- Email: [email protected]
Impingement-sheet mixing of liquids at unequal flow rates
Chemical Engineering and
229
Processing, 31 ( 1992) 229- 239
Impingement-sheet
mixing
of liquids
Robert J. Demyanovich”
and John R. Bourne
at unequal
flow rates
Technisch-Chemisches Laboratorium, ETH, 8092 Zurich (Switzerlund) (Received
December
2, 1991; in final form March
9. 1992)
Abstract Mixing of liquids by the impingement of two, thin sheets at unequal flow rates was studied using competitive, consecutive azo couplings that provide quantitative information about the micromixing. Liquid flow rate ratios of 6.0 and 12.0 were investigated at total flow rates ranging from 13-45 I min-‘. An expression for the energy dissipation rate in the impingement zone resulting from the inelastic collision was derived from the momentum balance and the geometry of the impinging sheets. For the experimental data, energy dissipation rates were calculated to lie in the range 6 900-97 500 W kg-‘. The major parameters affecting the micromixing rate of unequal, impinging sheets are the energy dissipation rate in the impingement zone (c,) and the volumetric flow rate ratio (x). Previously, the micromixing results for equal, impinging sheets were explained by a simple model assuming diffusion of the reactants within slabs of fixed thickness equal to the Kolmogorov microscale. Attempts to fit the experimental data for unequal, impinging sheets to a similar model, modified to account for volumetric flow rate ratio, resulted in a relatively poor correlation of the experimental results. However, a modification of the mixing modulus, to account for the effects of E on the diffusion length, yielded a reasonable correlation of the experimental data for unequal, impinging sheets to the model for equal, impinging sheets. Consequently, an expression was derived that relates the micromixing time for both equal and unequal, impinging sheets to a and E,.
Introduction
The impingement of thin sheets of liquids has been shown to yield rapid mixing of relatively large flow rates of liquids [ 11. For liquids with viscosities of 1.O cP, Demyanovich and Bourne have obtained diffusive mixing or micromixing times as low as 1 millisecond for flow rates up to 6 1 min-‘, and micromixing times of the order of 10 milliseconds for flow rates up to 76.0 1 min-’ [2]. All previous work with impingementsheet mixing, however, has been cohducted at equal flow rates for both liquids. Since the majority of liquid mixing operations on the industrial scale do not occur at a flow rate ratio of 1.0, experiments were conducted at different flow rate ratios in order to estimate the effect on the micromixing time for impinging sheets. Equal impinging sheets Since the impingement of liquid sheets has already been discussed in detail [ 1, 31, only a brief summary of the method will be provided here. Tt has been previ-
*Author to whom correspondence should be addressed at LiquiSheet Technologies, Inc., 4421 Gilbert St., Suite 204, Oakland, CA 9461 I, U.S.A.
0255-2701/92/$5.00
ously shown that within any sheet, liquid flows from the sheet origin outward along radial lines with no crossover of liquid between adjacent radial lines [3,4]. Once the sheet is formed in free space, the radial velocity of the liquid within the sheet remains constant up to ambient gas pressures of the order of 20 bar [5]. Since the drag of the surrounding gas on the free sheet was found to be negligible, there is no external force to reduce the sheet velocity. The equation of flow within the sheet is Q = q5R.w. Since the volumetric flow rate (Q) and the flow velocity (u) are constant, the area normal to flow (4fi) must also be constant. Since fi is constant, the sheet thickness is inversely related to its distance from the sheet origin. The sheet continues to thin until inertial forces are overcome by surface tension forces resulting in its breakup into droplets. Liquid sheets with flow rates of the order of several 1 min- ’ typically have residence times of the order of 5-10 milliseconds before they break up into droplets. Figure I illustrates the case of two, equal, single, fan-shaped sheets impinging at a contact angle of 2g. Before breakup of the single sheets into droplets, these sheets are impinged on one another to produce a mixture of the liquids that is also in the form of a thin continuous sheet. As shown in Fig. I, if the angle of impingement is acute (preferably less than 60”), the
CI 1992 ~ Elsevier Sequoia.
All rights reserved
230
\
R,
/-
Side View
Impingement Zone
Top View
unit mass of the liquids. Normally, E is difficult to measure because the mass in which the power is input cannot be accurately estimated. Often, the energy dissipation rate is determined after modelling the results of mixing experiments. Since the geometry of impinging sheets is well-defined, the energy dissipation rate resulting from impingement can be estimated. The calculation involves determining what power/mass is lost due to the inelastic collision of the sheets. The energy dissipation rate resulting from the inelastic impingement of two, equal, single sheets was previously given in ref. 2 as, v 3 cos p 6.I = Y 24
Fig. 1. Impingement of equal, single sheets at an angle of 2p. Top view illustrates sampling around the mixed sheet.
sheets are not destroyed at the impingement zone, but instead combine to form a mixed sheet. The impingement process can be described by momentum balances in the x and y directions. The nature of the collision (inelastic or elastic) was previously found to be a function of the angle of impingement [3]. At impingement angles less than 60”, the collision was shown to be inelastic with respect to the kinetic energy of the single sheets [3]. Hence, the kinetic energy associated with the single sheet II-velocity component, u,,, is released into the thin liquid sheets producing high rates of energy dissipation (> 105W kg-‘). High energy dissipation rates are generally required for the rapid mixing of two liquid streams. Micromixing
When two liquid streams are brought together so that some degree of turbulence is generated, small liquid elements called eddies or laminae are formed that are set in bulk motion. The average size of these eddies, known as the segregation length (L) of the mixture, depends on the liquid viscosity and on the intensity of the turbulence. The micromixing time (L’/D) of the two liquid streams is given by the time required for the components to diffuse over a distance equal to the segregation length. The energy dissipation rate (E) provides a measure of the intensity of the turbulence that is produced. The energy dissipation rate is related to the power input per
(1)
where &i is the energy dissipation rate in the impingement zone, u,, is the “mixing velocity”, and p is the half angle of impingement. This expression is entirely determined by the momentum balance and the geometry of the impinging sheets (see Fig. 1). Equation ( 1), however is only valid in the impingement zone. As the liquid flows away from the impingement zone, si decays rapidly since no additional energy is dissipated elsewhere in the mixed sheet. The rate of micromixing is limited by molecular diffusion between the dispersed eddies of the size of the segregation length. If the overall flow is laminar, the segregation length will be given by a size scale characteristic of the laminar flow (for impinging sheets this size scale would be the sheet thickness, s). If instead the flow is turbulent, the segregation length will be much smaller. For liquids with a viscosity of 1.0 cP, the Reynolds number in the impingement zone is turbulent, and hence, the Kolmogorov microscale provides a reasonable estimate of the segregation length in the mixed sheet [2]. The Kolmogorov microscale is an estimate of the smallest length scales in turbulent flows. At scales smaller than the Kolmogorov microscale, the flow is laminar, and therefore, diffusion of liquids at this scale provides the ultimate mechanism of mixing to the molecular level. The rate of micromixing is then limited by the time required for the components to diffuse to one another over a segregation length scale equal to the Kolmogorov microscale. The half-Kolmogorov length, 6, is given by, 6 = 0.5(U3/&,)“4
Substituting
eqn. (1) in eqn. (2) yields,
The smaller the Kolmogorov microscale of the turbulent flow, the faster the micromixing rate. For equal impinging sheets, eqn. (3) indicates that the mixing velocity. vY, and the kinematic viscosity, u, have the
231
E is the efficiency of pressure head conversion to kinetic energy by the impingement-sheet mixing device. Values of E range from 75580% over a pressure drop range of 0.25-3.0 bars and a viscosity range of 1-12 cP. The component velocities are, zi,Y= U, sin p
(7)
V IX
~
u,
cos
(8)
V2g
=
uz
sin D
V 2x
--
u2
cos
p
(9)
/3
(10)
The angle, IC,is the amount by which the mixed sheet is offset from the line bisecting the impingement angle, 2/L It is determined from the momentum balance:
Fig. 2. Geometry of unequal, impinging single sheets used for calculating the energy dissipation rate in the impingement zone.
tan ti = 0,/V,
influence on 6. The half-Kolmogorov thickness, however, is only weakly influenced by the single-sheet thickness at impingement. greatest
(11)
x direction:
Q, 01x+ Q,v,, = (Q, + Q&x
y direction:
Q, v~y-
Qzvzy= (Q, + Q&
rate for unequal sheets
P
(4)
P,V
q is the energy
dissipation rate in the impingement zone, P is the power lost due to the inelastic collision, m is the mass of the mixed sheet in the impingement zone, pm is the density of the mixed liquid, and V is the volume in the impingement zone. Figure 2 illustrates the impingement of two, unequal sheets at flow rates Q, and Q,, and pressure drops AP, and AP, respectively. For an inelastic collision, the power input to the- impingement zone is simply the difference between the kinetic energy/time of the mixeb sheet and the sum of the kinetic energy/time of both single sheets,
p=~~Q~~,2+~2Q,uz2 2
2
~rnQ,urn~ 2
(5)
(14)
The densities have been neglected since it is assumed that all liquids have the same density. In Fig. 2, in order to calculate the length of side AC, various angles must first be determined. Since line AD is parallel to line EB, the angle between ICand 7~must be equal to p.
