Mathematical modeling of single-line flow-injection analysis systems

Anal. Chem. 1988, 60,1700-1709

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Mathematical Modeling of Single-Line Flow- Injection Analysis Systems without Chemical Reaction S p a s D. Kolev' a n d E r n 0 Pungor* Institute of General and Analytical Chemistry, Technical University of Budapest, Gelle'rt te'r 4, H-1111 Budapest X I , Hungary

Mathematical models for slngle-line flow-Injection analysis (FIA) systems with syrlnge, hydrodynamic, and valve injection overcoming some substantlal drawbacks of the exlsting models for such flow systems (e.g., the constructive elements of the modeled systems are not considered as separate sections wllh their own geometrical dimensions and dlsperslon properties and the models are indifferent to the method of sample injection) are described. These models consider the slngle-line FIA systems as conslstlng of several connected in series tubular sections (Le., the injection section, the reactor, the measurement cell, and the tubes connecting these sectlons with the source bottle and the waste which are called fore- and aftsectlons). The dispersion of the analyte In each one of them Is described by the axially dbpersed plug flow model. The models proposed in this paper have been solved in the Laplace domain, and equations for calculation of the mean and the varlance of the concentration curves, detected in the measurement cell, have been derived. The adequacy of the mathematlcal models mentioned above has been checked experlmentally on slngle-line FIA manifolds with syringe, hydrodynamic, and valve lnjectlon of the analyte. The unknown Peclet numbers In the models have been determined by curve-fittlng using a simplex optimization procedure and numerical inversion of the Laplace domain solutions of the models. Falrly good agreement between the experimental results and the theoretical predlctlons has been observed, which shows that the hydraulk models outlined in this paper can be used quite successfully for the mathematical description and optimization of single-line FIA systems without chemical reactlon.

1. INTRODUCTION 1.1 P a t t e r n s of Flow in Single-Line Flow-Injection Analysis Systems a n d T h e i r Mathematical Description. Generally a single-line flow-injection analysis (FIA) system consists of connecting tubes (fore- and aftsection), injection device, reactor, and measurement cell (Figure 1). The connecting tubes are very often straight or bent empty tubes with arbitrary diameters that are different in the general case from the diameters of the other system's sections. Despite this fact in the existing models for FIA systems the connecting tubes have not been specified as flow sections with their own geometrical and dispersion characteristics (1-35). The injection device may be a syringe, some type of injection valve (14),or a hydrodynamic injection section (23). In the case of a valve or hydrodynamic injection, it is not considered separate from the reactor physical element of the FIA systems (4,lO-14,23-25,27,32,34). Only in the models based on the convective-diffusion equation has the contribution of the injection device, assumed as hydrodynamic injection section, Permanent address: Faculty of Chemistry, University of Sofia, Anton Ivanov Ave. 1, 1126 Sofia, Bulgaria.

Table I. Dispersion Models dispersion model

parameters

general dispersion model aclat = V(D.VC) - uvc general dispersion model for symmetrical pipe flow aclat = a [ ~ , ( a ~ / a x ) i / a+~(l/r)a[rD,(ac/ar)l/ar - u, (aclax) uniform dispersion model aclat = DLm(a2c/ax2) + (D,_/r)a[r(ac/ar)]/ar - U , (ac ax) dispersion plug flow model aclat = DLi(a2C/aX2) (D,/r)a[r(ac/ar)]/ar -

+

ii(ac/ax) axially dispersed plug flow model aclat = DL(a2c/ax2)- a(ac/ax)

to the overall analyte dispersion been taken into consideration ( I 5, 19, 20, 26, 28-31, 33). Usually the volume of the measurement cell is much smaller than that of the reactor and the injection section (10,12,21). For that reason in all mathematical models for FIA systems built until now, the dispersion of the analyte in this system's section has been neglected and the measurement cell has been excluded for the corresponding physical models. However, some experimental investigations have shown that the contribution of tubular conductivity (36) and photometric (37) cells to the shape of the output signal of the investigated flow systems is considerable. This result requires the inclusion of the measurement cell in the mathematical models for FIA systems. The disadvantages in the mathematical modeling of single-line FIA systems mentioned above reduce to a large extent the generality of the corresponding models and restrict considerably their applicability when some of the constructive elements are replaced or performance parameters are changed. For that reason the development of a mathematical model overcoming these drawbacks seems to be necessary for the proper description and effective optimization of the single-line FIA systems. 1.2 Choosing of the Most Appropriate Hydraulic Model f o r t h e Description of Single-Line F I A Systems. The approach used in chemical engineering for the mathematical description of all patterns of flow other than plug and backmix is the introduction of the so-called hydraulic models. They can be divided into four main classes: (1)dispersion models; (2) tanks-in-series models, known also as compartments or mixed-cell models; (3) combined models; and (4)empirical models (38-40). Usually the dynamic output signal of these flow systems when there is no chemical reaction between the analyte and the carrier stream resembles the response curves obtained from stimulus-response experiments (40) in straight empty tubes. In this case both the dispersion and the mixed-cell models can be applied for the description of the flow pattern. However, the elements of the sirigle-line FIA systems are very often tubular and for that reason the dispersion models (Table I) appear to be closer to the real physical situation. In the choice

0003-2700/88/0360-1700$01.50/00 1988 American Chemical Society

ANALYTICAL CHEMISTRY, VOL. 60, NO. 17, SEPTEMBER 1, 1988 T

Flgure 1. Schemes of FIA systems with syringe (a), delta-function (a), hydrodynamic (b), and valve (b) injection.

of the most appropriate dispersion model a compromise should be made between the contrary requirements for maximal possible mathematical simplicity and maximal possible precision in the description of the dispersion process. A qualitative analysis of the character of flow in the main sections of the single-line FIA systems shows that because of its complexity (e.g., bends, sudden enlargements, and contractions) models not using the exact velocity profile of the flow should be preferred. Such models are the dispersed plug flow and the axially dispersed plug flow model (Table I). Both of them are built up on the assumption that the velocity profile is flat. Considering these two models it seems that the use of the axially dispersed plug flow model is more attractive because it is simpler from a mathematical point of view, there are some theoretical and empirical equations for calculating its only parameter, i.e., the axial dispersion coefficient (21, 34), and it has been utilized quite successfully for different tubular process systems (38-40). By use of the only parameter of this hydraulic model or the corresponding dimensionless group (i.e., the Peclet number) as a criterion, reactors of different types (e.g., straight and coiled tube reactors, packed bed reactors, and single bead string reactors) can be compared with respect to their dispersion properties. This is not possible if models built for special types of reactors (e.g., chromatographic models for packed bed reactors) are used because they use different kinds of unique parameters.

