An Extension to the Incorporation Model of Micromixing and Its Use in

3460

Ind. Eng. Chem. Res. 2008, 47, 3460-3469

An Extension to the Incorporation Model of Micromixing and Its Use in Estimating Local Specific Energy Dissipation Rates M. Assirelli,‡ E. J. W. Wynn,§ W. Bujalski, A. Eaglesham,† and A. W. Nienow* Department of Chemical Engineering, UniVersity of Birmingham, B15 2TT, United Kingdom, and Huntsman Polyurethanes, Brussels, Belgium

The incorporation model of micromixing first developed conceptually and quantified by Villermaux and coworkers has been extended to cover both higher rates of micromixing (higher mean and local specific energy dissipation rates) and higher reaction rates (by using higher acid concentrations in the iodide-iodate model reaction scheme). The extended model has involved the use of Bader and Deuflhard’s semi-implicit discretization in the Bulirsch-Stoer method, which is especially suitable for stiff ordinary differential equations. Both exponential and linear rates of incorporation were considered, and polynomial equations for three acid concentrations for micromixedness ratios, R, from ∼1 to ∼100, were determined. It was shown that, though different acid concentrations gave different R values at the same feed position and agitation conditions, the micromixing time estimated from the model was constant (as it should be) with exponential incorporation. With linear incorporation, the micromixing time was much less and not constant and was therefore rejected for further analysis. Subsequently, it was shown that φ, the ratio of the local specific energy dissipation rate, T, to the average, jT, i.e., φ ) T/jT, was constant at the same reactant feed position except when fed into the region of (T)max close to the impeller. In this case, φ fell with increasing speed as the reactants were swept more rapidly from this region to regions of lower φ. By comparing estimates of φ from feeding a reactant at equivalent positions with a static pipe and one rotating with the impeller, it was found that Φ ) φpoint,max/φensemble,max was ∼2.7, in reasonable agreement with the value of ∼3 very recently obtained by Ducci and Yianneskis (AIChE J. 2005, 51, 2133) based on two-point laser Doppler anemometry (LDA) measurements. The absolute value of φensemble,max was rather high compared to the most recent estimates from LDA or particle image velocimetry (PIV), which may reflect some weakness in the model or in the quality of the chemical kinetics data. Introduction This paper is the fourth in a series related mainly to micromixing. In each of them, the iodide-iodate technique, developed earlier by Fournier et al.2 and Guichardon and Falk,3 has been used to characterize the micromixing. In the first two papers, the geometry employed was the typical baffled vessel agitated by a Rushton turbine. The first4 confirmed that feeding reactants with a fixed pipe close to the impeller, especially into the upper trailing vortex of a Rushton turbine, gave very much improved micromixing compared to surface feeding. The second paper5 showed that even better micromixing could be obtained by feeding at the same position but with the feed pipe rotating with the impeller, so that the feed always entered at the same position into the trailing vortex. The third paper considered micromixing in an equivalent unbaffled tank6 and showed a much better micromixing performance than anticipated, often giving a lower segregation index than in the baffled case at the same mean specific energy dissipation rate, jT (W/kg). In those papers, some use was made of an extension to the micromixing model developed by Villermaux and co-workers. Here, that extension is developed in more detail. In particular, for the numerical solution of the incorporation model, the Bulirsch* To whom correspondence should be addressed. E-mail: [email protected]. Phone: +44 121 414 5325. Fax: +44 121 414 5325. † Huntsman Polyurethanes. ‡ Current address: Institute of Chemical and Engineering Sciences, 1 Pesek Road, 627833 Singapore. § Current address: Fluent Europe Ltd., Sheffield Business Park, Sheffield S9 1XH, United Kingdom.

Stoer method for stiff equations7 is proposed as a better way to obtain high-accuracy solutions compared to the Runge-Kutta method of the fourth order proposed by Guichardon,8 and the two different versions are compared. Also, two incorporation functions, linear and exponential, are considered. It is shown that the exponential is much better than the linear, and that was the one used occasionally in the previous papers. The model development also enables further insight into micromixing to be obtained. Development of the Extension to the Earlier Model of Villermaux and Co-workers Falk and Villermaux9 introduced a generalized mixing model (GMM) of the Lagrangian type to analyze micromixing phenomena. The model assumes that an incoming reactant (into a stirred vessel, for example) is divided into a series of aggregates that interact with the bulk fluid but do not interact with each other. The mixing process is then based on four basic steps: erosion of the fresh fluid, dilution of the eroded material into the reacting cloud, incorporation of fluid from the bulk into the reacting cloud, and interaction (or diffusion) within the erosion and incorporation areas. The authors also presented examples showing the GMM model as a general case within which previously published models can be considered as special cases. They concluded that, if the incorporation process is considered the dominating mechanism, the GMM degenerated into the incorporation model and, using the appropriate functions, was shown to be equivalent to the engulfment model developed by Baldyga and Bourne.10 The reactions involved in the experimental method used in this work and in the modeling are as follows:

