Turbulent Jet Reactors

0263±8762/98/$10.00+0.00 € Institution of Chemical Engineers Trans IChemE, Vol 76, Part A, September 1998

TURBULENT JET REACTORS: Mixing Time Scales L. J. FORNEY and N. NAFIA School of Chemical Engineering, Georgia Institute of Technology, Atlanta, USA

R

eactor scale-up and numerical simulation of reactor performance require an estimate of the time scale of turbulent mixing. In the present paper, time scales for mixing within free turbulent jets are discussed. In particular, it is demonstrated that both the Kolmogoroff time (n/e)1/2 and eddy dissipation time k/e are proportional throughout the isotropic ¯ ow® eld of fully developed free jets that are commonly used in reactor design. The constant of proportionality between both times is shown to depend on the jet Reynolds number. Scaling laws are derived in the limiting case of either jet-to-tube velocity u/n ¡ 1 or u/n ¡ ¥ plus one intermediate value u/n ¡ 3. Both time scales are also compared with the yield from a mixing-controlled parallel reaction occurring within a free jet where it is demonstrated that the smaller Kolmogoroff value correlates the yield in the limit of a small scale reactant feed. The eddy dissipation time, however, is shown to correlate data for larger but ® xed reactant feed streams, but a complete correlation was not possible. Keywords: jet; reactor; mixing; selectivity

INTRODUCTION

turbulence, that is, not dominated by shear at a solid wall. For example, the large eddies formed by the motion of a mechanical impeller are not isotropic, but as the eddies decay transferring energy to smaller ¯ uid fragments, the smaller eddies formed become independent of the impeller geometry and eventually become nearly isotropic. In the latter case, the eddy spectrum depends only on the power supplied and on viscous dissipation. Several numerical models used to simulate reaction rates in turbulence use time scales de® ned in terms of local isotropy. For example, the mixing time (n/e)1/2 or Kolmogoroff time scale that appears in Table 1 (i.e., engulfment model) would appear to differ from the time scale or eddy dissipation time k/e used in other popular turbulent models. From Table 1, both the Interaction by Exchange with the Mean or IEM model developed by Villermaux and David5 and the eddy break-up model proposed by Spalding6 and re® ned by Magnusson and Hjertager7 use the dissipation time of the energy containing eddies k/e to characterize the time scale of mixing and reaction. To resolve the issue of time scales, the ratio of eddy dissipation time td -to-Kolmogoroff time tk is written as k td k = ne 1/2 = (1) tk (ne)1/2 e Since the turbulent kinetic energy per mass k = 32 (u9 )2 and the dissipation rate e = (u9 )3 /z e , substitution into equation (1) yields

The turbulent mixing of two miscible ¯ uids to promote chemical reactions is a common process in industrialized society. Effective use of turbulence can increase reactant contact and yield, which can signi® cantly reduce the cost of producing many chemicals. Common turbulent reactor designs are stirred tanks1,2 or tubular designs with jet injection3,4 . A number of mechanistic models have been developed to account for the essential features of real chemical reactors. These models contain at least one local mixing parameter or time scale whose magnitude determines the level of turbulent mixing. A list of several mixing models and their local time scales are tabulated in Table 1. Moreover, the scale-up of reactors also requires an estimate of the mixing time in relation to reaction times. In both instances, it would be useful and convenient to relate these mixing times to jet boundary conditions such as Reynolds number and geometry. In many common applications, the mixing models are applied in regions of isotropic turbulence. The purpose of this paper is to examine the relationship between the two time scales, that is, the Kolmogoroff and eddy dissipation times that appear in Table 1 for a common application of isotropic turbulence in a free jet, and to use these results to correlate the yield from fast mixing-controlled parallel reactions. SIMILARITY Isotropic turbulence occurs in a variety of large Reynolds number ¯ ows. Isotropic turbulence occurs downstream from a grid, in fully developed pipe ¯ ow and in jets and wakes. In fact, isotropy may often be found in the decay of free

td tk 3 where z 728

e

u9 z e = Re1/2 z n

(2)

is the average size of the energy containing eddies

TURBULENT JET REACTORS: MIXING TIME SCALES Table 1. Mixing models. Model Eddy break-up IEM Engulfment model