2. DEVELOPMENT OF MATHEMATICAL MODELS FOR SINGLE-LINE FLOW-INJECTION ANALYSIS SYSTEMS 2.1 Assumptions on Which the Models Are Based. In the present investigation each section of a single-line FIA system is assumed to be tubular and for that reason is characterized by its geometrical dimensions (i.e., diameter and length) and by its dispersion properties represented by the axial dispersion coefficient or the Peclet number. The syringe and the valve injection differ considerably from each other and for that reason it is expedient to consider single-line FIA systems with those types of injection devices separately. The hydrodynamic injection does not differ in principle from the valve injection and it will be considered as a special case of valve injection. The further assumptions on which the mathematical models for single-line FIA systems with syringe (Figure la) and valve (Figure l b ) injections are based on the following: (1)The dispersion of the analyte in each system’s section can be described by the axially dispersed plug flow model. This assumption excludes the treatment of double-humped peaks in the absence of chemical reaction. Such peaks have been obtained experimentally and predicted theoretically for short, straight empty tubular reactors (15,29,31, 33). The numerical simulations in ref 31 have given the condition for the double peaks to appear i.e. 0.01 < 7 e 0.1 (1) where 7 is reduced to molecular diffusion scale mean residence time (also called the Fourier number) which is defined as

= TD,L/v

1701

(2)

(The characteristic length L in ref 31 is equal to the reactor’s length xl.) Inequality 1 generally does not hold in the FIA systems used in the analytical practice. Besides that, as has been pointed out by Vanderslice et al. (15), the slightest turbulence due to mismatched fittings or some other nonideality in the flow is sufficient to smooth out the double humps. Probably because of these two reasons double peaks have been seldom obtained without chemical reaction (29)and the assumption made above does not limit essentially the applicability of the models based on it. (2) The connecting tubes, which are the so-called foresection and aftsection in the physical models, are infinite. This assumption is based on the fact that for a given “critical” length of the foresection and the aftsection of the modeled system the error due to the so-called “end effects” (Le., due to the fact that the lengths of these sections are not infinite in reality) is insignificant. In a theoretical investigation of ours (41) related to this problem it has been shown how the “critical” length, which has been found to be equal for both the foresection and the aftsection, depends on the other parameters of the flow system. (3) The time interval necessary for the flow to reach the condition of steady-state motion from the condition of rest existing before the sample injection in the FIA systems with valve and hydrodynamic injection can be neglected. It has been shown theoretically by us that the upper limit of the initial period starting from the moment of hydrodynamic or valve injection during which steady-state flow is established in a flow system consisting of m connected in series tubes could be calculated by the following formula (42): m

T r = 0.288(

k=l

l k / d a Z ) / ( v E lk/dk4) k=l

(3)

where i k and d k are the length and the diameter of the kth tube and u is the kinematic viscosity of the flow. With the help of eq 3, which has been experimentally confirmed (42),it has been shown that the time necessary for acceleration of the flow from condition of rest to that of steady-state motion is much shorter than the mean residence time of the analyte in the system. So long as the mean residence time is the time scale unit of the processes occurring in the FIA manifolds which are responsible for the formation of the output signal, the assumption mentioned above should be correct. (4)The mass transfer between the sample plug in the hydrodynamic injection section and the carrier solution in the adjacent sections (i.e., the foresection and the reactor) before the moment of injection can be neglected. The sample introduction into the hydrodynamic injection section (Figure l b ) is probably accompanied by induced circulation in the stopped flow a t the inlet and outlet points. This movement of the fluid will intensify the mass transport between the flowing sample stream and the stopped carrier solution, but since the diameters of the fore-section, the injection section, and the reactor are very small, this effect probably will be of no importance for the formation of the output signal. The induced circulation could be decreased if the flow rate in the sample circuit is diminished. In this case the time during which the sample plug remains in contact with the carrier solution before the injection is increased. If it is assumed that in this case the mass transfer is due only to molecular diffusion, then the concentration profile at both ends of the injection section will be described by the following equation (43): c/co = 0.5 erf~[x/(2D,t)’/~] where

(4)

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ANALYTICAL CHEMISTRY, VOL. 60, NO. 17, SEPTEMBER 1, 1988

c(0,x) = 0 for x > 0 = co for x d 0

(acc/aO)+ r c ( a c c / a x ) - (Yc/Pc)(d2Cc/dX2)= 0 (X, d

Simulations of eq 4 have shown that the molecular diffusion practically does not distort the initial concentration profile. In this case if the sample solution passes through the injection section at such a flow rate so that minimal disturbance in the stopped carrier stream occurs, assumption 4 is justified. (5) The analyte is uniformly distributed in the cross-section of the flow where the syringe injection is performed and no significant disturbance in the gross flow pattern of the system occurs during this process. When the sample is introduced in an FIA system by syringe the local disturbance of the flow is considerable because the diameters of the syringe's needle and of the tube into which the injection is performed are comparable. This local disturbance causes mixing of the sample with the carrier stream. For that reason it could be assumed that the analyte is uniformly distributed in this cross section of the flow. As far as the volume in which the local disturbance takes place is very small in comparison with the volume of the relevant section (e.g., foresection) of the flow system into which the injection is performed, it could be supposed that the gross flow pattern in this section as well as in the FIA manifold as a whole remains unchanged. (This assumption has been numerically proven and the results may be obtained from the authors.) (6) The axial dispersion coefficient and the molecular diffusion coefficient of the analyte do not depend on its concentration or on the concentration of other solutes in the sample or in the carrier solution. This assumption will be correct if the concentration of all solutes is low. In such a case the natural convection caused by density difference between the sample and the carrier solution will not play an important role in the dispersion of the analyte. Two types of detectors will be considered. The first one is measuring the mean concentration in a cross section of the flow in the measurement cell. For convenience, it is assumed that this cross section is situated in the middle of the measurement cell, i.e. at x , X,