Similar to equal, impinging sheets, the energy dissipation rate for the general case of unequal sheets can be estimated. The calculation involves determining what power/mass is lost in the impingement zone due to the collision, P .s+-zp m
(13)
The velocity of the mixed sheet, u,, is given by: U, = (6; + uy)OJ
Energy dissipation
(12)
7c=90”-p-w
(15)
r=90”-/!++
(16)
e=/?--rc
(17)
X=a+K
(18)
The thickness of each single sheet just prior to impingement is,
/
Q2
(20)
s2=(JG4A The sides BC and AB are given by, BC=s,/cos8
(21)
AB = YJ cos x
(22)
and,
where the subscripts m, 1 and 2 refer to the mixed sheet, single sheet No. 1, and single sheet No. 2, respectively. This analysis assumes that the densities of all liquids are equal. The velocities of each single sheet are calculated from the pressure drop using the following equation [ 31:
s,=AC=AB+BC
u = E(2 AP/p)“*
CD = s,(sin
(6)
(23)
where AC is equal to the mixed sheet thickness at the impingement zone. The length CD can be calculated since side AC (s,) and all angles of the triangle ACD are known, n/ sin 2fl)
(24)
232
The width of the impingement Ar =s,(sin
zone is given by,
7ccos S/sin 28)
(25)
Since in the experiments of this study, R 2 cc R, , the volume of the impingement calculated,
& z 4, and zone can be
V=#QR,S,AY
(26)
where s, and Ar are given by eqns. (23) and (25). Finally, for equal liquid densities and noting that Q,, = (bR,s,u,, the energy dissipation rate for unequal sheets is, E, =
%l
-
Q,u,'+
Q+z2-
Qmum*
Q, Ar
2
(27)
>
where U, and uZ are calculated from eqn. (6). The mixed sheet velocity, u,, is determined from eqn. (14). For the case of equal, single sheets impinged at an angle of 2p, eqn. (27) does reduce to eqn. (1) given earlier. For impinging, equal sheets, eqn. ( 1) directly reveals the effect of increasing sheet thickness (a) and ‘mixing velocity’ (a,) or pressure drop on ci. For unequal, impinging sheets, however, many operating conditions are possible, and the effects of various operating parameters on ei are not always apparent from eqn. (27). Figure 3 attempts to provide some perspective of the effect of volumetric flow rate ratio (CC)on the energy dissipation rate in the impingement zone of unequal
sheets. In Fig. 3 it is assumed that the pressure drop for both single sheets is equal. Wit is further assumed that the sum of the flow rates of both single sheets remains constant despite the variation in a, then the effect of CY on &i can be evaluated. Figure 3 does illustrate that Ed initially decreases rapidly with increasing a. At values of c( > 20, however, ci varies only slightly with increasing CL Figure 3 does show that the highest energy dissipation rates are achieved at volumetric flow rate ratios of 1 (note: values of CC4 1 in Fig. 3 are not possible since a general definition of c( is used in which the flow rate of any one of the two components is larger than the other).
Experimental The liquid-phase micromixing rate of unequal, impinging sheets was estimated using a test reaction whose product distribution is sensitive to the rate of diffusive mixing. The test reaction chosen was a series of competitive-consecutive, azo couplings [2, 61. The reaction between 1naphthol (A) and diazotized sulfanilic acid (B) proceeds as follows: kl A+B-R (28) k2 RfB-S (29) The product 2% X, = ~ CR+ 2cs
distribution,
X,, is given by, (30)
1200000
1000000
800000
0
0
5
10
15
20
25
30
35
40
45
!
a
Fig. 3. Variation of the calculated energy dissipation rare in the impingement zone for unequal, impinging sheets with the volumetric flow rate ratio. Fig. 3 assumes that the pressure drop of both single sheets is identical. Volumetric flow rates are in I min-‘, AP is in bar, and 6, has units of W kg-‘.
The quantities ca and cs are the concentrations of the monoazo dye (R) and the bisazo dye (S) respectively. The percentage of B in the form of S is given by the product distribution, X,, which was determined by spectrophotometric analysis after reaction was complete, and all B had been consumed. These azo couplings are competitive-consecutive reactions that are sensitive to diffusion effects (micromixing), and X, is a measure of the magnitude of these effects. The lower the relative value of X,, the faster the micromixing. However, the actual value of X, is dependent upon the relative rate constants of eqns. (28) and (29) and the concentrations of the reactants. A minimum value of X, is expected which corresponds to homogeneity of the reactants on the molecular scale before the second reaction has proceeded to any appreciable extent (for the reaction conditions discussed below, this value of X, is 0.0002). Figure 4 is an illustration of the experimental apparatus. The setup consists of the mixer as the primary unit, with auxilliary equipment of feed pumps, pressure gauges, rotameters, feed tanks, and a product holdup tank. A 500 litre tank was used to contain the l-naphthol, while a 50 litre tank was used for the diazotised
233
500 L Tank
50 L Tank
TO brain
Fig. 4. Experimental apparatus. CP, centrifugal pump; GP, gear pump; MS, mixed sheet; PG, pressure gauges; PT, product tank; R, rotameters.
sulfanilic acid. The diazo compound was pumped to the mixing device by an Ismatec gear pump capable of delivering a maximum flow rate of 5.5 1 min-’ at a maximum pressure drop of 4.0 bar. The coupler (l,naphthol) was pumped by a centrifugal pump capable of a maximum flow rate of 200 1 min-’ and a maximum pressure drop of 1.4 bar. Reactant flow rates were measured by rotameters that were calibrated using the ‘bucket and stopwatch’ weighing method. The reactants were pumped though the impingement-sheet mixer, formed into individual sheets, and impinged upon one another at a 30” angle in free space within the 500 litre product tank (details about the mixing device are given in ref. 1). After a certain distance from the mixing device (typically 20-30 cm) the mixed sheet broke up into droplets due to surface tension forces. The liquid fell to the bottom of the product tank and flowed to the drain. For all experiments the initial mole ratio of l-naphthol to diazotized sulfanilic acid was nominally 1.2. The rate constants for reactions 2X and 29 are approximately lo7 and 1900 1 mol-’ s-’ respectively. At a flow rate ratio of 6, the characteristic reaction times, which are based on the initial (k, cB,,-’ and (k,c,,)-’ concentration of B, were 0.004 and 20 ms respectively. At a flow rate ratio of 12, these characteristic reaction times were 0.002 and 10 ms, respectively. In contrast, a flow rate ratio of 1 yields characteristic reaction times of 0.02 and 100 ms respectively. Azo couplings were conducted at reactant viscosities of 1.0 CP and at 20 + 2 “C. Preparation of the reactants
as well as details on the spectrophotometric analysis of the products are given in ref. (7). The accuracy of the spectrophotometric analysis was about 0.005 at all values of 1,. Samples were collected in a small cylindrical vial that had a volume of IO cm3. As shown in Fig. 1, samples were taken at five different locations around the end of the mixed sheet or beyond it (distances equal to or greater than 20-30 cm, depending upon the experiment). At sampling distances (distance from the impingement zone) near the edge of the mixed sheet or beyond, no significant effect of sampling distance on the product distribution was noted (see Fig. 1). Hence, collection of the sample in a vial that stopped the forward momentum of the liquid did not affect the product distribution. It was previously observed that the distance from the impingement zone, at which samples were collected, did affect the value of the product distribution at increased viscosities but not at l.OcP [2]. The effect of unequal, reactant flow rates on the micromixing rate of impinging sheets was investigated at flow rate ratios (x = 9,/Q,) of 1.0, 6.0 and 12.0. The results at a flow rate ratio of 1.0 were taken from ref. 2. In all experiments, the nominal molar ratio of naphthol to diazotised sulphanilic acid was kept constant at 1.2, and the initial concentration of 1-naphthol was always 0.005 mol ll’. The corresponding initial concentrations of diazotized sulfanilic acid were 0.0042, 0.025 and 0.05 mol ll’ at flow rate ratios of 1.0, 6.0, and 12.0 respectively. The experimental parameters are presented in Table 1. Columns 2 and 3 of Table 1 list the volumetric flow rates of 1-naphthol (A) and diazotized sulfanilic acid (B) whereas columns 4 and 5 list the corresponding pressure drops. Column 7 lists the characteristic times, based on the initial concentration of B, of the second azo coupling reaction. Column 8 lists calculated values of the energy dissipation rate in the impingement zone as determined from eqn. (1) (equal, impinging sheets) or eqn. (27) (unequal, impinging sheets). Systematic
deviations
in the results
A mass balance check on the spectrophotometric analyses indicated that the right hand side of eqn. (31) was typically lo-20% less than the left hand side: CBO= CR+ 2cs
(31)
In the majority of the experimental samples, less B was found than was nominally fed to the impingement-sheet mixing device. It is important to note, however, that samples were collected at local positions at the edge of the mixed sheet (see Fig. 1). A single sample is, therefore, not necessarily representative of the mixture as a whole.
234 TABLE 1. Experimental d 1.0” 1.0” 1.0” 1.0” 1.O” 1.O” 6.0 60 6.0 6.0 12.0 12.0 12.0
parameters
QA (I min-‘)
Q, (1 mine’)
AP, (bar)
AP, (bar)
cBO(mol I-’ )
(&c,)
2.83 5.7 10.0 16.0 28.0 38.0
2.83 5.7 10.0 16.0 28.0 38.0
1.7 0.4 0.4 0.4 0.4 0.4
1.7 0.4 0.4 0.4 0.4 0.4
0.0042 0.0042 0.0042 0.0042 0.0042 0.0042
100 100 100 100 100 100
1.83 2.48 3.07 3.67 2.25 2.83 3.44
0.4 0.7 1.o 1.5 0.4 0.7 1.0
0.7 1.2 1.9 2.1 1.1 1.7 2.6
0 025 0.025 0.025 0.025 0.050 0.050 0.050
20 20 20 20 10 10 10
11.0 14.9 18.4 22.0 27.0 34.0 41.3
+(Wkg-‘)
-’ (ms)
401400 12500 9200 5700 3300 2400 14100 29200 56100 97500 6900 13300 24100
“from [2].
