Analysis of impingement mixing-reaction data: use of a lamellar model to generate fluid mixing information

302 Ind . Eng. Chem. R e s .

Values determined for AG at a given temperature, under the same experimental conditions, i.e., addition of SiOz and duration of isothermal heating, indicate that magnesium oxide obtained by 80% precipitation with dolomite lime is the most reactive; Le., the forsterite formation process is nearest to the equilibrium state. Magnesium obtained from seawater by 120% precipitation follows next, and then comes magnesium oxide p.a.

Differences in the value of AG are more conspicuous at higher temperatures.

The results of examination of AG, indicate that differences in values of AG, increase as differences in K ’x of various samples examined increase for the given temperature.

Differences in reactivity noted in the isothermal sintering process in the samples examined are due to differences in specific surfaces, chemical composition, and structural irregularities (dislocation density) between magnesium oxide obtained from seawater and magnesium oxide p.a.

The thermodynamic analysis of experimental results yields a better insight into the forsterite/spinel formation reaction and also into the sintering process with regard to the properties of products obtained, by comparing the results obtained with the theoretical values. The thermodynamic analysis, namely calculation of AG and AG,, thus provides a definite insight into the mechanism and

1989, 28, 302-315

rate of the forsterite/spinel formation process in the samples examined and makes it possible to predict the product properties in advance.

Acknowledgment

The authors thank the Republic Council for Scientific Research of Croatia for financial support (Contract 1.01.02.02.04, 1987).

Registry No. MgO, 1309-48-4; A1203, 1344-28-1; Si02, 7631- 86-9; forsterite, 15118-03-3; spinel, 1302-67-6.

Literature Cited Chung, F. H. J . Appl. Crystallogr. 1974, 7, 519-526. Chung, F. H. J. Appl. Crystallogr. 1975, 8, 17. Heasman, N. Gas Wurme Znt. 1979,28, 392. Hinz, W. Silikaty (Moscow) 1971, 75-97. Karapetyants, M. Kh. Chemical Thermodynamics; Leib, G., Transl.;

Kriiiianovskij, P. E.; Stern, 2. Yu. TeplofiziEeskie suoistua ne-

Petric, B.; Petric, N. Znd. Eng. Chem. Process Des. Deu. 1980, 19,

Schill, P. Silikaty (Moscow) 1982,26, 355.

Mir Publishers: Moscow, 1978.

metaliEeskih materialou; Energija: Leningrad, 1973.

329.

Received fo r review February 16, 1988 Revised manuscr ip t received July 15, 1988

Accepted August 5, 1988

Analysis of Impingement Mixing-Reaction Data: Use of a Lamellar Model To Generate Fluid Mixing Information

Henry A. Kusch and Julio M. Ottino* D e p a r t m e n t of Chemical Engineering, Univers i ty of Massachuse t t s , Amhers t , Massachuse t t s 01003

Dave M. Shannon Central Research Engineering Research Laboratory, Dow Chemical C o m p a n y , Mid land , Michigan 48640

Central to reaction injection molding (RIM) is the mixing by impingement of reacting monomer streams. Mixing occurs a t relatively low Reynolds numbers, O( lo2); however, little theoretical or experimental guidance exists for this case. Most previous studies focused on fast reactions in turbulent flows with little attention given to laminar or transitional flows. In this work, mixing is studied using simple chemical reactions with known kinetics. The data collected are mixing-dependent selectivity versus Reynolds number from competitive-consecutive azo-coupling reactions at complete conversion. Selectivity bounds on the experimental data points are developed to account for incomplete conversion of the limiting reactant to measurable products. Average fluid mixing information is backed out by matching solutions of a lamellar model to one set of experimental data. A sensitivity analysis is presented.

Reaction injection molding (RIM) is an industrially important process critically dependent upon mixing; poor mixing results in poor polymeric parts (Kolodziej et al., 1982). Virtually all current mixing designs are based on “head-on” impingement of two reactive monomer jets in a small cylindrical chamber. Usually, the Reynolds number, based on the more viscous jet diameter, is of the range 200-400 (Lee et al., 1980). However, in spite of commerical success, very little quantitative information about impingement mixing exists in the open literature, and retro- fitting to changes in the chemistry of the reacting system is often empirical. Experimental and modeling studies

* Author to whom correspondence should be addressed.

088S-5885/89/2628-0302$01.50/0

have been carried out to clarify the mixing process, but a clear picture has not yet emerged. Flow visualization reveals that mixing in the zone of impingement is intense, but intermittent and far from homogeneous, and that the mixing quickly decays to zero as a Poiseuille flow develops in the runner (Sandell et al., 1985). In fact, when contrasted with classical turbulence studies, mixing occurs at a relatively low Reynolds number. This does not simplify matters and indeed makes the analysis harder rather than simpler, since the theoretical guidance and intuition developed for highly turbulent flows, e.g., isotropy and ho- mogeneity, are far from being reasonable assumptions in this case. The possibility of analysis based on direct computational simulations seems at best remote. The

1989 American Chemical Society

Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 303

Table 1. Azo Dye Reactants and Products futility of a direct computational approach in resolving striation thickness scales (i.e., without averaging) becomes evident when contrasted with the recent numerical simulations of reactive mixing in shear layers (Riley et al., 1986)-a system far simpler than impinging jets-and the near impossibility of tracking deforming intermaterial areas in chaotic flows (Franjione and Ottino, 1987). Pre- vious theoretical studies of impingement mixing were based on simple models of stretching of striations (Lee et al., 1980) and turbulent scaling arguments at Kolmogorov scales (Tucker and Suh, 1980). These theoretical predictions have not been tested, and also, there are no studies focusing on the effect of impingement mixing on well- characterized simple chemical reactions. No theory to date can predict the existence of a critical Reynolds number(s).

The experimental studies are a bit more numerous, though far from conclusive: some studies have focused on direct visualization of the flow (Lee et al., 1980; Tucker and Suh, 1980; Sandell et al., 1985; Tyagi e t al., 1987), whereas others (Kolodziej et al., 1982; Nguyen and Suh, 1985) examined the polymer products to quantify the mixing. The results of flow visualization studies indicate that there is a “mixing transition” a t Reynolds numbers 0(102) whereas the examination of polymer samples indicates a distributed striation thickness morphology (Kolodziej et al., 1982). However, the analysis of the mixing problem in the polymer systems is complicated by gelation, viscosity effects, and interfacial mixing. Inter- facial mixing between RIM monomers has been observed by Fields et al. (1986) and Wickert e t al. (1987) and might be responsible for much of the fine scale mixing in polymer formulations. In fact, Ranz (1986) argued using order of magnitude calculations that mechanical mixing alone could not accomplish the mixing necessary for complete reaction in RIM. Obviously much work is needed to understand impingement mixing in the context of commercial RIM processes.

The objective here, however, is to minimize gelation, non-Fickian diffusion, and interfacial effects and to focus exclusively on the coupling between reactions and mixing. The fluid mechanics will be grossly simplified, but most of the improvements on this basic picture would result in quantitative and not qualitative changes. Although the reactions are not completely clean (there are some ques- tions about the kinetics and side reactions), they are the best ones currently in use and the most widely used in micromixing studies. Furthermore, they involve only low molecular weight reagents; therefore, the complications associated with polymerizations are eliminated (however, one advantage of polymerizations is lost, namely that the mixed structure is frozen due to gelation).

Experimental Section The reactions used in this work are the azo dye coupling

reactions first used for the evaluation of liquid-phase micromixing by Bourne and co-workers (Bourne et al., 1981). Neglecting side reactions, the azo dye materials react in a series-parallel sequence:

The reactants and products are listed in Table I. The first reaction is several orders of magnitude faster than the second reaction ( E = k 2 / k l = O(10-4)). When separate aqueous streams containing A and B are mixed, the products R and S me formed. The amount of each product formed is dependent on the mixing history and is a result

1-naphthol

N?f diazotized sulfanilic acid I

monoazo dye

bisazo dye * N2-oSo; SO;

of the simultaneous processes of diffusion and reaction which occur during micromixing. Therefore, when B is the limiting reagent (i.e., Nb/NBo > l), a well-mixed system will form predominantly the primary product, R; on the other hand, a poorly mixed system favors the secondary product, S. A measure of the distribution of B between the products is the selectivity, which is defined as the amount of B used to form S to the amount of B used to form both R and S (or initially present since B is the limiting reagent):

(3)

Since the first reaction is several orders of magnitude faster than the second reaction, the selectivity will vary between the approximate limits of 0 and 1, if B is the limiting reactant. X - 0 indicates “good” mixing, while X - 1 indicates “poor” mixing. Note, also, that if B is not the limiting reactant, then the selectivity a t the completion of the reaction is independent of the mixing.

The azo dye coupling reactions are particularly suited for micromixing studies because the product concentrations are relatively easy to analyze. Both products absorb light in the visible light region from 400 to 600 nm, and the reactants do not affect the analysis. This permits quantitative analysis of the final product composition by measurement of the absorbance spectrum for the mixture at several wavelengths. The wavelengths used for this work are 500, 510, 520, 580, 590, and 600 nm. The Lambert- Beer law is used for analysis of the product composition

(4)

where A , is the absorbance at wavelength X and d is the cell length. The extinction coefficients a t wavelength A, ERA and ES,, are tabulated by Bourne et al. (1985). The concentration of R and S are found by a least-squares data fit over all the wavelengths examined.

The reaction kinetics have been studied extensively by Bourne and co-workers (Bourne et al., 1981, 1985). Al- though the reactions have been extremely useful in mixing studies, there seems to be some confusion about the values

Ah = ERhCRd + Es,Csd

304 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989

Table 11. Rate Constants for the Azo Dye Coupling Reactions k l k2 A + B - R R + B - S

rl = klCAaCBb r2 = k2CBCCRd

authors conditions kl" k2" comments Bourne et al., 1981 pH 10, T = 298 K 7300 3.5 "original" rate constants, used by many

researchers for analysis of reactive micromixing data

Bourne et al., 1985 pH 10 3.338 X lo6 exp[-4257k/T] c = d = 1; a t T = 298 K, k2 = 2.09; a t T =

C, = C, = Co, pH 10, T = 293 K 735

C, # C,, pH 10, T = 293 K 18 180

"All rate constants have units m3 mol-' s-'.

of the rate constants and reaction orders. Bourne et al. (1981) first reported values of the rate constants, and the information was updated and improved by Bourne et al. (1985). Table I1 is a summary of the reported rate constants. As noted in Table 11, Bourne et al. (1985) found that when C, = CBo the total reaction order of the first reaction is -1.5. The data can also be fit with a total reaction order of 2 if the rate constant of the first reaction is concentration dependent. Alternatively, fitting all the data (C, # CBo and C, = C,,), they found the reaction order to be first in each reactant, but the rate constant was much higher and not concentration dependent. This anomalous behavior is explained by possible local pH in- homogeneities or incomplete mixing, and they concluded that kl - 12000 m3 mol-' s-l. When analyzing experimental data, the general assumption has been that since the first reaction is very fast it may indeed be treated as infinitely fast-the reactants cannot coexist-thereby the actual values of the rate constants need not be known (see e.g., Baldyga and Bourne (1984)). However, for this work the values kl = 18180 m3/(mol s)

will be used; see Table 11. We will examine the sensitivity of the results to changes in the rate constants in the Sensitivity Analysis section.

Apparatus and Procedure. Figure 1 is a schematic diagram of the apparatus used for the impingement mixing experiments. The two reactive dye components are stored under nitrogen in the reservoirs labeled A and B. In preparation for a mixing experiment, the delivery tubes are filled from the reservoirs. The delivery tubes are isolated from the reservoirs by valves, and a regulated nitrogen pressure from the ballast tanks is applied to the surface of the liquid in the delivery tubes.

The flow rates of A and B to the mixhead are controlled by the nitrogen ballast pressures and a selectable combi- nation of flow restricting tubes, arranged in parallel. This design permits a wide range of flow rates without the need for excessive variation of ballast pressures. A pressure- driven flow system was chosen to eliminate the possibility of flow pulsations which might be induced by mechanical pumping. The nitrogen ballast reservoirs are sufficiently large so that flow rate variations, due to pressure changes from expansion of the nitrogen gas, are minimal. The flow rates to the mixhead are calibrated before each experiment by removing the feed lines from the mixhead and collecting the fluid from a shot into beakers. The ballast pressures are adjusted to equilibrate the flow rates.

k2 = 1.64 m3/(mol s) E = 9.02 x 10-5 (5 )

_ - , -

293 K, k2 = 1.64 a + b = 1.54; if a = b = 1, then kl is

concentration dependent with 5300 < kl(Co) < 11300 and kl = 7300 a t Co = 0.0125 mol/m3

a = b = 1; not concentration independent

Pressure Transducers

Drawing not to scale

Mixhead

Figure 1. Schematic diagram of impingement mixing experimental apparatus.

Prior to initiation of a mixing experiment, all lines are filled with liquid up to the entrance of the mixhead. The mixhead and collection tube, however, are empty a t the start of each mixing experiment; similar to a real RIM shot. A mixing experiment is initiated by activation of the shot timer. This simultaneously opens two block valves allow- ing the A and B components to flow to the mixhead where they form the reacting mixture. Typically, two collection tube volumes (the collection tube volume is 10 cm3) of mixture are allowed to flow through the mixhead before the shot timer is deactivated and the material flows are stopped. The reacted mixture is collected as a mixed batch sample through a long thin tube into a large syringe, and its contents are analyzed by using a visible light spectro- photometer as previously described.

Figure 2 is a schematic diagram of the mixing head. Its configuration is similar to that found in many commerical impingement mixheads used for RIM. The incoming streams are directly opposed to one another, and their axes are perpendicular to the axis of the cylindrical mixing chamber. The back wall of the mixing chamber is flat and recessed slightly from the axes of the incoming streams to permit some recirculating flow. The collection tube fits tightly into the mixhead, and the transition from the mixhead to the collection tube is a smooth, constant bore to minimize downstream mixing effects. The mixhead is constructed from acrylic to permit visualization of flow patterns during mixing.

Experiments. The liquid jet Reynolds number is DVjet

Rej,, = - (6) V


a

D Q

O D Q

QQ QQQ,, $ Q

I _ _ _ _ . _ _ _ I

1.5 cm i Component

Delivery Line i .d. = 0.102 cm

1-1 0.5 cm

Figure 2. Schematic diagram of impingement mixhead. Gravity acts downward in the direction parallel to the axis of the mixhead.

Table 111. Exaerimental Parameters fluid viscosity, fi

fluid density, p

nozzle diameter, d

ratio of volumetric flow rates

initial stoichiometric ratio of reactants

temperature, T initial concentrations

A stream: 1 CP = 0.001 Pa s B stream: 1 CP = 0.001 Pa s A stream: 1000 kg/m3 B stream: 1000 kg/m3 A stream: 0.102 cm = 1.02 X m B stream: 0.102 cm = 1.02 X m QA/QB = 1.0

N,/N, = 1.0526

293 K I: c, = 0.05 mol/m3

11: c, = 0.1 moi/m3

111: c, = 0.2 mol/m3

C, = 0.0475 mol/m3

C B ~ = 0.095 mol/m3

CBo = 0.19 mol@

where D is the jet nozzle diameter, ujet is the average velocity of one of the liquid jets (in this case both are equal), and I, the kinematic viscosity, is the key mixing variable. In the experiments described here, only the average fluid jet velocity, ujet, was varied. The data, however, have been presented using the Reynolds number to facilitate com- parisons with other experiments. Table I11 lists the experimental parameters, and Figure 3 shows the variation of selectivity with Reynolds number for three initial concentrations. Generally, the data show the expected trends of decreasing selectivity (indicating better mixing) with increasing Reynolds numbers. There is a gradual decrease in selectivity a t jet Reynolds number above 100 and a sharp increase in selectivity at jet Reynolds numbers below 100. This result is encouraging since companion flow visualization studies of impingement mixing show a mixing transition at approximately the same Reynolds number (see also, Lee et al. (1980), Tucker and Suh (1980), Sandell et al. (1985), and Tyagi et al. (1987)). More specifically, the data set C 0.1 mol m-3 (Figure 3b) seems to asymptote in the'%iih Reynolds number limit to X - 0.01, while the other two data sets, C = 0.05 and 0.2 mol m-3 (Figure 3a and 3c, respectivela, asymptote to higher values. Trends between the data sets are not clear, and the data within each set are scattered. A critical examination of the data is necessary to establish bounds on the data points.

X

0.2 . m

m

m .

q . m

0 I I I I I I I I I

0 200 400 600 800

i I

0.4 4

Reynolds number b

Q

Q

Q

Q a Q Q

Q Q Q

Q

0 200 400 600 800 Reynolds number

C

0 -7 I I I I

0 200 400 600 800 Reynolds number

Figure 3. Experimental selectivity dependence on Reynolds number. Initial concentrations: (a) C, = 0.05 mol m-3, C , = 0.0475 mol m-9; (b) C, = 0.1 mol d , C, = 0.095 mol m-3; (c) CAo = 0.2 mol m-3, C, = 0.19 mol m".


(i x 1 - Table IV. Selectivity Bounds I 1 - 1 r

X Y 0.02 0.97 0.2 0.97 0.02 0.7 0.2 0.7 0.02 1.01 0.2 1.01 0.4 0.6

Xmin %X 0.0194 97 0.194 97 0.014 70 0.14 70 0.0202 101 0.202 101 0.24 60

x,, %X 0.0794 395 0.254 127 0.614 3070 0.74 370 0.0002 1 0.182 91 1.0 250

Selectivity Bounds. The error bounds on the data points (due to measurement errors, flow rate fluctuations, etc.) are typically on the order of the marks denoting the data points. Obviously those effects alone cannot account for the data spread. One possible reason for the data spread is due to the chemical system used to study the micromixing. The mass balance on species B, the limiting reagent, often does not close. There has been speculation about the reason for nonclosure (e.g., local pH effects and side reactions (Bourne et al., 1985)), but currently no concrete information exists. It is therefore useful to de- velop bounds on the selectivity to indicate the possible values of the selectivity if the missing B were indeed present. A measure of the conversion of B to measurable products, Y, is the ratio of B in the products R and S to the amount of B initially present:

(7)

The subscript denotes concentrations measured at reaction completion. The missing B is then (1 - Y)CB. A lower bound can be formed on the selectivity by ad$ing the missing B to species R:

Thus, the lower bound is simply the product of the selectivity and the conversion of B. For an upper bound, the missing B is reacted with product R to form additional product S:

There are two qualifiers to the above discussion. First, if the B conversion, Y, is greater than 1, then the subscripts max and mix are reversed. That is, Xmin is the upper bound because excess B is removed from R, thus forming a maximum. A similar argument can be used for the lower bound. Second, if there is more B missing than R present (C, < (1 - Y)C&), then the upper bound on the selectivity is 1, since there cannot be a negative amount of product R. Table IV shows the effects of incomplete conversion of B to measurable products corresponding to some representative selectivities and B conversions. The relative percentage of the upper bound can be very high if the selectivity is low, even if the conversion of B to measurable products is very high. The general trend in the experimental data is that a t high Reynolds numbers B conversions are good (although the relative percent error can still be large), while in the low Reynolds number region the B

0 200 400 600 800 Re! noid\ iiuinbei

Figure 4. Experimental selectivity dependence on Reynolds number with selectivity bounds due to incomplete conversion of the limiting reactant, B, to measurable products. Initial concentrations: C, = 0.1 mol m-3, CBo = 0.095 mol m-3.

conversions are typically poorer. One set of experimental data with selectivity bounds is shown in Figure 4.

The analysis above provides bounds on the data points but does not explain the data scatter. I t is therefore useful to discuss possible reasons for the data scatter before at- tempting an analysis. Impingement mixing is inherently a transient process. Typical shot times are short (O(1 s)), and flow visualization studies show complex time-varying mixing patterns. Thus, it is somewhat naive to expect that samples from different mixing experiments (at the same Reynolds number) would necessarily give identical results. These ideas have been exploited by workers dealing with turbulence. For example, Mehta and Tarbell (1987) used Bourne’s azo dye coupling reactions to study micromixing in a turbulent tubular reactor. The product concentration along the axis of the tube was found by using a fibre optic probe. They found it required 3 s of a real-time record of the signal to obtain a statistically stable value of the voltage (equivalent to absorbance) for each wavelength. The weight of the evidence, however, is that impingement mixing, a t Reynolds numbers of interest in RIM, is not a fully turbulent flow. Little guidance exists for studying mixing a t intermediate Reynolds number flows- transitional flows between steady laminar and homoge- nous, isotropic turbulence. The recent developments of dynamical systems theory, especially the study of chaotic mixing, seem promising, but the field is still in its infancy (for a review from a chemical engineering perspective, see Doherty and Ottino (1988)). A statistical approach toward data acquisition may be appropriate; however, the time penalty with the current apparatus and chemical system is too severe. Thus, for the time being, we will just keep these ideas in mind and work with the available data.

Lamellar Model A successful micromixing model must balance the de-

scription of coupled physicochemical hydrodynamic processes occurring on complex time-dependent structures with the engineer’s need for simple solutions, predictive capabilities, and physical insight. Of the many possible choices, the lamellar model seems to strike a good balance, successfully describing deformation, diffusion, and reaction


A = l , B = R = S = O , 7 = 0 , O < [ l 4 B = l , A = R = S = O , r = O , $ < t l l

a A / a t = aB/ag = aR/at = as/at = 0, 7 10, 4 = 0, 1 (19)

The dimensionless variables and parameters are defined as

in a form that is relatively easy to solve numerically. A brief review of the lamellar model follows. For background and development, see Ottino (1982); for applications to various reaction schemes, see Chella and Ottino (1984).