10.1021/ie070754n CCC: $40.75 © 2008 American Chemical Society Published on Web 12/12/2007

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3461 + H2BO3 + H S H3BO3

(1)

+ 5I- + IO3 + 6H S 3I2 + 3H2O

(2)

I2 + I- S I3

(3)

Small quantities of sulfuric acid were injected into the agitated tank containing a larger quantity of iodide and iodate ions buffered by sodium hydroxide and boric acid to give equimolar quantities of H3BO3 and H2BO3-. The injected sulfuric acid reacts almost instantaneously with the latter ion in the acidbase reaction 1. If the injected acid is mixed very quickly with the contents of the tank, then reaction 1 dominates with the small quantity of acid being neutralized by the excess of buffer. In contrast, if only a small proportion of the contents are micromixed with the injected acid, then the small amount of H2BO3- locally will be exhausted, leaving some H+ ions to undergo the Dushman reaction 2. When reaction 2 occurs, the iodine produced reacts to give triiodide ions that can be detected by spectrophotometry. For the model, it is assumed that the initial volume of injected fluid, Vφ2 , increases to a volume, V2(t), at time t because of the incorporation of some of the tank contents by micromixing. The resulting solution is assumed to be locally well-mixed but segregated from the remainder of the tank contents. Two alternative assumptions can be made concerning the rate of the incorporation. In the first, the volume increases linearly with time, the linear model, so that

V2(t) Vφ2

φ t dV2(t) V2 )1+ , ) tm dt tm

(4)

sented by fA, defined to be the total number of moles of A in the volume V2, made dimensionless by dividing by the total number of moles of H+ in the original injection,

fA )

Vφ2

()

g ) V2(t)/Vφ2

(11)

φ [H2BOdfH+ 3 ] 1 dg ) -6F2 dθ [H+] φ dθ

(12)

[I-] φ1 dg dfI) -5F2 - F3 + F′3 + + φ dθ [H ] dθ

(13)

2

2

φ [IO3 ] 1 dg ) -F2 + + φ dθ [H ] dθ

dfIO-3

(14)

2

dθ

) 3F2 - F3 + F′3

dθ

(5)

) F3 - F′3

(15)

(16)

with the dimensionless rates of reaction

+ 2 r2 ) k2[I-]2[IO3 ][H ]

(6)

F2 ) k2tm

log 10(k2) ) for µ < 0.166M for µ > 0.166M

]

(17)

(18)

F′3 ) k′3tm fI-3

(19)

and a dimensionless equivalent of eqs 4 or 5.

(7)

Finally, for the third reaction (eq 3), the rate was calculated from

r3 ) k3[I-][I2] - k′3[I3]

( ) ( )

[H+] φ2 4 (fH+fI-)2fIO-3 g

[H+] φ2 F3 ) k3tm (fI2fI-) g

where the rate constant was a function of the ionic strength, µ:

9.28105 - 3.664xµ 8.383 - 1.5112xµ + 0.23689µ

(10)

dfI-3

Again, tm is the micromixing time. The rates of the reactions have been carefully studied, and the previously published data3,11 were used in the present study. The acid-base reaction (eq 1) was assumed to be effectively instant. The rate of the second reaction (eq 2) was

{[

θ ) t/tm

The equations to be solved were, therefore,

dfI2

dV2(t) V2(t) t ) ) exp , tm dt tm

(9)

V2φ[H+] φ2

(where the subscripts on all concentrations are either 1 to denote the tank contents or 2 to denote the injected fluid). The benefit of using these quantities is that, unlike concentrations, they do not change merely due to dilution by incorporation of tank fluid. Their rates of change are, therefore, simpler to calculate. Dimensionless versions of time and volume were also used:

where tm is the micromixing time. Alternatively, the volume increases exponentially, the exponential model, so that

V2(t)

V2(t)[A]2

(8)

where k3 ) 5.9 × 106 m3 mol-1 s-1 and k 3′ ) 7.5 × 106 s-1. The justification for choosing these precise values for the rate constants is given elsewhere.3 The concentrations were simulated by solving the relevant differential equations, presented in a dimensionless form (in a similar way to Fournier et al.12): each species A was repre-