The thickness of the single sheets is not uniform and typically varies by approximately 20-30%) across the sheet spread angle [4]. This variation, however, is not random. The single sheets are generally thinner in the center than at the side free edges. Since the stream velocity of the single or mixed sheets is constant [3], thinner sections of a sheet have lower volumeric flow rates. Because the liquid in the mixed sheet is essentially in plug flow, any deviation in the local flow rate ratio at the impingement zone from the nominal flow rate ratio fed will affect the equality of eqn. (31). Further, the physical size differences between unequal sheets appears to influence the amount of B found in the spectrophotometric analyses. Thicker sheets are less affected by surface tension at the side free edges, and are probably more uniform in thickness than thinner sheets. For the unequal flow rate experiments, the lnaphthol (A) sheet was approximately 6 or 12 times thicker than the diazotized sulfanilic acid (B) sheet. Since the variation in the thickness of the B sheet was greater than that of the A sheet, it is not suprising that various amounts of B were found in the samples. Figure 5 plots the amount of B found by the spectrophotometric analysis as a function of angular sampling position at a volumetric flow rate ratio of 6. The curves shown are simple second degree polynomial fits to the data points. These curves clearly indicate that the lowest amount of B was usually found in samples taken at the center line (O” position). The amount of B found (c,,) generally increased as samples were collected at angular positions away from the center line towards the side free edges. Similar results were also obtained at a volumetric flow rate ratio equal to twelve. For the equal, flow rate experiments, however, the right hand and left hand sides of eqn. (31) were typically within +5X of each other [2]. Even though the single sheet thicknesses were not exactly uniform at impingement, both sheets were identical to one another.
1.00
0.95
0.90
0.85 s em
0.60
0" 0.75 0.70 0
0.65 0.60-r -1 60
-40
Angular
-20
0
Sampling
20
40
t3
Position
of Fig. 5. Variation of cBh/cBO as a function (see top view of Fig. I). Curves are second fits to rhe experimental data at constant rate. n 14 100 W kg-‘, 0 29 200 W kg-‘, I$ 97 500 W kg-‘.
sampling position order polynomial energy dissipation 56 100 W kg-l, 0
Any variation in thickness of one single sheet was matched by the other single sheet. Although the local flow rate within a single sheet may have varied by as much as 30%, this was true for both single sheets at identical positions. Thus, the local flow rate ratio was found to be within f 5% of the nominal flow rate ratio. Figures 6 and 7 plot the experimental values of the product distribution as a function of the concentration of B found in the spectrophotometric analysis (c,,) relative to the nominal concentration of B fed (ceo). At 01= 6 and 12, the values of X, increase as cBa/cB,, increases. At values of cBa/cBO less than 1.0, the local
235 0.14
0.12
0.10
cA~
cAo
cAo
0.00
X
A
S 0.06
(a)
t=o 0.04
I
I
FL._
I
I I
I
0.02
I I
0.00 0
‘Ao
I
0.60
I
o.io
oh0
oil0
1
-
1
Fig. 6. Variation in Fig. 5.
of ,I’, with c,,/c,,
A
at a = 6. Symbol
key is given
I I
I I
I I
I
=BS ‘=Bo
I
IS
A I
4
*
(b)
(1 +a)6
I I B
B
Fig. 8. Illustration of mixing model for unequal reactant flow rates. Figure (a) depicts diffusion into slabs of equal thickness where there are E or six times as many slabs of A as B. Figure (b) illustrates the modelling simplification where slab A is CLtimes as thick as slab B.
identical. Since NAO/NsO is usually held constant when modelling experimental micromixing results, values of X, at a corresponding abscissa values of 1.0 were used. These values of X, will be designated as X3. For u = 6, these values were extrapolated from Fig. 6, whereas in Fig. 7, values of X$ were interpolated.
0.08 1 0.06-
Modelling
0.04. 0.02-
0.00’ 0.50
0.60
0.70
0.80
0.90
1.00
1 .lC
%a‘=eo Fig. 7. Variation of X, with c~./c~,, at a = 12. q 6900 W kg-‘, 13 300 W kg’, x 24 100 W kg’.
0
molar ratio, (NAO/NBO)local, is greater than the nominal value of 1.2. It is generally the case that at constant energy dissipation rate, X, decreases with increasing N/WIN,,; a result that is confirmed in Figs. 6 and 7. The curves shown in Figs. 6 and 7 are linear fits to the data points at the different energy dissipation rates listed in Table 1. Although the curves are approximately parallel in each figure, the slopes at tc = 6 and 12 are not
of mixing results at unequal flow rates
Figure 8 illustrates the mixing model chosen. The model is similar to the one used earlier to explain the azo-coupling results at flow rate ratios of 1.0 [2]. Diffusion of reactants is assumed to occur within slabs of fixed thickness, 2L. Figure 8(a) depicts a situation where B is mixed with six times as much A. Since the energy dissipated does not differentiate between reactants, the size of the slabs of A and B are equal, but there are six times as many slabs of A. The question is, how are. these slabs distributed among each other? For the equal flow rate case, the slabs of A and B were assumed to alternate with no two slabs of A (or B) ever next to one another [2]. For the unequal flow rate case, the assumption is made that the slabs of A and B are distributed as given by the ratio of their volumetric flow rates. In Figure S(a) six slabs of A surround a single
236 TABLE 2. Model parameters CL 1.O” 1.O”
1.O”
1.O” 1.O” 1.O’ 6.0 6.0 6.0 6.0 12.0 12.0 12.0
QA (I min-‘)
Qe (1 min-‘)
Ei (W kg-‘)
P/D (IIIS)~
M
M/(1+ 4
S'(l+ a)‘/4D
2.83 5.1 10.0 16.0 28.0 38.0
2.83 5.7 10.0 16.0 28.0 38.0 1.83 2.48 3.07 3.67 2.25 2.83 3.44
40 1400 12500 9200 5700 3300 2400 14100 29200 56100 97500 6900 13300 24100
0.5
0.005
2.7 .3.2 4.0 5.3 6.2 2.6
0.027 0.032 0.040 0.053 0.062 0.128 0.089 0.064 0.049 0.366 0.264 0.196
0.0025 0.0135 0.0160 0.0200 0.0265 0.03 10 0.0183 0.0127 0.00914 0.00700 0.0282 0.0203 0.0151
0.5 2.7 3.2 4.0 5.3 6.2 31.9 22.1 15.9 12.3 156.3 109.0 84.5
11.o 14.9 18.4 22.0 27.0 34.0 41.3
I .8 1.3 1.o 3.7 2.6 2.0
“from [2]. bDiffuslve mixing time for CI= 1 only. For c( > 1, d2/Dsignificantly
underestimates
slab of B on both sides. Figure 8(b) shows that for modelling purposes, the effective thickness of the A slab can be taken as six times that of the B slab. For other volumetric flow rate ratios, the ratio of slab thicknesses would be equal to CL The unsteady-state equations for diffusion and reaction were previously given in ref. 8; the case presented here is simpler since the assumption is made that the slab thickness remains constant. The analysis is a moving boundary problem with initial concentrations of cAO and csO at t =O. The experimental data were non-dimensionalized as the ratio of a characteristic time scale of the diffusion process to a characteristic time scale of the second azo-couple reaction,
the diff‘usive mixing time
0.25-
0.20-
x;
0.15.
O.lO/ 0.05-
h4 = k,c,,L2/D
(32)
where M is the mixing modulus, k, is the rate constant of the second azo-coupling reaction, cn,, is the initial concentration of diazotized sulfanilic acid, and L is a representative half-slab thickness over which diffusion occurs. Under turbulent flow conditions in the impingement zone, L is equal to 6, the half-Kolmogorov thickness [2]. The modelling parameters are listed in Table 2. The flow rates, concentrations, and pressure drops from Table 1 are given as input data. For an unequal flowrate experiments, the energy dissipation rate was calculated from eqn. (27). For the experiments conducted at equal flow rates, eqn. (1) was used to calculate ai. Since all experiments conducted at a viscosity of 1.0 cP, where the Reynolds number in the impingement zone was high, the half-slab thickness for diffusion was equal to the half-Kolmogorov thickness, 6, and was calculated from eqn. (2) [2]. When compared with the M values at a flow rate ratio of 1.0, the M values for the unequal flow rate experiments are substantially larger.
/ o.oo! 1o-3
.A.
.,....I
A 10-z
M -= 1 +a
,,,,,,I
“1’
10-l
-4 100
W,3 D(l + a)
Fig. 9. Variation of X: with M/( 1 + a). Normalizing M by the factor 1 + a directly reveals the increase in product distribution as the volumetric flow rate ratio is increased. ( A) c( = 1, ( v) G(= 6,
(C) a = 12.
