Dynamics of Continuous Segmented Flow Analysis R. D. Begg Department of Mechanical Engineering, University of Glasgow, United Kingdom The paper describes a dynamic model for the Technicon system of continuous segmented flow chemical analysis. The final concentration distribution along the fluid segments in the reaction tube is related to certain flow parameters and the predicted distributions agree with those measured experimentally. Such a model is of use in investigations into the limitations on AutoAnalyzer throughput and has an application in the development of more accurate techniques for nonsteady-state analysis. The linear model provides a satisfactory representation of the operating characteristics, but it is noted that when certain reagents are involved the accuracy diminishes. Further tests and analysis improve the modeling for these fluids and a method is set out how such fluids can be identified and their nonlinear characteristics measured without resort to chemical analysis.
OF THE SEVERAL mechanized systems of chemical analysis, the most well established, particularly in hospital biochemistry laboratories, is the AutoAnalyzer (Technicon Limited, Chertsey, Surrey). Even with mechanization, the continuous increase in work of these laboratories puts great pressure on technicians and laboratory facilities. Thus any possibility of increasing output must be considered. Although the Technicon AutoAnalyzer is widely and successfully used, there seems to be little fundamental quantitative knowledge of the physics of the system. The object of the present investigation was to examine the working of these analyzers and to develop a mathematical model to represent the operating characteristics which would be of assistance in an experimental program aimed at isolating factors limiting analyzer performance. In the Technicon AutoAnalyzer, samples along with reagent are fed into a continuous tube as a series of short (ca. 1-inch) slugs separated by air bubbles. Some slugs of reagent alone are inserted into the tube between samples to increase the separation (Figure la). Over a long tube the total leakage of fluid around the bubbles is considerable and at the end of the tube the first slugs of the samples are diluted by leakage from the reagent slugs, which are themselves contaminated by the preceding sample slugs. The method known as equilibrium analysis depends on there being sufficient slugs in each sample so that a few at the end remain relatively undiluted and a quantitative measurement is made using these by colorimetric means. Thus the leakage rate between slugs places a limit on the maximum sampling rate which gives a satisfactory analysis. If sufficient sample slugs are used, there is no undiluted sample at the end to provide absolute analytical accuracy. In many instances, however, since the volume of each sample is constant, the ratios of the concentration in the final slugs of each sample (even though they do not represent the absolute concentration in the various samples) are taken to represent the ratios of the absolute concentrations. This is known as nonsteady-state analysis and its accuracy is dependent on the operating characteristic being linear and on several empirical corrections which can be made to the results. ANALYSIS
Linear Model. Figure l b shows conditions in an AutoAnalyzer tube at the interface between the reagent slugs and 854
ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
the slugs with both sample and reagent. The slugs A ” , A’, and A are of the reagent in which the concentration of sample substance is zero, while slugs B, C, D,E, etc. contain sample plus reagent and at entry to the tube contain full concentration of the substance. As the slugs pass along the tube, liquid quantities QA,QB, Qc, QD,and QE,leak across the air bubbles as shown. This leakage is caused by fluid from a slug wetting the walls of the tube and being picked up by the following slug. Two assumptions are made to derive equations to describe this situation. The leakage flow, inasmuch as it is affected by fluid properties, is determined only by conditions within the donor slug. Mixing within each slug takes a negligible time compared with the total traverse time; that is the time taken for fluid on the walls to mix into the recipient slug is neglected. Referring to Figure 16 and equating volume flows for slug B,
and equating concentration,
where C is the Concentration of sample substance in a slug I is the distance travelled along the tube Y is the volume of the slug Q is the volume of’leakage between adjacent slugs per unit length of tube traversed Since the concentration of all of the slugs A is unaltered by passage along the tube, CA and hence QAare constant, while CB, QB, and VB are all functions of I, the distance traveled. One more relationship between CB, QB, and VB is needed before a solution can be found. From physical considerations the most obvious relationship is that between C, the concentration, and Q, the leakage, that is the effect of the fluid characteristics on the leakage rate. To obtain an equation from the physics of the problem is a sizeable task as yet unsolved, but useful progress can be made by assuming certain empirical forms for this relationship. The simplest possible equations are arrived at by assuming Q independent of C, that is that the leakage from every slug in the system is the same. That is, QA = QB = Qc = Q (say) and it follows that dVB/dl = 0 so that VBis constant = V(say). Then Equations 1 and 2 combine to give : (3)
Also for slug C (4)
and similarly for the succeeding slugs. Solving Equation 3 with the initial condition CB = 1 at 1 = 0, and taking CA = 0 throughout
cB= e-11’
(5)
AIR BUBBLES
(1
('1
REAGENT
I1
F L U I D SLUGS
('1
SAMPLE
0
(1 t
0
I \\ \\
' (1 (1
REAGENT
(1
(1
REAGENT
(1
(1
SAMPLE +REAGENT
1.1.
