# The Effects of Mixing, Reaction Rates, and Stoichiometry on Yield for Mixing Sensitive Reactions—Part II: Design Protocols

Authors: Syed Imran A. Shah; Larry W. Kostiuk; Suzanne M. Kresta
Journal: International Journal of Chemical Engineering
 (2012)

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## Abstract

Competitive-consecutive and competitive-parallel reactions are both mixing sensitive reactions where the yield of desired product depends on how fast the reactants are brought together. Recent experimental results have suggested that the magnitude of the mixing effect may depend strongly on the stoichiometry of the reactions. To investigate this, a 1D, dimensionless, reaction-diffusion model was developed at the micromixing scale, yielding a single general Damköhler number. Dimensionless reaction rate ratios were derived for both reaction schemes. A detailed investigation of the effects of initial mixing condition (striation thickness), dimensionless reaction rate ratio, and reaction stoichiometry on the yield of desired product showed that the stoichiometry has a considerable effect on yield. All three variables were found to interact strongly. Model results for 12 stoichiometries are used to determine the mixing scale and relative rate ratio needed to achieve a specified yield for each reaction scheme. The results show that all three variables need to be considered when specifying reactors for mixing sensitive reactions.


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## Body

## 1. Introduction

Mixing-sensitive reactions are reactions which are particularly sensitive to the rate at which the reactants are brought together, that is, how fast they are mixed. These reactions are of two main types: the competitive-consecutive (C-C) reaction scheme, which involves two competing reactions where the second unwanted reaction consumes the desired product from the first reaction, and the competitive-parallel (C-P) reaction scheme, where two reactions compete for a limiting reagent, forming a desired and undesired product. The effects of mixing and relative reaction rates of the competing reactions have been investigated previously and it is known that mixing can affect the product distribution significantly. Past work has concentrated on the investigation of a single classical stoichiometry for each of the reaction schemes [1–25]. This work investigates whether the stoichiometry of the reaction plays a role in determining the maximum final yield of desired product and how the three reactor design variables—mixing, reaction, rate and stoichiometry interact.In a previous paper by Shah et al. [26], a model was developed to capture the effects of reaction stoichiometry, mixing, (characterized by the Damköhler number (Da)), and relative reaction rates, (characterized by a dimensionless reaction rate ratio (k2/k1)). General forms of the reactions as given in Table 1 were used to derive mass balance equations. From the equations a single Damköhler number that is common to both reaction types and all stoichiometries (Da) and a dimensionless reaction rate ratio (k2/k1) that is specific to each reaction type were obtained. Both Da and k2/k1 depend on the stoichiometric coefficients of the reaction scheme, making the potential impact of stoichiometry immediately evident. The expression for the general Damköhler number is:
(1)Da=k1′(ρTM)ϵLB2DB=k1′(ρT/M)ϵ(DB/LB2)=rateofdesiredreactionrateofdiffusion=(rateofdesiredreaction)*(diffusiontime).Table 1
General mixing sensitive reaction schemes.
Competitive consecutive (C-C)
Competitive parallel (C-P)
A
+
ϵ
B
→
k
1
′
α
P
A
+
ϵ
B
→
k
1
′
P
β
P
+
γ
B
→
k
2
′
S
C
+
γ
B
→
k
2
′
SThe effect of stoichiometry on the Damköhler number is given byϵ, the stoichiometric coefficient of the limiting reagent (B) in the desired reaction. The diffusion time in the smallest eddies is determined by the mixing scale, LB, the thickness of the smallest striation of concentration homogeneity, and the molecular diffusivity, DB, which completes the variables needed to determine the diffusion time at the smallest scales of segregation.The effect of the relative reaction rates of the competing reactions is also of interest. The model provides dimensionless reaction rate ratios for the C-C and C-P reactions as follows:(2)C-C:k2k1=[βα(α1+ϵ)β(ρTM)β+γ-ϵ-1]k2′k1′,(3)C-P:k2k1=[γϵ(ρTM)γ-ϵ]k2′k1′.These dimensionless reaction rate ratios are specific to the type of mixing sensitive reaction, C-C or C-P, and incorporate the effect of the relative reaction rates of the competing reactions(k2′/k1′) as well as the effect of stoichiometry. The model was validated for the classical reaction stoichiometry where all coefficients are equal to 1. This work extends the simulations to ten new stoichiometries where one or more of the coefficients are greater than one.For both the C-C and C-P simulations, the Damköhler number was varied from 0.01 to 10 000 in 100x increments, where 0.01 is the best and 10 000 is the worst mixing. The dimensionless reaction rate ratio (k2/k1) was varied from 1 to 0.00001 in 10x increments, where 1 is the worst ratio and 0.00001 is the best ratio. For the C-C reactions, ϵ was always 1 and α,β,γ were given a value of either 1 or 2. For the C-P reactions, ϵ and γ were either 1 or 2. Table 2 shows the different C-C reaction scheme stoichiometries investigated, with the corresponding dimensionless reaction rate ratios. Table 3 shows the same terms for the C-P reaction schemes.Table 2
Stoichiometries of reaction schemes and the corresponding dimensionless reaction rate ratio for the eight different C-C reactions. Da was always Da=k1′(ρT/M)(LB2/DB).
Case
Reaction scheme
ϵ
,
α
,
β
,
γ
k
2
/
k
1
1
A
+
B
→
k
1
′
2
P
1, 2, 1, 1
(
1
/
2
)
(
k
2
′
/
k
1
′
)
P
+
B
→
k
2
′
S
2
A
+
B
→
k
1
′
P
1, 1, 1, 1
(
1
/
2
)
(
k
2
′
/
k
1
′
)
P
+
B
→
k
2
′
S
3
A
+
B
→
k
1
′
P
1, 1, 2, 1
(
1
/
2
)
(
ρ
T
/
M
)
(
k
2
′
/
k
1
′
)
2
P
+
B
→
k
2
′
S
4
A
+
B
→
k
1
′
2
P
1, 2, 2, 1
(
ρ
T
/
M
)
(
k
2
′
/
k
1
′
)
2
P
+
B
→
k
2
′
S
5
A
+
B
→
k
1
′
2
P
1, 2, 1, 2
(
1
/
2
)
(
ρ
T
/
M
)
(
k
2
′
/
k
1
′
)
P
+
2
B
→
k
2
′
S
6
A
+
B
→
k
1
′
P
1, 1, 1, 2
(
1
/
2
)
(
ρ
T
/
M
)
(
k
2
′
/
k
1
′
)
P
+
2
B
→
k
2
′
S
7
A
+
B
→
k
1
′
P
1, 1, 2, 2
(
1
/
2
)
(
ρ
T
/
M
)
2
(
k
2
′
/
k
1
′
)
2
P
+
2
B
→
k
2
′
S
8
A
+
B
→
k
1
′
2
P
1, 2, 2, 2
(
ρ
T
/
M
)
2
(
k
2
′
/
k
1
′
)
2
P
+
2
B
→
k
2
′
STable 3
Stoichiometries of reaction schemes and the corresponding dimensionless reaction rate ratio and Damköhler number for the four different C-P reactions.
Case
Reaction scheme
ϵ
,
γ
k
2
/
k
1
Da
1
A
+
B
→
k
1
′
P
1, 1
k
2
′
/
k
1
′
k
1
′
(
ρ
T
/
M
)
(
L
B
2
/
D
B
)
C
+
B
→
k
2
′
S
2
A
+
B
→
k
1
′
P
1, 2
2
(
ρ
T
/
M
)
(
k
2
′
/
k
1
′
)
k
1
′
(
ρ
T
/
M
)
(
L
B
2
/
D
B
)
C
+
2
B
→
k
2
′
S
3
A
+
2
B
→
k
1
′
P
2, 1
(
1
/
2
)
(
ρ
T
/
M
)
-
1
(
k
2
′
/
k
1
′
)
k
1
′
(
ρ
T
/
M
)
2
(
L
B
2
/
D
B
)
C
+
B
→
k
2
′
S
4
A
+
2
B
→
k
1
′
P
2, 2
k
2
′
/
k
1
′
k
1
′
(
ρ
T
/
M
)
2
(
L
B
2
/
D
B
)
C
+
2
B
→
k
2
′
SWe caution the reader that this model is highly simplified. Elementary reactions involving 4 molecules do not exist, so the kinetic expressions for Cases 7 and 8 for the C-C reaction schemes and for Case 4 for the C-P reactions will assuredly have a different and more complex form for real reactions with similar stoichiometries. The mixing model is equally flawed when considering real systems. The simple striation model does not allow for variation of striation thickness with time as mixing progresses or between eddies; it includes no consideration of the effects of stretching and no mixing between layers. The dynamics of a more realistic model of turbulence would be much more complex. Keeping these limitations in mind, the goal of this work is to investigate whether a hypothesis that the reaction stoichiometry can have a significant impact on the course of mixing sensitive reactions is reasonable.