The difference results primarily from the higher initial concentration of B in the unequal flow rate cases. It was previously observed that plots of X, vs. M do not directly reveal the effect of 0: on X, [7]. At constant initial stoichiometric ratio, NAO/N,,, M does not stay constant as Y is increased, but instead increases directly with cBD(NAo/NBo = c~,_,~/c~o and M - cBo). Plots of X, vs. M/( 1 + cc), however, directly reveal the effect of volumetric flow rate ratio on the product distribution. Column 7 of Table 2 lists values of M/( 1 + a). Figure 9
237
plots values of XL extrapolated from Fig. 6 and interpolated from Fig. 7 at cn,/cn,, = 1.0 as a function of M( 1 + CL). The curves shown are for the fixed-slab model at values of CI= 1. 6 and 12. The results confirm that at constant values of M/( 1 + a), XL increases as u increases. When CI= I the fixed-slab model was previously shown to provide a good fit to the experimental data for equal, impinging sheets [2]. In Fig. 9, however, it is clear that, at c( = 6 and 12, the fixed-slab model does not provide as good a fit to the data. When c( = 6, the model underestimates experimental values of the product distribution, whereas when CY= 12, the model substantially overestimates experimental values The modelling strategy used previously and in this study attempts to utilize the simplest model possible to explain the results. Previous results from ref. 2 demonstrated that the simple, fixed-slab, diffusion model is applicable to impingement-sheet mixing of liquids at flow rate ratios of 1. The results from this investigation, however, illustrate that the simple, fixed-slab model is inadequate for the modelling of unequal, impinging sheets. It would appear from Fig. 9, that the larger the value of CI,the poorer the fit of the fixed-slab model to the experimental data. Probably, a more complex model that is truly based on physical considerations of turbulence, such as that given in ref. 8, would provide a better fit. The inadequacies of the fixed-slab model for the correlation of unequal, impinging sheets notwithstanding, a major focus of this work was to compare diffusive mixing times of unequal sheets with those of equal sheets. The quantity, fi2/D, was previously shown to provide a good estimate of the micromixing time of equal, impinging sheets under conditions where the impingement zone was in turbulent flow [2]. For unequal sheets, however, d2/D severely underestimates the diffusive mixing time. Nevertheless, it may be possible to estimate the micromixing time for unequal, impinging sheets by modifying the mixing modulus so that the fixed-slab model for equal, impinging sheets can be applied to unequal, impinging sheets. The time required for diffusive mixing ( > 90% completion) in a slab, containing both A and B, is approximately given by: t, - L’ID
(33)
where L is the half-slab thickness. For the case of equal, impinging sheets, L is equal to the half-Kolmogorov thickness, 6. The unequal, impinging sheet case is illustrated in Fig. 8(b). When NA,,/NBO- 1, diffusion of A and B will occur over a slab width of approximately (1 + x)S. Figure S(b) suggests that instead of setting L in eqn. (33) equal to 6 for approximating the micro-
0.154
bl/( 1+a) = k, %a (1+a) &4D Fig. 10. Variation of XL with A?/( 1 + a). This modified value allbws the unequal, impinging sheet experimental data to be roughly fitted by the equal, impinging sheet model. The half-slab thickness for diffusion used for the unequal, impinging sheet data is not equal to 6 but is instead equal to ( 1 + c()6/2. Symbol key is given in Fig. 9. mixing time for unequal. estimate should be, L
=
impinging
sheets,
a better
( 1 + 46 2
so that, (35) Column 8 of Table 2 lists values of the micromixing time based on (1 + ~)~6~/4D. Note, that when c( = 1, this expression reduced to 6’/D. At u = 6 and 12, values of the micromixing time calculated from (1 + ~)~a*/40 are significantly greater than those calculated from 6*/D. At CI= 6, the difference is approximately an order of magnitude, whereas at c( = 12, the difference is about a factor of 40. Do the experimental results justify using eqn. (35) as an estimate for the micromixing time of unequal, impinging sheets? Figure 10 plots X: against a modified mixing modulus, n?/( 1 + a). The curve shown is the simple, fixed-slab model with do= 1 (equal, impinging sheets). In Fig. 10 the slab half-thickness, L, is 6( 1 + LX)/ 2 and not 6.
G/( 1 + CL)=
D(l +tl)
= k,c,,( 1 + a)6’/4D
(36)
238
When a = 6 the fit of the equal, fixed-slab model to the experimental data is not superior to that of the unequal, fixed-slab model shown in Fig. 9. When a = 12, however, the fit is superior. More importantly, Fig. 10 allows for a direct comparison between micromixing times for equal and unequal impinging sheets. The diffusive mixing or micromixing time can be roughly but rapidly estimated using the following expression: 4Li ~ = -0.4917 In (1 +a)2
In E, - 1.2621
where td has units of seconds and .si has units of W kg-‘. Equation (37) is obtained from a plot of column 8 in Table 2 against column 4 in Table 2. Since the values of td were directly calculated from sit the regression coefficient for this fit is theoretically unity. However, since only four significant figures are shown, the values calculated from eqn. (37) deviate slightly from those listed in column 8. The energy dissipation rate in the impingement zone is calculated from eqn. (27) or eqn. ( 1). Equation (37) is valid for both unequal and equal impinging sheets under turbulent flow conditions in the impingement zone and for NAO/NBON 1. Finally, the diffusive mixing times of impinging sheets at various flow rate ratios and at constant energy dissipation rates (si) can be compared using the following expression, td2 -_=
c1
+
a2>2
tdl
c1
+a,)2
When x, = 1 and cc2= 6, eqn. (38) indicates that the diffusive mixing time of the unequal, impinging sheets will be a factor of 12.3 greater than that of the equal, impinging sheets. When or, and a2 equal 6 and 12 respectively, the ratio of diffusive mixing times is approximately equal to 3.5.
Conclusions
An expression was derived for the energy dissipation rate in the impingement zone (E,) of unequal, impinging sheets. Similar to the previously derived expression for ci of equal, impinging sheets, this expression is based upon the geometry of the impinging, unequal sheets and the momentum balance. Under similar operating conditions, a comparison of Ei values at different volumetric flow rate ratios (a) indicates a rapid decrease in si until a - 20, beyond which the rate of decrease in E, drops off markedly. Attempts were made to correlate the product distribution values of the azo couplings to a mixing modulus using a simple micromixing model based on diffusion of the reactants in fixed slabs. Although this model was
previously shown to provide adequate correlation when a = 1 (equal, impinging sheets), modifying the model to predict product distributions for a = 6 and 12 resulted in calculated values of the product distribution that were in poor agreement with experimental values. However, it was possible to estimate the micromixing time for unequal, impinging sheets by using the fixed-slab model at CI= 1 and modifying the mixing modulus. The diffusive mixing time in the mixing modulus was redefined as (1 + a)262/4D instead of ~3~10, as is the case for equal, impinging sheets. At constant diffusion coefficient and energy dissipation rate in the impingement zone, the micromixing time of unequal, impinging sheets is a factor of (1 + a)2/4 greater than the micromixing time for equal, impinging sheets. For the inelastic collision of impinging sheets, the major parameters affecting the speed of micromixing are the energy dissipation rate in the impingement zone and the volumetric flow rate ratio. Increasing energy dissipation rates decrease the micromixing time, whereas increasing volumetric flow rate ratios increase the micromixing time.
Nomenclature
initial concentration of A, mol 1-l equivalent concentration of B found in the specCl&, trophotometric analysis of R and S, mol I-’ of B, mol 1-l cl30 initial concentration of monoazo dye product, mol 1-l CR concentration concentration of bisazo dye product, mol 1-l diffusion coefficient, m2 s-r E efficiency of pressure head conversion to kinetic energy in impingement-sheet mixing device k, rate constant for first azo coupling reaction, 1 mol-’ s-’ k, rate constant for second azo coupling reaction, 1 molC’ s’ L characteristic length of diffusion or half-thickness of slab, /*m m mass flow rate, kg SS’ M mixing modulus; ratio of diffusion to reaction times M modified mixing modulus which essentially estimates the diffusive mixing time for unequal impinging sheets as if they were equal, impinging sheets. N A0 nominal molar amount of reactant A, mol NBO nominal molar amount of reactant B, mol P power dissipated at the impingement zone, W AP pressure drop used to produce single liquid sheet flow rate of reactant A. 1 min’ flow rate of reactant B, 1 min-’ iit em total liquid flow rate of mixed sheet equal to Q, + Q2, lmin-’ cAO
;;’
239
e
Ar R s
si 4n
SI s2
td
flow rate of single sheet, 1 min’ width of impingement zone, pm radial distance of single sheet to the impingement zone, m sheet thickness single sheet thickness, for equal impinging sheets, at impingement zone, pm mixed sheet thickness at impingement zone, pm thickness of single sheet No. 1 at impingement zone, pm thickness of single sheet No. 2 at impingement zone, pm characteristic time for diffusive mixing or micromixing, s velocity of single sheet, m SC’ velocity of mixed sheet, m SC’ x-component velocity of single sheet, m s-’ x-component velocity of mixed sheet, m s-’ mixing velocity; y-component velocity of single sheet, m SC’ y-component velocity of mixed sheet, m SC’ volume of impingement zone, m3 product distribution of azo coupling test reaction extrapolated or interpolated product distribution when csa /cBo = 1.
Greek letters
i 6 E K
volumetric flow rate ratio half-angle of impingement half-Kolmogorov thickness, pm energy dissipation rate, W kg-’ angle by which the mixed sheet is offset from the line bisecting the impingement angle
4 P An I)
spreading angle of thin liquid sheet liquid density, kg mm3 liquid density of mixed sheet, kg me3 kinematic viscosity, m2 s-’
Subscripts
1 2 m
single sheet No. 1 single sheet No. 2 mixed sheet
References 1 R. J. Demyanovich, Liquid mixing employing expanding, thinning liquid sheets, U.S. Pat. 4 735359, April 5, 1988. 2 R. J. Demyanovich and J. R. Bourne, Rapid micromixing by the impingement of thin liquid sheets. 2. Mixing study, Ind. Eng. Chem. Res., 28, (1989) 830. 3 R. J. Demyanovich and J. R. Bourne, Rapid micromixing by the impingement of thin liquid sheets. I. A photographic study of the flow pattern, Ind. Eng. Chem. Res., 28 (1989) 825. 4 N. Dombrowski and R. P. Fraser, A photographic investigation into the disintegration of liquid sheets, Phi/. Truns. R. sot., 247A (1954) 101. 5 N. Dombrowski and P. C. Hooper, The performance characteristics of an impinging jet atomizer in atmospheres of high ambient density, Fuel, 41 (1962) 323. 6 J. R. Boume, F. Kozicki and R. Rys, Mixing and fast chemical reaction I: Test reactions to determine segregation, Chem. Eng. Sci., 36 (1981) 1643. 7 J. R. Bourne, C. Hilber and G. Tovstiga, Kinetics of the azo coupling reactions between I-naphthol and diazotised sulphanilic acid, C/rem. Eng. Commun., 37 (1985) 293. 8 J. Baldyga and J. R. Bourne, A fluid mechanical approach to turbulent mixing and chemical reaction, Chem. Eng. Commun., 28 (1984) 231.
