Dispersion in Segmented Flow through Glass Tubing in Continuous-Flow Analysis: The Ideal Model L. R. Snyder” and H. J. Adler Technicon Instruments Corp., Tarrytown, N. Y. 1059 1
Sample dispersion or carryover in the flow of air-segmented liquid streams through open tubing is of broad analytical importance, and especially so in the technique of continuous-flow (AutoAnalyzer) analysis. A detailed model for the prediction of such dispersion under certain limiting conditions of practical interest is rederived, expanded, and experimentally verified. This so called “ideal” model assumes perfect mixing within each moving liquid segment.
Continuous-flow (CF) analysis is a general technique that was introduced by Skeggs ( I ) for carrying out automated chemical-reaction assays. CF analysis utilizes a flowing, airsegmented stream of reagents, interspersed by samples. As the samples traverse a length of tubing, they undergo reaction to a chromogen, which is then measured spectrophotometrically. In this way, virtually every kind of wet chemical analysis has been successfully automated (e.g., 2). The function of air-segmentation in CF analysis is to reduce longitudinal dispersion of sample along the flow path, which in turn decreases sample interaction (carryover) and increases analysis rate. A fundamental understanding of dispersion in segmented flow through open tubes is therefore essential t o t h e optimal design of typical CF systems. While a n exact expression for such longitudinal dispersion was developed 20 years ago by Golay for unsegmented flowing streams (31,no comparable treatment has been reported for dispersion in segmented liquid streams. Early attempts at describing dispersion in segmented-flow systems (e.g., 4-9) were limited to empirical relationships for the shape of sample curves following CF analysis. These equations were useful in correcting for sample carryover, but offered limited insight into the dispersion process. More recent studies (10-15) have provided a fundamental basis for describing dispersion in segmented flow, but have not been developed t o the point where it is possible t o predict dispersion as a function of experimental conditions. In the present two papers, we describe a detailed, essentially rigorous model of dispersion in segmented flow which allows t h e prediction of sample dispersion as a function of all important experimental parameters. T h e treatment is limited to flow through tubular elements, but this contribution to dispersion is of major importance in the design of high-speed CF systems.
two concentric tubes of plastic tubing, with provision for withdrawing a volume of liquid identical to that injected into the moving liquid segment. It was thus possible to inject a given volume of sample (dye) solution into a single segment, without changing the final volume of the segment. The injected dye is initially distributed unevenly within the liquid segment. If this poorly-mixed segment is introduced directly into the measuring coil G, the dispersion of dye during passage through the coil can be quite variable. This effect was eliminated by first allowing the dyed segment to flow through a short length of coiled Teflon tubing-the pre-mix coil of Figure 1. Because aqueous solutions do not wet the premix coil, no dispersion of dye occurs before the dyed segment reaches the first transverse colorimeter F. However complete mixing of dye within the segment does occur, as was seen by visual observation. The dye-concentrationprofile is monitored at the inlet and outlet of the measuring coil G, using homemade, linear-absorbance transverse photometers F and H. These external photometers did not contribute additional dispersion of dye. The inlet photometer F confirmed that all injected dye was contained within a single liquid segment; the outlet photometer H measured the resulting dispersion of dye after passage through coil G. A typical dye concentration profile as measured at photometer H is shown in Figure 2. Unless otherwise stated, experimental conditions were as given in Table I. Since Evans Blue (the dye of Table I) is a surfactant, the liquid surface tension varied with the concentration of dye, which led in turn to variations in liquid segment size during movement through the coil G (see discussion of 13).To minimize this effect, all solutions contained added surfactant (BRIJ-35in Table I), except where liquid-surface tension was varied.
THEORY Several workers (e.g., 10-15) have used essentially the same model to describe sample dispersion during segmented flow through a n open tube. These treatments begin either with a single sample-containing segment (10, 11) or-for the usual case in CF analysis-a sample step-function or slug consisting of a series of equivalent sample-containing segments (12-15).
