End Thermal Stresses in a Long Circular Rod

1968 ◽  
Vol 35 (2) ◽  
pp. 267-273 ◽  
Author(s):  
W. H. Chu ◽  
F. T. Dodge
Keyword(s):  
Temperature Gradient ◽  
General Solution ◽  
Thermal Stresses ◽  
Linear Interpolation ◽  
Temperature Gradients ◽  
Characteristic Functions ◽  
Large Temperature ◽  
End Effect ◽  
Title Problem ◽  
Characteristic Values

The title problem is solved by the method of collocation utilizing complex nonorthogonal characteristic functions. It is shown that the characteristic values can be obtained by repeated linear interpolation without much difficulty. Ten roots are given for the case of Poisson’s ratio equaling 0.3. For large temperature gradients, an example is given which shows high end stresses. The general solution due to the end effect dies down at the rate of exp (–2.722 z/a) or faster, but its magnitude depends on the steepness of the temperature gradient. This paper also shows that the Saint-Venant principle may not always be sufficient, that the end stress could be critical, and that, therefore, it should be calculated.

scholarly journals Metamorphism of Dry Snow as a Result of Temperature Gradient and Vapor Density Differences

1983 ◽  
Vol 4 ◽  
pp. 3-9 ◽  
Author(s):  
E. E. Adams ◽  
R. L. Brown
Keyword(s):  
Temperature Gradient ◽  
Heat Conduction Equation ◽  
Temperature Gradients ◽  
Large Temperature ◽  
Local Minima ◽  
Vapor Density ◽  
Snow Density ◽  
Local Maxima ◽  
The Difference

A heat conduction equation for the determination of the temperature profile in a snowpack is developed. The magnitude of the temperature gradient tends to increase as the snow surface is approached, with local minima through layers of high snow density and local maxima above and below these layers. Calculations are made of the difference in vapor density in the pore and over the ice grain surfaces which border the pore. In the presence of sufficient temperature and temperature gradient, faceted crystals will develop near the top of the pore, as ice is sublimed away from the surfaces in the lower region. There will be a reduction in the percentage of rounded grains as the faceted form develops. The process is demonstrated to be enhanced at warm temperatures and large temperature gradients in low density snow.

Modelling stress state in reaction columns

2003 ◽  
Vol 3 ◽  
pp. 72-81
Author(s):  
R.R. Mavlyutov ◽  
L.Н. Gorchakov ◽  
Sh.Sh. Galyaliev ◽  
N.M. Tuykin ◽  
A.G. Khakimov ◽  
...  
Keyword(s):  
Stress State ◽  
Thermal Stresses ◽  
Heat Removal ◽  
Temperature Gradients ◽  
Large Temperature ◽  
Simulation Approach ◽  
Negative Effects ◽  
Chemical Reactors ◽  
Thermal Regimes ◽  
Deformed State

The operation of chemical reactors, particularly those for producing benzene, is associated with the generation of a large amount of heat. Inaccurately chosen thermal regimes of performance in the reaction columns at large temperature gradients can be responsible for the occurrence of high thermal stresses that decrease the endurance of equipment elements. Therefore, it is quite topical to reveal, using the mathematical simulation approach, both positive and negative effects in relation to stress deformed state of equipment elements under different regimes of heat removal from the outside surface of the reaction columns.

scholarly journals A Note for Probabilistic Model of Polymer Crystallization in Temperature Gradients

Crystals ◽  
2019 ◽  
Vol 9 (10) ◽  
pp. 538
Author(s):  
Chunlei Ruan ◽  
Yunlong Lv
Keyword(s):  
Temperature Gradient ◽  
Crystallization Kinetics ◽  
Probabilistic Model ◽  
Polymer Crystallization ◽  
Temperature Gradients ◽  
Kinetics Model ◽  
Gradient Field ◽  
Conversion Degree ◽  
Avrami Equation ◽  
Large Temperature

A polymer crystallization kinetics model is the most important way to characterize the crystallization rate of polymers. Because polymers are poor heat conductors, the cooling of thick-walled shapes results in temperature gradients. Piorkowska (Piorkowska, E. J. Appl. Polym. Sci., 2002, 86: 1351–1362.) derived the probabilistic analytical model of polymer crystallization in temperature gradients based on the Avrami equation. However, there are some misunderstandings when using this model. Here, isotactic polypropylene (iPP) is chosen as a model polymer and its crystallization is studied in a temperature gradient field. Based on the results of the Monte Carlo method, the probabilistic model methodology is discussed. The results show that when the product has a large temperature gradient and a large temperature difference, the probabilistic model cannot be used directly; instead, it is necessary to use the average probabilistic model. This means that the sample should be divided into several smaller parts and the probabilistic model used separately for each small part. The values are then averaged to obtain the mean conversion degree of the melt into spherulites for the whole product. The effects of the division number are also discussed. The goal of the present paper is to better understand the polymer crystallization kinetics model in terms of temperature gradients.