229
Processing, 31 ( 1992) 229- 239
Impingement-sheet
mixing
of liquids
Robert J. Demyanovich”
and John R. Bourne
at unequal
flow rates
Technisch-Chemisches Laboratorium, ETH, 8092 Zurich (Switzerlund) (Received
December
2, 1991; in final form March
9. 1992)
Abstract Mixing of liquids by the impingement of two, thin sheets at unequal flow rates was studied using competitive, consecutive azo couplings that provide quantitative information about the micromixing. Liquid flow rate ratios of 6.0 and 12.0 were investigated at total flow rates ranging from 13-45 I min-‘. An expression for the energy dissipation rate in the impingement zone resulting from the inelastic collision was derived from the momentum balance and the geometry of the impinging sheets. For the experimental data, energy dissipation rates were calculated to lie in the range 6 900-97 500 W kg-‘. The major parameters affecting the micromixing rate of unequal, impinging sheets are the energy dissipation rate in the impingement zone (c,) and the volumetric flow rate ratio (x). Previously, the micromixing results for equal, impinging sheets were explained by a simple model assuming diffusion of the reactants within slabs of fixed thickness equal to the Kolmogorov microscale. Attempts to fit the experimental data for unequal, impinging sheets to a similar model, modified to account for volumetric flow rate ratio, resulted in a relatively poor correlation of the experimental results. However, a modification of the mixing modulus, to account for the effects of E on the diffusion length, yielded a reasonable correlation of the experimental data for unequal, impinging sheets to the model for equal, impinging sheets. Consequently, an expression was derived that relates the micromixing time for both equal and unequal, impinging sheets to a and E,.
Introduction
The impingement of thin sheets of liquids has been shown to yield rapid mixing of relatively large flow rates of liquids [ 11. For liquids with viscosities of 1.O cP, Demyanovich and Bourne have obtained diffusive mixing or micromixing times as low as 1 millisecond for flow rates up to 6 1 min-‘, and micromixing times of the order of 10 milliseconds for flow rates up to 76.0 1 min-’ [2]. All previous work with impingementsheet mixing, however, has been cohducted at equal flow rates for both liquids. Since the majority of liquid mixing operations on the industrial scale do not occur at a flow rate ratio of 1.0, experiments were conducted at different flow rate ratios in order to estimate the effect on the micromixing time for impinging sheets. Equal impinging sheets Since the impingement of liquid sheets has already been discussed in detail [ 1, 31, only a brief summary of the method will be provided here. Tt has been previ-
*Author to whom correspondence should be addressed at LiquiSheet Technologies, Inc., 4421 Gilbert St., Suite 204, Oakland, CA 9461 I, U.S.A.
0255-2701/92/$5.00
ously shown that within any sheet, liquid flows from the sheet origin outward along radial lines with no crossover of liquid between adjacent radial lines [3,4]. Once the sheet is formed in free space, the radial velocity of the liquid within the sheet remains constant up to ambient gas pressures of the order of 20 bar [5]. Since the drag of the surrounding gas on the free sheet was found to be negligible, there is no external force to reduce the sheet velocity. The equation of flow within the sheet is Q = q5R.w. Since the volumetric flow rate (Q) and the flow velocity (u) are constant, the area normal to flow (4fi) must also be constant. Since fi is constant, the sheet thickness is inversely related to its distance from the sheet origin. The sheet continues to thin until inertial forces are overcome by surface tension forces resulting in its breakup into droplets. Liquid sheets with flow rates of the order of several 1 min- ’ typically have residence times of the order of 5-10 milliseconds before they break up into droplets. Figure I illustrates the case of two, equal, single, fan-shaped sheets impinging at a contact angle of 2g. Before breakup of the single sheets into droplets, these sheets are impinged on one another to produce a mixture of the liquids that is also in the form of a thin continuous sheet. As shown in Fig. I, if the angle of impingement is acute (preferably less than 60”), the
CI 1992 ~ Elsevier Sequoia.
All rights reserved
230
\
R,
/-
Side View
Impingement Zone
Top View
unit mass of the liquids. Normally, E is difficult to measure because the mass in which the power is input cannot be accurately estimated. Often, the energy dissipation rate is determined after modelling the results of mixing experiments. Since the geometry of impinging sheets is well-defined, the energy dissipation rate resulting from impingement can be estimated. The calculation involves determining what power/mass is lost due to the inelastic collision of the sheets. The energy dissipation rate resulting from the inelastic impingement of two, equal, single sheets was previously given in ref. 2 as, v 3 cos p 6.I = Y 24
Fig. 1. Impingement of equal, single sheets at an angle of 2p. Top view illustrates sampling around the mixed sheet.
sheets are not destroyed at the impingement zone, but instead combine to form a mixed sheet. The impingement process can be described by momentum balances in the x and y directions. The nature of the collision (inelastic or elastic) was previously found to be a function of the angle of impingement [3]. At impingement angles less than 60”, the collision was shown to be inelastic with respect to the kinetic energy of the single sheets [3]. Hence, the kinetic energy associated with the single sheet II-velocity component, u,,, is released into the thin liquid sheets producing high rates of energy dissipation (> 105W kg-‘). High energy dissipation rates are generally required for the rapid mixing of two liquid streams. Micromixing
When two liquid streams are brought together so that some degree of turbulence is generated, small liquid elements called eddies or laminae are formed that are set in bulk motion. The average size of these eddies, known as the segregation length (L) of the mixture, depends on the liquid viscosity and on the intensity of the turbulence. The micromixing time (L’/D) of the two liquid streams is given by the time required for the components to diffuse over a distance equal to the segregation length. The energy dissipation rate (E) provides a measure of the intensity of the turbulence that is produced. The energy dissipation rate is related to the power input per
(1)
where &i is the energy dissipation rate in the impingement zone, u,, is the “mixing velocity”, and p is the half angle of impingement. This expression is entirely determined by the momentum balance and the geometry of the impinging sheets (see Fig. 1). Equation ( 1), however is only valid in the impingement zone. As the liquid flows away from the impingement zone, si decays rapidly since no additional energy is dissipated elsewhere in the mixed sheet. The rate of micromixing is limited by molecular diffusion between the dispersed eddies of the size of the segregation length. If the overall flow is laminar, the segregation length will be given by a size scale characteristic of the laminar flow (for impinging sheets this size scale would be the sheet thickness, s). If instead the flow is turbulent, the segregation length will be much smaller. For liquids with a viscosity of 1.0 cP, the Reynolds number in the impingement zone is turbulent, and hence, the Kolmogorov microscale provides a reasonable estimate of the segregation length in the mixed sheet [2]. The Kolmogorov microscale is an estimate of the smallest length scales in turbulent flows. At scales smaller than the Kolmogorov microscale, the flow is laminar, and therefore, diffusion of liquids at this scale provides the ultimate mechanism of mixing to the molecular level. The rate of micromixing is then limited by the time required for the components to diffuse to one another over a segregation length scale equal to the Kolmogorov microscale. The half-Kolmogorov length, 6, is given by, 6 = 0.5(U3/&,)“4
Substituting
eqn. (1) in eqn. (2) yields,
The smaller the Kolmogorov microscale of the turbulent flow, the faster the micromixing rate. For equal impinging sheets, eqn. (3) indicates that the mixing velocity. vY, and the kinematic viscosity, u, have the
231
E is the efficiency of pressure head conversion to kinetic energy by the impingement-sheet mixing device. Values of E range from 75580% over a pressure drop range of 0.25-3.0 bars and a viscosity range of 1-12 cP. The component velocities are, zi,Y= U, sin p
(7)
V IX
~
u,
cos
(8)
V2g
=
uz
sin D
V 2x
--
u2
cos
p
(9)
/3
(10)
The angle, IC,is the amount by which the mixed sheet is offset from the line bisecting the impingement angle, 2/L It is determined from the momentum balance:
Fig. 2. Geometry of unequal, impinging single sheets used for calculating the energy dissipation rate in the impingement zone.
tan ti = 0,/V,
influence on 6. The half-Kolmogorov thickness, however, is only weakly influenced by the single-sheet thickness at impingement. greatest
(11)
x direction:
Q, 01x+ Q,v,, = (Q, + Q&x
y direction:
Q, v~y-
Qzvzy= (Q, + Q&
rate for unequal sheets
P
(4)
P,V
q is the energy
dissipation rate in the impingement zone, P is the power lost due to the inelastic collision, m is the mass of the mixed sheet in the impingement zone, pm is the density of the mixed liquid, and V is the volume in the impingement zone. Figure 2 illustrates the impingement of two, unequal sheets at flow rates Q, and Q,, and pressure drops AP, and AP, respectively. For an inelastic collision, the power input to the- impingement zone is simply the difference between the kinetic energy/time of the mixeb sheet and the sum of the kinetic energy/time of both single sheets,
p=~~Q~~,2+~2Q,uz2 2
2
~rnQ,urn~ 2
(5)
(14)
The densities have been neglected since it is assumed that all liquids have the same density. In Fig. 2, in order to calculate the length of side AC, various angles must first be determined. Since line AD is parallel to line EB, the angle between ICand 7~must be equal to p.