+
Waste’
Air
. ...
n PUMP A
Liquid Liquid J
W
n 6,
Storage Coil C
Waste
EXPERIMENTAL
Sample Injection
The equipment used in the present studies is diagrammed in Figure 1. A continuous, air-segmented stream of liquid (chosen to have appropriate volumes of the individual air and liquid segments) is generated by the multichannel proportioning pump A (AutoAnalyzer
Pump-111;Technicon Instruments Corp.) and directed through the 3-way valve B-1 into the glass storage coil C. Once the storage coil and pre-mix coil E are filled with this stream, the 3-way valve B-1 is directed to waste, and valve B-2 is directed from waste to coil C (thus connecting pump and C). This allows the linear velocity of the segmented stream to be controlled independently of the volumes of liquid and air segments previously loaded into the storage coil. Sample is next injected into a single liquid segment at D. The injection module D consists of a syringe whose needle is inserted into
D Waste
Measuring Coil Figure 1. Schematic diagram of apparatus used in present study to measure sample dispersion in glass tubing
BI,B2, valves: E, mixing coil; F, H, transverse photometers ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
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Table I. Experimental Conditions for Dye-Dispersion Studies Measuring coils: Liquid: Sample: Other conditions:
0
4
8
12
16
Segment Number k
1219 cm length, 0.159 cm i.d., coiled with a 35-cm. diameter, positioned horizontally Water containing 0.1 vol % BRIJ-35 (nonionic surfactant), at 25 "C; q = 0.89 cP, u = 32 dynes/cma Evans Blue dye (C34H24NeO&Na4) linear velocity 1.09 cm/s, liquid segment volume 0.033 ml, air segment volume 0.010 ml, temperature 25 "C f 1 (ambient)
a For studies where liquid viscosity was varied, glycerol-water solutions were used, with viscosities taken from ( 2 4 ) ;for studies where liquid surface tension was varied, water-propanol solutions were used (no BRIJ-35), with surface tensions taken from (23).
Figure 2. Colorimeter output for dyed segment after passage through measuring coil
Initial Sample Segment
,Air.
Flow Direction
-'
Liquid
/
Figure 3. Segmented flow through an open tube
2 by No. 1occurs, and so on. Thus, in time, the sample originally contained in segment No. 0 spreads over several following segments, leading to dispersion of sample along the tube. Previous workers have made the following assumptions for the ideal model: 1)instantaneous mixing of film and segments, 2) constant dimensions of all segments and constant film thickness, 3) negligibly slow longitudinal diffusion in the film. One can then derive the following expression for the concentration of dye C k in the kth segment, following passage of dyed segments through a given tube:
C k / C = e-qqk/k! The assumptions of these previous workers led to a model defined here as the ideal model for dispersion in segmented flow through open tubes. This model is fairly accurate under certain limiting conditions, b u t previous workers have not developed the model sufficiently to allow ab initio predictions of sample dispersion as a function of experimental conditions. In this paper, we will carry this derivation to completion. Begg (13)has pointed out one practical exception to the ideal model (which he calls the linear model), so-called nonlinear dispersion arising from a change in liquid-surface tension as sample concentration varies. However the case of nonlinear dispersion is of minor interest generally, since such effects are counteracted in normal practice by addition of surfactant to the liquid stream. The ideal model assumes that, a t any time, the concentration of sample throughout a given liquid segment is uniform or constant; Le., mixing within the segment is at all times infinitely fast. However this assumption is never valid exactly, and it is often sufficiently poor to lead to serious breakdown of the ideal model. This leads to the definition in this paper and the development in the following paper (16) of a more general model: the non-ideal model for dispersion in segmented flow through open tubes. The Ideal Model (Previous Work). We will use the derivation and terminology of ( I 1 ) in the following discussion. The physical basis of this (ideal) model is illustrated by Figure 3. An initial dye (or sample) segment No. 0 is followed by undyed segments No. 1 , 2 , . . . Sample dispersion occurs because the liquid segments normally wet the inside wall of the tube (nonwetting liquids would give hydraulic problems), so t h a t a film of liquid phase follows each segment through the tube. An average thickness of this liquid film is assumed, df. The liquid film laid down by segment No. 0 has the same composition as that of segment No. 0, and this film is overtaken and mixed into segment No. 1,thereby transferring sample from segment No. 0 to No. 1.Similar contamination of segment No. 1018
ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
(1)
Here C is the initial dye concentration in segment No. 0 (before dispersion), and q is given by q =
VfIV,
q = 4dfL/Lsdt
(2) (Pa)
Vf is the total volume of film deposited from any segment during its passage through a given tube of length L and inside diameter d t , V , is the volume of a liquid segment, and L , is its length. The right-hand side of Equation 1 is the Poisson distribution function, which can be found in standard mathematical tables. The maximum in the resulting sample concentration profile (as in Figure 2) occurs in segments q and q - 1,or (q - '12) on average. For large values of q , the Poisson function approaches a Gaussian distribution (e.g., 1 7 ) , with a variance 02 equal to q. Next consider an initial series of sample-containing segments before dispersion; i.e., a sample-concentration stepfunction such that segments No. 0, -1, -2, . . . contain no sample initially, while segments No. 1, 2, . . . are dyed. Dispersion of sample in this case occurs around the edge of the step-function (k = O ) , to yield a sigmoid sample concentration profile that is given by summation of Equation 1 for every initial sample-containing segment. The result is a cumulative or integrated Poisson function (see 12-15), exactly analogous to Equation 1 for the case of a single initial sample segment. The relationship of dispersion from an initial sample segment and from an initial stepfunction is shown in Figures 4 A and 4 B , respectively. In each case the initial sample-concentration profile before dispersion is shown by the superimposed cross-hatched area. The standard deviations u for either Gaussian (Figure 4 A ) or integrated Gaussian (Figure 4 B ) curves are equal to q1I2. The displacement of the sample concentration maximum (Figure 4A) from k = -'/z is equal to q; the displacement of the 50%-maximum-concentration point (Figure 4 B ) from k = 0 is equal to q. For a finite sample
k
iBI
I
k-
Figure 5. Illustration of data reduction in present study Curve shown is calculated from Equation 4 with q = 8;derived values of qand u were 7.83 and 2.85, respectively k warh.in
(C I
k
Figure 4. Basic parameters in sample dispersion (A) Single sample-segment input. (6)infinite sample-step-function input. (C) Sample-multisegment (slug) input
step-function ("slug" input), corresponding to the usual case in C F analysis, t h e distribution shown in Figure 4C results. In each example of Figure 4, Gaussian distributions are shown. However, these can be replaced with Poisson distributions where q and u are small. I t should be noted that the quantities q and u are dimensionless; i.e., measured in units of segment = 3.5). number as in Figure 4A ( q % 12.5, u GZ The Ideal Model (Extension).Equations 1 and 2a define sample dispersion in terms of known variables ( L ,L,, d t ) and the film thickness df. If we can relate d f to measurable parameters, we will have completed the derivation of the ideal model. Recently Concus (18) has extended the earlier derivation of Levich (19) to arrive a t a theoretical value of d f for the case of perfectly-wetted (0' contact angle), small-diameter tubes: df = 0 . 6 7 ~ d t ( u 7 / 7 ) ~ / ~
(3)
Here u is the linear velocity of the segmented stream, 7 is the liquid voscosity, and y is the liquid surface tension. Experimental values of d f from direct measurement of film thickness (20, 21) are in rough agreement with Equation 3. However Equation 3 has so far not been applied to the prediction of dispersion in segmented-flow. The parameter q from Equation 2a can be combined with Equation 3 to eliminate d f a n d yield = 0.67 T L d t 2 ( ~ g / y ) 2 / 3 / V s
(4)
Equation 4 completes the description of the ideal model. It remains to verify Equation 4 and explore the consequences predicted by the ideal model.