## 2. Numerical Details

Simulation of the systems of five or six PDEs for the C-C and C-P reaction schemes, respectivelyr (shown in Tables4 and 5), was carried out using COMSOL 3.4, a finite element PDE Solver. It is worth noting that for the C-C reaction scheme only four of the equations are independent while for the C-P reaction scheme there are five independent equations. The 1D, transient, convection, and diffusion mass transport model was used, with the mass fractions for each species specified as the independent variables. The default Lagrangian-quadratic element was chosen. The specified 1D geometry line of unit length was split equally into two subdomains and a mesh of 2048 equally spaced elements with 2049 nodes was generated. The mesh was tested for grid independence, and it was found that 1024 elements was sufficient resolution to ensure repeatable results to within the required tolerance of the solver, which was set to 10−6. Since the geometry was only 1D and the computational cost was minimal, a finer mesh than the minimum required resolution was used. The total time taken per simulation was approximately 30 seconds. The boundary conditions were specified for no net mass transfer across the boundaries.Table 4
System of equations for competitive-consecutive reaction scheme, to be used with (1) and (2).
Species
Mass balance partial differential equation
A
∂
(
w
A
)
/
∂
t
*
=
(
∂
2
(
w
A
)
/
∂
x
*
2
)
-
[
Da
w
A
w
B
ϵ
]
B
∂
(
w
B
)
/
∂
t
*
=
(
∂
2
(
w
B
)
/
∂
x
*
2
)
-
ϵ
[
Da
w
A
w
B
ϵ
]
-
(
α
γ
/
β
)
[
Da
(
k
2
/
k
1
)
w
P
β
w
B
γ
]
P
∂
(
w
P
)
/
∂
t
*
=
(
∂
2
(
w
P
)
/
∂
x
*
2
)
+
(
1
+
ϵ
)
[
Da
w
A
w
B
ϵ
-
Da
(
k
2
/
k
1
)
w
P
β
w
B
γ
]
S
∂
(
w
S
)
/
∂
t
*
=
(
∂
2
(
w
S
)
/
∂
x
*
2
)
+
(
1
+
ϵ
+
(
α
γ
/
β
)
)
[
Da
(
k
2
/
k
1
)
w
P
β
w
B
γ
]
I
∂
(
w
I
)
/
∂
t
*
=
∂
2
(
w
I
)
/
∂
x
*
2Table 5
System of equations for competitive-parallel reaction scheme, to be used with (1) and (3).
Species
Mass balance partial differential equation
A
∂
(
w
A
)
/
∂
t
*
=
(
∂
2
(
w
A
)
/
∂
x
*
2
)
-
[
Da
w
A
w
B
ϵ
]
B
∂
(
w
B
)
/
∂
t
*
=
(
∂
2
(
w
B
)
/
∂
x
*
2
)
-
ϵ
[
Da
w
A
w
B
ϵ
]
-
ϵ
[
Da
(
k
2
/
k
1
)
w
c
w
B
γ
]
C
∂
(
w
C
)
/
∂
t
*
=
(
∂
2
(
w
C
)
/
∂
x
*
2
)
-
(
ϵ
/
γ
)
[
Da
(
k
2
/
k
1
)
w
c
w
B
γ
]
P
∂
(
w
P
)
/
∂
t
*
=
(
∂
2
(
w
P
)
/
∂
x
*
2
)
+
(
1
+
ϵ
)
[
Da
w
A
w
B
ϵ
]
S
∂
(
w
S
)
/
∂
t
*
=
(
∂
2
(
w
S
)
/
∂
x
*
2
)
+
(
ϵ
+
(
ϵ
/
γ
)
)
[
Da
(
k
2
/
k
1
)
w
c
w
B
γ
]
I
∂
(
w
I
)
/
∂
t
*
=
∂
2
(
w
I
)
/
∂
x
*
2Figure1 shows the initial conditions for the C-C and C-P schemes. For the C-C cases, the initial conditions were chosen such that all of the mass initially present could be converted to desired product P. This was done by specifying a ratio of A : B as 1 : 1 in all the simulations, with A and B being present in pure striations, wA0=wB0=1. For the C-P scheme, owing to the parallel nature of the reactions, the initial conditions were a bit more complicated. They were chosen such that either A or C could consume the entire limiting reagent B by themselves, that is, the initial ratios depended on the stoichiometric coefficients in the reaction scheme. In order to satisfy the constant mass concentration assumption of the model, it was necessary to include the inert in the C-P simulations. wB0 and wIZ0 were always set to 0.5, and wA0wC0,wIY0 were calculated accordingly.Initial conditions for (a) C-C and (b) C-P reaction scheme simulations.
(a)
(b)For most cases the limiting reagent,B, was completely consumed by Da·t*=500 so the transient simulation results are shown from Da·t*=0 to Da·t*= 500. Da·t* is equivalent to a dimensionless reaction time where Da·t*=t/τR, so running all simulations to Da·t*=500 is the same as running all simulations for 500 reaction times.The modelled equations allow for specification of the Damköhler number, the dimensionless reaction rate ratio and the stoichiometry. All possible combinations of the values of reaction rate ratio, Damköhler number, and stoichiometry for the C-C reaction scheme (stoichiometries given in Table2) and the C-P reaction scheme (stoichiometries given in Table 3) were investigated. This resulted in 192 converged cases for the C-C and 96 converged cases for the C-P reaction scheme.