For the purposes of this work, we can assume that the conservation equation for species i, assuming dilute solution, Fickean diffusion, constant density and diffusivity, is

(10)

Furthermore, we assume that the reaction and diffusion processes do not affect the fluid behavior so that the fluid mechanical and diffusion reaction problems are uncoupled. With the lamellar structure assumption, and striations characterized by a normal n, the species conservation equation becomes (Ottino, 1982)

aci/at + v.vci = D ~ V ~ C ~ + R;

where D is the stretching tensor (the symmetric part of the velocity gradient tensor). The stretching function is defined as

a(X,t) = D:nn (12)

and in general is a complex function of initial conditions (X) and time.

One of the advantages of this form of the species conservation equation is that the convective term can be eliminated by a change of variables. If position ( x ) and time ( t ) are transformed as

7 = t / t c E = x/s(t)

then the species conservation equation has the form

where tC is a characteristic time. The fluid mechanics enters in a multiplicative way, augmenting the diffusion coefficient through the q2 term:

SO t q = - = exp[ a(t’) dt’]

s ( t )

The effects of fluid stretching on diffusion are %fold; first, it creates intermaterial area for diffusion, and second, the stretching keeps the concentration gradients from decay- ing.

When applied to the azo dye coupling reactions, eq 13 results in four coupled nonlinear partial differential equations:

c, DI k2 1

SO2

DA

e = - t R = - P B = - A , = - CBO DA k1 1 cBo

tD = -

In general, these equations have to be solved numerically. In order to increase numerical stability and convergence, tc was chosen as either tR or tD, generally whichever is larger. To save computer time the average concentration of species S can be found from a mass balance, since S is not a reactant, and the species conservation equation need not be solved.

Three characteristic times describe the problem: diffusion and reaction times (tD and tR) defined above and a fluid mechanical time defined as the reciprocal of the time average of the stretching function:

(20) 1

tF(t) = - a(t)

a(t) = i j t a ( t ’ ) t o dt’

The three characteristic times form two dimensionless groups, the first and second Damkohler numbers:

D ~ I = tR/tF(t) (22)

D ~ I I = t ~ / t ~ (23)

The two Damkohler numbers fully characterize the problem, in the context of lamellar models, for a given reacting system. However, we note that the relationship between the dependent variables (conversion, final concentrations, etc.) and the independent variables or parameters (especially a@)) is not unique. For example, two different fluid mechanical histories, a(t) , can give the same final selectivity.

The lamellar model admits two limiting cases for the mixing-diffusion-reaction problem: perfectly mixed and diffusion controlled. The perfectly mixed limit corresponds to diffusion occurring much faster that reaction, Le., tR >> tD. The lamellar model then reduces to the equations of classical reactor analysis. The diffusion- controlled limit occurs when tD >> tR; reactants cannot coexist. Diffusion to the interface controls the rate of reaction. The fluid or mixing time, tF, however plays an important role in the transition between the diffusion- controlled and perfectly mixed limits. Let use now con- sider the perfectly mixed case ( X - 0).

Perfectly Mixed Limit. The rate expressions and initial conditions for the azo dye reactions are

dCA/dt = -kICACB

dCB/dt = -klCACB - ~ ~ C B C R

dC,/dt = klCACB - ~ ~ C B C R

dCs/dt = ~ ~ C B C R with initial and boundary conditions


CA = C,, CB = CB,, CR = Cs = 0, t = 0 (24)

Note that all the concentrations in this section only are volume-average concentrations. The equations are au- tonomous and independent of Cs, so the concentrations of B, R, and S may be found in terms of A ( E = k 2 / k l ) :

dCS CR - = -e- dCA CA

C&(l-f'cA' ~ C A c,=c,- + - 1 - € 1 - t

The time variation of A is given by the integral

Unfortunately this integral is nontrivial since CB is dependent on both CA and t. Fortunately, we are primarily interested in the asymptotic behavior, the selectivity a t complete conversion. The final concentration of A may be found from molar balances:

c, = CA + CR + CS

CBo = CB + CR + 2cS

Eliminating Cs and taking into account that B is the limiting reagent results in a nonlinear equation for the steady-state concentration of CA:

2c, - c B o = 2 C A , + CR(CA,)

where the subscript m denotes the steady-state concentration. The equation is easily solved recursively:

This equation may also be found from eq 28 by setting CB = 0. In dimensionless variables, we see that the final concentration of A is only dependent on the initial molar ratio and the ratio of the rate constants: - .I

A" I (33) A , = Il-%-m 2(1 - t) 1

L J

Thus, if OB is constant, in the perfectly mixed limit the dimensionless concentrations and selectivity, X, will be constant regardless of the actual values of C, and CFo. This result provides a check for the experimental data in the high Reynolds number limit since the initial molar ratio, OB, was the same for all experiments. The experimental selectivity data, however, converge at high Rey- nolds numbers to three different selectivities which do not show a clear trend, increasing or decreasing, with initial concentrations. The precision of our data does not allow for an explanation of these results. In the perfectly mixed

Table V. Lamellar Model Parameters DA = 1.14 X m2/s AB = DB/DA = 0.83

t D = sO2/DA = 913

k , = 18180 m3/(mol s)

so = 1.02 x m AR = = 0.60

OB = C,/C,, = 1.0526 4 = 0.5 (volume fraction stream A)

tR = l/(k,C&) e = 9.02 X

CB0' mol/m3 t R . Darr (=tn/tn)

I 0.0475 1.16 x 10-3 7.87 x 105 I1 0.095 5.79 x 10-4 1.58 X lo6 I11 0.19 2.90 x 10-4 3.15 X lo6

limit, the steady-state dimensionless concentrations are A , = 0.050184, B, = 0, R, = O.949632,Sm = O.OOO184, and the selectivity X = 0.000 387. At high values of stretching, the numerical data should asymptote to the same values.

Numerical Solution. The equations were solved by using the Method of Lines (MOL). This method is particularly convenient for stiff differential equations, such as those produced by fast-diffusion-controlled reactions @aII >> 1). The method reduces the nonlinear partial differential equations to a system of coupled ordinary differential equations by discretizing the spatial variables. The spatial derivatives are approximated by finite differences, and the concentration at each node is followed as an ordinary differential equation. The coupled nonlinear ordinary differential equations can easily be solved by using an integration package. There are two main considerations when implementing the MOL, the number of nodes and the integrator error tolerance. The number of nodes is picked high enough so that the numerical solution converges; 51 equally spaced nodes was suitable for the calculations reported here. The error tolerance is chosen to balance temporal integration error with spatial differencing error. If the tolerance is picked too small, the computer time is extremely high, while if it is too large, the error from the integrator may give erroneous results. We used a tolerance of 1 X with the DGEAR integrator of the IMSL library.

Results and Discussion Before a numerical solution can be implemented, a

number of parameters must be found, or estimated from knowledge of experimental conditions, correlations, or the literature. The parameters that are dependent on the experimental conditions are the temperature, initial concentrations, striation thickness, and volume fraction. The experiments were carried out a t 293 K and consist of three initial concentration data sets. There are several ways to refer to the data sets, by initial concentrations, reaction times, or, as done here, by the second Damkohler number. The volume fraction, 4, is taken as 0.5 because the flow rates and Reynolds numbers of the impinging jets were matched in all experiments. The choice of initial striation thickness is debatable; a t low Reynolds numbers, below the mixing transition, the mixhead diameter is a good choice, while a t high Reynolds numbers the nozzle diameter seems to be the suitable length scale (see flow visualization photographs in Lee et al. (1980), Tucker and Suh (1980), and Sandell et al. (1985)). We will take the initial striation thickness to be the feed nozzle diameter because we are primarily interested in the mixing at high Reynolds numbers. This choice will be discussed later. The diffusion coefficients were estimated by using the Wilke-Chang equation (Bird et al., 1960). Table V contains all of the parameters necessary for implementation of the model.

The final piece of necessary information is the evolution of the striation thickness s ( t ) , or, equivalently, the area stretch ~ ( t ) , or the stretching function a(t) . Since we do


of the lamellar model. To fit the experimental data, a relationship must be found between selectivity and Rey- nolds number. There are two options for the analysis. One option is to do the fit for all three data sets; the second option, which we have chosen, is to f i t one experimental data set and then test the data fit by making predictions for the other two data sets. The middle initial concentration data set, which corresponds to DuII = 1.58 x lo6, was chosen for the data fit because it had the most data points and least scatter in the high Reynolds number region. Nevertheless, the data fit is still difficult and im- precise because of the data scatter and large selectivity bounds. The data fit

log X = 2.2 - 1.5 log Re (37)

is shown with the data and selectivity bounds in Figure 8. Obviously many different fits are possible. If log X is eliminated between the numerical and experimental data fits, eq 35 and 37, then the functional relationship between stretching and jet Reynolds number is found as

log a = -4.4 + 4.2(log Re - 1.32)1/2 (38)

where a has units of inverse seconds. The Reynolds number must be greater than 21 so that the argument of the square root is positive.

The stretching-Reynolds number relationship can be tested by making predictions for the other two data sets, as shown in Figure 9. The predictions are done by sub- stituting the stretching-Reynolds number relationship, eq 38, into the selectivity-stretching relationships, eq 34 and 36. The prediction for DuII = 7.87 X lo5 is rather good, while the prediction for DuII = 3.15 X lo6 is low. There are many possible reasons for the underprediction (see Sensitivity Analysis section).

Comparison to Simale Theoretical Predictions. Another method of testing the stretching-Reynolds number relationship is to compare it to simple theoretical predictions. Lee et al. (1980) used ideas about stretching striations in an impingement mixhead to predict

DuII = 1.58 X lo6

2 Vjet

not know the details of the fluid mechanics of mixing in the mixhead, we will take a( t ) to be constant. This assumption is somewhat naive since we know the stretching is intense while the fluid is in the mixhead, but the stretching quickly decays to zero as the fluid leaves the mixhead. This assumption is only a first attempt at fitting the experimental data, and a more complicated form can be implemented later. However, we must be cautious when choosing a more complex stretching function because additional time scales may appear in the problem.

The Method of Lines is used to solve for the temporal concentration profiles across a single striation. Figure 5 shows a set of concentration profiles for the second Damkohler number 1.58 X lo6 and a( t ) = 1 s-l. Discon- tinuities or kinks in the profiles are apparent at short times. The calculation was done with 51 nodes; the profiles for calculations done with 101 nodes are slightly smoother; however, the solution has converged adequately, and the average concentrations are the same within the required tolerance. It is interesting that the A and B profiles overlap (many earlier works simplified the analysis and numberical solution by assuming that the first reaction was infinitely fast). At the end of the reaction, the final average dimensionless concentrations are A , = 0.034 82, B, = 0, R , = 0.45536, and from a mass balance S, = 0.00982 which corresponds to a selectivity of X = 0.041 35.

Each concentration profile calculation results in only one piece of information to compare with the experimental results: the selectivity a t complete conversion. To fit the experimental data, we need to know how the selectivity, a t reaction completion, changes with stretching. The variation of selectivity with stretching for all three initial conditions (second Damkohler numbers), with all other parameters held constant, is shown in Figure 6. The results are encouraging since they have the same form as the experimental data. The form of the curves is approximately hyperbolic, prompting a log-log plot of the numerical data, Figure 7. The lines on the Ieft side of Figure 7 (between s-l) are the limiting values of selectivity for the given second Damkohler numbers and DaI = 0 (a = 0). Higher values of selectivity are obtained with DuI < 0, but the physical picture is mechanical unmixing or striation thickness growing in time. The selectivity range of primary interest, for comparison to the high Reynolds number experiments, lies below about 0.3. The numerical data on the log-log plot are surprisingly linear in that region. Lines fitted through the selectivity data points give the following approximate relationships between stretching and selectivity

DuII = 7.87 x lo5 (34)

< X < loo, and LY =

log X = -1.65 - 0.83 log CY - O.O7(10g

log X = -1.38 - 0.79 log CY - O.O9(lOg a)2 DuII = 1.58 X lo6 (35)

log X = -1.12 - 0.74 log CY - 0.12(10g DuII = 3.15 X lo6 (36)

The fits are also shown in Figure 7. In the low-selectivity region, there seems to be another scaling relationship-the three data sets seem to be equally spaced. If the numerical data are replotted on a log-log plot as selectivity versus second Damkohler number with stretching constant, we obtain X - DUIIO.~~ for the low-selectivity data. This information is not directly relevant to the analysis, but it is relevant for analyzing the sensitivity of the results.

Data Fit. A relationship between the selectivity and stretching function, for each value of the second Damkohler number, was found from a numerical solution

4v 2

a L = ‘Jiet

(39) 1 + - t

2 v

where the subscript L denotes the authors Lee et al. This stretching function must be averaged over a residence time in the mixhead for comparison purposes, since the experimental stretching-Reynolds number relationship is implicitly a time-averaged value:

The residence time in the mixhead is proportional to the volume and inversely proportional to the total flow rate, 2Qjet. The volume will be parametrized by n, the number of mixhead diameters, D, in which mixing takes place:

The time average of the stretching function is then

Figure 10a compares the averaged Lee et al. prediction with different residence times (different n) to the exper-


I

' 1 e r

0 . 5 ;Ili

I I I I I I I I I I

I

Figure 5. Temporal dimensionless concentration profiles from solution of lamellar model. Parameters: Da1 = 5.79 X DUII = 1.58 X lo6, a = 1 s-l, AB = 0.83, AR = 0.60, pB = 1.0526, 6 = 9.02 X lo5, 4 = 0.5 (-) A, ( - - - ) B, ( - - - - ) R. (a) t = 0 s, (b) t = 1 s, (c) t = 2 8, (d) t = 3 s, and (e) t = 4 s.


4

0.2 4 f QA

Q El W I

1 A D

0 5 10 1s 20 25 a - sec-1

Figure 6. Selectivity variation with stretching at complete conversion from numerical solutions of lamellar model. Parameters: AB

(0) Dall = 7.87 x io5, (A) DaII = 1.58 x lo6, (0) DaII = 3.15 x lo6. = 0.83, AR = 0.60, & = 1.0526, c = 9.02 X lo5, 4 = 0.5, t~ = 9.13 S;

'On 1 n

\ \ \

l ( l - 3 I \ 1 I I I I

IO- ' IO-? 10-1 100 IO' 1 0 2 u - sec-'

Figure 7. Selectivity variation with stretching at complete conversion from numerical solutions of lamellar model on log-log plot. Parameters: AB = 0.83, AR = 0.60, = 1.0526, c = 9.02 X IO5, 6 = 0.5, tD = 913 s; (0) Dan = 7.87 X IO6, (A) Dan = 1.58 X IO6, (0) Dan = 3.15 X lo6. The lines fitted through the data points are given in the text, and the lines on the y axis for loo < X < correspond the selectivity for the respective second Damkohler numbers with no stretching.

imental stretching-Reynolds relationship, eq 38. The prediction gets better with an increasing number of mixhead diameters, but it seems to be an overprediction of stretching since flow visualization studies show laminar flow developing within 3-5 mixhead diameters downstream (this is also a clue that there are no additional mixing mechanisms, such as spontaneous emulsification at the interface). A possible reason for the overprediction is the implicit assumption in the model of Lee et al. that all the energy dissipated in the mixhead is directly used for stretching the striations. In actuality a significant percentage of the dissipated energy compresses the striations, occassionally making the striation thickness grow in time.