EXPERIMENTAL VERIFICATION OF THE IDEAL MODEL: FILM THICKNESS df AND THE DISPLACEMENT q OF THE SAMPLE DISTRIBUTION CURVE Data Reduction. I t was necessary to reduce data as in Figure 2 to give derived values of q and u. The procedure followed is illustrated in Figure 5. Comparison of derived values
621
'
I
n
"
"
'
Sampln Concentration IArbnrary U n m )
"
'
Sample Concentration (Arbitrary Units)
Flgure 6. Dependence of q and u on sample concentration Typical experimental plots, varying only dye concentration
of q and u with theoretical values, using data calculated from Equation 1, showed consistent agreement, except that all q values were low by 0.17 unit. This small and consistent discrepancy was corrected for by arbitrarily adding 0.17 to all derived values of q . Sample Concentration Effects. Derived values of q and CT were slightly dependent on sample concentration, presumably because of the surfactant properties of the dye. T o eliminate this variable in the present study, sample concentrations were varied and the resulting data extrapolated to zero concentration. This procedure is illustrated in Figure 6, where least-squares curves through the data give extrapolated values of = 6.3 and u = 3.03. Precision and Reproducibility. The precision of individual measurements of q and u was well within f5%, when measurements were carried out on the same tube within a short time-interval (e.g., Figure 6). Repeatability was poorer, when different tubes were compared over wide intervals in time. This is illustrated in Table 11, for repeat measurements of q and g as a function of velocity (different tubes, six months apart). In general, it was observed that reproducibility among tubes improved when all tubes were cleaned with detergent before use. Most of the data from the present study were obtained over a short interval, using the same tubes where possible. Experimental Verification of Equation 4. The ideal model assumes the validity of Equation 4. If we can show that Equation 4 is experimentally accurate, we will have verified the ideal model as well, since it is shown below that u2 approaches q under conditions favoring rapid mixing within individual liquid segments. ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1019
Table 11. Reproducibility of Measured Values of q and u for Two Different (Nominally Identical) Tubes, with Data Collected Six Months Aparta Tube A
Tube B
Velocity
4
U
9
U
0.8 1.7 3.3 5.0
6.5 10.5 14.1 17.0
2.98 4.32 5.49 7.17
5.4 8.4 13.5 17.5
2.82 4.12 7.10 10.4
" Experimental conditions as in Table I, except for velocity.
Beginning with the standard conditions of Table I (which approximate practical conditions for CF analysis), each of the parameters of interest was varied over a wide range, as indicated in Table 111. In order to test the applicability of Equation 4, it was found convenient to rearrange this relationship into the form qV,/* d t 2 L ( ~ a / y ) 2 = / 30.67 = D
(5)
Values of D could then be compared, as each of the different experimental parameters was varied. Average values of D are shown in the third column of Table 111, for variation of each parameter. I t is seen that D remains reasonably constant and equal to a value of about 0.50, for each of these 8 experimental series. The variability of D is exactly equivalent to the variability of calculated values of q (Equation 4), which is seen in Table I11 (column 3) to be about f10% (std dev). Thus Equation 4 accurately predicts values of q over a wide range in experimental conditions. A further test of Equation 4 is obtained by recasting Equation 4 into the form q = CLadtbucqdyeVsfV,g
(54
and solving for the coefficients a-g by multiple regression analysis of the q values summarized in Table 111. The fourth column of Table I11 compares experimental values of these different coefficients vs. the theoretical values of Equation 4. Thus for column length as a variable, the fourth column shows a 0.9 ( L O ) , which signifies that the derived value of the coefficient a in Equation 5a is 0.9, vs. a theoretical value of 1.0. In each case, reasonable agreement is seen between derived and theoretical dependencies of q on each experimental variable (column 4 of Table 111).
Finally, values of q can be plotted vs. the appropriate function of each variable, while other variables are held constant. Figure 7 summarizes this test of Equation 4, again with good agreement between experimental and theoretical values of q . Value of Din Equation 5; Wetting and Contact Angle. The value of D found experimentally is about 0.50 (Table III), vs. the value 0.67 predicted by Equation 4 (or 5). This may reflect a contact angle between glass and 0.1% BRIJIwater as liquid t h a t is greater than zero, since Equation 4 assumes a zero contact angle. Support for this theory is provided by the data of Table IV. As liquid surface tension y decreases, there is a general tendency toward lower values of D (low surface tension liquids wet glass less well). For most cases of practical interest, a value of D equal to 0.5 can be assumed; the data of Table IV suggest a value of D equal to 0.35 for most organic liquids, and a value of 0.6 for water without added surfactant. Effect of O t h e r Variables o n q. Table V shows the dependence of q on coil diameter; it is seen that q does not vary with coil diameter. Similar values of q were observed for both vertically and horizontally positioned tubes (other factors being equal). The ideal mode) predicts t h a t q will be independent of sample type. This is confirmed by the data of Table VI, where similar values of q are found for compounds of different molecular weight. P u m p pulsations were found to have an insignificant effect on values of q. Temperature was not studied as a variable, but it is expected that temperature effects can be predicted through their effect on liquid viscosity and surface tension. Sample concentration effects are expected to be generally minor (see Figure 6), particularly for compounds without surfactant properties. D I S P E R S I O N I N T H E IDEAL M O D E L P R E L I M I N A R Y COMPARISON W I T H EXPERIMENT The ideal model predicts that the dispersion of a sample as in Figure 4 is defined by a single parameter q , where q is both the retardation of the center of the original sample distribution (single sample-segment) and the variance u2 of the resulting (dispersed) sample concentration distribution. By analogy with the Martin model of chromatography (22),which is mathematically equivalent to the ideal model developed above for segmented flow in open tubes, we can predict what will happen as the weakest assumption of the ideal model fails; i.e., as mixing within the liquid segments becomes slow. As in chromatography, we expect t h a t sample displacement (or retention), equal to q , will remain constant, but sample dis-
Table 111. Summary of Experimental Studies of q (Conditions as in Table I unless Otherwise Noted") Variable studied Column length, L Velocity, u d t = 0.222 cm ( V , = 0.067) d t = 0.159 cm ( V , = 0.033) d t = 0.089 cm ( V , = 0.10,0.033) Viscosity, 9 Surface tension, y b Liquid segment volume, V, Air-bubble volume V,'
Variation of variable 150 < L
< 1220 cm
0.2 C u < 2.2 cm/s 0.5 C u C 5.0 1.3 C u < 13.0 0.9 C 7 < 2.2 CP. 49 < y 6 73 dynes/cm 0.01 < V , < 0.08 ml 0.005 C V , < 0.02 ml
D (Eq. 7)c
a 0.9 (1.0)
0.54 f 0.03 0.50 f 0.01 0.49 & 0.02 0.48 f 0.04 0.51 f 0.03 0.59 & 0.07 0.47 f 0.07 0.45 f 0.03
Test of Eq. 4d
I
b 2.0 (2.0) c 0.6 (0.7) d 0.8 (0.7) e - 0.4 (-0.7) f - 0.7 (-1.0) g 0.0 (0.0)
a See Table I of (26) for further details in experimental conditions. Surface tension values for water-propanol solutions taken from (23). Standard deviations are shown for the mean D values in each case. Regression analysis values of coefficients a-g from Equation 5a; theoretical values in parentheses (Equation 4).e In all cases, the length of air-bubbles exceeded 3/2 times the tube diameter d t , to provide complete occlusion of the tube by the bubble.
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ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
9
2.5 5
,
o
y 2o
0.0
400
200
600 L
800
ibl
1200
1000
i
ll 5o
///
5 / ,
L c I I 0.0
0.2
0.4
0.6
0.8
1.0
1.2
00
,
,
0.5
10
1.5
,
,
,
,
2.0
25
30
35
0.04
0.05
0.06
0.07
y2i3
1.6
1.4
2.0
.
1.5 -
0.0
0.01
0.02
0.03
lIV,
Y -2'3
Figure 7. Verification of Equation 4; conditions as in Table I, except for parameter varied
(a)Dependence of 9 on L, (b) Dependence of 9 on u,(c) Dependence of q on q. (d) Dependence of 9 on y. (e) Dependence of 9 on V,
Table IV. Effect of Liquid Composition on D of Equation
Table VI. Dependence of q on Sample Type"
7
D
Liquid Water No surfactant Surfactant Benzene Chloroform Methanol
Y (23)
Sample 0.59 3~ 0.07 0.45 & 0.54 0.35 0.41 0.28
49-730 32 28 27 23
KMn04 Evans Blueb Blue DextranC
35 13 4 Straight
3.0 3.1 3.0 2.9
158 960
6.1 6.6 6.1
16.8 21.5 21.5
2 x 106
7 = 0.9 CP
H* 1.4 1.3 1.4
3.7
Conditions of Table I, except tube length = 610 cm. See Equation 6 and related text.
persion will increase. T h a t is, u2 will no longer be equal t o q , but t o some larger value: u2 = H*q
u = 18
Table VII. Values of H* as a Function of Liquid Viscosity tl and Linear Velocity u
Table V. Sample Dispersion as a Function of Tube Coiling" 4
u=4
" Conditions as in Table I, unless otherwise stated. See Table I. Nonionic dye, Pharmacia Corp., Piscataway, N.J.
