[X] = the concentration of X V = control volume

A = the bounding surface area of the control volume n = outward-pointing normal vector v = velocity vector.

Because aeration is done only in the reactor, this is the place where growth of organisms occurs. Thus, although it is said that V is the volume of the control volume, in reality, this volume is simply the volume of the reactor.

d[X]/dt was previously given in Equation (15.71); combining this with Equation (15.73) after integrating over the volume of the reactor, produces d L[ X ] dV d [X] [ 5]

The convective rate of increase of [X] is

[X]V • ndA = (-Qo[X0] + (Qo - Qw)[Xe] + Qw[Xu]) (15.75)

Note that Qo[Xo] is preceded by a negative sign. This is because at point A, the direction of the velocity vector is opposite to the direction of the unit normal vector. Now, assuming the operation of the reactor is at steady state, the local derivative is equal to zero. Substituting everything into Equation (15.73),

Pm F^+TFÏ [X] V - kd [X] V = 0 + (- Qo [ Xo ] + ( Q0 - Qw)[XJ + Qw [Xu]) (15.76) Ks+ [ S]

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