Numerical Method The ordinary differential equations (ODEs) representing the rates of change of concentration were solved using a standard ODE integrator with adaptive step size. Specifically, Bader and Deuflhard’s semi-implicit discretization was used in the Bulirsch-Stoer method.7 This method is suitable for stiff ODEs. The adaptive step size was such that a smaller step would be required to replace any step whose estimated error magnitude in any of the variables was greater than 10-7 times that variable’s magnitude. One possible test of the overall accuracy was conservation of species, such as iodine atoms in iodate, iodide, iodine, and triiodide. For the example of iodine conservation, the errors are generally comparable to machine precision (using

3462

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 Table 2. Mean Specific Energy Dissipation Rates, EjT, for the Rushton Turbine N (rps)

jT (W/kg)

5.0 7.0 8.7 9.2

0.18 0.50 0.97 1.14

Table 3. Summary of the Initial Concentration of the Reactants Used reactants

concentration used (M)

[H3BO3] [NaOH] [KI] [KIO3] [H+]

Figure 1. Diagram of the vessel T29 equipped with baffles showing the location of the feed points.4

have values between 0 and 1 indicating the fraction of unwanted product, i.e., iodine. Therefore, the lower the value of XS, the better is the micromixing. The segregation index is defined as Xs ) Y/YST, where

Y)

Table 1. Coordinates of the Feed Locations

2r/D z/H

0.1818 0.0909 0.01167 0.00233 1.0, 2.0, 4.0

position

position

position

position

2.4 0.92

2.4 0.80

2.4 0.27

1.04 0.22

“double precision”, eight-byte variables throughout). This demonstrates that species conservation is present in the formulation of the equations; the errors are due to rounding only, and discretization errors are not apparent in this check. To simplify the evaluation of the Jacobian matrix, the effect of reactions on the ionic strength was neglected. This assumption is justified because the ionic strength changes slowly and has a small effect relative to other parameters. This point was confirmed by comparing the simulation results to those from a more laborious Runge-Kutta method, which did not use the Jacobian. The calculation of the reactions as a function of time continues until all the acid in the injected fluid has been neutralized. The integration of the ODEs tends to overshoot, resulting in small negative concentrations. Here, all negative concentrations were ignored and nonphysical negative reaction rates were set to zero. This procedure caused the overshoot to be small, because the adaptive-step-size routine detected the change in rate and reduced the step size to home in on the change. Experimental Section The experimental procedure has been reported in detail previously.4 In summary, the procedure is the iodide-iodate technique with successive injections into the 0.29 m diameter (T) baffled tank agitated by a Rushton turbine of diameter 1/3T at the points shown in Figure 1 at the coordinates given in Table 1 and midway between the baffles. The speeds (N (rev/s)) used and the mean specific energy dissipation rates (jT) are given in Table 2. A summary of the concentrations used are given in Table 3, where three different acid concentrations are indicated, each of which leads to different reaction rates. Following the addition of acid, the amount of I3 (determined by spectrophotometric analysis at λ ) 353 nm) at the end of the run gives an indication of the micromixing “efficiency” or selectivity at the particular feed position investigated. Further details are reported elsewhere.4,5,13 The micromixing efficiency is quantified by the segregation index, XS, which can

2Vtank[(I2) + (I3 )] Vinjection(H+)φ

(20)

and

YST )

φ 6(IO3) - φ φ 6(IO3 ) + (H2BO3 )

(21)

Thus, Y can be considered to be the ratio of moles of acid consumed by the reaction to give I2 over the total moles of acid injected, while YST is the yield of the reaction under total segregation (poor mixing) conditions, where the quantity of iodine formed is only due to the stoichiometric ratio of the reactants. Another parameter used is the micromixedness ratio, R, where

R)

1 - Xs Xs

(22)

It has been suggested that R can be considered as the ratio of the volume of the tank perfectly micromixed to the volume totally segregated.14,2 Combining the Model Predictions with the Experimental Results to Calculate Micromixing Parameters From the model, the final concentrations of the reactants, and therefore the parameters Xs and R as a function of the incorporation time assumed to be equal to the micromixing time, tm, can be calculated.12 Rousseaux et al.15 presented a summary of the results from such calculations as three power-law relations between tm and R, which corresponded to a particular set of conditions, namely, that the concentration [H+] was 1.0 mol/L and the exponential-growth model as in eq 5 was used. These relationships were

2 < R < 5, tm ) 0.73R-2.26

(23)

5 < R < 7, tm ) 0.82R-2.30

(24)

7 < R < 20, tm ) 0.158R-1.45

(25)

It was then assumed that the micromixing time, tm, evaluated

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3463

Figure 2. Results of the incorporation model, R vs tm, at three acid concentrations (a comparison of exponential and linear rates of incorporation).