Similar to equal, impinging sheets, the energy dissipation rate for the general case of unequal sheets can be estimated. The calculation involves determining what power/mass is lost in the impingement zone due to the collision, P .s+-zp m
(13)
The velocity of the mixed sheet, u,, is given by: U, = (6; + uy)OJ
Energy dissipation
(12)
7c=90”-p-w
(15)
r=90”-/!++
(16)
e=/?--rc
(17)
X=a+K
(18)
The thickness of each single sheet just prior to impingement is,
/
Q2
(20)
s2=(JG4A The sides BC and AB are given by, BC=s,/cos8
(21)
AB = YJ cos x
(22)
and,
where the subscripts m, 1 and 2 refer to the mixed sheet, single sheet No. 1, and single sheet No. 2, respectively. This analysis assumes that the densities of all liquids are equal. The velocities of each single sheet are calculated from the pressure drop using the following equation [ 31:
s,=AC=AB+BC
u = E(2 AP/p)“*
CD = s,(sin
(6)
(23)
where AC is equal to the mixed sheet thickness at the impingement zone. The length CD can be calculated since side AC (s,) and all angles of the triangle ACD are known, n/ sin 2fl)
(24)
232
The width of the impingement Ar =s,(sin
zone is given by,
7ccos S/sin 28)
(25)
Since in the experiments of this study, R 2 cc R, , the volume of the impingement calculated,
& z 4, and zone can be
V=#QR,S,AY
(26)
where s, and Ar are given by eqns. (23) and (25). Finally, for equal liquid densities and noting that Q,, = (bR,s,u,, the energy dissipation rate for unequal sheets is, E, =
%l
-
Q,u,'+
Q+z2-
Qmum*
Q, Ar
2
(27)
>
where U, and uZ are calculated from eqn. (6). The mixed sheet velocity, u,, is determined from eqn. (14). For the case of equal, single sheets impinged at an angle of 2p, eqn. (27) does reduce to eqn. (1) given earlier. For impinging, equal sheets, eqn. ( 1) directly reveals the effect of increasing sheet thickness (a) and ‘mixing velocity’ (a,) or pressure drop on ci. For unequal, impinging sheets, however, many operating conditions are possible, and the effects of various operating parameters on ei are not always apparent from eqn. (27). Figure 3 attempts to provide some perspective of the effect of volumetric flow rate ratio (CC)on the energy dissipation rate in the impingement zone of unequal
sheets. In Fig. 3 it is assumed that the pressure drop for both single sheets is equal. Wit is further assumed that the sum of the flow rates of both single sheets remains constant despite the variation in a, then the effect of CY on &i can be evaluated. Figure 3 does illustrate that Ed initially decreases rapidly with increasing a. At values of c( > 20, however, ci varies only slightly with increasing CL Figure 3 does show that the highest energy dissipation rates are achieved at volumetric flow rate ratios of 1 (note: values of CC4 1 in Fig. 3 are not possible since a general definition of c( is used in which the flow rate of any one of the two components is larger than the other).
Experimental The liquid-phase micromixing rate of unequal, impinging sheets was estimated using a test reaction whose product distribution is sensitive to the rate of diffusive mixing. The test reaction chosen was a series of competitive-consecutive, azo couplings [2, 61. The reaction between 1naphthol (A) and diazotized sulfanilic acid (B) proceeds as follows: kl A+B-R (28) k2 RfB-S (29) The product 2% X, = ~ CR+ 2cs
distribution,
X,, is given by, (30)
1200000
1000000
800000
0
0
5
10
15
20
25
30
35
40
45
!
a
Fig. 3. Variation of the calculated energy dissipation rare in the impingement zone for unequal, impinging sheets with the volumetric flow rate ratio. Fig. 3 assumes that the pressure drop of both single sheets is identical. Volumetric flow rates are in I min-‘, AP is in bar, and 6, has units of W kg-‘.
The quantities ca and cs are the concentrations of the monoazo dye (R) and the bisazo dye (S) respectively. The percentage of B in the form of S is given by the product distribution, X,, which was determined by spectrophotometric analysis after reaction was complete, and all B had been consumed. These azo couplings are competitive-consecutive reactions that are sensitive to diffusion effects (micromixing), and X, is a measure of the magnitude of these effects. The lower the relative value of X,, the faster the micromixing. However, the actual value of X, is dependent upon the relative rate constants of eqns. (28) and (29) and the concentrations of the reactants. A minimum value of X, is expected which corresponds to homogeneity of the reactants on the molecular scale before the second reaction has proceeded to any appreciable extent (for the reaction conditions discussed below, this value of X, is 0.0002). Figure 4 is an illustration of the experimental apparatus. The setup consists of the mixer as the primary unit, with auxilliary equipment of feed pumps, pressure gauges, rotameters, feed tanks, and a product holdup tank. A 500 litre tank was used to contain the l-naphthol, while a 50 litre tank was used for the diazotised
233
500 L Tank
50 L Tank
TO brain
Fig. 4. Experimental apparatus. CP, centrifugal pump; GP, gear pump; MS, mixed sheet; PG, pressure gauges; PT, product tank; R, rotameters.
sulfanilic acid. The diazo compound was pumped to the mixing device by an Ismatec gear pump capable of delivering a maximum flow rate of 5.5 1 min-’ at a maximum pressure drop of 4.0 bar. The coupler (l,naphthol) was pumped by a centrifugal pump capable of a maximum flow rate of 200 1 min-’ and a maximum pressure drop of 1.4 bar. Reactant flow rates were measured by rotameters that were calibrated using the ‘bucket and stopwatch’ weighing method. The reactants were pumped though the impingement-sheet mixer, formed into individual sheets, and impinged upon one another at a 30” angle in free space within the 500 litre product tank (details about the mixing device are given in ref. 1). After a certain distance from the mixing device (typically 20-30 cm) the mixed sheet broke up into droplets due to surface tension forces. The liquid fell to the bottom of the product tank and flowed to the drain. For all experiments the initial mole ratio of l-naphthol to diazotized sulfanilic acid was nominally 1.2. The rate constants for reactions 2X and 29 are approximately lo7 and 1900 1 mol-’ s-’ respectively. At a flow rate ratio of 6, the characteristic reaction times, which are based on the initial (k, cB,,-’ and (k,c,,)-’ concentration of B, were 0.004 and 20 ms respectively. At a flow rate ratio of 12, these characteristic reaction times were 0.002 and 10 ms, respectively. In contrast, a flow rate ratio of 1 yields characteristic reaction times of 0.02 and 100 ms respectively. Azo couplings were conducted at reactant viscosities of 1.0 CP and at 20 + 2 “C. Preparation of the reactants
as well as details on the spectrophotometric analysis of the products are given in ref. (7). The accuracy of the spectrophotometric analysis was about 0.005 at all values of 1,. Samples were collected in a small cylindrical vial that had a volume of IO cm3. As shown in Fig. 1, samples were taken at five different locations around the end of the mixed sheet or beyond it (distances equal to or greater than 20-30 cm, depending upon the experiment). At sampling distances (distance from the impingement zone) near the edge of the mixed sheet or beyond, no significant effect of sampling distance on the product distribution was noted (see Fig. 1). Hence, collection of the sample in a vial that stopped the forward momentum of the liquid did not affect the product distribution. It was previously observed that the distance from the impingement zone, at which samples were collected, did affect the value of the product distribution at increased viscosities but not at l.OcP [2]. The effect of unequal, reactant flow rates on the micromixing rate of impinging sheets was investigated at flow rate ratios (x = 9,/Q,) of 1.0, 6.0 and 12.0. The results at a flow rate ratio of 1.0 were taken from ref. 2. In all experiments, the nominal molar ratio of naphthol to diazotised sulphanilic acid was kept constant at 1.2, and the initial concentration of 1-naphthol was always 0.005 mol ll’. The corresponding initial concentrations of diazotized sulfanilic acid were 0.0042, 0.025 and 0.05 mol ll’ at flow rate ratios of 1.0, 6.0, and 12.0 respectively. The experimental parameters are presented in Table 1. Columns 2 and 3 of Table 1 list the volumetric flow rates of 1-naphthol (A) and diazotized sulfanilic acid (B) whereas columns 4 and 5 list the corresponding pressure drops. Column 7 lists the characteristic times, based on the initial concentration of B, of the second azo coupling reaction. Column 8 lists calculated values of the energy dissipation rate in the impingement zone as determined from eqn. (1) (equal, impinging sheets) or eqn. (27) (unequal, impinging sheets). Systematic
deviations
in the results
A mass balance check on the spectrophotometric analyses indicated that the right hand side of eqn. (31) was typically lo-20% less than the left hand side: CBO= CR+ 2cs
(31)
In the majority of the experimental samples, less B was found than was nominally fed to the impingement-sheet mixing device. It is important to note, however, that samples were collected at local positions at the edge of the mixed sheet (see Fig. 1). A single sample is, therefore, not necessarily representative of the mixture as a whole.
234 TABLE 1. Experimental d 1.0” 1.0” 1.0” 1.0” 1.O” 1.O” 6.0 60 6.0 6.0 12.0 12.0 12.0
parameters
QA (I min-‘)
Q, (1 mine’)
AP, (bar)
AP, (bar)
cBO(mol I-’ )
(&c,)
2.83 5.7 10.0 16.0 28.0 38.0
2.83 5.7 10.0 16.0 28.0 38.0
1.7 0.4 0.4 0.4 0.4 0.4
1.7 0.4 0.4 0.4 0.4 0.4
0.0042 0.0042 0.0042 0.0042 0.0042 0.0042
100 100 100 100 100 100
1.83 2.48 3.07 3.67 2.25 2.83 3.44
0.4 0.7 1.o 1.5 0.4 0.7 1.0
0.7 1.2 1.9 2.1 1.1 1.7 2.6
0 025 0.025 0.025 0.025 0.050 0.050 0.050
20 20 20 20 10 10 10
11.0 14.9 18.4 22.0 27.0 34.0 41.3
+(Wkg-‘)
-’ (ms)
401400 12500 9200 5700 3300 2400 14100 29200 56100 97500 6900 13300 24100
“from [2].
