Consider the problem of diffusion of carbon into an iron cannonball initially free of carbon. This process is often conducted in order to case-harden (i.e., produce a steel casing on) cannonballs. Suppose the cannonball is a sphere of radius r = b. Initially, the cannonball contains no carbon,

Question

Consider the problem of diffusion of carbon into an iron cannonball initially free of carbon. This process is often conducted in order to case-harden (i.e., produce a steel casing on) cannonballs. Suppose the cannonball is a sphere of radius r = b. Initially, the cannonball contains no carbon, but at t = 0 it is contacted with a carbon source that keeps the surface concentration at the constant value [latex]C_{s}[/latex]. We want to find the concentration distribution of carbon as a function of time.

Accepted Answer

Assuming that the concentration of carbon in the cannonball depends only on the radial distance r from the center of the ball, the concentration obeys the diffusion equation

[latex]\frac{\partial C }{\partial t} = D \frac{1}{r^{2} } \frac{\partial }{\partial r} (r^{2} \frac{\partial C}{\partial r} )[/latex] (10.8.24)

where D is the diffusivity of carbon in iron. The initial and boundary conditions for this problem are

C (r, 0) = 0 (10.8.25)

and

[latex]C (b, t) = C_{s} and \frac{\partial C}{\partial r} (0, t) = 0.[/latex] (10.8.26)

It is convenient to define [latex]\widetilde{C} (r, t) \equiv C (r, t) - C_{s},[/latex] such that [latex]\widetilde{C}[/latex] still satisfies
Eq. (10.8.24) and the initial and boundary conditions

[latex]\widetilde{C} (r,t - 0) = - C_{s}, \widetilde{C} (r=b, t) = 0, \frac{\partial \widetilde{C} }{\partial r} (r=b, t) = 0.[/latex] (10.8.27)

The Sturm-Liouville operator

[latex]Lu = - \frac{1}{ r^{2} } \frac{d}{dr} (r^{2} \frac{du}{dr} )[/latex] (10.8.28)

is self-adjoint for the boundary conditions in Eq. (10.8.27). Thus, we wish to solve the eigenproblem

[latex] -\frac{1}{ r^{2} } \frac{d}{dr} (r^{2} \frac{du}{dr} ) = \lambda u, u \in \mathcal{L}_{2} (0, b; r^{2} ),[/latex] (10.8.29)

with u(b) = 0 and [latex]\partial u (0) / \partial r = 0.[/latex] Using the transformation

[latex]u (r) = \frac{w(r)}{r} .[/latex]

the eigenproblem becomes

[latex]-\frac{d^{2} w }{d r^{2} } = \lambda w, w \in \mathcal{L}_{2} (0, b),[/latex] (10.8.30)

with

w(b) = 0 and w(0) = 0. (10.8.31)

The boundary condition for w(0) comes from the physical requirement that C < ∞ at the center of the cannonball and is equivalent to the condition [latex]\partial C / \partial r = 0 [/latex] at r = 0.

Equations (10.8.30) and (10.8.31) define a regular, self-adjoint Sturm- Liouville eigenproblem in [latex]\mathcal{L}_{2} (0, b)[/latex]. Thus, the eigenfunctions [latex]\phi _{n} , n = 1, 2, . . . , [/latex] for this problem form a complete orthonormal set in [latex]\mathcal{L}_{2} (0, b)[/latex]. Likewise, since [latex]\psi _{n} (r) = \phi _{n} (r) / r[/latex] is an eigenfunction of the operator in Eq. (10.8.29), it follows that [latex]\phi _{n} (r) / r, n = 1, 2, . . . ,[/latex] is a complete orthonormal set in [latex]\mathcal{L}_{2} (0, b; r^{2} ).[/latex]

The eigenfunctions and eigenvalues of Eq. (10.8.30) are

[latex]\phi {_{n} } (r) = \sqrt{\frac{2}{b} } \sin \frac{{n \pi r}}{b} , \lambda _{n} = (\frac{n \pi }{b} )^{2} , n = 1, 2, . . . .[/latex] (10.8.32)

Thus, if Eq. (10.8.24) is expressed as

[latex]\frac{\partial \widetilde{C} }{\partial t} = - DL \widetilde{C},[/latex] (10.8.33)

its solution is

[latex]\widetilde{C} = exp (-tDL) \widetilde{C}_{0} ,[/latex] (10.8.34)

or

[latex]C (r, t) = C_{s} + \sum\limits_{n = 1}^{\infty } exp (-t D (\frac{n \pi }{b} )^{2} ) \frac{\sqrt{2/b} }{r} \sin \frac{n \pi r}{b} imes \int_{0}^{b} \frac{\sqrt{2/b} }{r^{'} } \sin \frac{n \pi r^{'} }{b} (-C_{s} ) r^{'2} d r^{'} = C_{s} + C_{s} \sum\limits_{n = 1}^{\infty } (-1)^{n} exp (-(\frac{n \pi }{b} )^{2} Dt ) \frac{2b}{n \pi r} \sin \frac{n \pi r}{b} .[/latex] (10.8.35)

In Fig. 10.8.2, we show [latex]C / C_{s} [/latex] versus t (in units of b²/D) for r/b = 0.9, 0.5, and 0. A Mathematica program is included in the Appendix for this example.

Question 10.8.2: Consider the problem of diffusion of carbon into an iron can...

Related Answered Questions

Find Green’s function for the problem L u = d²u / dx² + 3 du / dx + 2u = f (x), -1 < x <1, (10.5.66) u (-1) - 2 u' (-1) = 0 u (1) + u' (1) = 0 (10.5.67) and solve the inhomogeneous problem for the cases (a) f (x) = x (10,5.68) and (b) f(x) = exp (-x²). (10.5.69) ...

Verified Answer:

Find Green’s function for the problem L u = -(1 + x)² d²u / dx² + 4 (1 + x) du / dx + 2u = f (x), 0 < x <1, (10.5.61) B1 u = u (0) = 0, B2 u = u (1) = 1. (10.5.62) ...

Verified Answer:

Use the method just described to find Green’s function g (x, y) for the differential expression Lu = – d² u/dx², (10.5.84) with B1 u ≡ u (0) = 0 B2 u ≡ u (1) = 0 . (10.5.85) ...

Verified Answer:

Find Green’s function for the problem d²u / dx² + u = x, u (0) = u’ (0) = 0. (10.5.42) ...

Verified Answer: