Consider a thin sheet of solid material with a large surface. One side of the material at r = O is insulated, so that the heat flux there is zero, while the other side at (dimensionless) r = 1 is kept at (dimensionless temperature) of 0 0. Initially, the temperature distribition in the thin sheet is f(x) = (1-22)/2. We wish to determine the unsteady-state temperature distribution within the sheet. (a) Write down the governing equation and the initial and boundary conditions for the (dimensionless) temperature 0, i.e., one-dimensional unsteady-state conduction described in class. (b) Derive the eigenvalues and eigenfunctions for the governing equation. (c) Derive the complete solution of the problem.
Consider a thin sheet of solid material with a large surface. One side of the material at r = O is insulated, so that the heat flux there is zero, while the other side at (dimensionless) r = 1 is kept at (dimensionless temperature) of 0 0. Initially, the temperature distribition in the thin sheet is f(x) = (1-22)/2. We wish to determine the unsteady-state temperature distribution within the sheet. (a) Write down the governing equation and the initial and boundary conditions for the (dimensionless) temperature 0, i.e., one-dimensional unsteady-state conduction described in class. (b) Derive the eigenvalues and eigenfunctions for the governing equation. (c) Derive the complete solution of the problem.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:2. Cooling of a Sheet of Materials
Consider a thin sheet of solid material with a large surface. One side of the material at r =
O is insulated, so that the heat flux there is zero, while the other side at (dimensionless) a =
1 is kept at (dimensionless temperature) of 0 = 0. Initially, the temperature distribition in
the thin sheet is f(x) = (1-x2)/2. We wish to determine the unsteady-state temperature
distribution within the sheet.
(a) Write dowm the governing equation and the initial and boundary conditions for the
(dimensionless) temperature 0, i.e., one-dimensional unsteady-state conduction described
in class.
(b) Derive the eigenvalues and eigenfunctions for the governing equation.
(c) Derive the complete solution of the problem.
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