Miscellaneous Exercises 361
andthecoefficientsshouldbedeterminedbya 0 = 2∫ 1
0f(ρ)ρdρ, an=∫ 1
(^0) ∫f(ρ)J^0 (λnρ)ρdρ
1
0 J^20 (λnρ)ρdρ
.
The functionf is known only roughly through experiment. Use the
numbers in the following table to finda 0 anda 1 by the trapezoidal rule
of numerical integration.ρ 0 0.1 0.2 0.3 0.4
f(ρ) 8.8 8.9 9.2 9.8 10.3
J 0 (λ 1 ρ) 10 0.964 0.858 0.696 0.493ρ 0.5 0.6 0.7 0.8 0.9 1.0
f(ρ) 11.2 12.0 13.1 14.1 14.8 15
J 0 (λ 1 ρ) 0.273 0.056 − 0. 135 − 0. 281 − 0. 373 − 0. 40334.In the article “Asymptotic analysis of intraparticle diffusion in GAC
batch reactors” [D.A. Lyn,Journal of Environmental Engineering, 122
(1996): 1013–1022], the author analyzes chemical diffusing into a spher-
ical particle, with a view to determining some parameter. The concen-
trationqis modeled in dimensionless variables by
1
r^2
∂
∂r(
r^2∂q
∂r)
=
∂q
∂t,^0 <r<^1 ,^0 <t,
q(r,t)bounded asr→ 0 ,
∂q
∂t(^1 ,t)=−D∂q
∂r(^1 ,t).
Separate the variables and find the eigenvalue problem, assuming that
q(r,t)=R(r)T(t).35.The eigenvalue problem that comes from Exercise 34 has a peculiar
boundary condition that prevents the eigenfunctions from being orthog-
onal. However, the author needs only the first terms of a series solution.
Find an equation for the eigenvalues. Confirm thatλ=0isasolution.
Find the next one numerically forD= 0 , 1 , 10.