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Research Papers

Direct and Inverse Solutions for Thermal- and Stress-Transients and the Analytical Determination of Boundary Conditions Using Remote Temperature or Strain Data

[+] Author and Article Information
A. E. Segall

C. Drapaca, D. Engels, T. Zhu, H. Yang

Engineering Science and Mechanics,  The Pennsylvania State University, 212 EES Building, University Park, PA 16803

J. Pressure Vessel Technol 134(4), 041011 (Jul 09, 2012) (9 pages) doi:10.1115/1.4006353 History: Received December 08, 2011; Revised February 27, 2012; Published July 09, 2012

From an analytical standpoint, a majority of calculations use known boundary conditions (temperature or flux) and the so-called direct route to determine internal temperatures, strains, and/or stresses. For such problems where the thermal boundary condition is known a priori, the analytical procedure and solutions are tractable for the linear case where the thermophysical properties are independent of temperature. On the other hand, the inverse route where the boundary conditions must be determined from remotely determined temperature and/or flux data is much more difficult mathematically, as well as inherently sensitive to data errors (i.e., ill-posed). When solutions are available, they are often restricted to a harsh, albeit unrealistic step change in temperature or flux and/or are only valid for relatively short time frames before temperature changes occur at the far boundary. While the two approaches may seem to be at odds with each other, a generalized direct solution based on polynomial temperature or strain-histories can also be used to determine unknown boundary conditions via least-squares determination of coefficients. Once the inverse problem (and unknown boundary condition) is solved via these coefficients, the resulting polynomial can then be used with the generalized direct solution to determine the thermal- and stress-states as a function of time and position. When used for both thick slabs and tubes, excellent agreement was seen for various test cases. In fact, the derived solutions appear to be well suited for many thermal scenarios, provided the analysis is restricted to the time interval used to determine the polynomial and the thermophysical properties that do not vary with temperature. Since temperature dependent properties can certainly be an issue that affects accuracy in these types of calculations, some recent analytical procedures for both direct and inverse solutions are also discussed.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 3

Transient temperature distribution across the radius of a cylinder subjected to an exponential heating on the internal surface with convection on the outer surface (a/b = 0.667 and Bi = 12)

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Figure 4

Comparison of transient hoop-stress distributions across the radius of a cylinder subjected to exponential heating on the internal surface and convection on the outer surface

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Figure 1

Conceptualization of the direct and inverse problems for (a) thick-walled cylinder thermally loaded on the internal surface and (b) semi-infinite slab with one adiabatic surface

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Figure 2

Comparison of inverse predictions and asymptotic thermal loading of an infinitely long and hollow cylinder as a function of the Fourier number (a/b = 0.667 and Bi = 12)

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Figure 11

Inverse predictions for a thermal diffusivity that is decreasing with temperature

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Figure 5

Comparison of the transient axial-stress distributions across the radius of a cylinder subjected to exponential heating on the internal surface and convection on the outer surface

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Figure 7

Comparison of inverse predictions and triangular thermal loading of an infinitely long and hollow cylinder as a function of the Fourier number (a/b = 0.667 and Bi = 12)

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Figure 8

Comparison of inverse predictions from a linear-to-constant down-shock as a function of the Fourier number for a semi-infinite slab

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Figure 9

Comparison of stress solutions for a linear-to-constant down-shock as a function of the Fourier number for a semi-infinite slab

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Figure 6

Comparison of transient radial-stress distributions across the radius of a cylinder subjected to exponential heating on the internal surface and convection on the outer surface

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Figure 10

Inverse predictions for a thermal diffusivity that is increasing with temperature

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