Planar approximation for diffusion in large cylinders and spheres
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Cylindrical Geometry
Planar Approximation for Diffusion in Large Cylinders and Spheres
C = C O + (C' - Co) i ' exp
v2)/(4Dt)] 2Dt
[--(r 2 +
o
Io(-~tt)
v dv
[2a]
J. U N N A M The diffusion solutions to cylindrical and spherical geometry systems are generally more complicated than the planar solutions. For large cylinders and spheres, the interface geometry is nearly planar on the diffusion scale. However, how large a cylinder or sphere should be and by how much a planar approximation deviates from the respective exact solution are important questions. The answers to these questions would be useful in checking a laboratory experiment or a failure in service. A single phase binary alloy system with constant diffusion coefficient (D) and zero flux boundary conditions is considered as an example. Figure 1 shows the nomenclature and schematic of the geometric models for planar, cylindrical, and spherical interfaces. The models consist of a B-rich region of composition C' and initial thickness (or radius) rl, and an A-rich region of composition C Oand initial thickness r3. The symbol r 2 designates the total thickness or radius considered, and is equal to r I + r 3. Diffusion between components A and B will result in a continuous composition variation as given by Fick's Second Law. For concentration independent diffusion coefficient (D) and zero-flux boundary conditions (J = 0), the volume fraction of element B at time t and distance r is given by the following equations. L2
Ct
__ C o
e x p ( - DtTr2n2)
[28]
I o is the modified Bessel function of the first kind of order zero, Jo and J ~are Bessel functions of the first kind of order zero and one, respectively, and or, is the n th root of J~(oLnrz) = 0. Equation [2a] is valid for r 3 ~ 6 v/-D~ and is well suited for short diffusion times. Equation [2b] is valid for all r I and r3, but converges rapidly only for small r v
Spherical Geometry
Ct - Co (
(r 1 + r] [r~ - r~ _erf \ ~ - ~ ] + erf ~ 2 ~ t )
C = Co + ~
r2~/~[exp(
--
E
(
(r'-r)2~ZlDt ]
(rl + r)2]]~
[3a]
f r~ + 2
Co) L r~
E e x p ( - D ta~)
r2r.=l
sin(otnr) x a2 sin2(ctnr2) [sin(anrl) - Otnrl cos(anr0]~.
r)) [la]
rz
j2--~,~2) ~nn ]
n = --oo
Co
n=oo
n=+oo
( e.(r +nr2 r) erf(" C = Co+(C'-
2rl
[~2
Co) r2+ r~ .=1
(e-Dta2) So(a.r) J '(a.r ,) ]
C = Co+ (C'-
Planar Geometry
C=Co+~
C = C o+ ( C ' -
+-
qT
n- I
-
[3b] where a. is the n th root of tan(a.r2) = Otnr2. Equation [3a] is valid for r 3 > 6v/-D~ and is well suited for short diffusion times. Equation [3b] is valid for all r~ and r 3, but converges rapidly only for small r 1.
Fl
sin( nrcr' I [mrr~] \ rz ] c~ ]
[lb] Planar
Equations [la] and [lb] are alternate forms of the exact solution, and Eq. [la] is well suited to short diffusion times while Eq. [lb] to long times.
J =0
Cylindrical
J =0-~
f
13
A
METALLURGICAL TRANSACTIONS A
J =0
co
r2
J. U N N A M is Visiting Professor, Department of Materials Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061. Manuscript submitted A
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