through the incorporation model is the same as that arising from the local specific energy dissipation rate, T,16 in terms of Kolmogoroff’s turbulence theory, i.e., using the relationship proposed by Baldyga and Bourne:10

tm ) 17.24

x

ν Τ

(26)

Since the local specific energy dissipation rate, T, is proportional to the mean specific energy dissipation rate, jT, φ ) T/jT can be calculated. However, it should be noted that the micromixedness ratio, R, and the micromixing time, tm, are essentially conceptual and model parameters. The same procedure has been followed here. The incorporation model was solved numerically as set out above for the same acid concentration of [H+] ) 1.0 M as previously reported, plus at 2.0 and 4.0 M and extended to values of R > 20 for both the exponential and the linear incorporation models. The present calculated values of micromixedness ratio and micromixing time based on the exponential model were compared with the values previously reported8,3 for the same reactant concentrations. The values were in excellent agreement, though the solutions did not match completely. When compared to the relationships summarized in eqs 23, 24, and 25 at a central point in each of the three ranges, for a fixed value of R, the value of tm differed by 6%, -4%, and -10%, respectively. These discrepancies can partly be attributed to the approximate nature of the fits in eqs 23, 24, and 25 (as an illustration of this, there is a 5% discontinuity in the transition at R ) 5). Disagreements could also be ascribed to the different numerical methodology employed here and to ambiguities in some of the initial kinetic parameters given in the literature11,12 necessary for the solution of the model. Figure 2 shows three solutions of the present model, plotting micromixedness ratio R versus micromixing time tm corresponding to each of the three sulfuric acid concentrations used assuming both an exponential incorporation or a linear function. From Figure 2, it can be seen that, at very low values of tm (very efficient mixing condition), a change in the acid concentration is reflected in a big change in micromixedness ratio, R,

Table 4. Summary of the Parameters Obtained from the Polynomial Regression of the Type y ) ax3 + bx2 + cx + d for Exponential Incorporation [H+]

a

b

c

d

R2

1.0 2.0 4.0

-0.089 -0.081 -0.065

0.965 0.926 0.832

-4.63 -4.73 -4.66

1.35 1.14 0.97

1.00 1.00 1.00

Table 5. Summary of the Parameters Obtained from the Polynomial Regression of the Type y ) ax3 + bx2 + cx + d for Linear Incorporation [H+]

a

b

c

d

R2

1.0 2.0 4.0

-0.114 -0.074 -0.032

1.054 0.908 0.800

-4.428 -4.386 -4.360

-0.801 -1.688 -2.477

1.00 1.00 1.00

and that the variation in R becomes progressively smaller with increasing micromixing time, tm. Clearly, the linear model also gives very different and much shorter micromixing times than the exponential model for the same R values, especially at high values of R. For example, for a typical value of R, such as 4.85 and acid concentration [H+] ) 1.0 M, the linear model suggests a micromixing time of ∼3.5 ms and the exponential model suggests one of ∼20 ms. Assirelli et al.4 showed experimentally that variation of the volume of the acid injected did not lead to different values of Xs (which further justified the adoption of the method of successive injection of acid used throughout all the work leading to this series of papers). Values of tank/acid volume ratio ranging from 250 to 1250 were tested here in the numerical solution of the two incorporation models. The values of Xs and R obtained did not vary significantly with the volume ratio with either the exponential or the linear models, in agreement with the earlier experimental work. In order to establish a better fit of the solutions of the incorporation model given in Figure 2, a polynomial regression was adopted here. For each solution to the model corresponding to a different concentration of sulfuric acid, a polynomial regression equation of the type y ) ax3 + bx2 + cx + d was obtained where y ) ln(tm) and x ) ln R. A summary of the empirical parameters obtained is listed in Table 4

3464

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008

Figure 3. Results at feed position at different agitator speeds: (a) variation of the micromixedness ratio, R, with the acid concentration and (b) comparison of the micromixing time, tm, at the two acid concentrations for the exponential and linear models.

for exponential rates of incorporation and in Table 5 for linear rates. Estimating the Values of Micromixing Times, tm, from the Micromixedness Ratio, r, Using the Present Models With the present model results, it is now possible to estimate values of tm, for all the acid concentrations used and for the wider range of R values measured in this work (based on the Xs values previously reported4,5), in comparison with those in the literature. Figure 3a shows the values of micromixedness ratio, R, for the feed position at the two sulfuric acid concentrations, [H+] ) 1.0 M and [H+] ) 2.0 M, and for the rotational speeds investigated. Contrary to the trend obtained with the segregation index, Xs,4 the micromixedness ratio increases with increasing rotational speed and decreasing acid concentration. In other words, as the rate of mixing is enhanced by the increasing agitator speed, the proportion of the vessel that is well-mixed is increased. Similarly, if the rate of reaction decreases, the overall effect is the same.