Mixing Time Spalding6 David and Villermaux5 Bourne and Baldyga13

(k/e) (k/e) (n/e)1/2

Moreover, an additional relation from two alternative assumptions is also necessary, namely, similarity between turbulence and mean ¯ ow properties such as turbulent energy u9 2 = c1 u2x

and u9 is the root-mean-square turbulent velocity. Hence, the dissipation and Kolmogoroff time scales are proportional provided the eddy Reynolds number Rez is constant in the ¯ ow ® eld. Hinze8 states that complete similarity would exist in geometrically similar ¯ ows if one velocity and one length scale are suf® cient to produce an identical velocity distribution in terms of reduced coordinates. Isotropic turbulence that occurs in self-preserving ¯ ows, such as the decay behind a grid or in free turbulent jets, exhibits complete similarity such that the eddy Reynolds number Rez is a constant. Davies9 , p. 58, for example, provides a relationship between the turbulent kinetic energy k and dissipation rate e for fully developed ¯ ow in a pipe e/k 3 u/d

(3)

in which case Rez 3 Re7/8 . (4) The question addressed in the present paper is whether a similar expression exists downstream from the nozzle that relates constant values of the eddy Reynolds number Rez to the nozzle Reynolds number Re. Such an expression would allow one to determine the proportionality between the dissipation and Kolmogoroff time scales throughout the jet ¯ ow® eld and, in fact, to either interchange the time scales in mixing models with no loss in accuracy or to scale-up jet performance in reactor design applications. Similarity in Free Jets The classical theories of Boussinesq, Prandtl and Reichardt for a round free jet consider similarity relations of the form (Hinze8 p. 521) ux x1 a p = f (z) (5a) u d and ur x1 a n = g(z) (5b) u d and x1 a s u9 2 = u2 h(z) (5c) d where the reduced radial coordinate r/d (5d) z = x a q. 1 d Here x = 2 a is the position of the origin of similarity. Hinze also demonstrates that substitution of equations (5a)±(5d) into conservation of mass and momentum expressions will not provide similar solutions unless one imposes the limiting cases of either u/v ¡ 1 or u/v & 1.

(6)

or, alternately, constant eddy viscosity ne 3 u9 z 524). Thus, ne = c2

e

(Hinze8 , p. (7)

where c1 , c2 are constants. Here, c2 depends on the nozzle diameter and velocity but both constants are independent of spacial coordinates x, z and z e is the size of the energy containing eddy. Both assumptions of equations (6) or (7) are realistic for the limiting case u/v & 1 since conservation of mass and momentum suggest ux 3 (x 1 a)2 1 and the jet radius r1/2 (x) 3 x 1 a. Additional experimental evidence provides u9 3 ux and z e 3 r1/2 (x) (Davies9 , p. 72). These similarity solutions are not so simple, however, for the limiting case u/v ¡ 1 when the jet is not completely similar (Hinze8 , pp. 524, 525). The two limiting cases u/v ¡ 1 and u/v & 1 are now considered separately. Case u/v & 1 In this case, the free jet is introduced into a quiescent ambient ¯ uid. The integral mass and momentum balance equations along with the additional assumption of equation (6) provide a completely similar solution for equations (5a)±(5d), where 2p = 2n = s = 2 2 and q = 1. Thus, u9 3 (x 1 a)2 1 or the turbulent kinetic energy k 3 (x 1

a)2 2 .

Moreover, since r1/2 (x) 3 x 1 tion becomes e 3 (x 1

Trans IChemE, Vol 76, Part A, September 1998

729

a)2

(8) a 3 z e , the turbulent dissipa-

4

(9)

and from equations (1) and (2), both the eddy Reynolds number Rez and the ratio of eddy dissipation time td -toKolmogoroff time tk are constant. Thus, td c (10) tk 3 3 where one would expect that the constant c3 depends on the nozzle Reynolds number as shown later in the Numerical Results section. Moreover, both time scales are proportional and the ratio is independent of distance from the jet nozzle. The two additional assumptions in equations (6) and (7) are also shown to be equivalent for the limiting case u/v & 1. Case u/v = 1 In this case, the free jet is introduced isokinetically into the ambient ¯ uid. Unlike the previous case, the integral mass and momentum balance equations along with the additional assumption of equation (6) provide an incompletely similar solution (Hinze8 , p. 524) where in equations (5a)±(5d), p = 2 2/3, q = 1/3, s = n = 2 4/3. Thus, u9 3 (x 1 a)2 2/3 and the turbulent kinetic energy k 3 (x 1

a)2

4/3

Moreover, since r1/2 (x) 3 (x 1

(11) a)

1/3

3 z e , the turbulent

730

FORNEY and NAFIA

dissipation becomes e 3 (x 1

a)2

7/3

.

(12)

Thus, the eddy Reynolds number Rez 3 (x 1

a)2

1/3

(13)

and the ratio of eddy dissipation time td -to-Kolmogoroff time tk becomes td (x 1 tk 3

a)2

1/6

(14)

or is nearly independent of distance from the jet ori® ce. Figure 2. Centreline decay of a conserved inert tracer.

NUMERICAL RESULTS Fluid dynamic simulations were performed with a commercial computational ¯ uid dynamic (CFD) code (Fluent version 4.3). The turbulent transport processes were evaluated with the standard two-equation k 2 e turbulence model of Launder and Spalding10 . In order to determine if the solution procedure was independent of the grid size and correctly modelled turbulent dispersion, a simulation was performed on the decay of an inert conserved scaler from the data of Singh11 . The present 30 ´ 200 axisymmetric grid and solution procedure were adequate to predict turbulent dispersion from a coaxial jet, including the region of ¯ ow establishment. A schematic of the axisymmetric jet is shown in Figure 1 and the excellent correlation between experimental and numerical results are indicated in Figure 2. Case u/v & 1 Numerical solutions are provided along the jet centreline for the turbulent kinetic energy k and dissipation rate e in Figure 3. The jet nozzle occurs at (x/d)j = 1.6 and the ¯ ow is established at the end of the potential core at x/d = 5.0. The origin of similarity occurs at x/d = 0.6 (a/d = 1.0). Since the axial coordinate x is measured over the entire grid, one seeks spacial dependence on the coordinate x 2 d

x a x = 2 1 d j d d

0.6.

and e 3 (x/d 2

0.6)2

4

for x/d > 5.0. Likewise in Figure 4, the dissipation time k/e 3 (x/d 2

0.6)2

(16)

and Kolmogoroff time (n/e)1/2 3 (x/d 2

0.6)2

(17)

for x/d > 5.0. Finally, the eddy Reynolds number Rez and ratio of eddy dissipation-to-Kolmogoroff time td /tk are plotted in Figure 5 and, as noted, are roughly constant in the established region of jet ¯ ow. Case u/v = 1 In this case, the numerical solutions differ somewhat from theoretical predictions where in Figure 6, one obtains k 3 (x/d 2

0.6)2

1

(18)

(15)

Numerical solutions for both k and e in Figure 3 conform to the classical theory since k 3 (x/d 2

0.6)2

2

Figure 1. Schematic of axisymmetric jet. Also, small reactant feed tube of diameter df % d.

Figure 3. Numerical centreline values for k (top) and e (bottom). Jet ori® ce is x/d = 1.6. Formulas are for established ¯ ow, x/d > 5.0.

Trans IChemE, Vol 76, Part A, September 1998

TURBULENT JET REACTORS: MIXING TIME SCALES

Figure 4. Numerical centreline values for eddy dissipation time (top) and Kolmogoroff time (bottom). Jet ori® ce at x/d = 1.6. Formulas are for established ¯ ow, x/d > 5.0.

in contrast to theory that predicts k 3 (x/d 2 Likewise, in Figure 6, one obtains e 3 (x/d 2

0.6)2

2

0.6)2

4/3

.

731

Figure 6. Numerical centreline values for k (top) and e (bottom). Jet ori® ce is x/d = 1.6. Formulas are for established ¯ ow, x/d > 6.0.