" Water-propanol solutions
Coil diameter, cl, cm
Mol wt
(6)
Here H * is (for t h e moment) a n empirical correction factor always greater than one. Experimental measurement of q and H * values (see following paper 16) confirms Equation 6 in a qualitative sense. Thus H * values less than 1are not observed. Similarly, for conditions favoring rapid or complete mixing (lower liquid velocities u or viscosities v), H* values approach 1. For less propitious conditions, where mixing is expected t o
a
u = 1.1cm/s
U
H*
t
H*
0.5 1.6 3.3 5.0
1.2 2.0 3.7 6.1
0.9 1.3 2.2
1.5 2.2 4.4
Other conditions as in Table I.
be slow, H * increases and eventually becomes large with respect t o 1. These general trends are illustrated by the representative data of Table VI1 (from 16).
SUMMARY A rigorous theory for the ideal model has been developed, and shown to accurately describe sample displacement ( 9 ) values. T h e ideal model ( H * = 1)also gives a good description of dispersion ( u values) under conditions where intra-segment mixing is rapid (e.g., low values of u and 7). In the following paper (16),the present treatment is generalized to allow acANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
1021
curate prediction of dispersion in both slow- and fast-mixing the theory Of dispersion in segsystems, thereby mented flow through open tubes.
SYMBOLS See list a t end of (16),which applies to both papers.
LITERATURE CITED (1) L. J. Skeggs. Am. J. Clin. Pathol., 28, 311 (1957). (2) “Technicon Bibliography, 1967-75”. Technicon carp,, ~ ~~ , y , , 1975. (3) M, J, E, slay, in Chromatography, 1958jt,D, H, Des.,,, Ed, Academic Press, New York, 1958. p 36. (4) R . Thiers and K. Oglesby, Clin. Chem. ( Winston-Salem, N.C.), 10, 246 (1 964). (5) R. Thiers, R. Cole, and W. Kirsch, Clin. Chem. ( Winston-Salem, N.C.), 13, 451 (1967). (6) M. Evanson, G. Hicks, and R . Thiers. CM. Chem. ( Winston-Salem, N.C.). 16, 606 (1970). (7) R. Thiers, M. Jevn and R . Wildermann. Clln. Chem. ( Winston-Salem, N.C.), ’ 16, 832 (1970); (8)W. H. C. Walker, C. A. Pennock, and G. K. McGowan, Ciin, Chim. Acta, 27, 421 (1970).
(9) A. L. Chaney, in “Automation in Clinical Chemistry, 1967, Symposium I ” , N. B. Scova et ai., Ed., Mediad, Inc.. White Plains, N.Y., 1968, p 115. (10) J. Hrdina, 6th Colloq. Amino Acid Analysis, Technicon Corp.. Monograph No. 3, 1967. ( 1 1) G. Ertingshausen, H. J. Adler, and A. S. Reichler, J. Chromatogr., 42,355 (1969). (12) R . Thiers. A. Reed and K. Delander, Clin. Chem. ( Winston-Salem, N.C.), 17, 43 (1971). (13) R. Begg, Anal. Chem., 43, 854 (1971). (14) R . Begg. Anal. Chem., 44, 631 (1972). (15) W. H. C. Walker and K. R . Andrew, Clln. Chlm. Acta, 57, 181 (1974). (16) L. R . Snyder and H. J. Adler, Anal. Chem., 48, 1022 (1976). and Chem. Eng. Soc., 5, 258~(1956). , ~(17) A. Klinkenberg ~ ~ F. F. Sjenitzer, t ~ ~ (18) P. J. Concus, J. Phys. Chem., 74, 1818 (1970). (19) V. J. Levich, “Physicochemical Hydrodynamics”, Prentice Hall, Englewood Cliffs, N.J., 1962, p 681.
L; Fcifi;;ba,odt~~~ :i:g: ::?&{;,
~ ~ , ~ ; ; ~ (1974), ~ ~ , ~ (22) A. J. p. and L. M, Synge9 J.* 359 1358 (lg4’). (23) R. C. Weast, Ed., “Handbook of Chemistry and Physics”, 52nd ed.,Chemical Rubber Publishing Co., Cleveland, Ohio, 1971, p F-29. of Chemistry and Physics,,,31st &,, (24) c, D, Hodgeman, Rubber Publishing Co., Cleveland, Ohio, 1949, p 1768.
RECEIVED for review December 19,1975. Accepted February 18, 1976.
Dispersion in Segmented Flow through Glass Tubing in Continuous-Flow Analysis: The Nonideal Model L. R. Snyder” and H. J. Adler Technicon lnstruments Corp., Tarrytown, N.Y. 7059 7
The “ideal” model of sample dispersion in the flow of airsegmented liquid streams through open tubing is expanded here to include the effects of slow mixing within moving liquid segments. The resulting “nonideal” model allows predictlon of sample dispersion in segmented flow over a broad range of experimental conditions. Comparlson of experimental dispersion data with values calculated from the nonldeal model shows excellent agreement. The design of continuous-flow ( AutoAnalyzer) systems for minimum dispersion and sample interaction, and/or maximum analysis rates can now be done theoretically, rather than empirically as in the past.
In ( I ) , we described the development and experimental verification of a rigorous treatment of sample dispersion during segmented flow through open tubes: the so-called ideal model. The ideal model allows the approximate prediction of sample dispersion as a function of experimental conditions in continuous-flow (CF) or AutoAnalyzer analysis. This model is relatively accurate under conditions that allow near-complete mixing of sample within each liquid segment; however, it becomes less reliable when experimental conditions change so as to favor incomplete mixing. Here we expand the ideal model to correct for slow mixing, arriving finally a t the nonideal model. We also demonstrate its validity and ability to accurately predict dispersion over a range of conditions of interest in continuous-flow analysis.
EXPERIMENTAL E x p e r i m e n t a l conditions a n d procedures are described fully ( I 1. Unless otherwise noted, t h e standard conditions o f T a b l e I of (I) were used in each experiment. Precision a n d r e p r o d u c i b i l i t y are discussed in ( I ) .
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ANALYTICAL CHEMISTRY, VOL. 48, NO. 7, JUNE 1976
THEORY A detailed, rigorous treatment of mixing within individual segments during segmented flow through open tubing has so far not been reported. Horvath e t al. ( 2 , 3 )have reported experimental data for intra-segment mixing during flow through enzyme-coated tubes, and correlated their results in terms of conventional engineering parameters such as the Nusselt and Reynolds numbers. However, their studies were directed toward radial mixing a t steady state, rather than axial dispersion. Here we will pursue an alternative approach, based on the theory of dispersion in chromatography. Bolus Flow. An understanding of mixing in segmented flow must begin with the characteristic bolus flow pattern that is observed within moving segments ( 2 , 4 , 5 ) .Figure 1 shows a schematic of this flow pattern, which is readily observable by adding dye at different points within the moving liquid segment. If dye is added a t the center A or leading edge B of the moving segment, the color of the dye is immediately dispersed into a characteristic figure-8 pattern, shown schematically by After this the crosshatched portion of Figure 1 (labeled ‘‘1”). figure-8 pattern is achieved, dye moves more slowly into adjacent regions or streamlines (“2”, “3”, etc.). Eventually the mixing of dye within the segment is complete, so that the dye is uniformly distributed. Thus, longitudinal mixing across the segment is rapid, while radial mixing or mass transfer is slow. A Hypothetical Analog of Mixing in Bolus Flow. The visual examination of dyed segments as above suggests the existence of discrete-essentially laminar-streamlines or currents ( 1 , 2 , . , . as in Figure 1).Mixing must then occur by movement of dye from one parallel current to its adjacent neighbor, by some combination of molecular diffusion plus convective mixing. This process appears conceptually similar
~
,
3
