Analysis of solids
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The document analyzes transient heat conduction in one-dimensional infinite and semi-infinite solids. It discusses the exact and approximate analytical solutions to the one-dimensional transient heat conduction equation for a plane wall, long cylinder and sphere. It presents the non-dimensionalization of the problem which reduces the number of independent variables from eight to three. It also discusses the analytical solutions for transient heat conduction in a semi-infinite solid under various boundary conditions like constant surface temperature and convection, involving error functions. Dimensionless temperature distributions and charts are presented.
Analysis of solids
- 1. ANALYSIS OF INFINITE AND SEMI- INFINITE SOLIDS - Dinesh Chandra pant 1 (IIT Roorkee)
- 2. 2 TRANSIENT HEAT CONDUCTION IN LARGE PLANE WALLS, LONG CYLINDERS, AND SPHERES WITH SPATIAL EFFECTS We will consider the variation of temperature with time and position in one-dimensional problems such as those associated with a large plane wall, a long cylinder, and a sphere. Schematic of the simple geometries in which heat transfer is one-dimensional. Transient temperature profiles in a plane wall exposed to convection from its surfaces for Ti >T∞.
- 4. 4 Non-dimensionalization reduces the number of independent variables in one- dimensional transient conduction problems from 8 to 3, offering great convenience in the presentation of results.
- 5. 5 Exact Solution of One-Dimensional Transient Conduction Problem
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- 7. 7 The analytical solutions of transient conduction problems typically involve infinite series, and thus the evaluation of an infinite number of terms to determine the temperature at a specified location and time.
- 8. 8 Approximate Analytical and Graphical Solutions Solution with one-term approximation The terms in the series solutions converge rapidly with increasing time, and for τ > 0.2, keeping the first term and neglecting all the remaining terms in the series results in an error under 2 percent.
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- 10. 10 (a) Midplane temperature Transient temperature and heat transfer charts (Heisler and Grober charts) for a plane wall of thickness 2L initially at a uniform temperature Ti subjected to convection from both sides to an environment at temperature T∞ with a convection coefficient of h.
- 13. 13 TRANSIENT HEAT CONDUCTION IN SEMI- INFINITE SOLIDS Schematic of a semi-infinite body. Semi-infinite solid: An idealized body that has a single plane surface and extends to infinity in all directions. The earth can be considered to be a semi-infinite medium in determining the variation of temperature near its surface. A thick wall can be modeled as a semi-infinite medium if all we are interested in is the variation of temperature in the region near one of the surfaces, and the other surface is too far to have any impact on the region of interest during the time of observation. For short periods of time, most bodies can be modeled as semi-infinite solids since heat does not have sufficient time to penetrate deep into the body.
- 14. 14 Transformation of variables in the derivatives of the heat conduction equation by the use of chain rule. Analytical solution for the case of constant temperature Ts on the surface error function complementary error function
- 15. 15 Error function is a standard mathematical function, just like the sine and cosine functions, whose value varies between 0 and 1.
- 17. 17 Dimensionless temperature distribution for transient conduction in a semi-infinite solid whose surface is maintained at a constant temperature Ts.
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- 20. 20 Variation of temperature with position and time in a semi-infinite solid initially at temperature Ti subjected to convection to an environment at T∞ with a convection heat transfer coefficient of h.
- 21. 21 Thank you !!