The eddy Reynolds number and ratio of eddy dissipationto Kolmogoroff time td /tk are plotted in Figure 8 which are both roughly constant in the established region of the ¯ ow. It is important to note that the ratio of times

(19)

in contrast to theory that predicts e 3 (x/d 2 0.6)2 7/3 . The dissipation time scale k/e, however, plotted in Figure 7 yields (20) k/e 3 (x/d 2 0.6) which conforms to theory, while the Kolmogoroff time in Figure 7 (n/e)1/2 3 (x/d 2

0.6)

differs from theory where (n/e)1/2 3 (x/d 2

(21) 0.6)7/6 .

Figure 5. Eddy Reynolds number Rez and ratio of times td /tk versus distance on jet centreline. Jet ori® ce x/d = 1.6. Formulas are for established ¯ ow, x/d > 5.0.

Trans IChemE, Vol 76, Part A, September 1998

Figure 7. Numerical centreline values for eddy dissipation time (top) Kolmogoroff time (bottom). Jet ori® ce at x/d = 1.6. Formulas are for established ¯ ow, x/d > 6.0.

732

FORNEY and NAFIA td /tk 3 (x/d 2 0.6)2 1/6 according to theory, or one would expect, at most, a decrease of only 15% over the range of 5 < x/d < 15. Numerical results with the k 2 e turbulent model, however, predict a far smaller change. The present case u/v = 1, of course, is far less important than the former case u/v ¡ ¥ in terms of reactor applications. An additional surface plot of Rez is provided in Figure 10. Eddy Reynolds Number

Figure 8. Eddy Reynolds number Rez and ratio of times td /tk versus distance on jet centreline. Jet ori® ce x/d = 1.6. Formulas are for established ¯ ow, x/d > 6.0.

A summary plot of the constant eddy Reynolds number Rez achieved downstream from the nozzle versus the jet Reynolds number Re is indicated in Figure 9. Here, values of the jet-to-tube velocity ratio u/v are superimposed on the plot to illustrate a transition from one asymptote at isokinetic ¯ ow u/v = 1 to another in the limit u/v ¡ ¥. Tangent lines were drawn at three values of u/v = 1, 3, 20 in Figure 9 to provide simple functional forms between Rez and Re that would cover the entire range of Re. These relationships between Rez and Re are summarized below, Rez = b1 Re8/3 , 11/6

Rez = b2 Re

,

for u/v ¡ 1

(22a)

for u/v ¡ 3

(22b)

and Rez = b3 Re,

for u/v ¡ ¥

(22c) 2 10

Figure 9. Constant eddy versus jet Reynolds numbers.

where from Figure 9, b1 = 6.75 ´ 10 , b2 = 4.65 ´ 102 6 and b3 = 0.089. These results indicate that both the Kolmogoroff time tk and dissipation time td are interchangeable throughout the jet ¯ ow® eld provided proper account is made of the boundary condition or jet Reynolds number Re. Thus, substitution of either equations (22a, b, c) into equation (2) would relate the magnitude of the Kolmogoroff time in the established region of the turbulent jet to the jet

Figure 10. Surface plot of eddy Reynolds number Rez . u/v = 3.0, Re = 67, 650. Jet ori® ce at x/d = 1.6.

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TURBULENT JET REACTORS: MIXING TIME SCALES

Reynolds number Re and eddy dissipation time or one obtains, td = 2.6 ´ 102 5 Re4/3 , tk td = 2.2 ´ 102 3 Re11/12 , tk

u/v ¡ 1

(23a)

u/v ¡ 3

(23b)

and td = 0.3 Re1/2 , tk

u/v ¡ ¥

(23c)

733

Restricting the following discussion to large jet velocities u/v & 1, the eddy dissipation time td k z e e 3 u9 or multiplying numerator and denominator by u9 , substituting equation (5c) with s = 2 2 for u9 2 and Rez = z e u9 /n, one obtains on the jet centreline where h(z = 1) = 1 k n Re e 3 z u2

x1

a d

2

.