## 3. Results and Discussion

The results obtained for the simulations are profiles of mass fraction for each of the species over the spacex* for all dimensionless times t*. Samples of these results were given in Shah et al. [26] (Figures 2 and 3, resp.).Plots of yield ofP versus Da for decreasing k2/k1 ratios for the C-C cases.
k
2
/
k
1 = 1
k
2
/
k
1 = 0.1
k
2
/
k
1 = 0.01
k
2
/
k
1 = 0.001
k
2
/
k
1 = 0.0001
k
2
/
k
1 = 0.00001Plots of Yield ofP versus k2/k1 for two sample C-C stoichiometries. The curves represent the different Da. Curves for Da=0.01 lie exactly under the curves for Da=1.
C-C stoichiometry Case 2:A+B→k1′P; P+B→k2′S
(b)
C-C stoichiometry Case 7:A+B→k1′P; 2P+2B→k2′SSince the main process objective is to maximize the production of desired productP, the profiles of mass fraction of P are of most interest. These profiles were integrated over the domain to obtain the total mass of desired product and the instantaneous yield using the formula:
(4)YP=massofspeciesPatt*maxmassofPobtainable=∫-0.50.5wPdx*(t*)0.5wBo(1+1/ϵ).FollowingYP over time gives the progression of yield over time. The total production of P as time progresses can be observed in a plot of YP versus dimensionless time Da·t*.Figures 4 and 5 in Shah et al. [26] show examples of such plots.Plots of yield ofP versus k2/k1 for various Da at Da·t*=500 for C-C cases.
Da = 1
(b)
Da = 100
(c)
Da = 10000Figure 5
Plot of yield ofP versus  k2/k1 for Da =10000 at Da·t*=50000 for C-C cases.The final yield ofP at Da·t*=500 was plotted against Da and k2/k1 to assess the effects of each of the variables on the final yield of desired product. Representative figures are presented here. The results (YP versus Da and k2/k1) make up a 3D surface for each stoichiometry. Slices of the results are presented as follows: a base case set of six slices of YP versus Da is presented to show the effects of mixing (Da) and dimensionless reaction rate ratio (k2/k1) on yield of desired product. Where a point needs to be clarified, the results are replotted by taking a slice along the k2/k1 axis or by expanding the time variable. The results are then revisited to focus on the effect of stoichiometry and how it interacts with mixing and reaction rate. The same set of plots is repeated for the C-P reaction schemes. The results are then summarized to show a possible design specification space of Da and k2/k1 for a specified yield.


### 3.1. Competitive-Consecutive (C-C) Reaction

Figures2(a) to 2(f) are semilog plots of the final YP versus Damköhler number for decreasing dimensionless reaction rate ratio, k2/k1. The curves on each of the plots represent the eight C-C stoichiometry cases. The effects that are of interest are the effects of the Damköhler number (mixing), reaction rate ratio k2/k1 (chemical kinetics), and the reaction stoichiometry (Cases 1 to 8). The Damköhler number is plotted on the x-axis and the results are first presented from this perspective. Next, the effect of ratio k2/k1 is observed by comparing graphs (a) to (f). Finally the effect of stoichiometry (Cases 1 to 8) within each graph is reviewed and summarized. At each step, selected results are replotted either to illustrate a point more clearly or to provide a perspective on the data.


#### 3.1.1. The Effect of Damköhler Number (Da): The Well-Mixed Limit

The plots in Figure2 show a decrease in yield of desired product with increasing Damköhler number. This trend is true for all stoichiometries and at all values of k2/k1. A larger Damköhler number represents worse mixing (larger striations) and a smaller Damköhler number represents improved mixing (smaller striations). The yield for Da=1 and Da=0.01 is the same for all stoichiometries at all k2/k1. The yield at Da=100 decreases for the cases with unfavourable dimensionless reaction rate ratios (k2/k1≥0.01) but remains for same for the favourable k2/k1 ratios (k2/k1≤0.001). By Da=10000 the yield always decreases, regardless of the value of k2/k1.The data is replotted in Figures3(a) and 3(b) for the best (Case 7) and worst (Case 2) stoichiometry cases to confirm the YP versus k2/k1 trends. This shows that the results for Da=0.01 and Da=1 lie exactly on top of one another. When Da is increased from 1 to 100 the curves diverge over part of the domain: for Case 2 when k2/k1≥0.001 and for Case 7 when k2/k1≥0.01. It is concluded that Da=1 is the well-mixed limit for large k2/k1, while Da=100 is the well-mixed limit for small k2/k1 and that the meaning of “large k2/k1” depends on the stoichiometry.


#### 3.1.2. The Effect of Dimensionless Reaction Rate Ratio (k2/k1): Can Chemistry Solve the Problem?

Looking at the changes in Figure2 from (a) to (f), as k2/k1 decreases the yield increases. This result is as expected, since a smaller k2/k1 gives a slower undesired side reaction. As k2/k1 is decreased it is also clear that the curves for the different stoichiometric cases approach, collapsing onto one curve by k2/k1=0.0001. The results are replotted to directly compare the effect of k2/k1 on the 8 cases in Figure 4(a):  Da=1 (well mixed), (b) Da=100 and (c)Da=10000 (badly mixed).Y
P increases to 1 as k2/k1 decreases for all stoichiometries, with the 8 cases collapsing at about k2/k1=0.001 for the well-mixed cases and at k2/k1=0.0001 for Da=10000, the badly mixed case. The yield of the badly mixed case is about half of what was obtained in the well-mixed results. This brings up an interesting question: “Is it possible to get perfect yield of desired product, that is, YP~1, for Da=10000, a badly mixed case?”. The simulations for the Da =10000 cases were rerun for a hundred times longer to Da·t*=50000 instead of  Da·t*=500. Figure 5 shows the results of these simulations. To obtain a perfect yield with insufficient mixing, both a 100x longer reaction time and 100x better chemistry are required. This clearly illustrates the need for an improved understanding of mixing.


#### 3.1.3. Effect of Stoichiometry

Returning for a final time to Figure2 there is a clear effect of stoichiometry on the final yield of desired product, and the effect of stoichiometry shows interactions with both the mixing scale and the dimensionless reaction rate ratio.In Figure2(a)  (k2/k1=1) the curves are widely separated, coming closer together as Da increases and the effect of bad mixing becomes dominant. There is a clear distinction between stoichiometries, with Cases 7 and 8 having the highest yields and therefore the most favourable stoichiometry, and Cases 1 and 2 having the lowest yields, meaning they have the least favourable stoichiometries. Cases 1 and 2 require only two molecules for the side reaction, while Cases 7 and 8 require four molecules. This places a mass transfer and collision probability limitation on the side reaction for Cases 7 and 8, hence making them more favourable for the desired reaction. Cases 3–6 all consume 3 molecules in the undesired reaction, making the mass transfer limit similar; however, at the large values of k2/k1, Cases 3 and 4 have a much lower yield of P than Cases 5 and 6, forming two bands.Comparing the stoichiometries of Cases 1–4 with Cases 5–8, for the first 4 cases, 1 molecule of B is consumed in the second reaction, and in the last 4, 2 molecules of B are consumed. Comparing Case 3 to Case 6 and Case 4 to Case 5, the desired reactions are identical, whereas the stoichiometric coefficients on the undesired reaction are different. Of particular interest is the larger stoichiometric coefficient on reagentB in Cases 5 and 6. An examination of the local concentration profiles showed that B is the limiting reagent locally in the second reaction. Since Cases 5 and 6 require two molecules of the limiting reagent in the undesired reaction, making it harder for the reaction to proceed, it makes sense that the yield of P will be higher for Cases 5 and 6 as compared to Cases 3 and 4.Now comparing the stoichiometries within pairs, in three of the four pairs producing moreP in the first reactions leads to an increase in P consumption in the second reaction and a drop in yield. In other words, at large k ratios, the two reactions compete effectively for reagents. Under these conditions, when less P is produced in the desired reaction, the side reaction is slowed down. The trend is reversed for Cases 3 and 4. These are the only cases where more P is consumed than B in the undesired reaction, so in this case it appears that the mass transfer limit on P relative to B overwhelms the effect of producing more P.These trends continue in Figure2(b)  (k2/k1=0.1) and Figure 2(c) (k2/k1=0.01). For k2/k1≤0.001 the curves collapse onto each other and the effect of stoichiometry vanishes. For the very bad mixing condition (Da=10000) that was allowed to proceed for a very long time, Figure 5, the effect of stoichiometry creeps back into the picture, affecting yield at all the way to k2/k1=0.00001.