1

(Y = eff(t) ( - ; ; ) I 2 (45)

The energy dissipation can be estimated as

At high Reynolds numbers, the contribution of the laminar flow in the runner is less than 5% of the jets, so it may be neglected. The kinetic energy of the jets should be dissipated during the residence time in the mixhead, and the viscous dissipation can be estimated as the loss of kinetic energy divided by the residence time:

4p vje:d2 nD3

E, = -

The functional form of the stretching function is

(47)

where the subscript OM denotes the order of magnitude estimate of the stretching. A comparison between the


a Table VI. Sensitivity Analysis Results: Diffusion Parameters

0 I I I I I I I 1

(I 200 400 600 XO(1 Reynolds number

Figure 9. Prediction of selectivity dependence on Reynolds number: (a) C, = 0.05 mol m-3, C, = 0.0475 mol m-3 (Dan = 7.87 X lo6); (b) C , = 0.2 mol m-3, C, = 0.19 mol m-3 (DaII = 3.15 X lo6). The line is the prediction.

experimental stretching-Reynolds number relationship and the order of magnitude estimate with several values of the residence times and stretching efficiencies is shown in Figure lob. It is difficult to choose good values of the residence time and stretching efficiencies a priori to ensure a good match. Overall the order of magnitude estimate seems to be an overprediction, because the efficiencies are smaller and residence times larger than expected for a close match to the experimental results. However, there seems to be an approximate confirmation of the predicted pow- er-function dependence on the Reynolds number.

Sensitivity Analysis. The results presented are dependent on the choice of several parameters. Some, such as initial concentrations, mixhead geometry, and Reynolds numbers, are "robust"; there is little doubt of the actual values. Others, referred to as "questionable" parameters, such as diffusion coefficients, rate constants, and striation thickness considerations, enter through the model and are fundamentally less certain. The sensitivity analysis will concentrate on the questionable parameters and the high

Base Case Parameters reaction: kl = 18 180 f = 9.02 x io-5 t R = 5.79 x lo4 s

m3/(mol S) diffusion: DA = 1.14 X 10-0 AB = 0.83

AR = 0.60 t p = 0.5 s fluid a = 2 s-1

m2/s tD = 1.58 x lo4 s

mechanics: Da, = 1.16 X D~rr = 1.58 X lo6 parameter changes X % diff

base case DA = 1.25 X 10-0 mz/s, AB = 0.75, AR = 0.55

(increase DA lo%, DB and DR base case) DA = 1.03 X lo* mz/s, AB = 0.92, AR = 0.67

(decrease DA lo%, DB and DR base case) DA = 1.14 X lo* m2/s, AB = 0.91, AR = 0.60

(increase DB 1070, DA and DR base case) DA = 1.14 X lo4 mz/s, AB = 0.75, AR = 0.60

(decrease DB lo%, DA and DR base case)

(increase DR lo%, DA and DB base case) DA = 1.14 X lo* m2/s, AB = 0.83, AR = 0.54

(decrease DR lo%, DA and DB base case) DA = 1.14 X loF0 mz/s, AB = 1, AR = 1

DA = 1.14 X lo4 mz/s, AB = 0.83, AR = 0.66

0.022 49 0.022 37

0.022 61

0.021 97

0.023 08

0.023 07

0.021 85

0.024 43

~~

-0.5

0.5

-2.3

2.6

2.6

-2.8

8.6

Table VII. Sensitivity Analysis Results: Reaction Parameters Base Case Parameters

reaction: kl = 18180 f = 9.02 x io-' t R = 5.79 x io-4 s m3/(mol S)

diffusion: DA = 1.14 X lo4 AB = 0.83

AR = 0.60 fluid a = 2 s-1 t F = 0.5 s

m2/s t~ = 1.58 X 10" s

mechanics: DUI = 1.16 X DUII = 1.58 X lo6 parameter changes X 70 diff

base case kl = 19998 m3/(mol s) t = 8.21 X lo-'

(increase kl 1070, kz base case) k, = 16362 m3/(mol s), t = 1.00 X

(decrease kl lo%, kz base case) kl = 18 180 m3/(mol s), e = 9.92 X 10"

(increase kz lo%, kl base case) kl = 18 180 m3/(mol s), e = 8.12 X lo-'

(decrease k2 lo%, kl base case) k l = 19998 m3/(mol s), c = 9.02 X lo-'

(increase kl and kz 10%) kl = 16362 m3/(mol s), e = 9.02 X 10"

(decrease k, and kz 10%) kl = 7300 ms/(mol s), e = 4.79 X lo4

(original values) kl = 7300 m3/(mol s), e = 1.44 X

0.022 49 0.022 46

0.022 55

0.024 63

0.020 33

0.024 57

0.020 41

0.047 84

0.023 54

-0.1

0.3

9.5

-9.6

9.3

-9.2

113

4.7

Reynolds number region, since that is the region of primary interest in RIM.

Although the problem formulation and solution are fa- cilitated by casting the lamellar model equations in dimensionless form, the sensitivity analysis becomes complicated because dimensional parameters often appear in several dimensionless parameters. For example, if we want to isolate the effect of increasing the first rate constant, k l , by lo%, the first Damkohler number and the ratio of the rate constants, t, must be decreased 9% and the second Damkohler number increased 10%. Furthermore, because of the multiplicity of parameters, it is difficult to gain physical insight when discussing changes in dimensionless variables. Therefore, although the equations are solved in dimensionless form, we will discuss the sensitivity of the results by studying the effects of changes in full dimensional form. To further simplify matters, we select a case study corresponding to DaI = 1.16 X DaII = 1.58 X lo6, and a = 2 s-l (the Reynolds number is 375). Specific results are shown in Table VI (diffusional parameters) and Table VI1 reaction parameters). The following discussion summarizes the major conclusions. The diffusion param-


a

/ /

/ /

/ /

/ /

/ /

/

20

15

- u x 10 I

tl

5

0

0 200 400 600 Reynolds number

/ b /

/ /

/ /

/ /

/ /

/ ,/' / ,/'

/ / ,/'

/ / ,/'

/ ,/ /

/ ,/' / /'

0 200 400 600 Reynolds number

Figure 10. Comparison of experimental stretching-Reynolds number dependence to simple theoretical predictions. The solid line is the experimental result. (a) Averaged Lee predictions with (- - -1 n = 3, (----) n = 5, (-e-) n = 12. (b) Order of magnitude predictions with (---) n = 4 and eff = 3%, (----) n = 7 and eff = 2%, (-*-) n = 10 and eff = 1%.