The thickness of the single sheets is not uniform and typically varies by approximately 20-30%) across the sheet spread angle [4]. This variation, however, is not random. The single sheets are generally thinner in the center than at the side free edges. Since the stream velocity of the single or mixed sheets is constant [3], thinner sections of a sheet have lower volumeric flow rates. Because the liquid in the mixed sheet is essentially in plug flow, any deviation in the local flow rate ratio at the impingement zone from the nominal flow rate ratio fed will affect the equality of eqn. (31). Further, the physical size differences between unequal sheets appears to influence the amount of B found in the spectrophotometric analyses. Thicker sheets are less affected by surface tension at the side free edges, and are probably more uniform in thickness than thinner sheets. For the unequal flow rate experiments, the lnaphthol (A) sheet was approximately 6 or 12 times thicker than the diazotized sulfanilic acid (B) sheet. Since the variation in the thickness of the B sheet was greater than that of the A sheet, it is not suprising that various amounts of B were found in the samples. Figure 5 plots the amount of B found by the spectrophotometric analysis as a function of angular sampling position at a volumetric flow rate ratio of 6. The curves shown are simple second degree polynomial fits to the data points. These curves clearly indicate that the lowest amount of B was usually found in samples taken at the center line (O” position). The amount of B found (c,,) generally increased as samples were collected at angular positions away from the center line towards the side free edges. Similar results were also obtained at a volumetric flow rate ratio equal to twelve. For the equal, flow rate experiments, however, the right hand and left hand sides of eqn. (31) were typically within +5X of each other [2]. Even though the single sheet thicknesses were not exactly uniform at impingement, both sheets were identical to one another.
1.00
0.95
0.90
0.85 s em
0.60
0" 0.75 0.70 0
0.65 0.60-r -1 60
-40
Angular
-20
0
Sampling
20
40
t3
Position
of Fig. 5. Variation of cBh/cBO as a function (see top view of Fig. I). Curves are second fits to rhe experimental data at constant rate. n 14 100 W kg-‘, 0 29 200 W kg-‘, I$ 97 500 W kg-‘.
sampling position order polynomial energy dissipation 56 100 W kg-l, 0
Any variation in thickness of one single sheet was matched by the other single sheet. Although the local flow rate within a single sheet may have varied by as much as 30%, this was true for both single sheets at identical positions. Thus, the local flow rate ratio was found to be within f 5% of the nominal flow rate ratio. Figures 6 and 7 plot the experimental values of the product distribution as a function of the concentration of B found in the spectrophotometric analysis (c,,) relative to the nominal concentration of B fed (ceo). At 01= 6 and 12, the values of X, increase as cBa/cB,, increases. At values of cBa/cBO less than 1.0, the local
235 0.14
0.12
0.10
cA~
cAo
cAo
0.00
X
A
S 0.06
(a)
t=o 0.04
I
I
FL._
I
I I
I
0.02
I I
0.00 0
‘Ao
I
0.60
I
o.io
oh0
oil0
1
-
1
Fig. 6. Variation in Fig. 5.
of ,I’, with c,,/c,,
A
at a = 6. Symbol
key is given
I I
I I
I I
I
=BS ‘=Bo
I
IS
A I
4
*
(b)
(1 +a)6
I I B
B
Fig. 8. Illustration of mixing model for unequal reactant flow rates. Figure (a) depicts diffusion into slabs of equal thickness where there are E or six times as many slabs of A as B. Figure (b) illustrates the modelling simplification where slab A is CLtimes as thick as slab B.
identical. Since NAO/NsO is usually held constant when modelling experimental micromixing results, values of X, at a corresponding abscissa values of 1.0 were used. These values of X, will be designated as X3. For u = 6, these values were extrapolated from Fig. 6, whereas in Fig. 7, values of X$ were interpolated.
0.08 1 0.06-
Modelling
0.04. 0.02-
0.00’ 0.50
0.60
0.70
0.80
0.90
1.00
1 .lC
%a‘=eo Fig. 7. Variation of X, with c~./c~,, at a = 12. q 6900 W kg-‘, 13 300 W kg’, x 24 100 W kg’.
0
molar ratio, (NAO/NBO)local, is greater than the nominal value of 1.2. It is generally the case that at constant energy dissipation rate, X, decreases with increasing N/WIN,,; a result that is confirmed in Figs. 6 and 7. The curves shown in Figs. 6 and 7 are linear fits to the data points at the different energy dissipation rates listed in Table 1. Although the curves are approximately parallel in each figure, the slopes at tc = 6 and 12 are not
of mixing results at unequal flow rates
Figure 8 illustrates the mixing model chosen. The model is similar to the one used earlier to explain the azo-coupling results at flow rate ratios of 1.0 [2]. Diffusion of reactants is assumed to occur within slabs of fixed thickness, 2L. Figure 8(a) depicts a situation where B is mixed with six times as much A. Since the energy dissipated does not differentiate between reactants, the size of the slabs of A and B are equal, but there are six times as many slabs of A. The question is, how are. these slabs distributed among each other? For the equal flow rate case, the slabs of A and B were assumed to alternate with no two slabs of A (or B) ever next to one another [2]. For the unequal flow rate case, the assumption is made that the slabs of A and B are distributed as given by the ratio of their volumetric flow rates. In Figure S(a) six slabs of A surround a single
236 TABLE 2. Model parameters CL 1.O” 1.O”
1.O”
1.O” 1.O” 1.O’ 6.0 6.0 6.0 6.0 12.0 12.0 12.0
QA (I min-‘)
Qe (1 min-‘)
Ei (W kg-‘)
P/D (IIIS)~
M
M/(1+ 4
S'(l+ a)‘/4D
2.83 5.1 10.0 16.0 28.0 38.0
2.83 5.7 10.0 16.0 28.0 38.0 1.83 2.48 3.07 3.67 2.25 2.83 3.44
40 1400 12500 9200 5700 3300 2400 14100 29200 56100 97500 6900 13300 24100
0.5
0.005
2.7 .3.2 4.0 5.3 6.2 2.6
0.027 0.032 0.040 0.053 0.062 0.128 0.089 0.064 0.049 0.366 0.264 0.196
0.0025 0.0135 0.0160 0.0200 0.0265 0.03 10 0.0183 0.0127 0.00914 0.00700 0.0282 0.0203 0.0151
0.5 2.7 3.2 4.0 5.3 6.2 31.9 22.1 15.9 12.3 156.3 109.0 84.5
11.o 14.9 18.4 22.0 27.0 34.0 41.3
I .8 1.3 1.o 3.7 2.6 2.0
“from [2]. bDiffuslve mixing time for CI= 1 only. For c( > 1, d2/Dsignificantly
underestimates
slab of B on both sides. Figure 8(b) shows that for modelling purposes, the effective thickness of the A slab can be taken as six times that of the B slab. For other volumetric flow rate ratios, the ratio of slab thicknesses would be equal to CL The unsteady-state equations for diffusion and reaction were previously given in ref. 8; the case presented here is simpler since the assumption is made that the slab thickness remains constant. The analysis is a moving boundary problem with initial concentrations of cAO and csO at t =O. The experimental data were non-dimensionalized as the ratio of a characteristic time scale of the diffusion process to a characteristic time scale of the second azo-couple reaction,
the diff‘usive mixing time
0.25-
0.20-
x;
0.15.
O.lO/ 0.05-
h4 = k,c,,L2/D
(32)
where M is the mixing modulus, k, is the rate constant of the second azo-coupling reaction, cn,, is the initial concentration of diazotized sulfanilic acid, and L is a representative half-slab thickness over which diffusion occurs. Under turbulent flow conditions in the impingement zone, L is equal to 6, the half-Kolmogorov thickness [2]. The modelling parameters are listed in Table 2. The flow rates, concentrations, and pressure drops from Table 1 are given as input data. For an unequal flowrate experiments, the energy dissipation rate was calculated from eqn. (27). For the experiments conducted at equal flow rates, eqn. (1) was used to calculate ai. Since all experiments conducted at a viscosity of 1.0 cP, where the Reynolds number in the impingement zone was high, the half-slab thickness for diffusion was equal to the half-Kolmogorov thickness, 6, and was calculated from eqn. (2) [2]. When compared with the M values at a flow rate ratio of 1.0, the M values for the unequal flow rate experiments are substantially larger.
/ o.oo! 1o-3
.A.
.,....I
A 10-z
M -= 1 +a
,,,,,,I
“1’
10-l
-4 100
W,3 D(l + a)
Fig. 9. Variation of X: with M/( 1 + a). Normalizing M by the factor 1 + a directly reveals the increase in product distribution as the volumetric flow rate ratio is increased. ( A) c( = 1, ( v) G(= 6,
(C) a = 12.