#### 3.1.4. Summary of Results for the C-C Reaction Scheme

For competitive-consecutive reactions, we have the following.(a)
Da≤1 is the well-mixed limit when k2/k1≥0.01 and Da≤100 is the limit when k2/k1<0.01.(b)
k2/k1≤0.001 with Da≤100 provides yields >95% at  Da·t*=500.(c)
The effect of stoichiometry can be large if thek2/k1 is unfavourable. This effect vanishes at k2/k1≤0.001 for short reaction times and good mixing. At longer reaction times with bad mixing conditions, the effect of stoichiometry reappears and requires a much smaller value of k2/k1 to eradicate.(d)
The most favorable stoichiometry is one where the desired reaction consumes fewer molecules than the undesired reaction, thus placing a mass transfer limitation on the undesired reaction.


### 3.2. Competitive-Parallel (C-P) Reaction

The analysis proceeds through the same questions posed for the C-C reaction.(i)
What is the well mixed limit (Da)?(ii)
How much can chemistry contribute (k2/k1)?(iii)
Is the stoichiometry important?The base data is presented in Figure6, with selected data replotted in Figures 7‒9.Plots of yield ofP versus Da for decreasing k2/k1 ratios for the C-P cases.
k
2
/
k
1 = 1
(b)
k
2
/
k
1 = 0.1
(c)
k
2
/
k
1 = 0.01
(d)
k
2
/
k
1 = 0.001
(e)
k
2
/
k
1 = 0.0001
(f)
k
2
/
k
1 = 0.00001Plots of yield ofP versus k2/k1 for two sample C-P stoichiometries. The curves represent the different Da. Curves for Da=0.01 lie exactly under the curves for Da = 1.
C-P stoichiometry Case 1:A+B→k1′P; C+B→k2′S
(b)
C-P stoichiometry Case 3:A+2B→k1′P; C+B→k2′S


#### 3.2.1. Effect of Damköhler Number (Da)

The plots in Figure6 show a large drop in YP when Damköhler number increases from 100 to 10000 with the exception of Case 3 at k2/k1=1 where the yield increases steadily with increasing Da. Case 3 is the only stoichiometry where the desired reaction consumes more molecules than the undesired reaction, so when the reaction rates are equal and mixing is good, the mass transfer limitation on the desired reaction will push the yield toward the secondary reaction. Since the reactions are parallel, there is no barrier to the production of S over P.For the C-P reaction pair, yield is constant forDa≤100, giving a well-mixed limit of Da=100. Figures 7(a) and 7(b) confirm this result for Cases 1 and 3, respectively. The curves show the four values of Da investigated. The curves for Da=0.01 and Da=1 lie exactly on top of each other and the curve for Da=100 is very close, but increasing Da to 10 000 drops the yield significantly. These results are very similar to the C-C results. It is concluded that Da=100 is the well-mixed limit for C-P reactions.


#### 3.2.2. Effect of Dimensionless Reaction Rate Ratio (k2/k1)

Looking at Figure6 from the perspective of reaction rate ratio, as k2/k1 is decreased, the curves for all the stoichiometries move upwards, increasing the yield for all cases at all values of Da. As expected, a smaller k2/k1 increases the yield of desired product. For Cases 1 and 2 the maximum yield is approximately 1 and for Cases 3 and 4 it is 0.88. The curves for Cases 1 and 2 collapse at k2/k1=0.001 (Figure 6(d)) and the curves for Cases 3 and 4 collapse at k2/k1=0.0001 (Figure 6(e)). The largest change with improving k2/k1 is seen for Case 3, while Case 2 improves the least. These trends are confirmed in Figure 8, showing a persistent effect of stoichiometry on the maximum attainable yield. The poorly mixed condition (Da=10 000) may not have reached completion at Da·t*=500 since the yield is only about 50%.Plots of yield ofP versus k2/k1  for various Da at Da·t*=500 for C-P cases.
Da = 1
(b)
Da = 100
(c)
Da = 10,000Figure 9
Plot of yield ofP versus  k2/k1for Da=10000 at Da·t*=50000 for C-P cases.In Figure9 the poorly mixed condition is allowed to progress 100x longer. At this point, the yield can increase to above 90% but only for a 100x smaller k2/k1 ratio. The curves seem to collapse at k2/k1=0.00001, but at lower k2/k1’s the effect of stoichiometry persists, even at long times. Once again, 100x more residence time and 100x better chemistry are required to eliminate all the effects of 100x worse mixing.


#### 3.2.3. Effect of Stoichiometry

Figures6(a) to 6(f) illustrate a significant effect of stoichiometry for the C-P case. At k2/k1=1 (Figure 6(a)) the difference in yield for the different stoichiometries is enormous, and there is a clear distinction between the most favourable and least favourable reaction stoichiometries (Case 2 and Case 3, respectively). This difference decreases as k2/k1 decreases (Figures 6(b) to 6(f)). For the C-P scheme, the effect of stoichiometry persists even at k2/k1=0.00001, where the difference in yield between Cases 1 and 2 and Cases 3 and 4 is more than 10%. This is different from the C-C stoichiometries where the effect of stoichiometry disappeared by k2/k1≤0.001.Referring to Table3, Cases 1 and 4 require the same number of molecules for both reactions, while Case 2 favors the desired reaction (fewer molecules) and Case 3 favors the side reaction. Case 3 is the only case of all 12 stoichiometries considered where the desired reaction consumes more molecules than the side reaction, imposing a mass transfer and collision probability limitation on the desired reaction. This explains the dramatically different results observed for Case 3 and suggests that this class of reaction stoichiometry must be handled more carefully than the others.


#### 3.2.4. Summary of Results for the C-P Reaction Scheme

For competitive-parallel reactions, we have the following.(a)
Da≤100 is the well-mixed limit for all k2/k1.(b)
For all cases with favourable stoichiometries, atDa·t*=500, k2/k1 ≤ 0.001 provides the maximum yield, as for the C-C reaction scheme, but for the C-P reaction scheme the maximum yield depends on the stoichiometry. Case 3, the only case with an unfavourable stoichiometry, requires k2/k1≤0.0001 to achieve maximum yield.(c)
Stoichiometry affects the maximum obtainable yield to some degree for all mixing conditions and allk2/k1.(d)
As for the C-C scheme, increasing the number of molecules in the side reaction imposes a mass transfer limitation on that reaction, making conditions more favorable for the desired reaction. Case 3 in the C-P scheme clearly illustrates that when the desired reaction consumes more molecules than the undesired reaction, it is more difficult to achieve a high yield of desired product.