(25)

Finally, substituting equation (24) into equation (25), one obtains

MIXING TIME SCALES The ¯ uid properties of inertia, convection, viscosity and molecular diffusion provide different time scales for turbulent mixing. For example, it can be shown that the time scale due to the combined effects of viscosity and molecular diffusion at the jet exit would be tD 3 l2k/D 3 d 2 Re2 3/2 /D for jet velocities that are large relative to the surroundings where lk is the Kolmogoroff length (see appendix). As is shown below, the Kolmogoroff time due to the effects of viscosity and convection is tk 3 td /Re1/2 3 d Re2 1/2 /u for the same circumstances. Thus, one obtains a ratio of diffusion-to-Kolmogoroff times for liquids, tD n = Sc & 1. tk 3 D

(24a)

Likewise, the ratio of the diffusion-to-eddy dissipation time, where the latter is proportional to d/u, can be written in the form tD Sc >1 td 3 Re1/2

(24b)

or, as before, tD > td for most liquid jets. In principle, although the magnitude of all three time scales tD > td & tk could be related to jet boundary conditions for various jet-to-tube velocity ratios, the analysis below is restricted to the smaller of the three, that is, the dissipation and Kolmogoroff times td and tk , respectively. This is in the interest of brevity and because of the sparsity of experimental data.

td 3

d x1 a 2 . u d

(26)

Thus, the dissipation time td = k/e increases with distance squared from the point of established ¯ ow (x/d > 5.0 in Figure 4) with a characteristic shape for large u/v but whose magnitude is scaled by the quantity d/u, as is demonstrated by the numerical results in Figure 4. Similar results can be obtained for arbitrary jet-to-tube velocity ratios, that is, a magnitude scaled by d/u but with pro® les of different shape, as shown in Figure 7 for u/v = 1. Substituting equations (22c) and (26) into equation (2), the magnitude of the dissipation time td is proportional to the Kolmogoroff time tk in the form t d 1/2 n1/2 x 1 a tk 3 d1/2 3 3/2 Re u d

2

for u/v & 1. It should be noted that tk in equation (27) would have the same shape with distance downstream as td in equation (26) but that tk is scaled by the factor Re2 1/2 , as con® rmed by both numerical curves in Figure 4. Similar pro® les are illustrated in Figure 7 where k/e 3 d/u((x 1 a)/d) for u/v = 1 and the scaling factor for tk is Re2 4/3 . The results of a similar analysis for the magnitude of both the dissipation and Kolmogoroff time scales for velocity ratios u/v = 1 and 3 are tabulated in Table 2.

CORRELATION OF REACTION DATA Reaction data were chosen for a mixing sensitive parallel

Table 2. Mixing times scales for round turbulent jets.

Re Rez

u/v ¡ 1.0

u/v ¡ 3.0

u/v ¡ ¥

4q/pnd ud/n

4q/pnd ud/n

4q/pnd ud/n

4.65 ´ 102 6 Re11/6

0.089 Re

6.75 ´ 102

10

Re8/3

3

td

3d /q d/u

3d /q d/u

3 3d /q d/u

tk

4/3 3td /Re

11/12 3td /Re

1/2 3td /Re

d 13/3 n4/3 3 q7/3

d 47/12 n11/12 3 q23/12

d 7/2 n1/2 3 q3/2

n4/3 3u7/3 d1/3

d 1/12 n11/12 3 u23/12

d 1/2 n1/2 3 u3/2

Trans IChemE, Vol 76, Part A, September 1998

(27)

3

734

FORNEY and NAFIA

reaction scheme from Baldyga, et al.12 of the form A1

B2

22

¡R

R1

B2

22

¡S

AA 1

B2

22

¡Q

k1

k2

k3

(28)