=

+

XI

(xZ

- x,)/2 =

~1

+ Ax/2

(5)

This type of detector is approximated by an optical detector. The second type of detector, considered in this study, is measuring the average concentration in the whole volume of the measurement cell i.e.

Such type of detector might be a conductivity one. In the further considerations for the sake of brevity these two types of concentrations will be named as local (c), and average (caJ concentrations, respectively. On the basis of the assumptions made above, the mathematical description of the single-line FIA systems already discussed will be outlined. 2.2 FIA System with Syringe Injection. An FIA system with syringe injection is presented schematically in Figure la. It consists of four sections connected in series, namely: foresection, reactor, measurement cell, and aftsection. The injection point x,, is situated in the foresection. This flow system is described by the following set of partial differential equations in dimensionless quantities and variables:

(aC,/dfl)

+ ya(dCa/dx)

x d X,)

(9)

- (ya/Pa)(d2Ca/dX2) = O

(X, G X) (10) where yj (j = f, r, c, a) are coefficients derived in a previous paper of ours (44) which make possible the use of eq 7-10 for the description of flow systems comprised of tubular sections with various diameters. When the diameters are equal, all coefficients yjare unity. The corresponding initial and boundary conditions of eq 7-10 are C,(X,O) = C,(X,O) = CC(X,O)= C,(X,O) = 0

(Ila)

Cf(o-,0) = C,(0+,0)

(Ilb)

C,(0-,0) - (l/Pf)[dcf(o-,0)/dxl = C,(O+,O) - (l/P,)[dC,(0+,0)/dXl ( I l c )

C,(X1-,0) = CC(X1+,0)

(Ild)

C,(X1-,0) - ( l / ~ , ) [ ~ c , ( x , - , ~ ) / d = xl Cc(Xl+,fl)- (l/Pc)(dcc(xl+,O)/dxl (1le) Cc(Xz-,O) = Ca(X,+,B)

(11f-l

Cc(X,-,O) - U/PJ [dC,(Xz-,0) /dXl = Ca(Xz+,0) - (l/pa)[dCa(X,+,B)/dXI ( I l g ) Cf(-a,O) = finite Ca(+m,O) = finite

(Ilh) (lli)

The input signal $(S) (eq 7) is assumed to be rectangular function with dimensionless time span equal to K = 1 for 0 d 0 d K = 0 for K < 0 and K > 0 (12)

$(e)

(Equations 7-10, their initial and boundary conditions, and the input signal in dimensional form may be obtained from the authors.) By means of the Laplace transform eq 7-10 have been reduced to a system of four ordinary linear differential equations which have been solved analytically. (The detailed solution may be obtained from the authors.)

Cm5= 2q,[l - exp(-Kp)]

X

[qa + q c - (qa - q c ) exp(-f',q,AX)] exp[-Pf(O.5 q,)Xo + Pc(0.5 - qc)AX/2 + P,(o.5 - q r ) X 1 l / ( A s ~ )(13) C a y " = 2q17C[l - exp(-Kp)1((0.5 + qc)(qa + qc) - (0.5 qc)(qa - qC) exp(-2PCqcAX) - 2qc(0.5 qa) e x ~ t P ~ ( 0-. 5 qc)AXll exp[P,(0.5 - q,)Xl - Pf(0.5- qf)Xol/(AXAsp2)

+

(14) where

As = K(qf + qr)[(qa + qc)(qr + q c ) - (4a- q c )

X

( q , - 4c) exP(-2PcqcAX)I - (9, - nf)[(qa + Qc)(Qr - Q c ) (qa - qc)(qr + qc) e x ~ ( - 2 ~ ~ q , A Xexp(-2P1q,XJ )l (15)

If the concentration is normalized-with respect to the total amount of the injected analyte (i.e., C$ = (?,S/K) and the limit of for K 0 is found, then the expression obtained describes the Laplace transform of the detected concentration profile in the case of the delta function input signal. The corresponding equations for the local and the average concentrations are the following:

e,.

-

C,* = C , s p / [ 1

- exp(-Kp)]

(16)

[ 1 - exp(-Kp) J

(17)

and

C,,6

= Ca:p/


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Table 11. Geometrical Dimensions of the F I A Manifolds Used valve injection length, m diameter, mm

section foresection" injection section reactor

0.4950 0.1750 0.2660 0.5490 1.0650 1.6200 0.5820 0.5300 0.0023 0.0150 0.5250

tubular measmt cell thin-layer measmt cellb aftsectionO

0.609 0.609 0.599 0.598 0.601 0.611 0.330 0.821 0.501 0.850 0.584

hydrodynamic injection length, m diameter, mm

syringe injection length, m diameter, mm

0.4950 0.2020 0.5520

0.535 0.536 0.535

0.5110

0.536

0.5520

0.535

0.0023

0.501

0.0023

0.501

0.5250

0.584

0.5250

0.584

for the straight portions of the foresection and the aftsection. The cross section of the thin-layer measurement cell is assumed to be circular. a Only

and hydrodynamic injection (Figure lb) consists of five partial differential equations in dimensionless quantities and variables similar to eq 7-10 (i.e., in eq 7 the right-hand side is equal to 0 and one more equation similar to eq 8-10 is included for the description of the dispersion process in the injection section). The initial conditions are the following: Cf(X,O) = C,(X,O) = Cc(X,O) = Ca(X,O) =O (2la) C,(X,O) = 1 Figure 2. FIA manifolds with hydrodynamic injection in injecting (a) and loading (b) position and with syringe injection (c): (1-4) siliconerubber tubes; (5, 6) T pieces; (7) measurement cell; (8) thermocouple; (9) syringe; lines within a circle parallel and perpendicular to those outside the circle represent open and closed position of a sillconerubber tube, respectively.

(21b)

while the boundary conditions are of the same type as eq lla-lli. The equations for the Laplace transforms of the local (C,) and the average (Cav)concentrations are the following:

The equations for the Laplace transforms of the local (eq 13 and 16) and the average (eq 14 and 17) concentrations are cumbersome expressions and their analytical inverse transformation is in the general case extremely difficult if not impossible (45). But there is a possibility of finding directly from them some important quantities characterizing the concentration curve measured by the detector. These quantities are the statistical moments of the concentration curve. They can be calculated by the following relationship, proposed by Van Der Laan (46): pj

=

(-l)j

lim(dJC/dd)

(18)

P+

Among them the two most important and frequently used moments are the first moment about the origin pl and the second moment about the mean a2. The former moment, known as the mean (eq 19), defines the center of gravity of the concentration curve while the latter one, known as the variance (eq 20), characterizes its width (40). If the curve has Gaussian shape its mean is equal to 1 and its base-line-tobase-line time calculated at concentration equal to 1% of the maximal concentration at curve's peak is 6.07a (47) p1 =

JSCO do

where

In the case of hydrodynamic injection, if the injection section is part of the foresection, the equations mentioned above can be simplified (i.e., Pi = Pf and yi = rf). If the concentration is normalized with respect to the whole injected quantity of the analyte and after that its limit for Xe 0 is determined, then the ideal case of delta-function injection is obtained.

-

3. EXPERIMENTAL SECTION

By use of eq 18 and 20 the relations for the mean and the variance have been derived. (The equations for the mean and the variance for FIA systems with syringe, delta-function, valve, and hydrodynamic injection may be obtained from the authors.) 2.3 F I A System with Valve o r H y d r o d y n a m i c Injection. The mathematical description of an FIA system with valve

3.1 Flow Systems. Three single-line FIA systems, all with

straight tube reactors but different with respect to the method of sample introduction (i.e., valve, hydrodynamic, and syringe injection) have been investigated experimentally. 3.1.1 F I A System w i t h Valve Injection. The main geometrical dimensions of this as well as of the other experimental manifolds (i.e., with hydrodynamic and with syringe injection) used in this work are summed up in Table 11.

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A rotary injection valve (Type OE-320, LABOR MIM, Hungary) with external sample loop was used. The latter was a bent Teflon tube with internal diameter equal to 0.609 mm. It was assumed that the whole sample volume filling the external loop and some of the valve’s internal bores were situated in the external loop only. On that basis the length of the external loop, which corresponds in this case to the injection section in the model (Figure lb), was recalculated (Table 11). The foresection, the aftsection, and the reactor also were Teflon tubes. The internal diameters of all tubes were determined from the volume of bidistilled water with which they can be filled. The tracer, which was a standard solution of potassium chloride, was detected conductometrically by two different home-made conductivity measurement cells. The one is tubular and consists of two platinum disk electrodes (diameter and width 5.0 mm and 1.0 mm, respectively) with bores in the center (0.5 mm id.) divided by a Teflon spacer. The other is a thin-layer cell and consists of two flat, parallel stainless steel electrodes again divided by a Teflon spacer. The flow system was kept a t 20.0 f 0.1 “C by means of a thermostating system consisting of a thermostat and polyethylene tubes (15.0 mm id.) connected in series. Water a t the above temperature was circulated through them. Each tubular section of the manifold was installed along the axis of a polyethylene tube with appropriate length. The source bottle was placed in the basin of the thermostat. The temperature of the flow was controlled by a thermocouple. The value of volumetric flow rate in the manifold was governed by the head of pressure of the fluid in the source bottle. Nitrogen was used for that purpose. The flow rate was determined by collecting and weighing the effluent of the manifold over intervals of 5 min. The use of a source bottle under pressure instead of a peristaltic pump assured flow without pulsations. The absolute nitrogen pressure in the source bottle during all experiments was less than 2 atm. The calculations based on Henry’s law (48) have shown that under such pressure and at 20 OC the molar concentration of nitrogen in the carrier solution (bidistilled water) should not exceed 1.7 X At such a low concentration of nitrogen it can be expected that neither the viscosity of the carrier solution nor the diffusion constant of potassium chloride will change substantially. The use of nitrogen under pressure has not caused the formation of nitrogen bubbles in the flow. The lengths of the foresection and the aftsection (Table 11) have been chosen in such a way, that the so-called “end effects” are insignificant (41). 3.1.2 FIA System with Hydrodynamic and Syringe Injection. This manifold was constructed on the basis of the manifold with valve injection by replacing the valve and the reactor with a configu; Ition of three stainless steel tubes connected in series by two T pieces, made from Perspex (Figure 2). One end of this configuration was attached to the measurement cell while the other one was joined by means of a short silicone-rubber tube to the Teflon tube used as the foresection in the manifold with valve injection. The Teflon tube downstream of the thermocouple as well as the third conduits of the two T pieces (the other two conduits of each of the T pieces were attached to the stainless steel tubes) were connected with short silicone-rubber tubes. When necessary these silicone-rubber tubes were closed by clipping. When syringe injection was performed, the syringe needle was connected with the third conduit of the T piece situated closer to the measurement cell (Figure 2c). The third conduit of the other T piece was closed. The thermostating of flow was done in the same way as in the manifold with valve injector. The bores of the T pieces had diameters equal to 0.50 mm and volumes approximately equal to 0.2 pL. In the framework of the physical models presented in Figure 1,they were considered as parts of the flow system’s sections with which they connect. The geometrical dimensions of the manifold with hydrodynamic and syringe injection are presented in Table 11. 3.2 Reagents and Apparatus. Standard solutions of potassium chloride with concentrations in the range 2.00 x 10” to 5.00 X lo4 M were used for both injection and calibration. The carrier solution was bidistilled water.