Figure 3b shows the corresponding values of micromixing time, tm, with their respective error bars for the two concentrations of acid, calculated for both the linear and exponential incorporation models. In the exponential case, the same values of micromixing time, tm, within experimental error, at both acid concentrations, were obtained. On the contrary, tm values calculated for the linear model are significantly different (their error bars never overlap) at the two acid concentrations. Logically, it is to be expected that, since varying the acid concentrations leads to different values of the chemical reaction rate, the segregation index, Xs, and the micromixedness ratio, R, which depend on the balance between that rate (which varies with concentration but is independent of agitation conditions) and the rate of mixing (which is independent of concentration but varies with agitation conditions), should depend on the concentration. On the other hand, the micromixing time, tm, a fluid dynamic parameter, should vary with agitation conditions but be independent, within experimental error, of the acid concentration used and, consequently, of the



kinetics of the system. Figure 3b suggests that concept is matched for the exponential incorporation model but not for the linear one. Figure 4 and Figure 5 show the same parameters represented in Figure 3 but for the feed positions and , respectively. Again, the micromixing time, tm, is independent of the reactant concentration used at a fixed agitator speed for the exponential incorporation model, and the values of tm became smaller with increasing agitator speed and when feeding closer to the agitator, i.e., as T increases in going from feed point to , for the exponential growth model. On the other hand, again, tm seems to be dependent on the reactant concentration used when assuming the linear model. Overall, these results lead to the conclusion that adoption of such a linear incorporation model for the reactive cloud is inappropriate, and the use of the exponential incorporation model is recommended. Finally, a similar analysis, but only using the exponential incorporation model as justified above, was undertaken on the Xs data obtained from experiments undertaken with the feed at

position using a modified Rushton turbine (MRT). This modification enabled acid addition to be made continuously into the trailing vortex, so that T is always at or close to its maximum value, by means of a feed pipe rotating with the agitator.5 The results are shown in Figure 6. Figure 6a shows the R values (higher than those in Figures 3-5 for the same acid concentrations but again with a lower value for the stronger acid), while Figure 6b shows that the exponential incorporation model again gives the same micromixing time, tm, independent of the acid concentration but less than that with all the other feed modes. Estimating the Local Specific Energy Dissipation Rates, ET, and O, the Ratio of the Local ET to EjT, from the Present Exponential Model On the basis of the micromixing time, tm, the local T has been calculated via eq 26. Using jT values given in Table 2, φ

3466



) T/jT values have been determined. The values are reported in Table 6 for all the static feed pipe experiments for [H+] ) 1.0 M. While the local tm values fall with increasing agitator speed as shown earlier, and hence the T values rise, the φ values do not show any trend (except very close to the impeller, and this is discussed further below). Essentially, for positions to , the values of φ are constant independent of speed within experimental error, as has been found when determining φ by LDA17 or PIV.18 Generally, they are of the order expected, being significantly less than 1 at positions and and well above 1 at position .17 Determining φ by laser-based techniques is not a trivial task18 and requires certain significant assumptions and extensive data treatment. Indeed, there is a very significant range of values reported using such techniques, especially close to the impeller. Nevertheless, all the values shown in Table 6 are in reasonable accord with the trends reported from LDA measurement, e.g., from Geisler et al.17

Table 7 and Table 8 show the values of the ratio φ evaluated at static feed positions and , respectively, for two acid concentrations, and again, all the values estimated are in good agreement with each other. Table 9 and Table 10 are for the two feed modes where the position chosen for the acid addition is the best estimate for the location of (T)max, with either a static pipe (point ) or one moving so that it always maintains that position relative to the impeller (the MRT), respectively. These two cases are worthy of further consideration. First, in each case, there are random variations at the different acid concentrations, so that, at a particular speed, φmax may be greater at either concentration. This greater degree of scatter is related to the model, which leads to a greater sensitivity of tm to R at high R values (see Figure 2) and, hence, T and φ. In addition, experimental scatter has a much bigger impact close to the impeller where measured Xs values are the lowest and in the limit as Xs f 0, R f ∞, so that small errors in Xs become magnified again in determining


Figure 6. Results with continuously feeding at position with the modified Rushton turbine at different agitator speeds: (a) variation of the micromixedness ratio, R, with the acid concentration and (b) comparison of the micromixing time, tm, at the two acid concentrations for the exponential model 2. Table 6. Values of ET and O ) ET/EjT for [H+] ) 1.0 M for All Static Feed Pipe Experiments position

position

N (rps)