where k1 & k3 > k2 . Equal concentrations of components A and AA were discharged through a nozzle of diameter d into a much larger tank containing the same components. The rate limiting reactant B at ¯ owrate QB was introduced through a feed pipe of small diameter df % d located at 1.75d downstream from the nozzle exit and 0.37d from the centreline as illustrated in Figure 1 by Baldyga, et al.12,2 This novel geometry provides an excellent means of investigating the effects of turbulence on fast reactions by comparing the yield of product Q downstream as determined by reaction (27), nozzle geometry, ¯ uid properties, ¯ ow rates and stoichiometry. For example, to isolate the physical effects of turbulence on the product yield of Q or XQ , all of the data chosen were subject to the same stoichiometry (i.e., equal reactant concentrations between experiments and total volume and mole ratios of A to B). Small Reactant Feed Micromixing controlled values of XQ were recorded by Baldyga et al.2 with a small jet nozzle of diameter d = 0.008 m and feed tube df = 0.001 m. Jet velocities were varied from 4 to 11 m s2 1 giving a range of jet Reynolds number 36, 000 < Re < 100, 000. Data were recorded at two values of the kinematic viscosity for pure aqueous solutions (n = 0.89 ´ 102 6 m2 s2 1 ) and with an additive hydroxyethyl cellulose (n = 0.62 ´ 102 6 m2 s2 1 ). The small feed tube introduced the reactant B into the jet at constant QB # 4 ml min2 1 . (CB = 60 mol m2 3 ) which was suf® ciently small to eliminate the effects of the magnitude of QB on the yield XQ . The yield XQ recorded with the small jet nozzle is plotted in Figure 11 versus the Kolmogoroff time scale tk 3 (d/u) (Re2 1/2 ). It is evident that tk is the appropriate time scale for these micromixing controlled data. Assuming the rate constant k3 = 125 m3 mol2 1 s2 1 , the Damkohler number

Figure 11. Micromixing-controlled yield of dye Q from 2-naphthol versus Kolmogoroff time scale. Small jet diameter, d = 0.008 m. Flowrate of sulphanilic acid QB # 4 ml min2 1 .

Figure 12. Micromixing-controlled yield of dye Q from 2-naphthol versus Damkohler number. Same data as Figure 11; k3 = 125 m3 mol2 1 s2 1 , cBo = 60 mol m2 3.

based on the Kolmogoroff scale, de® ned as Da =

k3 cBo d u Re1/2

(29)

would cover the range 0.01 # Da < 0.23 for the data as illustrated in Figure 12. The issue of scaleup to larger jet nozzles d is discussed in the appendix.

Large Reactant Feed If the ¯ owrate of B from the small feed tube at a ® xed position in the jet is increased, the scale of the B containing eddies increases as LC = (QB /u)1/2 (Baldyga et al.2 ) and the important mixing time scale shifts from the Kolmogoroff to the eddy dissipation time d/u as evident by the correlation in Figure 13. One might expect that the transition from the Kolmogoroff to the dissipation time would occur when the scale of the feed Lc becomes much larger than the Kolmogoroff length lk = (n3 /e)1/4 or Lc /lk & 1. However, there are not enough experimental data to con® rm this hypothesis. All data for the yield XQ that appeared to be sensitive to QB for a larger but ® xed value of QB = 12 ml min2 1 are plotted in Figure 13. This illustrates that the eddy dissipation

Figure 13. Jet mixing-controlled yield of dye Q from 2-naphthol versus eddy dissipation time scale. Flowrate of sulphanilic acid from small feed tube is large but constant, QB = 12 ml min2 1 .

Trans IChemE, Vol 76, Part A, September 1998

TURBULENT JET REACTORS: MIXING TIME SCALES

735

where the turbulent dissipation rate e = (u9 )3 /z

(2A)

e

If the jet velocity is large relative to the surroundings, then the centreline pro® le for the rms turbulent velocity from equation (5c) with h(z) = 1 and s = 2 2 is ux 3 u9 3 u

x1

a

2 1

.

d

Likewise, the size of the energy containing eddies z equation (7) and Davies9 , p. 70 indicates that z Figure 14. Jet mixing-controlled yield of dye Q from 2-naphthol versus eddy Kolmogoroff time scale. This illustrates that reaction is not controlled by micromixing. Same data as Figure 13.