### 3.3. Phase Plots of Variables for C-C And C-P Reaction Schemes for the Purpose of Design

The purpose of design graphs, in the context of mixing sensitive reactions, is to provide a direct method of predicting the yield of desired product without extensive calculations and investigation of the different variables. For our purposes, design graphs would incorporate the effects of stoichiometry, mixing, andk2/k1 and the combinations of the three that could possibly give a certain specified yield of desired product. The purpose of the graphs would be to assist in the design of the following problem.“If one has a certain mixing sensitive reaction and a target yield of desired product in mind, what range of values of dimensionless reaction rate ratio and Damköhler would be required to ensure that the target yield of desired product will be obtained? If one of the variables is inadequate, is it still possible to obtain the target yield? If it is possible, how much do the other variables need to change to accommodate for that inadequacy, if at all?”The figures should allow for a design space to be established within which decisions can be made based on different criteria and restrictions.Figure10 and Figure 11 show the required Damköhler number (x-axis) versus the dimensionless reaction rate ratio (y -axis) for yields of desired product of (a) 85% or more, (b) 95% or more, and (c) 99% or more at a time of  Da·t*=500. They are set up such that the axis intersection (origin) represents the most favourable conditions for both k2/k1 and Damköhler number. The figures have marked regions of required k2/k1 and Da where a specified yield can be obtained for different stoichiometries. These figures are essentially a filtered summary of the results presented in the previous sections. The effect of stoichiometry quickly becomes evident as the specified yield is increased. For the C-C cases, the requirements on the k2/k1 and Da become more stringent as YP increases. For C-P Cases  3 and 4, yields greater than 90% were unachievable for  Da·t*=500.Design spaces for yield ofP for Da and k2/k1  at  Da·t*=500 for C-C cases.
Y
P is at least 0.85 (85%)
Y
P is at least 0.95 (95%)
Y
P is at least 0.99 (99%)Design spaces for yield ofP for Da and k2/k1 at    Da·t*=500  for C-P cases.
Y
P is at least 0.85 (85%)
Y
P is at least 0.95 (95%)
Y
P is at least 0.99 (99%)In order to link these results to physical process variables, the diffusion time is replaced with an analogous mixing time scale for turbulent flow. The Batchelor time scale gives the diffusion time for a Kolmogorov sized eddy and is numerically equal to the Kolmogorov time scale, which is the lifetime of the same smallest eddy before the momentum is diffused by viscosity. Replacing the molecular diffusion time in a slab in our definition of Da with the expression for the Kolmogorov time scale in a turbulent eddy gives the following:(5)Da=k1′(ρTM)ϵ(νεD)1/2
where ν is the kinematic viscosity and εD is local rate of dissipation of turbulent kinetic energy (TKE) per mass. Rearranging to solve for εD gives the following:
(6)εD=ν(k1′(ρT/M)ϵDa)2.This expression proposes a relationship between the rate of dissipation of turbulent kinetic energy (εD), the reaction kinetics of the desired reaction (k1′), and the reaction stoichiometry (ϵ) which might be used to compare bench scale performance with the Da and reaction rate ratio from the phase plot. For example, for a C-C reaction of the type Case 1 (ϵ=1), using typical values for water (ν=1×10-6 m2/s, ρT=1000kg/m3, M=18kg/kmole), εD for Da = 1 becomes
(7)εD=3.09×10-3k1′2.This gives a first-order estimate for the required level of dissipation at the bench scale. Since the model is based on a static striation thickness rather than the dynamic stretching and cutting field present in both chaotic and turbulent flows, this should be considered a starting point for further investigations in the context of more accurate models of mixing and diffusion.


## 3.1. Competitive-Consecutive (C-C) Reaction

Figures2(a) to 2(f) are semilog plots of the final YP versus Damköhler number for decreasing dimensionless reaction rate ratio, k2/k1. The curves on each of the plots represent the eight C-C stoichiometry cases. The effects that are of interest are the effects of the Damköhler number (mixing), reaction rate ratio k2/k1 (chemical kinetics), and the reaction stoichiometry (Cases 1 to 8). The Damköhler number is plotted on the x-axis and the results are first presented from this perspective. Next, the effect of ratio k2/k1 is observed by comparing graphs (a) to (f). Finally the effect of stoichiometry (Cases 1 to 8) within each graph is reviewed and summarized. At each step, selected results are replotted either to illustrate a point more clearly or to provide a perspective on the data.


### 3.1.1. The Effect of Damköhler Number (Da): The Well-Mixed Limit

The plots in Figure2 show a decrease in yield of desired product with increasing Damköhler number. This trend is true for all stoichiometries and at all values of k2/k1. A larger Damköhler number represents worse mixing (larger striations) and a smaller Damköhler number represents improved mixing (smaller striations). The yield for Da=1 and Da=0.01 is the same for all stoichiometries at all k2/k1. The yield at Da=100 decreases for the cases with unfavourable dimensionless reaction rate ratios (k2/k1≥0.01) but remains for same for the favourable k2/k1 ratios (k2/k1≤0.001). By Da=10000 the yield always decreases, regardless of the value of k2/k1.The data is replotted in Figures3(a) and 3(b) for the best (Case 7) and worst (Case 2) stoichiometry cases to confirm the YP versus k2/k1 trends. This shows that the results for Da=0.01 and Da=1 lie exactly on top of one another. When Da is increased from 1 to 100 the curves diverge over part of the domain: for Case 2 when k2/k1≥0.001 and for Case 7 when k2/k1≥0.01. It is concluded that Da=1 is the well-mixed limit for large k2/k1, while Da=100 is the well-mixed limit for small k2/k1 and that the meaning of “large k2/k1” depends on the stoichiometry.


### 3.1.2. The Effect of Dimensionless Reaction Rate Ratio (k2/k1): Can Chemistry Solve the Problem?

Looking at the changes in Figure2 from (a) to (f), as k2/k1 decreases the yield increases. This result is as expected, since a smaller k2/k1 gives a slower undesired side reaction. As k2/k1 is decreased it is also clear that the curves for the different stoichiometric cases approach, collapsing onto one curve by k2/k1=0.0001. The results are replotted to directly compare the effect of k2/k1 on the 8 cases in Figure 4(a):  Da=1 (well mixed), (b) Da=100 and (c)Da=10000 (badly mixed).Y
P increases to 1 as k2/k1 decreases for all stoichiometries, with the 8 cases collapsing at about k2/k1=0.001 for the well-mixed cases and at k2/k1=0.0001 for Da=10000, the badly mixed case. The yield of the badly mixed case is about half of what was obtained in the well-mixed results. This brings up an interesting question: “Is it possible to get perfect yield of desired product, that is, YP~1, for Da=10000, a badly mixed case?”. The simulations for the Da =10000 cases were rerun for a hundred times longer to Da·t*=50000 instead of  Da·t*=500. Figure 5 shows the results of these simulations. To obtain a perfect yield with insufficient mixing, both a 100x longer reaction time and 100x better chemistry are required. This clearly illustrates the need for an improved understanding of mixing.


### 3.1.3. Effect of Stoichiometry

Returning for a final time to Figure2 there is a clear effect of stoichiometry on the final yield of desired product, and the effect of stoichiometry shows interactions with both the mixing scale and the dimensionless reaction rate ratio.In Figure2(a)  (k2/k1=1) the curves are widely separated, coming closer together as Da increases and the effect of bad mixing becomes dominant. There is a clear distinction between stoichiometries, with Cases 7 and 8 having the highest yields and therefore the most favourable stoichiometry, and Cases 1 and 2 having the lowest yields, meaning they have the least favourable stoichiometries. Cases 1 and 2 require only two molecules for the side reaction, while Cases 7 and 8 require four molecules. This places a mass transfer and collision probability limitation on the side reaction for Cases 7 and 8, hence making them more favourable for the desired reaction. Cases 3–6 all consume 3 molecules in the undesired reaction, making the mass transfer limit similar; however, at the large values of k2/k1, Cases 3 and 4 have a much lower yield of P than Cases 5 and 6, forming two bands.Comparing the stoichiometries of Cases 1–4 with Cases 5–8, for the first 4 cases, 1 molecule of B is consumed in the second reaction, and in the last 4, 2 molecules of B are consumed. Comparing Case 3 to Case 6 and Case 4 to Case 5, the desired reactions are identical, whereas the stoichiometric coefficients on the undesired reaction are different. Of particular interest is the larger stoichiometric coefficient on reagentB in Cases 5 and 6. An examination of the local concentration profiles showed that B is the limiting reagent locally in the second reaction. Since Cases 5 and 6 require two molecules of the limiting reagent in the undesired reaction, making it harder for the reaction to proceed, it makes sense that the yield of P will be higher for Cases 5 and 6 as compared to Cases 3 and 4.Now comparing the stoichiometries within pairs, in three of the four pairs producing moreP in the first reactions leads to an increase in P consumption in the second reaction and a drop in yield. In other words, at large k ratios, the two reactions compete effectively for reagents. Under these conditions, when less P is produced in the desired reaction, the side reaction is slowed down. The trend is reversed for Cases 3 and 4. These are the only cases where more P is consumed than B in the undesired reaction, so in this case it appears that the mass transfer limit on P relative to B overwhelms the effect of producing more P.These trends continue in Figure2(b)  (k2/k1=0.1) and Figure 2(c) (k2/k1=0.01). For k2/k1≤0.001 the curves collapse onto each other and the effect of stoichiometry vanishes. For the very bad mixing condition (Da=10000) that was allowed to proceed for a very long time, Figure 5, the effect of stoichiometry creeps back into the picture, affecting yield at all the way to k2/k1=0.00001.