T (W/kg)

φ

T (W/kg)

φ

5.0 7.0 8.7 9.2

0.03 0.14 0.27

0.15 0.29 0.28

0.06 0.16 0.45

0.35 0.33 0.47

position

Table 7. Values of O ) ET/EjT for Position at the Acid Concentration [H+] ) 1.0 and 2.0 M N (rps)

[H+] ) 1.0 M φ

[H+] ) 2.0 M φ

5.0 7.0 8.7

0.35 0.33 0.47

0.37 0.34 0.45

position

T (W/kg)

φ

T (W/kg)

φ

1.00 3.95 5.71 6.63

5.7 8.0 5.9 5.8

24.3 52.2 58.2 63.3

138 106 60 55

T and φ. Also, close to the impeller, the actual value of T is very sensitive to position, so that any slight discrepancy in the placing of the feed pipe between runs would again lead to potentially large real changes in T and φ. It is suggested that the somewhat random variations of φmax with different acid concentrations arise for these three reasons. If the results with agitator speed for each feed mode are now compared, there is a progressive fall in φ with increasing speed. This trend is probably because, with increasing speed, the

Table 8. Values of O ) ET/EjT for Position at the Acid Concentration [H+] ) 1.0 and 2.0 M N (rps)

[H+] ) 1.0 M φ

[H+] ) 2.0 M φ

5.0 7.0 8.7 9.2

5.7 8.0 5.9 5.8

7.8 6.5 6.0 5.7

reactants are more rapidly swept away from the chosen region of highest T very close to the impeller to lower values, so that the locally “average T” experienced by the reactants is less.

3468


Table 9. Values of Oensemble,max ) (ET)ensemble,max/EjT for Position at the Acid Concentration [H+] ) 1.0 and 4.0 M N (rps)

[H+] ) 1.0 M φensemble,max

[H+] ) 4.0 M φensemble,max

138 106 60 55 90

201 88 88 47 106

5.0 7.0 8.7 9.2 mean

Table 10. Values of Opoint,max ) (ET)point,max/EjT for the MRT at the Acid Concentration [H+] ) 1.0 and 4.0 M φpoint,max [H+]

N (rps) 5.0 7.0 8.5 9.2 mean

) 1.0 M

[H+] ) 4.0 M

304 345 275 168 273

371 281 198 141 247

Table 11. Φ ) (ET)point,max/(ET)ensemble,max ) (Omrt/O4) at the Different Agitator Speeds Based on the Current Reaction Data and Model N (rps)

φ

5.0 7.0 8.5/8.7 9.2 mean

1.99 3.22 3.17 3.02 2.66

For example, in the case of the rotating pipe, the feed will be into a region with a velocity close to that of the tip speed, as measured by LDA or PIV.1,18,20 Assuming that the local velocity is the tip speed and that the added acid is immediately accelerated to that speed, then during the micromixing time of about 1 ms (Figure 6), it will move (πDN)tm, i.e., ∼3 mm. Given the small size of the zone17,20 of (T)max, convection of reactants to zones of lower T seems quites feasible, leading to the fall in estimated T with increasing speed. Comparing now the two feed modes, φ with the rotating pipe always feeding into (T)max is in every case greater than that with the static pipe where T varies cyclically from this maximum value to lower ones with the passage of each impeller blade. Until recently, all work on micromixing had used a static feed pipe to give chemical reaction data for positions close to the impeller as a result of the “locally average” flow and any estimate of the local T was similarly an “average”. So the data presented here comparing the “locally averaged maximum T” from the fixed point , i.e., φ4 or φensemble,max, with that from the MRT, i.e., φmrt or φpoint,max, is an extension of that first presented by Assirelli et al.5 However, at the time of going to press with that paper, the only literature data from LDA or PIV on the maximum value of T was of the ensemble average type, (T)ensemble,max. Though there was considerable information on the angle-resolved flow field, it was not of sufficient accuracy to enable angle-resolved or point values, φpoint,max to be determined. Very recently, Ducci and Yianneskis1 have measured both ensemble-averaged and angle-resolved T very close to the Rushton impeller in the trailing vortex, thus enabling Φ ) φpoint,max/φensemble,max based on two point LDA measurements to be determined. A maximum value of Φ of ∼3 at an angle θ ≈ 15° after the blade is indicated, while the values found here with addition at θ ≈ 20° are presented in Table 11 for each agitator speed with an overall average of ∼2.7. These results are in very good agreement given the experimental and data treatment difficulties in both the LDA and chemical techniques. It is worth noting that, when eqs 23-25 from Rousseaux et al.15 were used outside their recommended range in the earlier

