vol 97 (3) ◽  
pp. 1060-1066
Author(s):  
P. F. Thomason
Keyword(s):  
Thermal Stress ◽  
Steady State ◽  
Metal Cutting ◽  
Thermal Stresses ◽  
Heat Conduction Problem ◽  
Equivalent Stress ◽  
Cutting Tools ◽  
Thermoelastic Stress ◽  
Plastic State ◽  
Steady State Heat

Closed form expressions for the steady-state thermal stresses in a π/2 wedge, subject to constant-temperature heat sources on the rake and flank contact segments, are obtained from a conformal mapping solution to the steady-state heat conduction problem. It is shown, following a theorem of Muskhelishvili, that the only nonzero thermal stress in the plane-strain wedge is that acting normal to the wedge plane. The thermal stress solutions are superimposed on a previously published isothermal cutting-load solution, to give the complete thermoelastic stress distribution at the wedge surfaces. The thermoelastic stresses are then used to determine the distribution of the equivalent stress, and this gives an indication of the regions on a cutting tool which are likely to be in the plastic state. The results are discussed in relation to the problems of flank wear and rakeface crater wear in metal cutting tools.

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