9.8 MOdELing rEactiOns WitH tHE PLUg-fLOW WitH
9.8.1 Pseudo-first-Order reaction rate Law with the Pfd Model
( z z z) ( z z z)
A z C Q C C A J J A z r
t
∂
∂ +∆ +∆
⋅ ∆ ⋅ = ⋅ − + ⋅ − + ⋅ ∆ ⋅
We employ the definitions of change, write Q as the product of vs and A, divide through by the cross-sectional area of the reactor, use the dispersion analog of Fick’s law to define ,J
and take the limit of the expression as Δz→0 to arrive at a second- order partial differential equation:
2 s 2
C C C
v D r
t z z
∂ ∂ ∂
∂ = − ∂ + ∂ +
Although consideration of the unsteady state case would present interesting and chal- lenging opportunities for intellectual pursuit, herein we will consider only the steady- state case. Most reactions in environmental systems that can be considered flow reactors are well approximated by the steady-state solution. When we set C
t
∂
∂ to 0, our mass balance will lead to a second-order ordinary, rather than a second-order partial, differential equation:
2
0 sdC d C2
v D r
dz dz
= − + + (9.17a)
In order that we may directly employ the dispersion number obtained from the RTD analyses, we will normalize the result in a manner similar to that accomplished for the conservative tracer:
ξ τ
ξ − + =
2
2 0
s
D d C dC
v L d d r (9.17b)
We may substitute our pseudo-first-order rate law into Equations 9.17a and 9.17b.
Although Equation 9.18a preserves the structure of the relation, in much of the previous literature, we find that the form of Equation 9.18b has been used. Of
importance, the dimensionless group
s
D
v L is the dispersion number (its inverse is the Peclet number) employed in much of the literature in characterizing degrees of dispersion in both engineered and natural reactors. We will use Equation 9.18a in our analyses employing the PFD model:
2
2 s 0
d C dC
D v k C
dz − dz − ′ = (9.18a)
2 s 2
' 0
D d C dC
v L dξ − dξ −τk C = (9.18b)
Wylie (1966) presents the methodology to develop a general, closed-form solution to Equation 9.18a:
1 2
1 R z 2 R z
Cz = c e +c e (9.19)
where
2 2
s s
1 2
4 4
2 and 2
s s
v v kD v v kD
R R
D D
+ + − +
= =
In order to obtain the particular solution to Equation 9.18a, two boundary conditions are necessary. Hulbert (1944) proposed that the concentration at the influent must be continuous and that the reaction would cease at the effluent leading to the two conditions:
( 0) in
( )
; 0
z z L
C C dC
= dz
= = =
These were examined by Danckwerts (1953) and later by Wehner and Wilhelm (1956). Rather than concentration (state variable) continuity at the inlet, flux conti- nuity was proposed, leading to the alternative set of boundary conditions:
s s ( 0 )
( )
( 0 )
; 0
z z L
z
dC dC
v C D v C
dz = + = − dz = −
− = =
The resultant discontinuity of the state variable (concentration) at the influent was deemed necessary in the light that the mass rate of target reactant entering the reactor be constant across the inlet boundary. This set of boundary conditions was accepted by Levenspiel
(1972, 1999) and later by Fogler (2005) and Froment et al. (2011). The resultant closed- form analytic solution can be found in these chemical reaction engineering textbooks.
Unfortunately, the accepted boundary conditions are not correct. We cannot abide a discontinuity in the state variable at the influent and thus must conclude that the inlet boundary condition is flawed. Further, we can certainly reason that at the exit of a short reactor, the reaction is not complete and thus the outlet boundary condition is also flawed.
Our subsequent analyses seek to define the profiles in target reactant concentration through the reactor. Were we to envision the overall reactor as a set of successively longer reactors, we would find ourselves successively zeroing the exit spatial derivative of reac- tant concentration for each implementation of the solution. Weber and DiGiano (1996) reasoned that the spatial derivative of concentration tends to zero at z = ∞, a correct asser- tion, but of little use in our analyses herein. We propose that both state variable and flux continuity must be obeyed at both the inlet and outlet boundaries of the reactor. We con- sider the reaction on the z = 0+ and z = L– planes at the inlet and outlet of the reactor, respectively. A schematic representation of the inlet and outlet boundaries and associated processes is presented in Figure 9.9.