OA
'A
'A
A ' '0 L E A K A G E FLOWS
'C
QD
1. b.
Figure 2. Variation of concentration along sample
Figure 1. Segmented flow in AutoAnalyzer tube
Using this value for CBin Equation 4 with Cc = 1 at 1 = 0,
cc = e-l/A(l+
k)
Distribution predicted from linear model Typical nonlinear distribution 0 0 0 Measured distribution after flow through 40-ft glass heating coil Measured distribution after flow through 80-ft glass heating coil
------
la. Arrangement of sample and reagent slugs l b . Leakage flows due to slug velocity
++ +
(6)
Continuing to solve the equations for the concentration variation in the slugs following C provides the useful solution that the concentration of the ( N 1)th slug along the sample is : = e - f / ~ ( l+ + [Z/XZ J V / X N %)1 (7)
makes some allowance for change in Q with change in concentration is the equation
Q
+
c,.
where X
=
V .is a constant to be regarded as a path length
-
Q
constant similiar in concept to a time constant. The series in the bracket is that for e''A truncated after the ( N 1)th term. Thus as N .--, m , CN 1 for all finite I and A. This indicates that if sufficient slugs of constant size are considered, there will always be some at the end which are almost undiluted. Figure 2 shows the predicted variation of concentration of sample with slug number for various values of tube length. This parameter is made nondimensional as I j X where X is the length constant. From this graph it can be seen that for short tubes ( e . g . ,when I / X = 1) the 5th slug is still almost pure after a complete traverse. However, for I / X = 6 (a tube 6 times as long), the 14th slug has to be taken to obtain the same purity at the end of the tube. It follows from the expression for C.,,that, . provided the total volume of the sample slugs in the reaction tube is kept constant, the degree of dilution of the last slug is almost independent of the number of slugs into which the sample is divided. For, if S, is the total volume of sample in the slugs and N the number of slugs, then X = S , / Q N and the concentration reaches its final value when additional terms in the truncated exponential series make a negligible difference to the sum.
+
KiC
+ Kz
(9)
Depending on the relative values ascribed to KI and Kz, either a high or a low dependence on C can be simulated. The constants Kl and K2 are associated with the fluids involved. Equations 1 and 2 remain valid, but since QA and QB are no longer equal, the volume of each slug changes as it passes along the tube. If the leakage flow increases with concentration, then the slugs whose concentration is diminishing will shrink while those whose concentration is increasing will lengthen, Substituting Equations 9 and 2 into 1 and simplifying
For the first slug in a sample, CAis a constant; but for slugs thereafter, the equivalent of CAis a complex function of length. This renders a direct analytic solution to find the concentration of the Nth slug impossible. However, useful information can be derived from the equation if it is transformed into one describing the variation in the slug length. This equation is
Integrating once using the substitution y V" log"
vo
reducing to the condition S, >> IQ which is independent of N , the number of slugs, provided it is greater than about 6. Nonlinear Model. The assumption that Q is independent of C is obviously an oversimplification in view of well known experimental facts that some reagents behave much better than others in AutoAnalyzers. An empirical model which
=
=
d VB dl
-
-. + Constant = CAKi- 1 4-KZdV, dl
where Vo is the slug - volume at I = 0. At this point two sets of initial conditions can be specified as illustrated on Figure 3. Representing a large volume of sample plus reagent followed by a much smaller volume slug of reagent solution (Figure 3a). For the first slug only: CA = 1 for all 1, C, = 0 for I = 0. ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
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SAMPLE
+
REAGENT
REAGENT
0
C,
ca
0 --t
QA
OB
3.a.
REAGENT
SAMPLE, REAGENT
0
CA
CB
4
OA
0 4
OB
3 b.
Figure 3. Configuration for nonlinear analysis and tests 3a. Initial conditions: CB = 0 for I = 0; Ca = 1 throughout 36. Initial conditions: CB = 1 for I = 1; Ca = 0 throughout
Substituting these conditions, VB
logVa
LENGTH O F TUBE
K1 =
~
-
-
dl KI - KZ
- --
dvB a t zero and infinity can be From this values of V B and dl found. VB VBand V Oand d= -Kl. dl
Atl=O,
Representing a large volume of reagent alone followed by a smaller slug of sample plus reagent, (Figure 36) again for the first slug only: C A = 0 for all I , CB = 1 for 1 = 0. And substituting these conditions,
and again the values of VBand d VBcan be found at I dl atl= w ~
=
0 and
VB = Voand-d VB = -Kl. dl
Atl=O,
AtI= b From these results, simple tests can be developed to establish nonlinear behavior in fluids and also to measure the constants Kl and Kz directly. Neither test involves any chemical analysis. EXPERIMENTAL Measurement of Nonlinear Constants. Figure 3 shows two configurations, corresponding to the two sets of initial conditions described earlier, which can be used to measure nonlinear behavior. If in either case the small slug changes In length after being pumped through a length of tube, then the parameter K1 is not zero and the leakage is a function of concentration. Moreover by plotting the length of the slug at various intermediate positions along the tube, the initial
gradient
(%
=
TRAVERSED ( F E E T )
Figure 4. Variation of slug length with tube length traversed __ Biuret slug: initial conditions 3b - - Saline slug: initial conditions 3b
-KI or + Kl can be found along with the
From these, both KI and Kzcan be calculated. As the extent of the nonlinearity decreases, the accuracy with which KZ ( = Q of the linear model) can be found decreases and the test fails for completely linear systems when chemical methods based on Reference 1 can be used to find KZaccurately. A number of tests have been made of substances and reagents used in AutoAnalyzer work. The procedure was to pump the bubble/slug arrangements of Figures 3a and 3b through one or more standard 40-foot heating coils in series, measuring the length of the small slug a t various distances along the tube. During these tests, considerable efforts were needed to keep the slug velocity constant while the slug traversed the tube, but the flow had to be stopped at regular intervals to permit the small slug to be measured and some transient velocity conditions were unavoidable. Another problem encountered was the variation in the internal diameter of the tube of the glass coils. These had to be calibrated by passing a slug of water through, preceded by a large volume of water, and measuring changes in the length of the small slug. Since there were no changes in concentration involved, the changes in the slug length were taken to indicate the variation of the tube bore. This also provided some insight into the accuracy obtainable, and repeatability within 2 or 3 was achieved. RESULTS AND DISCUSSION
Shown on Figure 2 are experimental measurements of individual slug concentrations after passage through two different lengths of standard Technicon tubing. Each set of concentration points on the graph indicates a value of l/X for the particular system, and it can be seen that doubling the tube length doubles the apparent value of l/X, which is typical of a linear system. The predictions of concentration distribution of Figure 2 have been compared with measured distributions from many laboratory analysis circuits and good agreement has been found in most instances, each with its own indicated value of I/h. However, several important exceptions have been observed which suggest that the assumptions made in developing the model are incomplete for certain reagents. A typical exceptional distribution is shown on Figure 2 as a dash line. Figure 4 shows two typical results from the measurement of (1) R. E. Thiers, R. R. Cole, and W. J. Kirsch, C[i/i. Cliem., 13, 451 (1967).
