This equation simplifies in case the medium is

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Unformatted text preview: tion term is: ∫ Σ (r )φ (r, t ) dV a V and the leakage term is: ˆ ∫ J (r, t ) ⋅ nds = ∫ ∇ ⋅ J (r, t ) dV s V where we converted the surface integral to a volume integral by use of Gauss’ Theorem or the divergence theorem. Substituting for the different terms in the balance equation we get: 1 ∂φ ( r , t ) dV = ∫ S ( r, t ) dV − ∫ Σ a ( r )φ ( r, t ) dV − ∫ ∇ ⋅ J ( r, t ) dV ∂t V V V ∫v V or: ⎛ 1 ∂φ ( r , t ) ⎞ − S + Σ aφ + ∇ ⋅ J ⎟ dV = 0 ∂t ⎠ V ∫⎜ v ⎝ (25) Since the volume V is arbitrary we can write: 1 ∂φ ( r, t ) = −∇ ⋅ J − Σ aφ + S v ∂t (26) We now use the relationship between J and ϕ (Fick’s law) to write the diffusion equation: 1 ∂φ ( r, t ) = ∇ ⋅ [ D( r )∇φ ( r, t )] − Σ a ( r )φ ( r, t ) + S ( r, t ) v ∂t (27) This equation is the basis of much of the development in reactor theory using diffusion theory. 6. THE HELMHOLTZ EQUATION The diffusion equation (27) is a partial differential equation of the parabolic type. It also describes physical phenomena in heat conduction, gas diffusion, and material diffusion. This equation simplifies in case the medium is uniform or homogeneous such that D and Σ a do not depend on the position r as: 1 ∂φ ( r, t ) − D ∇2φ ( r, t ) + Σ a φ ( r, t ) = S ( r, t ) v ∂t (28) where we used the fact that the divergence of the gradient leads to the Laplacian operator: ∇.∇ = ∇ 2 The Laplacian operator ∇ 2 depends on the coordinate system used: Cartesian: ∇2 ≡ ∂2 ∂2 ∂2 + 2+ 2 ∂x 2 ∂y ∂z (29) Cylindrical: ∇2 ≡ 1 ∂ ⎛ ∂ ⎞ 1 ∂2 ∂2 +2 ⎜r ⎟ + 2 r ∂r ⎝ ∂r ⎠ r ∂θ 2 ∂z (30) Spherical: ∇2 ≡ 1 ∂2 ∂⎛...
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