scholarly journals Metamorphism of Dry Snow as a Result of Temperature Gradient and Vapor Density Differences

1983 ◽  
Vol 4 ◽  
pp. 3-9 ◽  
Author(s):  
E. E. Adams ◽  
R. L. Brown
Keyword(s):  
Temperature Gradient ◽  
Heat Conduction Equation ◽  
Temperature Gradients ◽  
Large Temperature ◽  
Local Minima ◽  
Vapor Density ◽  
Snow Density ◽  
Local Maxima ◽  
The Difference

A heat conduction equation for the determination of the temperature profile in a snowpack is developed. The magnitude of the temperature gradient tends to increase as the snow surface is approached, with local minima through layers of high snow density and local maxima above and below these layers. Calculations are made of the difference in vapor density in the pore and over the ice grain surfaces which border the pore. In the presence of sufficient temperature and temperature gradient, faceted crystals will develop near the top of the pore, as ice is sublimed away from the surfaces in the lower region. There will be a reduction in the percentage of rounded grains as the faceted form develops. The process is demonstrated to be enhanced at warm temperatures and large temperature gradients in low density snow.

Thermal Stresses in Bodies Exhibiting Temperature-Dependent Elastic Properties

1952 ◽  
Vol 19 (3) ◽  
pp. 350-354
Author(s):  
H. H. Hilton
Keyword(s):  
Temperature Gradient ◽  
Elastic Properties ◽  
Plane Stress ◽  
Material Properties ◽  
Thermal Stresses ◽  
Temperature Gradients ◽  
Temperature Dependent ◽  
Steady State Temperature ◽  
Specific Temperature ◽  
Walled Cylinder

Abstract Expressions are derived for thermal stresses and strains due to a steady-state temperature gradient in a thick-walled cylinder and a circular thin plate, made of a material having temperature-dependent elastic properties. Two numerical examples are computed for specific temperature gradients and temperature-dependent elastic properties, which yield results showing that the maximum thermal stresses are appreciably lower and the maximum thermal strains are larger than the corresponding values obtained for temperature-independent properties. The validity of the thermal plane-stress assumptions is investigated and it is shown that such solutions, regardless of whether the material properties are temperature-dependent or constant, are only approximations. The smaller the temperature gradient the more closely are the plane-stress assumptions satisfied.

Role of substrate shape on thermal energy transmission in robotized wire and arc additive manufacturing

2019 ◽  
Vol 25 (7) ◽  
pp. 1285-1294 ◽  
Author(s):  
Rong Li ◽  
Jun Xiong
Keyword(s):  
Additive Manufacturing ◽  
Temperature Gradient ◽  
Thermal Energy ◽  
Molten Pool ◽  
Temperature Gradients ◽  
Maximum Temperature ◽  
Large Temperature ◽  
Energy Transmission ◽  
Content Type ◽  
Thin Walled

Purpose The purpose of this study is to present how the thermal energy transmission of circular parts produced in robotized gas metal arc (GMA)-based additive manufacturing was affected by the substrate shape through finite element analysis, including distributions of thermal energy and temperature gradient in the molten pool and deposited layers. Design/methodology/approach Three geometric shapes, namely, square, rectangle and round were chosen in simulation, and validation tests were carried out by corresponding experiments. Findings The thermal energy conduction ability of the deposited layers is the best on the round substrate and the worst on the rectangular substrate. The axial maximum temperature gradients in the molten pool along the deposition path with the round substrate are the largest during the deposition process. At the deposition ending moment, the circumferential temperature gradients of all layers with the round substrate are the largest. A large temperature gradient usually stands for a good heat conduction condition. Altogether, the round substrate is more suitable for the fabrication of circular thin-walled parts. Originality/value The predicted thermal distributions of the circular thin-walled part with various substrate shapes are helpful to understand the influence of substrate shape on the thermal energy transmission behavior in GMA-based additive manufacturing.