### 3.1.4. Summary of Results for the C-C Reaction Scheme

For competitive-consecutive reactions, we have the following.(a)
Da≤1 is the well-mixed limit when k2/k1≥0.01 and Da≤100 is the limit when k2/k1<0.01.(b)
k2/k1≤0.001 with Da≤100 provides yields >95% at  Da·t*=500.(c)
The effect of stoichiometry can be large if thek2/k1 is unfavourable. This effect vanishes at k2/k1≤0.001 for short reaction times and good mixing. At longer reaction times with bad mixing conditions, the effect of stoichiometry reappears and requires a much smaller value of k2/k1 to eradicate.(d)
The most favorable stoichiometry is one where the desired reaction consumes fewer molecules than the undesired reaction, thus placing a mass transfer limitation on the undesired reaction.


## 3.1.1. The Effect of Damköhler Number (Da): The Well-Mixed Limit

The plots in Figure2 show a decrease in yield of desired product with increasing Damköhler number. This trend is true for all stoichiometries and at all values of k2/k1. A larger Damköhler number represents worse mixing (larger striations) and a smaller Damköhler number represents improved mixing (smaller striations). The yield for Da=1 and Da=0.01 is the same for all stoichiometries at all k2/k1. The yield at Da=100 decreases for the cases with unfavourable dimensionless reaction rate ratios (k2/k1≥0.01) but remains for same for the favourable k2/k1 ratios (k2/k1≤0.001). By Da=10000 the yield always decreases, regardless of the value of k2/k1.The data is replotted in Figures3(a) and 3(b) for the best (Case 7) and worst (Case 2) stoichiometry cases to confirm the YP versus k2/k1 trends. This shows that the results for Da=0.01 and Da=1 lie exactly on top of one another. When Da is increased from 1 to 100 the curves diverge over part of the domain: for Case 2 when k2/k1≥0.001 and for Case 7 when k2/k1≥0.01. It is concluded that Da=1 is the well-mixed limit for large k2/k1, while Da=100 is the well-mixed limit for small k2/k1 and that the meaning of “large k2/k1” depends on the stoichiometry.


## 3.1.2. The Effect of Dimensionless Reaction Rate Ratio (k2/k1): Can Chemistry Solve the Problem?

Looking at the changes in Figure2 from (a) to (f), as k2/k1 decreases the yield increases. This result is as expected, since a smaller k2/k1 gives a slower undesired side reaction. As k2/k1 is decreased it is also clear that the curves for the different stoichiometric cases approach, collapsing onto one curve by k2/k1=0.0001. The results are replotted to directly compare the effect of k2/k1 on the 8 cases in Figure 4(a):  Da=1 (well mixed), (b) Da=100 and (c)Da=10000 (badly mixed).Y
P increases to 1 as k2/k1 decreases for all stoichiometries, with the 8 cases collapsing at about k2/k1=0.001 for the well-mixed cases and at k2/k1=0.0001 for Da=10000, the badly mixed case. The yield of the badly mixed case is about half of what was obtained in the well-mixed results. This brings up an interesting question: “Is it possible to get perfect yield of desired product, that is, YP~1, for Da=10000, a badly mixed case?”. The simulations for the Da =10000 cases were rerun for a hundred times longer to Da·t*=50000 instead of  Da·t*=500. Figure 5 shows the results of these simulations. To obtain a perfect yield with insufficient mixing, both a 100x longer reaction time and 100x better chemistry are required. This clearly illustrates the need for an improved understanding of mixing.


## 3.1.3. Effect of Stoichiometry

Returning for a final time to Figure2 there is a clear effect of stoichiometry on the final yield of desired product, and the effect of stoichiometry shows interactions with both the mixing scale and the dimensionless reaction rate ratio.In Figure2(a)  (k2/k1=1) the curves are widely separated, coming closer together as Da increases and the effect of bad mixing becomes dominant. There is a clear distinction between stoichiometries, with Cases 7 and 8 having the highest yields and therefore the most favourable stoichiometry, and Cases 1 and 2 having the lowest yields, meaning they have the least favourable stoichiometries. Cases 1 and 2 require only two molecules for the side reaction, while Cases 7 and 8 require four molecules. This places a mass transfer and collision probability limitation on the side reaction for Cases 7 and 8, hence making them more favourable for the desired reaction. Cases 3–6 all consume 3 molecules in the undesired reaction, making the mass transfer limit similar; however, at the large values of k2/k1, Cases 3 and 4 have a much lower yield of P than Cases 5 and 6, forming two bands.Comparing the stoichiometries of Cases 1–4 with Cases 5–8, for the first 4 cases, 1 molecule of B is consumed in the second reaction, and in the last 4, 2 molecules of B are consumed. Comparing Case 3 to Case 6 and Case 4 to Case 5, the desired reactions are identical, whereas the stoichiometric coefficients on the undesired reaction are different. Of particular interest is the larger stoichiometric coefficient on reagentB in Cases 5 and 6. An examination of the local concentration profiles showed that B is the limiting reagent locally in the second reaction. Since Cases 5 and 6 require two molecules of the limiting reagent in the undesired reaction, making it harder for the reaction to proceed, it makes sense that the yield of P will be higher for Cases 5 and 6 as compared to Cases 3 and 4.Now comparing the stoichiometries within pairs, in three of the four pairs producing moreP in the first reactions leads to an increase in P consumption in the second reaction and a drop in yield. In other words, at large k ratios, the two reactions compete effectively for reagents. Under these conditions, when less P is produced in the desired reaction, the side reaction is slowed down. The trend is reversed for Cases 3 and 4. These are the only cases where more P is consumed than B in the undesired reaction, so in this case it appears that the mass transfer limit on P relative to B overwhelms the effect of producing more P.These trends continue in Figure2(b)  (k2/k1=0.1) and Figure 2(c) (k2/k1=0.01). For k2/k1≤0.001 the curves collapse onto each other and the effect of stoichiometry vanishes. For the very bad mixing condition (Da=10000) that was allowed to proceed for a very long time, Figure 5, the effect of stoichiometry creeps back into the picture, affecting yield at all the way to k2/k1=0.00001.


## 3.1.4. Summary of Results for the C-C Reaction Scheme

For competitive-consecutive reactions, we have the following.(a)
Da≤1 is the well-mixed limit when k2/k1≥0.01 and Da≤100 is the limit when k2/k1<0.01.(b)
k2/k1≤0.001 with Da≤100 provides yields >95% at  Da·t*=500.(c)
The effect of stoichiometry can be large if thek2/k1 is unfavourable. This effect vanishes at k2/k1≤0.001 for short reaction times and good mixing. At longer reaction times with bad mixing conditions, the effect of stoichiometry reappears and requires a much smaller value of k2/k1 to eradicate.(d)
The most favorable stoichiometry is one where the desired reaction consumes fewer molecules than the undesired reaction, thus placing a mass transfer limitation on the undesired reaction.