time is the correct mixing time in this instance. For example, when the data from Figure 13 are replotted versus the Kolmogoroff time in Figure 14, where tk is observed to be of the same magnitude as in Figure 11, the data are not correlated, as expected. It should be noted that the use of a small reactant feed tube in an external turbulent ¯ ow ® eld, as illustrated in Figure 1 potentially introduces another length scale at large ¯ owrate QB (and dimensionless group) that are necessary to correlate product yield. Attempts to overcome this additional requirement, however, have had some success2 . CONCLUSIONS In the present paper it is demonstrated that the Kolmogoroff time (n/e)1/2 and eddy dissipation time k/e are proportional throughout the ¯ ow® eld of round turbulent jets that are commonly used in reactor design. The constant of proportionality between both times is shown to depend on the nozzle Reynolds number. It is also demonstrated that the smaller Kolmogoroff time scale correlates the yield from mixing-controlled parallel reactions when one reactant is introduced slowly into the jet (small scale reactant feed QB ) and that the results are independent of QB . Moreover, the level of micromixing does not depend explicitly on either the Schmidt number, Sc, Reynolds number, Re or the energy dissipation rate, e. Increasing the feed rate of one reactant, however, (large scale reactant feed QB ) introduces an additional length scale (or time scale) that depends on QB . Under the latter circumstance the data could only be correlated with the larger eddy dissipation time scale and at ® xed QB . This suggests that, in general, a three-dimensional plot may be necessary for the selectivity or XQ = f (Da, G) where G depends on QB for large feed rates QB and Da is the Damkohler number based on the eddy dissipation time. More work must be done to understand the latter case. APPENDIX Mixing Time Scales The Kolmogoroff length is lk = (n3 /e)1/4 Trans IChemE, Vol 76, Part A, September 1998

(1A)

e

3d

x1

a

.

d

(3A) e

from

(4A)

Thus, substituting equations (3A) and (4A) into the dissipation rate equation (2A), one obtains e 3 (u2 /d)

x1

a

2 4

(5A)

d

Hence, the molecular diffusion time through the Kolmogoroff length tD 3 l2k/D from equations (1A) to (4A) is of the form tD 3

n3/2 d 1/2 x 1 a 2 . D u3/2 d

(6A)

Likewise, the Kolmogoroff time tk can be written in the form given by equation (27) or, d 1/2 n1/2 x 1 a 2 tk 3 3/2 . u d

(7A)

Therefore, the ratio of diffusion-to-Kolmogoroff time tD /tk is independent of position from the nozzle and the constant magnitude of the ratio is tD n/D = Sc tk 3

(8A)

Similar arguments can be made for the ratio of diffusionto-eddy dissipation time tD /td , where the latter is de® ned in equation (26) td 3

d x1 a u d

2

(9A)

or one obtains tD Sc 3 td Re1/2 .

(10A)

It should be noted that if Sc ~ 103 for most liquids, then Sc/Re1/2 > 1 if the nozzle Reynolds number Re < 106 . Scale-up Consistent with the assumption of local isotropy and selfpreservation within free turbulent jets, one assumes that features of isotropic turbulence are determined by the ® nescale structure. Thus, the ratio of Kolmogoroff-to-¯ ow time tk /td from equations (7A) and (9A) is very small. The same quantity can also be interpreted as the ratio of the reaction

736

FORNEY and NAFIA

zone length-to-reactor size since ux tk 1 3 %1 ux td Re

(11A)

and is independent of axial position. Similar arguments for the fractional change in the turbulent dissipation rate e over the reaction zone length provide 1 de 1 Dxk 3 e dx Re

(12A)

which is also independent of axial position where Dxk = ux tk . From the discussion above, one concludes that changes in feed point along the jet axis would have no effect on the shape of the selectivity pro® le for any jet Reynolds number Re. Moreover, similar conclusions regarding scale-up to larger jets can be made provided that Re is large. Baldyga and Bourne14 derived an expression similar to equation (11A) based on dimensional arguments for a ® xed feed point in a stirred tank. The data in the latter work indicate a constant selectivity provided Re > 20, 000 for both large and small tanks. For additional information see the work of Bourne and Yu15 who demonstrate similar selectivity pro® les in stirred tanks for similar values of the Damkohler number when the tank volume was increased by a factor of 30 and the feed position was ® xed near the impeller discharge. Some scatter in their experimental data was evident, however, for a feed position in the suction region of the impeller. In the latter case, the principle of local isotropy and self-preservation along a streamline are likely to be disturbed by the action of the impeller. NOMENCLATURE