The conductance of the flow was measured by a Lambda 1 (Gynkotek, FRG) digital conductometer. It was connected to an OmniScribe strip chart recorder (Series D-5000, Houston Instruments, USA). The response of the recorder is less than 0.33 s to within f l % of full scale, independent of line frequency and voltage. This time compared to the mean residence time of fluid in the systems under the experimental conditions used (from 2.94 to 71.64 s and in most of the cases being greater than 15.0 s) is relatively small. For that reason the response time of the recorder will be of importance only in cases of a very small mean residence time. The potential of the thermocouple was measured by a 3468A H P multimeter (Hewlett-Packard, USA). Thermostat U15” (GDR) was used for keeping the flow at constant temperature. The filling of the injection loop of the valve with tracer solution was done by aspiration with a LKB (12000 Varioperpex, Sweden) peristaltic pump. A HP-226 personal desk-top computer (Hewlett-Packard, USA) was utilized for processing of the experimental results. The recorded strip chart recorder response curves were digitized by a HP-9111A graphics tablet (Hewlett-Packard, USA). 3.3 Experimental Investigation of the Flow Systems. 3.3.1 Experimental Technique. The stimulus-response technique (40) was used for identification of the unknown parameters in the models described in the present paper. 3.3.2 Experimental Procedure. (a) Injection of Tracer. The valve injection was performed manually after filling the sample loop with tracer solution using the peristaltic pump. The hydrodynamic injection was performed in the following way: (1) The carrier stream circuit is stopped by closing of silicone-rubber tubes 1and 4 (Figure 2b). (2) Silicone-rubber tubes 2 and 3 are opened and the injection section stretching between the two T-pieces is filled a t low flow rate with standard tracer solution by aspiration with the peristaltic pump (Figure 2b). (3) Silicone-rubber tubes 2 and 3 are closed and silicone-rubber tube 4 is opened (Figure 2a). (4) The hydrodynamic injection itself is executed by opening silicone-rubber tube 1 (Figure 2c) with which the movement of the carrier solution is restored. The syringe injection (Figure 2c) was performed manually in a way so that the volumetric flow rate of the stream does not increase by more than 1.5%. One microliter of standard tracer solution was injected during each experiment. The point of injection was placed at X = 0 (Figure la). (b) Calibration Curves. Calibration curves were obtained on the basis of 20 standard solutions of potassium chloride (2.00 X to 8.00 X lo4 M) used as carrier solutions. The dependence between concentration and conductance has been described by polynomial regression equation of the second power. It has been found that the flow rate does not affect the concentration dependence of the conductance. 3.3.3 Response Curves. Tracer response curves for the manifolds with valve, hydrodynamic, and syringe injection and with the tubular conductivity cell were obtained at different flow rates (i.e., 0.2-2.0 mL min-’). In the FIA system with valve injection, the length and the diameter of the reactor were also varied (Le., 0.26-1.62 m and 0.33-0.82 mm, respectively). For the same manifold some of the response curves were detected by the thin-layer conductivity cell. 3.4 Processing of the Experimental Response Curves. 3.4.1 Unknown Models’ Parameters. The unknown parameters of the models described in this paper are the Peclet numbers of the different sections (Figures 1). Taking into consideration the geometrical dimensions of the experimental FIA manifolds a simplification has been made that reduces the number of the unknown parameters for each one of the two models by two. According to the simplification the values of the Peclet numbers of the foresection and the aftsection are calculated according to the theory of Taylor and Aris ( 4 S 5 1 ) P = aL/(D, + d2a2/192D,) (25) This assumption has been made on the basis of the following considerations: Let us consider an FIA system with hydrodynamic injection and equal diameters of all system sections. If the distance between the points of injection and detection is long enough so that the


dispersion is diffusion controlled, then eq 25 can be applied for calculating the Peclet numbers of all flow system sections which in this case are equal. If some of the sections except for the foresection and the aftsection are replaced, then it can be expected that the dispersion properties of the remaining two sections will be unchanged (Le., eq 25 is still valid for them) though the new sections may have totally different Peclet numbers. On this basis it can be concluded that if the foresection and the aftsection are straight empty tubes, then their Peclet numbers can be calculated according to the Taylor-Aris theory. The considerations made above assume that the effects related to possible differences between the diameters of the system’s sections will not play a significant role in the dispersion process in the foresection and in the aftsection. In FIA systems the conditions (Le., geometrical dimensions and flow rates) are such that d2zi2/(192D,) is much greater than D,. For that reason instead of eq 25, the so-called Taylor equation (49) has been used, i.e. PT = 48rD,L/~ = 487 (26) where IJ = (rd2/4)ais the volumetric flow rate. 3.4.2 Parameter Identification. There are two different approaches for the determination of the unknown model parameters from the experimental response curves (i.e., system output signals). These are the moments method and curve fitting (40, 52-54).

