Page 52 - Introduction to chemical reaction engineering and kinetics

34 Chapter 2: Kinetics and Ideal Reactor Models

[2] Properties may change continuously in the direction of flow, as illustrated for cA
in Figure 2.4.
[3] In the axial direction, each portion of fluid, no matter how large, acts as a closed
system in motion, not exchanging material with the portion ahead of it or behind
it.
[4] The volume of an element of fluid does not necessarily remain constant through
the vessel; it may change because of changes in T, P and rtt, the total number of
moles.

2.4.2 Material Balance; Interpretation of ri

Consider a reaction represented by A + . . . + products taking place in a PFR. Since
conditions may change continuously in the direction of flow, we choose a differential
element of volume, dV, as a control volume, as shown at the top of Figure 2.4. Then the
material balance for A around dV is, from equation 1.5la (preceding equation 2.3-3):
FA - (FA + dF,) + r,dV = dn,ldt (2.4-1)
(for unsteady-state operation)

FA - (FA + dF,) + r,dV = 0 (2.4-2)
(for steady-state operation)

From equation 2.4-2 for steady-state operation, together with the definitions pro-
vided by equations 2.3-5 to -7, the interpretations of rA in terms of FA, f~, 5, and CA,
corresponding to equations 2.3-8 to -11, are3

(-rA) = -dFA/dV = -dF,/qdt (2.4-3)
= FAod fAldV (2.4-4)
= - v,dtldV (2.4-5)
= -d(c,q)ldV (2.4-6)

These forms are all applicable whether or not the density of the fluid is constant
(through the vessel).
If density is constant, equation 2.4-6 takes the form of equation 2.2-10 for constant
density in a BR. Then, since q is constant,

(-rA) = -dc,/(dV/q) (2.4-7)
= -dc,ldt (constant density) (2.2-10)

where t is the time required for fluid to flow from the vessel inlet to the point at which
the concentration is CA (i.e., the residence time to that point). As already implied in
equations 2.4-7 and 2.2-10, this time is given by

3For comparison with the “definition” of the species-independent rate, r, in footnote 1 of Chapter 1, we have
the similar result:

r(PF’R) = rilvi = (llVi)(dFi/dV) = (llv,q)(dFildt) (2.4-3a)