## 3.2. Competitive-Parallel (C-P) Reaction

The analysis proceeds through the same questions posed for the C-C reaction.(i)
What is the well mixed limit (Da)?(ii)
How much can chemistry contribute (k2/k1)?(iii)
Is the stoichiometry important?The base data is presented in Figure6, with selected data replotted in Figures 7‒9.Plots of yield ofP versus Da for decreasing k2/k1 ratios for the C-P cases.
k
2
/
k
1 = 1
(b)
k
2
/
k
1 = 0.1
(c)
k
2
/
k
1 = 0.01
(d)
k
2
/
k
1 = 0.001
(e)
k
2
/
k
1 = 0.0001
(f)
k
2
/
k
1 = 0.00001Plots of yield ofP versus k2/k1 for two sample C-P stoichiometries. The curves represent the different Da. Curves for Da=0.01 lie exactly under the curves for Da = 1.
C-P stoichiometry Case 1:A+B→k1′P; C+B→k2′S
(b)
C-P stoichiometry Case 3:A+2B→k1′P; C+B→k2′S


### 3.2.1. Effect of Damköhler Number (Da)

The plots in Figure6 show a large drop in YP when Damköhler number increases from 100 to 10000 with the exception of Case 3 at k2/k1=1 where the yield increases steadily with increasing Da. Case 3 is the only stoichiometry where the desired reaction consumes more molecules than the undesired reaction, so when the reaction rates are equal and mixing is good, the mass transfer limitation on the desired reaction will push the yield toward the secondary reaction. Since the reactions are parallel, there is no barrier to the production of S over P.For the C-P reaction pair, yield is constant forDa≤100, giving a well-mixed limit of Da=100. Figures 7(a) and 7(b) confirm this result for Cases 1 and 3, respectively. The curves show the four values of Da investigated. The curves for Da=0.01 and Da=1 lie exactly on top of each other and the curve for Da=100 is very close, but increasing Da to 10 000 drops the yield significantly. These results are very similar to the C-C results. It is concluded that Da=100 is the well-mixed limit for C-P reactions.


### 3.2.2. Effect of Dimensionless Reaction Rate Ratio (k2/k1)

Looking at Figure6 from the perspective of reaction rate ratio, as k2/k1 is decreased, the curves for all the stoichiometries move upwards, increasing the yield for all cases at all values of Da. As expected, a smaller k2/k1 increases the yield of desired product. For Cases 1 and 2 the maximum yield is approximately 1 and for Cases 3 and 4 it is 0.88. The curves for Cases 1 and 2 collapse at k2/k1=0.001 (Figure 6(d)) and the curves for Cases 3 and 4 collapse at k2/k1=0.0001 (Figure 6(e)). The largest change with improving k2/k1 is seen for Case 3, while Case 2 improves the least. These trends are confirmed in Figure 8, showing a persistent effect of stoichiometry on the maximum attainable yield. The poorly mixed condition (Da=10 000) may not have reached completion at Da·t*=500 since the yield is only about 50%.Plots of yield ofP versus k2/k1  for various Da at Da·t*=500 for C-P cases.
Da = 1
(b)
Da = 100
(c)
Da = 10,000Figure 9
Plot of yield ofP versus  k2/k1for Da=10000 at Da·t*=50000 for C-P cases.In Figure9 the poorly mixed condition is allowed to progress 100x longer. At this point, the yield can increase to above 90% but only for a 100x smaller k2/k1 ratio. The curves seem to collapse at k2/k1=0.00001, but at lower k2/k1’s the effect of stoichiometry persists, even at long times. Once again, 100x more residence time and 100x better chemistry are required to eliminate all the effects of 100x worse mixing.


### 3.2.3. Effect of Stoichiometry

Figures6(a) to 6(f) illustrate a significant effect of stoichiometry for the C-P case. At k2/k1=1 (Figure 6(a)) the difference in yield for the different stoichiometries is enormous, and there is a clear distinction between the most favourable and least favourable reaction stoichiometries (Case 2 and Case 3, respectively). This difference decreases as k2/k1 decreases (Figures 6(b) to 6(f)). For the C-P scheme, the effect of stoichiometry persists even at k2/k1=0.00001, where the difference in yield between Cases 1 and 2 and Cases 3 and 4 is more than 10%. This is different from the C-C stoichiometries where the effect of stoichiometry disappeared by k2/k1≤0.001.Referring to Table3, Cases 1 and 4 require the same number of molecules for both reactions, while Case 2 favors the desired reaction (fewer molecules) and Case 3 favors the side reaction. Case 3 is the only case of all 12 stoichiometries considered where the desired reaction consumes more molecules than the side reaction, imposing a mass transfer and collision probability limitation on the desired reaction. This explains the dramatically different results observed for Case 3 and suggests that this class of reaction stoichiometry must be handled more carefully than the others.


### 3.2.4. Summary of Results for the C-P Reaction Scheme

For competitive-parallel reactions, we have the following.(a)
Da≤100 is the well-mixed limit for all k2/k1.(b)
For all cases with favourable stoichiometries, atDa·t*=500, k2/k1 ≤ 0.001 provides the maximum yield, as for the C-C reaction scheme, but for the C-P reaction scheme the maximum yield depends on the stoichiometry. Case 3, the only case with an unfavourable stoichiometry, requires k2/k1≤0.0001 to achieve maximum yield.(c)
Stoichiometry affects the maximum obtainable yield to some degree for all mixing conditions and allk2/k1.(d)
As for the C-C scheme, increasing the number of molecules in the side reaction imposes a mass transfer limitation on that reaction, making conditions more favorable for the desired reaction. Case 3 in the C-P scheme clearly illustrates that when the desired reaction consumes more molecules than the undesired reaction, it is more difficult to achieve a high yield of desired product.


## 3.2.1. Effect of Damköhler Number (Da)

The plots in Figure6 show a large drop in YP when Damköhler number increases from 100 to 10000 with the exception of Case 3 at k2/k1=1 where the yield increases steadily with increasing Da. Case 3 is the only stoichiometry where the desired reaction consumes more molecules than the undesired reaction, so when the reaction rates are equal and mixing is good, the mass transfer limitation on the desired reaction will push the yield toward the secondary reaction. Since the reactions are parallel, there is no barrier to the production of S over P.For the C-P reaction pair, yield is constant forDa≤100, giving a well-mixed limit of Da=100. Figures 7(a) and 7(b) confirm this result for Cases 1 and 3, respectively. The curves show the four values of Da investigated. The curves for Da=0.01 and Da=1 lie exactly on top of each other and the curve for Da=100 is very close, but increasing Da to 10 000 drops the yield significantly. These results are very similar to the C-C results. It is concluded that Da=100 is the well-mixed limit for C-P reactions.


## 3.2.2. Effect of Dimensionless Reaction Rate Ratio (k2/k1)

Looking at Figure6 from the perspective of reaction rate ratio, as k2/k1 is decreased, the curves for all the stoichiometries move upwards, increasing the yield for all cases at all values of Da. As expected, a smaller k2/k1 increases the yield of desired product. For Cases 1 and 2 the maximum yield is approximately 1 and for Cases 3 and 4 it is 0.88. The curves for Cases 1 and 2 collapse at k2/k1=0.001 (Figure 6(d)) and the curves for Cases 3 and 4 collapse at k2/k1=0.0001 (Figure 6(e)). The largest change with improving k2/k1 is seen for Case 3, while Case 2 improves the least. These trends are confirmed in Figure 8, showing a persistent effect of stoichiometry on the maximum attainable yield. The poorly mixed condition (Da=10 000) may not have reached completion at Da·t*=500 since the yield is only about 50%.Plots of yield ofP versus k2/k1  for various Da at Da·t*=500 for C-P cases.
Da = 1
(b)
Da = 100
(c)
Da = 10,000Figure 9
Plot of yield ofP versus  k2/k1for Da=10000 at Da·t*=50000 for C-P cases.In Figure9 the poorly mixed condition is allowed to progress 100x longer. At this point, the yield can increase to above 90% but only for a 100x smaller k2/k1 ratio. The curves seem to collapse at k2/k1=0.00001, but at lower k2/k1’s the effect of stoichiometry persists, even at long times. Once again, 100x more residence time and 100x better chemistry are required to eliminate all the effects of 100x worse mixing.