eters are the diffusivities of A, B, and R and the initial striation thickness (striation thickness considerations are postponed until the end of the section). The sensitivity analysis indicates that the results are largely insensitive to the diffusivity of A and only slightly more sensitive to the diffusivities of B and R. Errors of 10% in any of the diffusivities changed the selectivity by less than 3 % . Changing the ratio of the diffusion coefficients to unity-this is a typical choice in micromixing analyses- while keeping the diffusivity of A constant increases the selectivity by almost 9%. Analysis shows also that the selectivity is insensitive to small changes in the first rate constant but very sensitive to small changes in the second rate constant. There seems to be a large uncertainty in the rate constants as outlined in Table 11. The effects of the rate constant uncertainty can be explored by using the "original" kinetic constants (Bourne et al., 1981). The result is greater than 100% difference in selectivity from

Monodisperse Distribution

A B A B A B A B

SB I-

SA unit cell

Bidisperse distribution A B A B A B A B

- unit cell

Figure 11. Monodisperse and bidisperse striation thickness distributions.

the base case parameters. The difference is primarily due to the differences in the second rate constant. Obviously, the correct values of the rate constants are very important in micromixing studies, especially when the ratio of the rate constants is O(104). Further evaluation must wait until more complete information about the reaction kinetics of the primary reactions and side reactions becomes available. The overall picture which emerges is that the results are fairly robust to small uncertainties and the experimental stretching-Reynolds number relationship is limited by the precision of the data fit, which in turn is hindered by the data scatter and selectivity bounds.

Fluid mixing enters lamellar models through the deformation of striations in time. At large Reynolds numbers, the initial striation thickness is picked to be the diameters of the impinging jets. Large Reynolds numbers imply large stretching and low Reynolds numbers small stretching. However, a t low Reynolds numbers, the suitable characteristic initial length of segregation is no longer the jet diameter, but the mixhead diameter, which means a different initial striation thickness and second Damkohler number are appropriate. Thus, there is an inconsistency, which can be partially avoided by putting less weight on the high-selectivity or low-stretching data in both the numerical and experimental data. The intermediate or transition region, between high and low Reynolds number and stretching, is even more troublesome because there must be a distribution of length scales. In fact, this partially explains the data scatter in the intermediate Rey- nolds number region. Different fluid elements will have different characteristic lengths of segregation, so even if they undergo comparable stretching, the selectivities will be different. The cumulative effect may result in experimental data scatter.

In the formulation of the lamellar model the deforming striations are assumed to be represented by a single length scale, the striation thickness. In reality there must be a distribution of striation thicknesses. Other initial con- figurations can be studied (e.g., a deforming checkerboard in two dimensions or dispersed drops in three dimensions); however, there is compelling evidence that a t small scales-the scales a t which the reaction and diffusion are occurring-there is a single dominant direction due to local stretching (see Corcos and Sherman (1984), and Riley et al. (1986)). However, the assumption of a monodisperse striation thickness is questionable. Fields and Ottino

314 Ind. Eng. Chem. Res., Vol. 28, NO. 3, 1989

1 , , , ,i

i

( 1 j I I I I I I I I I I

1 0 5 I 0

Figure 12. Variation of selectivity with degree of bidispersity of striation thickness. The degree of bidispersity is defined in the text. The parameters are DuI = 1.16 X DUII = 1.58 X lo6, a = 2 s-l,

(1987) studied the effects of a striation thickness distribution in a lumped parameter model of an unpremixed polymerization and found isolation of monomers due to formation of impenetrable polymer layers and rapid con- sumption of the thinnest striations. Their results are difficult to generalize, so it is useful to look at a simple variation of the monodisperse striation thickness distribution: a bidisperse striation thickness distribution. The idea is illustrated in Figure 12. A monodisperse striation thickness distribution has one length scale for each of the reactants, and the striation thickness is defined as the average of the two. The bidisperse striation thickness distribution has two length scales for each reactant, SA1 and sA2, and sB1 and sB2. For comparison purposes, the average of the bidisperse reactant layers must equal the length scales of the monodisperse striations. Simple measures of the distributions are the ratios of the reactant layers length scales which we will define as eA and 0 B :

This measure is only useful for bidisperse striation; statistical measures would be necessary to characterize a real striation thickness distribution. Without any loss of gen- erality, we can restrict 81 < 1 (I = A, B), and the bounds on the distribution are therefore zero and one. Further, we assume that the distributions for each reactant are equal. For 0 = 1, the bidisperse striation thickness distribution reduces to the monodisperse case. The other limit, 8 = 0, also reduces to a monodisperse distribution. The difference is that the striation thickness will be twice the average, so the effective second Damkohler number will be 4 times the actual value with 0 = 1. Thus, for any average property Z

Although the idea of a bidisperse striation thickness is simple conceptually, it is much more computationally in- tensive since the symmetries of the monodisperse case are destroyed and periodic boundary conditions with a t least 4 times as many nodes are necessary for the implementation of numerical solutions. The dashed lines in Figure 11 show a unit cell for the mono- and bidisperse striation thickness distributions, across which the lamellar model equations must be solved.

AB = 0.83, AR = 0.60, PB = 1.0526, c = 9.02 X lo5, @ = 0.5.

@A = SAI/SAZ OB = SBI/SBZ (49)

Z(DU1, DUII, e=o) = Z(DU1, ~ D U I I , 8=1) (50)

The base case for the sensitivity analysis is still the same, and the parameter being varied is now the distribution, 0. The results, shown in Figure 12, show that a striation thickness distribution (i.e., diffusion will be relatively more important) gives higher selectivities. However, it is sur- prising that there is a maximum in the selectivity, away from the limits of zero and one. The gain in the selectivity is not very large, however, even for wide distributions, primarily because the stretching is large enough to overcome the extra diffusion resistance. At smaller values of the stretching, larger variations in the selectivity are expected. The conclusion is that distributions do not affect the selectivity in a substantial manner, especially for large value of the stretching.

In summary, we conclude the following: (1) The experimental results are hindered by data scatter

and incomplete conversion of the limiting reactant to measurable products.

(2) The stretching dependence on Reynolds number, found from matching one experimental data set to solutions of a lamellar model, is consistent with the other two data sets, within experimental uncertainty. The initial concentrations differ by a factor of 2 among each data set.

(3) Both the model of Lee et al. (1980) and an order of magnitude calculation overpredict significantly the stretching in an impingement mixhead. (4) The modeling results are sensitive to uncertainty in

the second rate constant but are insensitive to uncertainties in diffusivities or the first rate constant.

( 5 ) A variation in the striation thickness distribution does not affect the predictions at high Reynolds numbers under the conditions studied.

In spite of these shortcomings, the investigation of mixing using reactions as a probe of the fluid mechanics holds substantial promise, and further analysis and re- finements might provide information inaccessible by other means.

Acknowledgment

Company Cooperative Research Program. This research was supported by the Dow Chemical

Registry No. A, 90-15-3; B, 121-57-3; R, 574-69-6; S, 99701- 05-0.

Literature Cited Baldyga, J.; Bourne, J. R. A Fluid Mechanical Approach to Turbu-

lent Mixing and Chemical Reaction. Part 111. Computational and Experimental Results for the New Micromixing Model. Chem.

Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; Wiley: New York, 1960.

Bourne, J. R.; Hilber, C.; Tovstiga, G. Kinetics of the Azo Coupling Reactions between 1-Naphthol and Diazotised Sulphanilic Acid. Chem. Eng. Commun. 1985,37, 296-314.