z

a c d df Da D k Lc e

q QB r Re Rez Sc td tk tD u ux ur u9 v

origin of similarity, m reactant concentration, mol m2 3 jet diameter, m feed diameter, m Damkohler number molecular diffusivity, m2 s2 1 rate constant, m3 mol2 1 s2 1 scale of B eddies, m size of energy containing eddies, m volume ¯ owrate in jet, m3 s2 1 reactant feed of B, mz min2 1 radial coordinate, m nozzle Reynolds number eddy Reynolds number Schmidt number eddy dissipation time, s Kolmogoroff time, s molecular diffusion time, s nozzle ¯ uid velocity, m s2 1 axial ¯ uid velocity, m s2 1 radial ¯ uid velocity, m s2 1 root-mean-square turbulent velocity, m s2 ambient tube velocity, m s2 1

x XQ

axial coordinate, m selectivity of product Q

Greek k e n ne z lK

Symbols turbulent kinetic energy, m2 s2 2 turbulent energy dissipation, m2 s2 kinematic viscosity, m2 s2 1 eddy viscosity, m2 s2 1 reduced radial coordinate Kolmogoroff time, s

3

REFERENCES 1. Patterson, G. K., 1990, Simulating turbulent-® eld mixers with multiple second-order chemical reactions, AIChEJ, 36: 1457. 2. Baldyga, J., Bourne, J. R., Dubuis, B., Etchells, A. W., Gholop, R. V. and Zimmermann, B., 1995, Jet reactor scale-up for mixing-controlled reactions, Trans IChemE, 73 (A5): 497. 3. Brodkey, R. S., 1974, Turbulence in Mixing Operations, Academic Press, New York. 4. Forney, L. J., Na® a, N. and Vo, H. X., 1996, Optimum jet injection in a tubular reactor, AIChE J, 42: 3113. 5. David, R. and Villermaux, J., 1987, Interpretation of micromixing effects on fast competing reactions in semi-batch stirred tanks by a simple interaction model, Chem Eng Commun, 54: 333. 6. Spalding, D. B., 1971, Mixing and chemical reaction in steady con® ned turbulent ¯ ames, 13th Int Symp on Combustion, (The Combustion Institute) p. 649. 7. Magnussen, B. F. and Hjertager, B. H., 1976, On mathematical modeling of turbulent combustion with special emphasis on soot formation and combustion. 16th Symp on Combustion, (The Combustion Institute) p. 719. 8. Hinze, J. O., 1987, Turbulence, (McGraw-Hill). 9. Davies, J. T., 1972, Turbulence Phenomena, (Academic Press). 10. Launder, B. E. and Spalding, D. B., 1972, Mathematical Models of Turbulence, (Academic Press). 11. Singh, M., 1973, Chemical reactions in one and two dimensional turbulent ¯ ow systems, PhD Thesis, (Carnegie Mellon Univ, Pittsburgh, PA). 12. Baldyga, J., Bourne, J. R. and Zimmermann, B., 1994, Investigation of mixing in jet reactors using fast, competitive-consecutive reactions, Chem Eng Sci, 49: 1937. 13. Baldyga, J. and Bourne, J. R., 1984, A ¯ uid mechanical approach to turbulent mixing and chemical reaction, Chem Eng Commun, 20: 259. 14. Baldyga, J. and Bourne, J. R., 1988, Calculations of micromixing in inhomogeneous stirred tank reactors, Chem Eng Res Des, 66: 33. 15. Bourne, J. R. and Yu, S., 1994, Investigations of micromixing in stirred tank reactors using parallel reactions, Ind Eng Chem Res, 33: 41±55.

ACKNOWLEDGEMENT The authors wish to thank Herschel Reese and Dr Hanh Vo for the corporate contribution of computer hardware from Dow Corning Corp. The donation of CFD software from Fluent Inc. (Dr Jayathu Murphy) is also gratefully acknowledged.

ADDRESS Correspondence concerning this paper should be addressed to Dr L. J. Forney, School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia, GA 30332, USA. 1

The manuscript was received 6 November 1997 and accepted for publication after revision 19 March 1998.

Trans IChemE, Vol 76, Part A, September 1998