## 3.2.3. Effect of Stoichiometry

Figures6(a) to 6(f) illustrate a significant effect of stoichiometry for the C-P case. At k2/k1=1 (Figure 6(a)) the difference in yield for the different stoichiometries is enormous, and there is a clear distinction between the most favourable and least favourable reaction stoichiometries (Case 2 and Case 3, respectively). This difference decreases as k2/k1 decreases (Figures 6(b) to 6(f)). For the C-P scheme, the effect of stoichiometry persists even at k2/k1=0.00001, where the difference in yield between Cases 1 and 2 and Cases 3 and 4 is more than 10%. This is different from the C-C stoichiometries where the effect of stoichiometry disappeared by k2/k1≤0.001.Referring to Table3, Cases 1 and 4 require the same number of molecules for both reactions, while Case 2 favors the desired reaction (fewer molecules) and Case 3 favors the side reaction. Case 3 is the only case of all 12 stoichiometries considered where the desired reaction consumes more molecules than the side reaction, imposing a mass transfer and collision probability limitation on the desired reaction. This explains the dramatically different results observed for Case 3 and suggests that this class of reaction stoichiometry must be handled more carefully than the others.


## 3.2.4. Summary of Results for the C-P Reaction Scheme

For competitive-parallel reactions, we have the following.(a)
Da≤100 is the well-mixed limit for all k2/k1.(b)
For all cases with favourable stoichiometries, atDa·t*=500, k2/k1 ≤ 0.001 provides the maximum yield, as for the C-C reaction scheme, but for the C-P reaction scheme the maximum yield depends on the stoichiometry. Case 3, the only case with an unfavourable stoichiometry, requires k2/k1≤0.0001 to achieve maximum yield.(c)
Stoichiometry affects the maximum obtainable yield to some degree for all mixing conditions and allk2/k1.(d)
As for the C-C scheme, increasing the number of molecules in the side reaction imposes a mass transfer limitation on that reaction, making conditions more favorable for the desired reaction. Case 3 in the C-P scheme clearly illustrates that when the desired reaction consumes more molecules than the undesired reaction, it is more difficult to achieve a high yield of desired product.


## 3.3. Phase Plots of Variables for C-C And C-P Reaction Schemes for the Purpose of Design

The purpose of design graphs, in the context of mixing sensitive reactions, is to provide a direct method of predicting the yield of desired product without extensive calculations and investigation of the different variables. For our purposes, design graphs would incorporate the effects of stoichiometry, mixing, andk2/k1 and the combinations of the three that could possibly give a certain specified yield of desired product. The purpose of the graphs would be to assist in the design of the following problem.“If one has a certain mixing sensitive reaction and a target yield of desired product in mind, what range of values of dimensionless reaction rate ratio and Damköhler would be required to ensure that the target yield of desired product will be obtained? If one of the variables is inadequate, is it still possible to obtain the target yield? If it is possible, how much do the other variables need to change to accommodate for that inadequacy, if at all?”The figures should allow for a design space to be established within which decisions can be made based on different criteria and restrictions.Figure10 and Figure 11 show the required Damköhler number (x-axis) versus the dimensionless reaction rate ratio (y -axis) for yields of desired product of (a) 85% or more, (b) 95% or more, and (c) 99% or more at a time of  Da·t*=500. They are set up such that the axis intersection (origin) represents the most favourable conditions for both k2/k1 and Damköhler number. The figures have marked regions of required k2/k1 and Da where a specified yield can be obtained for different stoichiometries. These figures are essentially a filtered summary of the results presented in the previous sections. The effect of stoichiometry quickly becomes evident as the specified yield is increased. For the C-C cases, the requirements on the k2/k1 and Da become more stringent as YP increases. For C-P Cases  3 and 4, yields greater than 90% were unachievable for  Da·t*=500.Design spaces for yield ofP for Da and k2/k1  at  Da·t*=500 for C-C cases.
Y
P is at least 0.85 (85%)
Y
P is at least 0.95 (95%)
Y
P is at least 0.99 (99%)Design spaces for yield ofP for Da and k2/k1 at    Da·t*=500  for C-P cases.
Y
P is at least 0.85 (85%)
Y
P is at least 0.95 (95%)
Y
P is at least 0.99 (99%)In order to link these results to physical process variables, the diffusion time is replaced with an analogous mixing time scale for turbulent flow. The Batchelor time scale gives the diffusion time for a Kolmogorov sized eddy and is numerically equal to the Kolmogorov time scale, which is the lifetime of the same smallest eddy before the momentum is diffused by viscosity. Replacing the molecular diffusion time in a slab in our definition of Da with the expression for the Kolmogorov time scale in a turbulent eddy gives the following:(5)Da=k1′(ρTM)ϵ(νεD)1/2
where ν is the kinematic viscosity and εD is local rate of dissipation of turbulent kinetic energy (TKE) per mass. Rearranging to solve for εD gives the following:
(6)εD=ν(k1′(ρT/M)ϵDa)2.This expression proposes a relationship between the rate of dissipation of turbulent kinetic energy (εD), the reaction kinetics of the desired reaction (k1′), and the reaction stoichiometry (ϵ) which might be used to compare bench scale performance with the Da and reaction rate ratio from the phase plot. For example, for a C-C reaction of the type Case 1 (ϵ=1), using typical values for water (ν=1×10-6 m2/s, ρT=1000kg/m3, M=18kg/kmole), εD for Da = 1 becomes
(7)εD=3.09×10-3k1′2.This gives a first-order estimate for the required level of dissipation at the bench scale. Since the model is based on a static striation thickness rather than the dynamic stretching and cutting field present in both chaotic and turbulent flows, this should be considered a starting point for further investigations in the context of more accurate models of mixing and diffusion.


## 4. Conclusions

The three effects of initial mixing condition, dimensionless reaction rate ratio, and reaction stoichiometry were investigated in detail using a 1D, transient, reaction-diffusion model for competitive-consecutive (C-C) and competitive-parallel (C-P) reactions. It was found that smaller values of Damköhler number and dimensionless reaction rate ratio are desirable to maximize the yield of desired product for both the C-C and C-P reaction schemes. It was also found that the stoichiometry of the reaction can affect the final yield of desired product considerably and needs to be taken into consideration in the design of reactors for such reactions. The following limits were found for each of the variables.(i)
Da≤1  is the well-mixed limit for the C-C reaction when k2/k1≥0.01. With k2/k1≤0.01, the well-mixed limit can be relaxed to Da≤100.(ii)
Da≤100 is the well-mixed limit for all k2/k1 for the C-P reaction.(iii)
k2/k1≤0.001 with Da≤100 provides the maximum yield at Da·t*=500 for both the C-C and the C-P reactions with favourable stoichiometries. For the C-P case with the most unfavourable stoichiometry k2/k1≤0.0001 is required.(iv)
The effect of stoichiometry is significant. Where the secondary reaction consumes more molecules than the primary reaction, it is subject to a mass transfer constraint which favors the primary reaction. For the C-P scheme the effect of stoichiometry is persistent, affecting the final yield by more than 10% even for a well-mixed case with favorablek2/k1.The model used here is intended to illustrate the importance of the previously uninvestigated effect of stoichiometry on the yield of desired product and to provide general forms of the Damköhler number and reaction rate ratio for C-C and C-P reactions. The results are not intended to give accurate predictions of yield but to illustrate the trends in a quantitative way. It is hoped that the results will guide development of experimental reaction schemes which serve as mixing probes and development of more realistic reactor models which incorporate mixing, reaction kinetics, and stoichiometry.



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*Source: 654321-2012-10-09.xml*