An infinite cylindrical rod (radius a) is initially at temperature T = To and its surface temperature is reduced to T = 0. Determine the time required for the cylinder to cool. To do this: (a) Write down the defining equations and scale the problem to reduce it to the form: —(rT,), = T₂ with T(r,0) = 1, and T(1,t) = 0, T(0, t) finite. (b) Proceed as in lectures to obtain a Bessel function expansion of the solution and evaluate the coefficients so that the initial condition is satisfied. (c) Plot out the solution as a function of r at t = = 0 to see how many terms are required for a good answer. (d) Plot out the temperature at r = 0 and thus determine the cooling time in unscaled
An infinite cylindrical rod (radius a) is initially at temperature T = To and its surface temperature is reduced to T = 0. Determine the time required for the cylinder to cool. To do this: (a) Write down the defining equations and scale the problem to reduce it to the form: —(rT,), = T₂ with T(r,0) = 1, and T(1,t) = 0, T(0, t) finite. (b) Proceed as in lectures to obtain a Bessel function expansion of the solution and evaluate the coefficients so that the initial condition is satisfied. (c) Plot out the solution as a function of r at t = = 0 to see how many terms are required for a good answer. (d) Plot out the temperature at r = 0 and thus determine the cooling time in unscaled
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:An infinite cylindrical rod (radius a) is initially at temperature T = To and its surface
temperature is reduced to T = 0. Determine the time required for the cylinder to cool.
To do this:
(a) Write down the defining equations and scale the problem to reduce it to the form:
-—-{(rTr)r
(b) Proceed as in lectures to obtain a Bessel function expansion of the solution and
evaluate the coefficients so that the initial condition is satisfied.
=
= T with T(r, 0) = 1, and T(1, t) = 0, T(0, t) finite.
(c) Plot out the solution as a function of r at t = 0 to see how many terms are
required for a good answer.
(d) Plot out the temperature at r = 0 and thus determine the cooling time in unscaled
terms as a function of radius etc.
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