In the present investigation curve fitting has been chosen because it overcomes the following two substantial drawbacks of the moments method: First, numerical errors associated with the computation of the statistical moments of the response curves used by the moments method can be quite large. Moments calculations weight heavily the “tail” portion of experimental response curves, where the ratio of the signal to the noise is less favorable and measured curves are the least informative. The weighting of the ‘‘tail” increases greatly as the order of the moment increases. Second, the moments method assumes in advance that the model is an excellent fit to the data; therefore no statistical information concerning the quality of the model fit is obtained from the fitting process. It seems that the moments method is suitable only in those cases where the fit of the model to the data is knwon ahead of time to be quite good and where the model incoporates only a few parameters so that the use of the higher moments becomes unnecessary ( 5 4 ) . Another important reason used in choosing curve fitting is the fact that the models developed by us are multiparameter. For the identification of these parameters according to the moments method, moments of higher order are necessary. In this connection it should be mentioned that the corresponding Laplace domain solutions are cumbersome expressions and their subsequent differentiation for the calculation of the required moments (eq 13) and especially of those of higher order is extremely complicated. A simplex optimization method (55),which is regarded as very suitable for the cases of multiparameter problems, has been used for curve fitting. The function minimized by this procedure has been the square root of the mean squared error between the experimental and the theoretical response curves

The algorithm of Nelder and Mead (56) has been utilized. The “tail” of the experimental response curve, as its least informative portion, has not been processed during the curvefitting procedure. Only that part of the response curve where the concentration has been greater than 5% of the maximal concentration at its peak has been used for identification of the models’ parameters. 3.4.3 Calculation of the Theoretical Response Curves. The Laplace domain solutions of the models for single-line FIA systems have been numerically inversed by means of the methods employing expansion of the Laplace transform into a series of Chebyshov polynomials of the first kind and into a Fourier sine series (45). Using the axially dispersed plug flow model in cases when analytical solution exists (i.e., for double infinite, singly infinite, and closed vessel (45))the “critical” value of the Peclet

1705

number necessary for deciding which of the two numerical methods mentioned above is to be preferred for a given value of the Peclet number has been found. The models described in this paper include in the general case four or five different Peclet numbers so that it should be pointed out which one of them is to be considered for choosing the most suitable numerical inversion method. It has been assumed that the Peclet number that can be used for that purpose is the one that affects most strongly the output signal of the system. For that reason the sensitivity of the models to the Peclet numbers of the different sections has been investigated. It has been observed how the output signal of the system changes when each one of the Peclet numbers is varied. 3.4.4 Calculation of the Statistical Moments. The first and the second moments about the origin and the second moment about the mean for each experimental response curve have been calculated by using the coefficients of the corresponding cubicspline interpolation function (57). This method has been applied instead of the frequently used trapezoid rule because it assures higher precision of calculation (57). 3.4.5 Computer Program. A computer program written in BASIC 2.0 and run on a HP-226 personal desk-top computer has been used for performing all the calculations described above. 4. RESULTS AND DISCUSSION 4.1 Determination of the “Critical”Value of the Peclet Number. I t has been found that in the cases of P 5 10, the numerical inversion method based on expansion of the Laplace transform into a series of Chebyshov polynomials of the first kind should be preferred, whereas for P > 10, the method based on the Fourier sine series expansion should be preferred. 4.2 Sensitivity of the Model for the FIA System with Valve Injection to the Peclet Numbers of the System’s Sections. The model for the FIA system with valve injection has been used for the simulations because besides the sections of an FIA system with syringe injection, it includes one more section, i.e., the injection section. The parameters of the system simulated have been chosen to be typical for the FIA manifolds more frequently used in the practice (v = 1.0 mL min-l; x , = -0.20 m; x1 = 1.0 m, and x 2 - x1 = 2.0 mm). By variation of the values of the different Peclet numbers and comparison of the calculated output signals, the sensitivity of the model to these Peclet numbers has been investigated. The results obtained (Figure 3) show that the axial dispersion in the reactor, characterized by the corresponding Peclet number, affects most strongly the system’s response. When the Peclet number of the measurement cell has been varied, the simulated output signals have differed very slightly from each other. For that reason the corresponding concentration curves are not presented graphically. On the basis of these results, it has been concluded that the reactor’s Peclet number should be used as a criterion for selecting the more appropriate method for numerical inverse Laplace transformation. 4.3 Experimental Results. The results obtained from processing of the experimental tracer response curves may be obtained from the authors. 4.3.1 Experimental Proof for the Validity of the Models. Figure 4 illustrates the agreement between the experimental and the theoretical response curves obtained by curve fitting. The value of the diffusion coefficient of potassium m2 s-l (58). The kinematic chloride a t 20 O C is 1.77 X viscosity for the same temperature is 1.004 X 10+ m2 s-l (58). The following results confirm the validity of the models proposed in the present study: (1)In the range of flow velocities and reactor lengths where the Taylor-Aris theory holds (15, 59), i.e. 7

> 0.8

(28)

the values for the reactor’s Peclet number determined by curve fitting and according to the Taylor equation (26) are very close

1706


o

1 2 0 1 2 3 0 1 ~ 30 1 2 3 4 5 8 (C)

'1 0

1

2

3

9

(d) - - pa-10

Figure 3. Output signals calculated for different values of the Peclet number of the foresectlon (a), the injection section (b), the reactor (c), and the aftsection (d) [the Peclet numbers not varied in the simulations are equal to 20; v = 1.0 mL min-'; x , = -0.2 m; x 1 = 1.0 m; x 2 x , = 0.2 mm].