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ANALYTICAL CHEMISTRY, VOL. 43, NO. 7, JUNE 1971
the change in slug volume, using the configuration of 36. The results have been corrected for variations in the tube bore. Graph A, where the small slug was a dilute solution of Biuret, is typical of a substance with a nonlinear characteristic. From the graph, with V o = 1 X A, where A is the crosssectional area of the tube in in.2: Ratio initial: final slug volume = e + K 1 / K 2= 1.5 and the initial gradient dVjdl = -Kl = -0.00083 X A i r ~ . ~ / i n ThusK1 . = 0.00083 AinSa/in., K2 = 0.00276 A in.3/in and K,/KZ = 0.4, indicating that there is nonlinear behavior. Graph B, where the small slug contained saline solution, shows no change in slug volume, KI and K2 cannot be determined, and behavior is linear. Measurements of this kind confirmed that, of the reagents tested, the hydroxides of sodium and potassium exhibited the largest nonlinearity. Others, such as potassium ferricyanide and sodium tartrate behaved similarly but to a lesser extent, while many other substances were found to exhibit no departure from linearity. The results so far obtained have established a fairly successful dynamic model for the Technicon AutoAnalyzer which has been used to demonstrate various characteristics of the measurement system. Figure 2 shows that the variation of concentration along a sample is “S” shaped, with a considerable portion of the graph concave upward. This fact is well established in practice and is shown by Thiers et al. in Figure 2 A of their report ( I ) . Such a graph can be represented only by a simple exponential over the latter part of the range; over the initial part, the error involved in such an approximation is considerable. It is this error which has led Thiers et al. to postulate an unexplained “lag phase” in order to reconcile their exponential model of the characteristics with the “S” curves. If a model is to be of value in predicting the concentration maxima for nonsteady-state working, it must draw its information from the first slugs of the sample and not from those slugs at the end whose concentration is to be predicted. By predicting the concentration of individual slugs, the model emphasizes how much the debubbling process degrades the information available from the analyzer-if not for normal usage, then certainly from the point of view of nonsteady-state working-and an increase in the quality of the information could result in reduced sample volumes and increased throughput. The matter hinges on the balance between increased complexity of recording and greater throughput. As demonstrated on Figure 2 the model is linear, that is the apparent l / X for a 40-foot coil is half that for an 80-foot coil. Thus if the tailing characteristics for other circuit elements can be measured in terms of equivalent lengths (Z/X), then the equivalent length of the elements in series can be obtained by simple addition of the elemental I/h’s to predict the overall concentration distribution at the end. Work is in progress to measure the l/X for the various circuit elements used in the AutoAnalyzer under a normal range of operating conditions.
Although the methods and models which have been developed enable nonlinear behavior to be detected and measured, the lack of an analytic solution for the nonlinear equations involved prevents any immediate use being made of the results to calibrate or improve nonlinear steady-state methods. Moreover, although the nonlinear model appears to be relevant to the individual fluids which have been tested, its application to reacting mixtures in which concentrations are continuously varying has still to be investigated. Work is now in hand to obtain numerical solutions to the equations for various forms of the relationship between concentration and leakage (Equation 9). It is envisaged that these will have several applications. First, it will be possible to develop a series of charts similar to Figure 2 but for various values of Kl/K2. Using the appropriate chart from these, it will be possible to compare alternative circuits for an analytic method to determine which has potentially the greatest throughput (minimum sample volume necessary). Second, these charts will provide a more accurate method correcting for nonsteady-state working than a linear approximation by providing the correct shape of concentration distribution along the sample to correspond to the measured values of Kl and K2. Third, it is hoped to develop this numerical solution into fast computer algorithm which would permit a substantial decrease in the sample volume required for any desired accuracy by using nonsteady-state working in conjunction with an on-line data acquisition system. In addition to these numerical solutions, the investigation is being continued to extend the model to be suitable for substances, such as protein solutions, where the mixing characteristics cannot be assumed to be instantaneous as is assumed in the basic linear model. CONCLUSION
From the results and experience so far obtained, it is felt that the linear model provides a real insight jnto AutoAnalyzer dynamics, especially from the point of view of further laboratory automation, and that it is capable of extension to describe most of the phenomena of continuous flow analysis. ACKNOWLEDGMENT
The author wishes to acknowledge the many helpful discussions he has had with Adam Fleck of Glasgow Royal Infirmary who is heading the experimental approach to the problem, and the support accorded the project by J. D. Robson and H. G . Morgan of the University of Glasgow. RECEIVED for review August 14, 1970. Accepted January 8, 1971.
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