and time t, [X] has the dimension of M/L . The factors <p has the dimensions of (M/M)(M/L3) = M/L3; thus, <p corresponds to [X]. In the theorem <p is a point value, while [X] is an average value inside the reactor. [X] then correspond to the value of <p that is averaged. Therefore, d\M<pdV = diM(<pp)aVedV = dJMftX-] dV = d[X] \MdV = d[X]V (90) dt dt dt dt dt
The volume of the reactor is constant, so V may be taken out of the differential:
dt dt dt
Now, let us determine the expression for fA<pv • n dA. This is the convective derivative and it only applies to the boundary. In the case of the reactor, there are two portions of this boundary: the inlet boundary and the outlet boundary. Let the inflow to the reactor be Q; this will also be the outflow. Note that Q comes from v • n dA and, because the velocity vector and the unit vector are in opposite directions at the inlet, Q will be negative at the inlet. At the outlet, because the two vectors are in the same directions, Q will be positive. The concentration at the outlet will be the same as the concentration inside the tank, which is [X]. Thus, letting the concentration at the inflow be [Xo]
<pv • n dA = (- Q)[Xo] (at the inlet) + (+ Q)[X] (at the outlet)
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