Figure 9.9 schematic representation of the entrance and exit boundaries of a reactor visual- ized using the plug-flow with dispersion (Pfd) model. representations of the transport and reactive processes within elements of thickness dz and area Ax on the reactor sides of the entrance and exit planes are shown.
Discontinuities in dispersion occur across the inlet and outlet boundaries and the magnitude of dispersion is considered constant throughout the reactor:
0 0; 0
z z L z L
D = − =D = + = D +≤ ≤ − =D
Reaction processes begin at the reactor side of the inlet boundary and cease at the reactor side of the outlet boundary. Material balances on the target reactant across the inlet and outlet boundaries (AX is area) are written:
X X X
0 0
s s
X
v v dC
A C A C A D A dzr
− dz
+
= − +
s s
X X X X L
L
v dC v
A C A D A dzr A C
dz − +
− +
=
For small reactant abundances, the total fluid flow rate is virtually constant through the entire reactor, hence also across the inlet and outlet boundaries. State variable continuity (target reactant concentration) across the boundaries must also be preserved:
X s 0 X s 0 and X s X s
L L
v v v v
A − A + A − A +
=
=
0 0
[ ]C − =[ ] an [ ]C + d C L− =[ ]C L+
Thus target reactant flux across the inlet and outlet boundaries is conserved:
0 0 and
s s s s
Xv Xv Xv L Xv L
A C − A C + A C − A C +
= =
The gradient in concentration then is attributable solely to the reaction process, which begins and ends at the same positions as does the dispersion process:
0
0 and 0
L
dC dC
D dzr D dz r
dz + dz −
= − + − + =
Were we to examine the conditions at the inlet and outlet boundaries of an ideal PFR, we would find the concentration gradients at both boundaries to be directly related to the respective reaction rates. We rearrange this result and employ Hulbert’s inlet boundary condition to yield the boundary conditions necessary to the particular solu- tion of Equation 9.18a.
( 0) in
( )
z ; z L
z L
dC dz
C C r
dz D
= = = = =
The concentration gradient at the effluent is indeed non-zero. Unfortunately, our result is not yet usable, since for quantitation we would necessarily know our effluent concentration to compute the reaction rate. Moreover, we are unable to assign a numerical value to dz. Of significant use, however, is the ratio of the concentration gradient at the effluent to that at the influent, for pseudo-first order kinetics, equal to the ratio of the respective concentrations
+ + + +
− −
− −
′
′
⋅ −
= = = =
⋅ −
0 0 0 0 0
L L L
L L
dCdz dzD r r k C C
dC dz r r k C C
dz D
(9.20)
We employ this ratio at steady state such that
L
dC dz
has a constant value, K. We now have the means to develop the particular solution to Equation 9.18a. We set the derivative of concentration at z = L equal to K to obtain a relation for c1:
2
1 2
2 in 1
1 2
R L
R L R L
K R C e
c R e R e
= −
−
We employ the inlet concentration at z = 0 to obtain a second relation for c2:
1
1 2
2 in 1 in 1
1 2
R L
R L R L
C R e K
c C c
R e R e
= − = −
−
We still have the matter of evaluating K with which to deal. One means to approach this solution is to employ the ideal PFR solution for a given set of conditions to ini- tialize K. The concentration gradient at the influent will be constant. Given an estimate for K, the reactor concentration profile can be solved. A new estimate of K can be obtained using the previous approximations of the exit concentration and inlet concentration gradient:
out
( 0 ) 1 in 1
i
z i i
dC C
K dz = + − C −
= ⋅
The solution is, of course, iterative and would cease when the relative change in K reaches an acceptably small value. The iterative process involves only the positions z = 0 and z = L. Once converged, we have the concentration value at the outlet and, if desired, we may produce the profile across the reactor using the alternative boundary condition C|z = L = Cout. An alternative particular solution is obtained once we can specify the concentration (type 1) boundary condition at both inlet and outlet:
2 1
1 2 1 2
out in in out
1 ; 2 in 1
R L R L
R L R L R L R L
C C e C e C
c c C c
e e e e
− −
= = − =
− −
Most conveniently, for numerical development of the internal concentration profile through the reactor for the applicable steady-state conditions, the general solution (Equation 9.19) is left as is, and the integration constants are defined by applying the boundary conditions. We can, of course, develop the concentration profile through the reactor merely by incrementing the value of L from zero to the targeted overall flow path length.