The difference results primarily from the higher initial concentration of B in the unequal flow rate cases. It was previously observed that plots of X, vs. M do not directly reveal the effect of 0: on X, [7]. At constant initial stoichiometric ratio, NAO/N,,, M does not stay constant as Y is increased, but instead increases directly with cBD(NAo/NBo = c~,_,~/c~o and M - cBo). Plots of X, vs. M/( 1 + cc), however, directly reveal the effect of volumetric flow rate ratio on the product distribution. Column 7 of Table 2 lists values of M/( 1 + a). Figure 9
237
plots values of XL extrapolated from Fig. 6 and interpolated from Fig. 7 at cn,/cn,, = 1.0 as a function of M( 1 + CL). The curves shown are for the fixed-slab model at values of CI= 1. 6 and 12. The results confirm that at constant values of M/( 1 + a), XL increases as u increases. When CI= I the fixed-slab model was previously shown to provide a good fit to the experimental data for equal, impinging sheets [2]. In Fig. 9, however, it is clear that, at c( = 6 and 12, the fixed-slab model does not provide as good a fit to the data. When c( = 6, the model underestimates experimental values of the product distribution, whereas when CY= 12, the model substantially overestimates experimental values The modelling strategy used previously and in this study attempts to utilize the simplest model possible to explain the results. Previous results from ref. 2 demonstrated that the simple, fixed-slab, diffusion model is applicable to impingement-sheet mixing of liquids at flow rate ratios of 1. The results from this investigation, however, illustrate that the simple, fixed-slab model is inadequate for the modelling of unequal, impinging sheets. It would appear from Fig. 9, that the larger the value of CI,the poorer the fit of the fixed-slab model to the experimental data. Probably, a more complex model that is truly based on physical considerations of turbulence, such as that given in ref. 8, would provide a better fit. The inadequacies of the fixed-slab model for the correlation of unequal, impinging sheets notwithstanding, a major focus of this work was to compare diffusive mixing times of unequal sheets with those of equal sheets. The quantity, fi2/D, was previously shown to provide a good estimate of the micromixing time of equal, impinging sheets under conditions where the impingement zone was in turbulent flow [2]. For unequal sheets, however, d2/D severely underestimates the diffusive mixing time. Nevertheless, it may be possible to estimate the micromixing time for unequal, impinging sheets by modifying the mixing modulus so that the fixed-slab model for equal, impinging sheets can be applied to unequal, impinging sheets. The time required for diffusive mixing ( > 90% completion) in a slab, containing both A and B, is approximately given by: t, - L’ID
(33)
where L is the half-slab thickness. For the case of equal, impinging sheets, L is equal to the half-Kolmogorov thickness, 6. The unequal, impinging sheet case is illustrated in Fig. 8(b). When NA,,/NBO- 1, diffusion of A and B will occur over a slab width of approximately (1 + x)S. Figure S(b) suggests that instead of setting L in eqn. (33) equal to 6 for approximating the micro-
0.154
bl/( 1+a) = k, %a (1+a) &4D Fig. 10. Variation of XL with A?/( 1 + a). This modified value allbws the unequal, impinging sheet experimental data to be roughly fitted by the equal, impinging sheet model. The half-slab thickness for diffusion used for the unequal, impinging sheet data is not equal to 6 but is instead equal to ( 1 + c()6/2. Symbol key is given in Fig. 9. mixing time for unequal. estimate should be, L
=
impinging
sheets,
a better
( 1 + 46 2
so that, (35) Column 8 of Table 2 lists values of the micromixing time based on (1 + ~)~6~/4D. Note, that when c( = 1, this expression reduced to 6’/D. At u = 6 and 12, values of the micromixing time calculated from (1 + ~)~a*/40 are significantly greater than those calculated from 6*/D. At CI= 6, the difference is approximately an order of magnitude, whereas at c( = 12, the difference is about a factor of 40. Do the experimental results justify using eqn. (35) as an estimate for the micromixing time of unequal, impinging sheets? Figure 10 plots X: against a modified mixing modulus, n?/( 1 + a). The curve shown is the simple, fixed-slab model with do= 1 (equal, impinging sheets). In Fig. 10 the slab half-thickness, L, is 6( 1 + LX)/ 2 and not 6.
G/( 1 + CL)=
D(l +tl)
= k,c,,( 1 + a)6’/4D
(36)
238
When a = 6 the fit of the equal, fixed-slab model to the experimental data is not superior to that of the unequal, fixed-slab model shown in Fig. 9. When a = 12, however, the fit is superior. More importantly, Fig. 10 allows for a direct comparison between micromixing times for equal and unequal impinging sheets. The diffusive mixing or micromixing time can be roughly but rapidly estimated using the following expression: 4Li ~ = -0.4917 In (1 +a)2
In E, - 1.2621
where td has units of seconds and .si has units of W kg-‘. Equation (37) is obtained from a plot of column 8 in Table 2 against column 4 in Table 2. Since the values of td were directly calculated from sit the regression coefficient for this fit is theoretically unity. However, since only four significant figures are shown, the values calculated from eqn. (37) deviate slightly from those listed in column 8. The energy dissipation rate in the impingement zone is calculated from eqn. (27) or eqn. ( 1). Equation (37) is valid for both unequal and equal impinging sheets under turbulent flow conditions in the impingement zone and for NAO/NBON 1. Finally, the diffusive mixing times of impinging sheets at various flow rate ratios and at constant energy dissipation rates (si) can be compared using the following expression, td2 -_=
c1
+
a2>2
tdl
c1
+a,)2
When x, = 1 and cc2= 6, eqn. (38) indicates that the diffusive mixing time of the unequal, impinging sheets will be a factor of 12.3 greater than that of the equal, impinging sheets. When or, and a2 equal 6 and 12 respectively, the ratio of diffusive mixing times is approximately equal to 3.5.
Conclusions
An expression was derived for the energy dissipation rate in the impingement zone (E,) of unequal, impinging sheets. Similar to the previously derived expression for ci of equal, impinging sheets, this expression is based upon the geometry of the impinging, unequal sheets and the momentum balance. Under similar operating conditions, a comparison of Ei values at different volumetric flow rate ratios (a) indicates a rapid decrease in si until a - 20, beyond which the rate of decrease in E, drops off markedly. Attempts were made to correlate the product distribution values of the azo couplings to a mixing modulus using a simple micromixing model based on diffusion of the reactants in fixed slabs. Although this model was
previously shown to provide adequate correlation when a = 1 (equal, impinging sheets), modifying the model to predict product distributions for a = 6 and 12 resulted in calculated values of the product distribution that were in poor agreement with experimental values. However, it was possible to estimate the micromixing time for unequal, impinging sheets by using the fixed-slab model at CI= 1 and modifying the mixing modulus. The diffusive mixing time in the mixing modulus was redefined as (1 + a)262/4D instead of ~3~10, as is the case for equal, impinging sheets. At constant diffusion coefficient and energy dissipation rate in the impingement zone, the micromixing time of unequal, impinging sheets is a factor of (1 + a)2/4 greater than the micromixing time for equal, impinging sheets. For the inelastic collision of impinging sheets, the major parameters affecting the speed of micromixing are the energy dissipation rate in the impingement zone and the volumetric flow rate ratio. Increasing energy dissipation rates decrease the micromixing time, whereas increasing volumetric flow rate ratios increase the micromixing time.
Nomenclature
initial concentration of A, mol 1-l equivalent concentration of B found in the specCl&, trophotometric analysis of R and S, mol I-’ of B, mol 1-l cl30 initial concentration of monoazo dye product, mol 1-l CR concentration concentration of bisazo dye product, mol 1-l diffusion coefficient, m2 s-r E efficiency of pressure head conversion to kinetic energy in impingement-sheet mixing device k, rate constant for first azo coupling reaction, 1 mol-’ s-’ k, rate constant for second azo coupling reaction, 1 molC’ s’ L characteristic length of diffusion or half-thickness of slab, /*m m mass flow rate, kg SS’ M mixing modulus; ratio of diffusion to reaction times M modified mixing modulus which essentially estimates the diffusive mixing time for unequal impinging sheets as if they were equal, impinging sheets. N A0 nominal molar amount of reactant A, mol NBO nominal molar amount of reactant B, mol P power dissipated at the impingement zone, W AP pressure drop used to produce single liquid sheet flow rate of reactant A. 1 min’ flow rate of reactant B, 1 min-’ iit em total liquid flow rate of mixed sheet equal to Q, + Q2, lmin-’ cAO
;;’
239
e
Ar R s
si 4n
SI s2
td
flow rate of single sheet, 1 min’ width of impingement zone, pm radial distance of single sheet to the impingement zone, m sheet thickness single sheet thickness, for equal impinging sheets, at impingement zone, pm mixed sheet thickness at impingement zone, pm thickness of single sheet No. 1 at impingement zone, pm thickness of single sheet No. 2 at impingement zone, pm characteristic time for diffusive mixing or micromixing, s velocity of single sheet, m SC’ velocity of mixed sheet, m SC’ x-component velocity of single sheet, m s-’ x-component velocity of mixed sheet, m s-’ mixing velocity; y-component velocity of single sheet, m SC’ y-component velocity of mixed sheet, m SC’ volume of impingement zone, m3 product distribution of azo coupling test reaction extrapolated or interpolated product distribution when csa /cBo = 1.
Greek letters
i 6 E K
volumetric flow rate ratio half-angle of impingement half-Kolmogorov thickness, pm energy dissipation rate, W kg-’ angle by which the mixed sheet is offset from the line bisecting the impingement angle
4 P An I)
spreading angle of thin liquid sheet liquid density, kg mm3 liquid density of mixed sheet, kg me3 kinematic viscosity, m2 s-’
Subscripts
1 2 m
single sheet No. 1 single sheet No. 2 mixed sheet
References 1 R. J. Demyanovich, Liquid mixing employing expanding, thinning liquid sheets, U.S. Pat. 4 735359, April 5, 1988. 2 R. J. Demyanovich and J. R. Bourne, Rapid micromixing by the impingement of thin liquid sheets. 2. Mixing study, Ind. Eng. Chem. Res., 28, (1989) 830. 3 R. J. Demyanovich and J. R. Bourne, Rapid micromixing by the impingement of thin liquid sheets. I. A photographic study of the flow pattern, Ind. Eng. Chem. Res., 28 (1989) 825. 4 N. Dombrowski and R. P. Fraser, A photographic investigation into the disintegration of liquid sheets, Phi/. Truns. R. sot., 247A (1954) 101. 5 N. Dombrowski and P. C. Hooper, The performance characteristics of an impinging jet atomizer in atmospheres of high ambient density, Fuel, 41 (1962) 323. 6 J. R. Boume, F. Kozicki and R. Rys, Mixing and fast chemical reaction I: Test reactions to determine segregation, Chem. Eng. Sci., 36 (1981) 1643. 7 J. R. Bourne, C. Hilber and G. Tovstiga, Kinetics of the azo coupling reactions between I-naphthol and diazotised sulphanilic acid, C/rem. Eng. Commun., 37 (1985) 293. 8 J. Baldyga and J. R. Bourne, A fluid mechanical approach to turbulent mixing and chemical reaction, Chem. Eng. Commun., 28 (1984) 231.