Bourne, J. R.; Kozicki, F.; Rys, P. Mixing and Fast Chemical Reactions-I. Test Reactions to Determine Segregation. Chem. Eng. Sci. 1981, 36, 1643-1648.

Chella, R.; Ottino, J. M. Conversion and Selectivity Modifications due to Mixing in Unpremixed Reactors. Chem. Eng. Sci. 1984,

Corcos, G. M.; Sherman, F. S. The Mixing Layer: Deterministic Models of a Turbulent Flow. Part 1. Introduction and the Two Dimensional Flow. J . Fluid Mech. 1984, 139, 29-65.

Doherty, M. F.; Ottino, J. M. Chaos in Deterministic Systems: Strange Attractors, Turbulence, and Applications in Chemical Engineering. Chem. Eng. Sci. 1988, 43, 139-183.

Fields, S. D.; Ottino, J. M. Effect of Striation Thickness Distribution on the Course of an Unpremixed Polymerization. Chem. Eng. Sci. 1987, 42, 459-465.

Fields, S. D.; Thomas, E. L.; Ottino, J. M. Visualization of Interfacial Urethane Polymerizations by Means of a New Microstage Reac- tor. Polymer 1986, 27, 1423-1432.

Eng. Commun. 1984,28, 259-281.

39, 551-567.

Ind. Eng. Chem. Res. 1989,28, 315-324 315

Franjione, J. C.; Ottino, J. M. Feasibility of Numerical Tracking of Material Lines and Surfaces in Chaotic Flows. Phys. Fluids 1987,

Kolodziej, P.; Macosko, C. W.; Ranz, W. E. The Influence of Imp- ingement Mixing on Striation Thickness Distribution and Prop- erties in Fast Polyurethane Polymerization. Polym. Eng. Sci.

Lee, L. J.; Ottino, J. M.; Ranz, W. E.; Macosko, C. W. Impingement Mixing in Reaction Injection Molding. Polym. Eng. Sci. 1980,20,

Mehta, R. V.; Tarbell, J. M. An Experimental Study of the Effect of Turbulent Mixing on the Selectivity of Competing Reactions. AIChE J. 1987, 33, 1089-1101.

Nguyen, L. T.; Suh, N. P. Effect of High Reynolds Number on the Degree of Mixing in RIM Processing. Polym. Process. Eng. 1985, 3(1,2), 37-56.

Ottino, J. M. Description of Mixing with Diffusion and Reaction in terms of the Concept of Material Surfaces. J. Fluid Mech. 1982,

Ranz, W. E. Analysis of Reaction Processes in Which Microscopic

30, 3641-3643.

1982,22, 388-392.

868-874.

114, 83-103.

Heterogeneities Appear: Scale-up and Scale-Down of Polymeri- zation Reactions. Ind. Eng. Chem. Fundam. 1986,25, 561-565.

Riley, J. J.; Metcalfe, R. W.; Orszag, S. A. Direct Numerical Simu- lations of Chemically Reacting Turbulent Mixing Layers. Phys. Fluids 1986, 29, 406-422.

Sandell, D. J.; Macosko, C. W.; Ranz, W. E. Visualization Technique for Studying Impingement Mixing a t Representative Reynolds Numbers. Polym. Process. Eng. 1985, 3(1,2), 57-70.

Tucker, C. L., 111; Suh, N. P. Mixing for Reaction Injection Molding. I. Impingement Mixing of Liquids. Polym. Eng. Sci. 1980, 20,

Tyagi, A.; Wood, P. E.; Hrymak, A. N. Fluid Mechanics of Jet Impingement Mixing. Presented at the 1987 AIChE Annual Meeting, New York, paper 34b.

Wickert, P. D.; Ranz, W. E.; Macosko, C. W. Small-scale Mixing Phenomena During Reaction Injection-Molding. Polymer 1987,

87 5-886.

28, 1105-1110.

Received for review April 25, 1988 Accepted October 3, 1988

I

A Study of Hydrogen Bonding in Alcohol Solutions Using NMR Spectroscopy

Anne M. Karachewski, Marianne M. McNiel,+ and Charles A. Eckert" Department of Chemical Engineering, University of Illinois, Urbana, Illinois 61801

The thermodynamics of hydrogen-bonded systems are best represented by chemical or combined physical-chemical theories of solutions. Although this entails the disadvantage of a large number of parameters, this limitation can be overcome by independent measurement of physically meaningful parameters. Previously we have shown how NMR chemical shift data and limiting activity coefficient measurements could make such direct determination for highly solvated systems. Here we extend these methods to associated systems. Data and mathematical models are presented for alcohol- hydrocarbon systems. The results demonstrate the use of NMR to predict thermodynamic properties such as VLE and enthalpies.

Phase equilibria data are often needed for design purposes in the chemical, petrochemical, and pharmaceutical industries. Usually for complicated or highly nonideal systems, direct experimental measurements are performed over the entire composition range. Predictive methods for phase equilibria would be more useful than direct experimental measurements especially when quick estimates are needed.

Prediction of phase equilibria for strongly complexing or hydrogen-bonding mixtures has remained a challenge. Most solution theories for correlating or predicting phase equilibria attempt to explain all solution nonidealities in terms of nonspecific physical intermolecular forces. Such solution theories are not very sensitive to the association models and because of their empirical nature do not allow cross-prediction of thermodynamic properties (Wilson, 1964; Renon and Prausnitz, 1968). Solution theories based on chemical theory have also been used to represent phase equilibria of hydrogen-bonding mixtures and have been more successful (Dolezalek, 1908; Kretschmer and Wiebe, 1954; Renon and Prausnitz, 1967). Physical effects can be included in chemical theory models but usually require one or more additional adjustable parameters (Chen and Bagley, 1978; Nath and Bender, 1983; Zong et al., 1984). The primary disadvantage of such chemical theory models is the large number of adjustable parameters that must

* To whom all correspondence should be addressed. Present address: Air Products and Chemicals, Inc., Allentown,

PA 18195.

0888-5885/89/2628-0315$01.50/0

be obtained especially for systems exhibiting a large degree of association (Acree, 1984). If such parameters can be measured separately and directly, as by spectroscopic methods, the use of these models is much more attractive.

Alcohol-hydrocarbon solutions have been chosen as a starting point in the study of associated systems. The large excess properties and immiscibility exhibited by alcohol mixtures is a direct consequence of hydrogen bonding. Hydrogen bonding gives rise to specific interactions between atoms or functional groups. Formation of hydrogen bonds modify many chemical and physical properties in- cluding partial molar volumes, viscosities, IR intensities, and NMR chemical shifts (Acree, 1984). Changes in these physical properties can be related to equilibrium constants for complex formation and subsequently used for prediction of phase behavior.

In this work, nuclear magnetic resonance (NMR) is used to determine association equilibria for alcohol-hydrocarbon solutions and to relate experimental measurements to equilibrium constants for complex formation (McNiel, 1985, 1987; Eckert et al., 1986; Karachewski, 1988). Equilibrium constants are obtained by fitting chemical shift data to various chemical-physical association models. The use of NMR as a tool for the study of hydrogen bonding has become increasingly more common in recent years (Foster and Fyfe, 1965; Bruno et al., 1983). For- mation of a hydrogen bond causes a large change in the shielding of a proton donor by lowering the effective electron density, causing a shift of the resonance signal to lower fields. For aromatic molecules, the signal is shifted

0 1989 American Chemical Society

Documents

Analysis of impingement mixing-reaction data: use of a lamellar model to generate fluid mixing information