paper by Assirelli et al.5 to estimate Φ, a value of 3.5 was obtained, i.e., the agreement with the work of Ducci and Yianneskis,1 which was not available at the time, was slightly worse. Finally, the absolute values of φensemble,max/jT available in the literature should be compared with those estimated here from the current extension of the incorporation model. Table 9 gives an average value of just less than 100. This value is rather high compared to the literature, though the range there is very large. The first estimate of φensemble,max/jT was by Cutter,19 who used a tracer/photographic technique and gave a value of ∼70. More recently, Schafer,20 using the more sophisticated LDA approach, gave a value of ∼52. The most recent estimate was from Ducci and Yianneskis,1 again using two-point LDA. They expressed the ensemble average of (T)max in the form (T)ensemble,max/N3D2 ) 12. Thus, for D/T ) 1/3 and assuming that Po ) 5 (a value was not given in the paper), φensemble,max/jT ) 51, in very good agreement with Schafer.20 All these values are significantly less than the value estimated here, but given the potential errors from the two different sets of experiments and the data treatment associated with the two ways of making the measurements, the agreement is not totally unacceptable. On the other hand, it may be that the basic assumption in the incorporation model, that the micromixing time is equivalent to that estimated from Kolmogoroff’s theory of locally isotropic turbulence (eq 26),15,16 is incorrect. The errors involved in the two totally different ways of arriving at (T)max and φmax are probably too great to resolve this question from this work. In addition, of course, there is always the possibility that the kinetics of the complex reaction scheme are not described accurately in the relationships in eqs 1-3 and 5-8 for all the concentrations used. On the other hand, the ability of the model to produce unified values of φ for each injection position would not support this possibility. Conclusions The incorporation model of micromixing has been extended to cover both shorter micromixing times due to higher mean and local specific energy dissipation rates and higher reaction rates. It has been solved more efficiently and accurately by using Bader and Deuflhard’s semi-implicit discretization in the Bulirsch-Stoer method, which is especially suitable for stiff ordinary differential equations. Both exponential and linear rates of incorporation were considered, and polynomial equations for three acid concentrations for the iodide-iodate reaction scheme were determined. It was shown, based on previously reported experimental segregation indices, Xs,4,5 that, though different acid concentrations gave different micromixedness ratios, R, at the same feed position and agitation conditions, the micromixing time estimated from the model was constant (as it should be) with exponential incorporation. With linear incorporation, the micromixing time was much less and not constant and was, therefore, rejected for further analysis. Thus, it is recommended that the model using exponential incorporation be used. Subsequently, based on the assumption that the micromixing time from the model was equivalent to that based on Kolmogoroff’s theory of locally isotropic turbulence, it was shown that φ, the ratio of the local specific energy dissipation rate, T, to the average, jT, i.e., φ ) T/jT, was constant, independent of agitator speed, at the same reactant feed position, except when, based on earlier LDA and PIV studies, the position was into the region of (T)max close to the impeller. In the latter case, φ fell with increasing speed, and it is suggested that this is because the reactants were swept more rapidly from this region to regions of lower φ.


By comparing estimates of φ from feeding at the point with the highest local specific energy dissipation rate with a static pipe (to give φensemble,max4) and one rotating with the impeller (to give an angularly resolved φpoint,max at a position ∼20° behind an impeller blade5), it was found that Φ ) φpoint,max/φensemble,max was ∼2.7. This value is in reasonable agreement with the value of ∼3 very recently obtained by LDA measurements for the first time by Ducci and Yianneskis.1 The absolute value of φensemble,max was rather high compared to the most recent estimates from LDA or PIV, which may reflect some weakness in the original model or the kinetics on which the whole analysis is based. Acknowledgment One of us (M.A.) acknowledges financial support for this project by Huntsman Polyurethanes Europe BVBA (Belgium) and the School of Chemical Engineering in Birmingham (U.K.). The authors also thank Bob Sharpe, especially for the fabrication of the MRT impeller, and other workshop staff for their general technical assistance. Nomenclature A ) generic species or reagent a, b, c, d ) constants B ) baffles width (m) C ) clearance off the bottom (m) D ) impeller diameter (m) fA ) dimensionless number of moles of species A g ) incorporation function H ) liquid height (m) k2 ) kinetic constant of second reaction (M-4 s-1) k3 ) forward kinetic constant of third reaction (M-1 s-1) k′3 ) reverse kinetic constant of third reaction (s-1) K3 ) equilibrium coefficient (M-1) N ) impeller speed (s-1) ri ) reaction rate of the reaction i (M/s) R2 ) correlation coefficient t ) time (s) tm ) micromixing time (s) T ) tank diameter or temperature (m) or (K) Vφ2 ) initial volume of injected fluid (m3) V2(t) ) volume of injected fluid at time t (m3) Vinjection ) volume of injected acid (m3) Vtank ) tank volume (m3) Xs ) segregation index Y ) actual yield of undesired product YST ) maximum yield of undesired product Greek Symbols R ) micromixedness ratio T ) local specific energy dissipation rate (m2 s-3) jT ) mean specific energy dissipation rate (m2 s-3) φensemble,max ) (T)ensemble,max/jT φpoint,max ) (T)point,max/jT Φ ) (φpoint,max/φensemble,max) µ ) ionic strength (M)

