Derivation of the equation of continuity, in S19.1 the species equation of continuity is derived by making a mass balance on a small rectangular volume ∆x ∆y ∆z fixed in space. (a) Repeat the derivation for an arbitrarily shaped volume element V with a sufficiently smooth fixed boundary S. Show that the species mass balance can be written as use the Gauss divergence theorem to convert the surface integral to a volume integral, and then obtain Eq. 19.1-6. 

(b) Repeat the derivation using a region of fluid contained within a surface, each point of which is moving with local mass average velocity.

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J Psdv = -| (n - n,ds + J r,av