Figure 4. Experimental (- - -) and theoretical (-) response curves obtained at different flow rates in the manifolds with (a) tubular conductivity cell, valve injection and reactor's length 0.549 m, (b) tubular conductivity cell and hydrodynamic injection, (c) tubular conductivity cell and syringe injection, and (d) thin-layer conductivity cell and valve injection.

to each other (within 3% error). (2) For the FIA system with hydrodynamic injection, the Peclet number of the injection section does not differ substantially from the predictions of the Taylor theory (the mean relative error is less than 1.5%). This should be expected because in the experimental manifolds the injection section

and the foresection are straight tubes with almost equal diameters. For that reason the injection section may be regarded as part of the foresection. Besides that it has been already assumed that Taylor's equation (26) is valid for both the foresection and the aftsection. The result mentioned above confirms at the same time the validity of the assumption that


Taylor’s equation can be used for the foresection and the aftsection. (3) In the manifold with valve injection the length and the diameter of the reactor have been varied while in all experiments the injection section (i.e., the valve’s internal bores and external loop) has remained unchanged. For that reason its axial dispersion coefficient has had to remain constant when the flow rate has not been changed. The results obtained show that for close flow rates and different geometrical dimensions of the reactor, the corresponding values of the valve’s axial dispersion coefficient do not differ substantially from each other. The above results show that although the estimated Peclet numbers of the flow systems’ sections could be correlated with each other to some extent, as is frequently the case with such type of problems, their values agree quite well with the theoretical predictions and with the physical situation in the flow systems. 4.3.2 Axial Dispersion in the Valve Injector and in the Reactor. In all experiments the Peclet number of the valve injection section has been found to be greater than that predicted by eq 26 (i.e., PT).Probably this is due to the fact that the injection section has not been a straight empty tube. The nonuniformities of the flow there caused by bends, sudden expansions, and contractions have induced more intensive radial mixing, thus decreasing the axial dispersion. As has been already pointed out when T > 0.8 the experimentally determined values for the reactor’s Peclet number have agreed fairly well with the values calculated by Taylor’s equation. For T < 0.8 the experimental P, values have been found to be greater than those predicted by eq 26. This result agrees with the experimental data obtained in ref 21. 4.3.3 Delay in the Output Signal. It has been found that when the detection has been performed by the tubular conductivity cell, the theoretical response curves have lagged behind the experimental curves. Such effect has been observed also in other experimental investigations of single-line FIA systems (15,19,29,35)and its magnitude could be expressed by the factor of delay f This factor has varied from 0.800 to 0.932 depending on the reactor’s diameter (15,29). In the experiments carried out in this work, f has varied in broader ranges (0.430-1.000). By multiple regression analysis an empirical equation for calculating f as a function of the Reynolds number (7.83 < Re < 68.66) and the Fourier number T has been derived

f = 0.740Re0.151~0.387 for 0.058 < T C 0.8 = 1.0 for T > 0.8

(30)

The mean relative error in calculating f by eq 30 for T < 0.8 is 4.60%. The existence of this effect is probably due to a large extent to the fact that the electric field between the electrodes in the tubular conductivity cell has not been homogeneous. It has to be more intensive near the measuring surface of the electrodes than a t the core of the measurement cell. I t can be expected that this nonhomogeneity of the electric field will lead to distortion of the detected output signal when the tracer (i.e., potassium chloride) is not uniformly distributed in the cross section of the flow in the measurement cell. Obviously, for large T values, when the dispersion process is diffusion controlled and the concentration gradient in radial direction is small, the distortion of the output signal will be less pronounced. In the experiments such a situation has existed when the flow rate has been relatively small so that T has been greater than 0.8. In this case very good agreement between the experimental and the theoretical output signals has been

1707

obtained (Figure 4) and the value off has been very close to 1. For higher flow rates the factor of delay f has been less than 1and the deviations between the best fits and the corresponding experimental response curves (Figure 4) have been more considerable. Probably this result has been caused by the distortion of the system’s output signal. This conclusion has been additionally confirmed by the fact that the zeroth moment of the experimental response curves, which in the ideal case should be unity, has been less than 1 (Le., 0.770 < po < 1.00). Similar results have been obtained when instead of the flow rate the reactor’s diameter has been changed. For instance, when the diameter has been 0.330 mm, the value off has been very close to 1,while for the same flow velocities when d, = 0.821 mm, f has varied from 0.770 to 0.850. For the FIA manifold with a thin-layer cell, f has been approximately equal to 1,which could be explained by the fact that the electric field between parallel flat electrodes is almost homogeneous. The deviations between the experimental response curves and their best theoretical fits (Figure 4d) have been probably caused mainly by the measurement cell geometry and the electrode material, which has been stainless steel. To a small extent and mainly for small mean residence times, the response time of the recorder can also contribute to the delay in the appearance of the experimental curves. On the basis of the considerations above it can be said that most probably the adequacy of the models could be improved to some extent in one of the following ways: if the models describe the real physical picture in the measurement cell; if the measurement cell is constructed in a way to approach the assumptions on which the models are based (e.g., to have homogeneous electric field in the case of conductometric detection). The former possibility is accompanied with serious difficulties from a mathematical point of view concerning the description of the electric field in the measurement cell and the radial distribution of the analyte in it. In connection with this it seems that the latter possibility is to be preferred. 4.3.4 Comparison between the Method of Moments and Curve Fitting. The data obtained for the mean and the variance of the response curves show that the difference between the theoretical and the experimental values increases with the increase of the order of the moment and with the decrease of the value of the Fourier number. The former relationship could be predicted on the basis of the conclusions made when the parameter identification methods have been discussed while the latter one probably is connected with the distortion of the output signal at lower T values (i.e., T < 0.8). The method of moments has been applied for determination of the Peclet numbers of the reactor and of the measurement cell in the manifold with syringe injection. For that purpose the corresponding equations for the mean and the variance have been used. The equation for the mean contains only one of the two unknown parameters mentioned above and its solution should give it. This parameter is the Peclet number of the measurement cell. The calculations have shown that very often the solution of this equation gives unreasonable values for it (e.g., negative values). This unexpected result can be explained by the fact that the error in the determination of the mean is considerable with respect to the very strong dependence of P, on pl. For that reason the values of the measurement cell’s Peclet number, necessary for calculation of the reactor’s Peclet number, have been taken directly from the data determined by curve fitting. The reactor’s Peclet number calculated from the equation for the variance differs considerably from the corresponding values determined by curve fitting (Table 111). This result confirms the conclusions

