ABSTRACT
The multiple reactions involve parallel, series, and mixed reactions. They consist of complex reactions in which the specific rates of each reaction should be determined. They often occur in industrial processes. These reactions can be simple, elementary, irreversible and reversible, or even nonelementary. Some cases are:
Crotonoaldehyde Butiraldehyde Butanol
Parallel reactions
Series-parallel reactions
Series reaction
Fischer Tropsch
O O OH
+
+
H C ³ H C³ H C³
+H ²
+H ²
k ¹
k ²
k ¹
k ¹
k ²
k ²
C + 2H O
CO
CO
CO
CO
+
+
+
CO22H O
2H O
2H O
CH4 +2H3
2H2 +[C H ]n n2n
6.1 SIMPLE REACTIONS IN SERIES
Consider here the simplest irreversible and first-order reactions. If the reaction is consecutive or in series of the type:
A k1−−−−→R k2−−−−→ S
−dCA dt
= k1CA (6.1)
dCR dt
= k1CA − k2CR (6.2)
dCS dt
= k2CR (6.3)
Considering that in the beginning of the reaction there is only pure reactant, we have:
CA + CR + CS =CA0 (6.4) Defining new variables:
ϕA = CACA0 (6.5)
and analogously ϕR and ϕS, the initial concentration CA0 is always used as a reference. Making the time dimensionless (θ = k1t) and substituting these new variables into Equations 6.1 and 6.4, we obtain the following solutions:
ϕA = e−θ (6.6)
ϕR = 1(κ − 1) [e −θ − e−χθ] (6.7)
ϕS = 1 + 1(χ − 1)e −χθ − χ
(χ − 1)e −θ (6.8)
Considering that
χ= k2 k1
where χ is a parameter of the specific rate. By solving Equations 6.6-6.8, we obtain ϕA, ϕR, and ϕS as a function of θ and
hence the concentrations of each component as a function of the time, represented by the kinetic curves shown in Figure 6.1. The curves show that the concentration profile of A decreases exponentially, and A totally disappears when θ →∞. On the other hand, the concentration of R increases initially and then decreases, since R will be formed over time and transformed into S. Note that the curve of R shows a maximum and depends on the parameter χ, relating the specific rates of the reactions. The time corresponding to this concentration can be determined as follows:
dCR dt
= 0 or dϕR dθ
= 0 (6.9)
Differentiating Equation 6.7, we obtain:
dϕR dθ
= 1 (κ − 1) [χ e
−χθ − e−θ] = 0 (6.10)
Consequently, we can determine the time corresponding to this maximum, i.e.:
θmax = ln χ(χ − 1) (6.11)
Substituting θmax into Equation 6.7, we obtain the maximum concentration of R:
ϕRmax = exp(−χθmax) (6.12)
Substituting θmax (Equation 6.11) into Equation 6.12, we finally obtain:
ϕRmax =χ[χ/(1−χ)] (6.13)
or
CRmax CA0
=χ[χ/(1−χ)] (6.14)
Observe that ϕRmax or the maximum concentration of the intermediate CRmax changes, when the parameter χ is varied. This means that if the specific rate k2 of
1 tration of the intermediate product R and the corresponding time tmax decrease. Figure 6.1b shows the behavior of the concentration of R as a function of χ.