ν ) kinematic viscosity (m2 s-1) θ ) dimensionless time Literature Cited (1) Ducci, A.; Yianneskis, M. Direct determination of energy dissipation in stirred vessels with two-point LDA. AIChE J. 2005, 51, 2133. (2) Fournier, M. C.; Falk, L.; Villermaux, J. A new parallel competing reaction system for assessing micromixing efficiency-Experimental approach. Chem. Eng. Sci. 1996, 51, 5053. (3) Guichardon, P.; Falk, L. Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part I: Experimental procedure. Chem. Eng. Sci. 2000, 55, 4233. (4) Assirelli, M.; Bujalski, W.; Nienow, A. W.; Eaglesham, A. Study of micromixing in a stirred tank using a Rushton turbine. Comparison of feed positions and other mixing devices. Chem. Eng. Res. Des. 2002, 80, 855. (5) Assirelli, M.; Bujalski, W.; Eaglesham, A.; Nienow, A. W. Intensifying micromixing in a semi-batch reactor using a Rushton turbine. Chem. Eng. Sci. 2005, 60, 2333. (6) Assirelli, M.; Bujalski, W.; Eaglesham, A.; Nienow, A. W. Macroand micro-mixing studies in an unbaffled vessel agitated by a Rushton turbine. Chem. Eng. Sci. 2008, 63, 35. (7) Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.; Cambridge University Press: New York, 1992. (8) Guichardon, P. Caracterisation chimique du micromelange par la reaction iodure-iodate. Ph.D. Thesis, L’Institut national polytechnique de Lorraine, Nancy, France, 1996. (9) Falk, L.; Villermaux, J. A generalized mixing model for initial contacting of reactive fluids. Chem. Eng. Sci. 1994, 49, 5127. (10) Baldyga, J.; Bourne, J. R. Turbulent mixing and chemical reactions; John Wiley & Sons: Chichester, U.K., 1999. (11) Guichardon, P.; Falk, L.; Villermaux, J. Characterisation of micromixing efficiency by the iodide-iodate reaction system. Part II: Kinetic study. Chem. Eng. Sci. 2000, 55, 4245. (12) Fournier, M. C.; Falk, L.; Villermaux, J. A new parallel competing reaction system for assessing micromixing efficiency-Determination of micromixing time by a simple mixing model. Chem. Eng. Sci. 1996, 51, 5187. (13) Assirelli, M. Micromixing studies in turbulent stirred baffled and unbaffled Vessels agitated by a Rushton turbine: An experimental study; University of Birmingham: Birmingham, U.K., 2004. (14) Villermaux, J. Micromixing phenomena in stirred reactors. In Encyclopedia of Fluid Mechanics; Gulf Publishing: Houston, TX, 1986; p 707. (15) Rousseaux, J. M.; Falk, L.; Muhr, H.; Plasari, E. Micromixing efficiency of a novel sliding-surface mixing device. AIChE J. 1999, 45, 2203. (16) Schaer, E.; Guichardon, P.; Falk, L.; Plasari, E. Determination of local energy dissipation rates in impinging jets by a chemical reaction method. Chem. Eng. J. 1999, 72, 125. (17) Geisler, R.; Krebs, R.; Forschner, P. Local turbulent shear stress in stirred vessels and its significance for different mixing tasks. Inst. Chem. Eng. Symp. Ser. 1994, 136, 243. (18) Sharp, K. V.; Adrian, R. J. PIV study of small-scale flow structure around a Rushton turbine. AIChE J. 2001, 47, 766. (19) Cutter, L. A. Flow and turbulence in a stirred tank. AIChE J. 1966, 12, 35. (20) Schafer, M. Charakterisierung, auslegung und verbesserung des makro- und mikromischens in geru¨hrten beha¨ltern. Ph.D. Thesis, Nu¨rnberg, Germany, 2001.

ReceiVed for reView May 29, 2007 ReVised manuscript receiVed September 28, 2007 Accepted October 1, 2007 IE070754N

An Extension to the Incorporation Model of Micromixing and Its Use in

Recommend Documents