1708


Table 111. Comparison between the Moments Method (a) and Curve Fitting (b) T

P, (a)

P, (b)

0.916 0.469 0.174 0.123

47.29 29.44 22.73 58.01

44.71 24.19 11.87 9.52

made when the moments method and curve fitting have been previously discussed. I t should be taken into consideration that the great differences between the Peclet numbers calculated by the two methods at low T values are due not only to the unreliable “tail” portion of the tracer response curve (the use of which is one of the main drawbacks of the moments method) but also to the distortion in the output signals caused by the nonhomogeneous electric field in the measurement cell. For that reason a t higher T values the deviations between the moments method and curve fitting are much smaller. 5. CONCLUSIONS The analytical-experimental mathematical models for single-line FIA systems developed and experimentally confirmed in the present work show the following three main advantages: (1) They take into consideration the differences in the dispersion properties and the geometrical dimensions between the sections of the described flow systems. (2) They consider various ways of analyte injection and detection. (3) The models for FIA systems with detectors measuring the local or the average concentration are general also with respect to the type of the reactor used. The parameter identification procedure based on curve fitting can be utilized for the determination of the diffusion coefficients of the injected particles (e.g., ions, molecules) when the Taylor theory is valid for the reactor. The mathematical models outlined in this work could be used also for investigation of the flow pattern in chromatographic manifolds and in process systems composed of tubular sections. After the introduction of the necessary additional terms (e.g., chemical reaction terms, separation terms) in the equations comprising the models mentioned above, the models can be applied for the description of flow analysis and process systems in the case of chemical reaction or for the modeling of the processes in chromatography. ACKNOWLEDGMENT The authors thank Kllra T6th for valuable discussion and helpful remarks. NOMENCLATURE A = function defined by eq 15 and 24 c = concentration of the analyte in the flow [kmol m-3] c, = concentration of the analyte in sample [kmol m-3]* C = c V,/ ( Vinjco), dimensionless analyte concentration C = Laplace transform of C d = tube diameter [m]* D = dispersion coefficient in the general dispersion model [m2 s-11 DL, = axial dispersion coefficient in the general dispersion model for symmetrical pipe flow [m2 s-l] DR, = radial dispersion coefficient in the general dispersion model for symmetrical pipe flow [m2 s d ] DLm= axial dispersion coefficient in the uniform dispersion model [m2 s-l] D , = radial dispersion coefficient in the uniform dispersion model [m2 s-l] DL‘ = axial dispersion coefficient in the dispersed plug flow model [mz s-l] DR = radial dispersion coefficient in the dispersed plug flow -model [m2 SKI]

DL = axial dispersion coefficient in the axially dispersed plug flow model [m2 D, = molecular diffusion coefficient [m2 s-’1 f = factor of delay F = square root of the mean squared error k = time span of syringe injection [SI K = ku/ V,, dimensionless time span of syringe injection 1 = length of the kth section in a flow system [m] L = -xo + x1 + 5 x / 2 for the system with syringe or deltafunction injection (Figure la) or x1 + A x / 2 for the system with hydrodynamic or valve injection (Figure lb), characteristic length [m] m = number of connected in series tubular sections in a flow system np = number of processed points from the tracer response curve p = Laplace complex variable P = uL/D,, Peclet number q = [0.25 + p / ( + y P ) ] ” 2 r = radial position [m] Re = Ud/u, Reynolds’ number t = time [SI Tr = Transitional time defined by eq 3 [SI u = linear flow rate [m s-l] u = mean linear flow rate [m s-l] u,= linear flow rate for symmetrical pipe flow [m s-l] u = volumetric flow rate [m3 V = characteristic volume if the diameter along the characteristic length L is constant and equal to d [m3] Vlnl = injected sample volume [m3] V , = (x/4)(-x0df2 + xld? + Axd,2/2) for the system with synringe or delta-function injection (Figure l a ) or ( x / 4)(x,d,2 + AxdC2/2)for the system with hydrodynamic or valve injection (Figure lb); characteristic volume [m3] x = axial distance [m] X = x / L ; dimensionless axial distance y = distance along the y axis of a rectangular coordinate system [m] z = distance along the z axis of a rectangular coordinate system [ml Greek letters Y =

vt/v

xo) = Dirac delta-function [m-’1 6 ( X - X,) = L6(x - xo); dimensionless Dirac delta-function Ax = x 2 - x,; length of the measurement cell [m]* AX = A x / L ; dimensionless length of the measurement cell Q = tv/V,; dimensionless time pi = ith moment about the origin of the output signal o = kinematic viscosity [m2 s-’3 o2 = variance of the output signal 7 = Fourier number (reduced to molecular diffusion scale mean residence time) $ ( t ) = rectangular input signal $(e) = ( Vt/u)$(t);dimensionless rectangular input signal

6(x

-

Subscripts 0 = the point of syringe or delta-function injection 1 = the beginning of the measurement cell 2 = the end of the measurement cell a = the aftsection A = the time of appearance of the response curve av = the average concentration in the measurement cell c = the measurement cell calcd = theoretically calculated value e = the beginning of the injection section exptl = experimentally obtained value f = the foresection i = the injection section k = the kth point of the tracer response curve m = the middle of the measurement cell and the local concentration r = the reactor T = Peclet number calculated by Taylor’s equation Superscripts


h = hydrodynamic injection n = normalized concentration s = syringe injection u = valve injection 6 = delta-function injection

* These physical quantities are given in the Paper in other, u9dy in the literature, units -, pL, mL min-’) for making the experimental conditions and results easier for direct evaluation.

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RECEIVED for review May 18,1987. Resubmitted December 3, 1987. Accepted March 14, 1988. Support for this research came from Project No. MTA-KKA 499/85 of the Hungarian Academy of Sciences.

Mathematical modeling of single-line flow-injection analysis systems

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