Transient Thermal Analysis of Electrical Machine

2009 ◽  
Author(s):  
Ahmad K. Sleiti
Keyword(s):  
Thermal Analysis ◽  
Temperature Gradient ◽  
Thermal Stresses ◽  
Electric Machines ◽  
Temperature Gradients ◽  
Electrical Machine ◽  
Transient Thermal Analysis ◽  
Operation Conditions ◽  
Severe Reduction ◽  
Transient Thermal

Transient thermal analysis of electric machine under realistic operation conditions and thermal losses is studied. A symmetrical portion of the stator and rotor is modeled and all thermal losses and cooling boundary conditions are applied according to operational duty cycle. It is found that there is a temperature gradient across the stator of more than 30 °C, across the rotor of more than 70 °C and across the whole machine of more than 100 °C. These temperature gradients could cause high thermal stresses and lead to severe reduction in the machine life. It is extremely important in future designs to consider reducing the temperature gradients by optimizing the design of the electric machines through advanced cooling techniques and strategies.

scholarly journals The principle of minimum entropy production and snow structure

2004 ◽  
Vol 50 (170) ◽  
pp. 342-352 ◽  
Author(s):  
Perry Bartelt ◽  
Othmar Buser
Keyword(s):  
Mass Transfer ◽  
Temperature Gradient ◽  
Entropy Production ◽  
Surface Curvature ◽  
The State ◽  
Temperature Gradients ◽  
Curvature Radius ◽  
Minimum Entropy ◽  
Snow Layer ◽  
Minimum Entropy Production

AbstractAn essential problem in snow science is to predict the changing form of ice grains within a snow layer. Present theories are based on the idea that form changes are driven by mass diffusion induced by temperature gradients within the snow cover. This leads to the well-established theory of isothermal- and temperature-gradient metamorphism. Although diffusion theory treats mass transfer, it does not treat the influence of this mass transfer on the form — the curvature radius of the grains and bonds — directly. Empirical relations, based on observations, are additionally required to predict flat or rounded surfaces. In the following, we postulate that metamorphism, the change of ice surface curvature and size, is a process of thermodynamic optimization in which entropy production is minimized. That is, there exists an optimal surface curvature of the ice grains for a given thermodynamic state at which entropy production is stationary. This state is defined by differences in ice and air temperature and vapor pressure across the interfacial boundary layer. The optimal form corresponds to the state of least wasted work, the state of minimum entropy production. We show that temperature gradients produce a thermal non-equilibrium between the ice and air such that, depending on the temperature, flat surfaces are required to mimimize entropy production. When the temperatures of the ice and air are equal, larger curvature radii are found at low temperatures than at high temperatures. Thus, what is known as isothermal metamorphism corresponds to minimum entropy production at equilibrium temperatures, and so-called temperature-gradient metamorphism corresponds to minimum entropy production at none-quilibrium temperatures. The theory is in good agreement with general observations of crystal form development in dry seasonal alpine snow.

Conjugated Heat Transfer in Skirt Hot Box in Pressure Vessels

2006 ◽  
Author(s):  
Hossein Shokouhmand ◽  
Manoochehr Bozorgmehrian
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Temperature Gradient ◽  
Thermal Stresses ◽  
Weld Line ◽  
Pressure Vessels ◽  
Conductive Heat ◽  
Air Pocket ◽  
Cyclic Operation ◽  
Conjugated Heat Transfer

Pressure vessels are common equipment in oil, gas and petrochemical industries. In a hot containing fluid vessel, excessive temperature gradient at junction of skirt to head (weld line), can cause unpredicted high thermal stresses; Thereby fracture of the vessel may occur as a result of cyclic operation. Providing a hot box (air pocket) in crotch space is a economical, applicable and easy mounted method in order to reduce the intensity of thermal stresses. Natural convection due to temperature difference between the wall of pocket, will absorb heat near the hot wall (head of the vessel) and release that near the cold wall (skirt of the vessel), then the skirt wall conducts heat to the earth as a fin. This conjugated heat transfer removes the temperature gradient boundary at welded junction. This phenomena will lead the temperature gradient on the weld line from a sudden to smooth behavior, thereby the skirt-head junction, that is a critical region, could be protected from excessive thermal stresses. In this paper the profit of hot box and conjugated heat transfer in cavity has been demonstrated experimentally. As a result it is shown that the conductive heat transfer through the skirt (which acts as a fin) ensures the continuation of natural convection in the box. Also the governing equations has been solved numerically and compared with experimental results.

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