ANALYTICA CHIMICA ACTA

ELSEVIER Analytica Chimica Acta 346 (1997) 285-297

Monitoring time-varying concentrations in sample streams by multiple input chromatography

" a *

D.J . L o u w e r s e a, H . C . S m l t ' , M . K a l j u r a n d b

a Laboratory for Analytical Chemistry, University of Amsterdam, Nieuwe Achtergracht 166, 1018 WV Amsterdam, Netherlands b Department of Analytical Chemistry, Institute of Chemistry, Akadeemia tee 15, Tallinn EEO026, Estonia

Accepted 10 December 1996

Abstract

The possibility of monitoring time-varying concentrations with conventional chromatography or correlation chromato- graphy (CC) is restricted. The highest frequency that can be monitored is, due to Shannon, limited for conventional chromatography to (2xchromatogram length) -l , and for CC, due to a moving average effect, to (sequence length) -1. A method is presented in which considerable higher frequencies can be monitored. Theoretically the power spectral density of the chromatographical peak shape limits the upper frequency that can be monitored. The method originates from CC. A pseudo-random binary sequence (PRBS) is used as injection pattern. The detector signal however is not correlated or deconvoluted with a PRBS as in CC. The time-varying concentrations of the components are described as functions of time and a number of parameters, e.g. parameters of a Fourier series. These parameters then are optimised in a non-linear fitting procedure. A detector signal is calculated using the parameters, the known PRBS and known peak shapes; and the squared differences with the real detector signal are minimised. The results from simulated experiments and practical experiments are shown.

Keywords: Process monitoring; Multiple input chromatography

I . Introduct ion

Monitoring time-varying concentrations with mul- tiple input chromatography (MIC) has been a point of discussion and subject of research from the very start of correlation chromatography (CC). Already in their first paper concerning CC [1], Izawa et al. pointed out that with this correlation technique it is possible ' to enable continuous measurement of each component of injected sample gas'. Several researchers have been dealing with the subject ever since [2-13]. In 1969,

*Corresponding author.

0003-2670/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PI I S0003-2670(97)00 138-4

Godfrey and Devenish [2] described a situation in which the, so-called, correlogram (a virtual chroma- togram calculated by cross-correlating the detector signal with the pseudo-random input signal) was distorted very severely due to a changing concentra- tion. Also Annino and Bullock [3,4] noticed the effects of a changing sample concentration and described situations leading to a distorted correlogram. Kaljur- and and K~illik [5-7,9] contributed substantially to this subject by applying CC for pyrolysis gas chromato- graphy and thermochromatography. Mitra and Bose [13] also described a situation with changing compo- nents concentrations. They however only showed the

2 8 6 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

result of the deconvolution procedure before and after, and not during the system change.

Applying CC for monitoring time-varying concen- trations will always produce correlation noise (noise present in the correlogram that is not originating from detector noise) as a consequence of these variations. In CC the detector signal y(t) can be described as the impulse response h(t) convoluted with the input func- tion x(t), plus detector noise n(t). The impulse response h(t) can be compared with a noise free chromatogram originating from a single injection and the input function x(t) mostly is a pseudo-random binary sequence (PRBS).

y(tl) = x(/1) * h(t2) + n(/1), (1)

where * is the convolution parameter, tl is runtime of the process, and t2 is chromatogram time.

The correlogram ht(t) can be calculated either by deconvoluting y(t) with x(t) by means of a Fourier or a Hadamard transformation, or by cross-correlating y(t) with x(t). The noise in the correlogram originating from detector noise, n'(t), is reduced when it is compared with the detector noise present in a single injection chromatogram,

h'(t) = h(t) + n'(t). (2)

A stationary impulse response h(t) is required for CC. If h(t) is not stationary due to fluctuating con- centrations, the deconvolution or correlation proce- dure will calculate an average h'(t). The resulting correlogram will be erroneous. Errors will show up as correlation noise. The level of the correlation noise is related to the deviations of the components ' con- centration. In several cases the correlation noise level can be predicted [6]. Kaljurand and Ktillik review this subject extensively in their book [14]. It is very difficult if not impossible to suppress correlation noise in practice. The frequency characteristics of this noise are equal to the frequency characteristics of the com- ponent peaks. Common filtering techniques cannot be used as they equally reduce the correlation noise and the component peaks, there is no gain in signal-to- noise ratio (S/N).

The deconvolution or correlation procedure average the concentration levels of the components over a period of one PRBS. When the concentration of a certain component varying in time is monitored with CC by continuously calculating the correlogram and

extracting the peak information of the component, the obtained signal will be comparable with the original signal filtered by a moving average with a time window of one PRBS. When a signal is filtered with a moving average especially high frequencies are reduced. This process can be demonstrated more easily in the Fourier domain. The Fourier transformed (FT) signal consists of amplitudes and phase-shifts as a function of the frequency. Filtering a signal with a moving average is equal to multiplying the FT of the signal with the FT of a moving average which is a sinc function. Fig. 1 shows the Power spectral Density (PSD) (a squared form of the FT signal) of a moving average to illustrate the effect. When the signal of the monitored component includes a periodicity equal to the period length of a sequence (frequency 1 in Fig. 1), this periodicity will almost completely be filtered out. When in conventional chromatography the sampling interval is equal to the chromatogram length, accord- ing to Shannon a periodicity larger than twice this interval can be monitored. This corresponds to a maximum frequency which can be monitored at 0.5 in Fig. 1. Compared to conventional chromatography, CC can monitor faster variations, but not more than approximately twice as fast.

This is in contrast to what could be expected when only the (average) injection frequency is considered in combination with the spectral properties (e.g. the PSD) of a chromatographic peak shape. The clock

p 0

W

e

r

i 2 3 4

frequency

Fig. 1. PSD of a moving average. With a moving average length equal to a PRBS sequence length, the frequency scale is 1/(PRBS sequence). The highest frequency that can be monitored with CC is approximately 1/(PRBS sequence).

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 287

period of a PRBS (being the length of a PRBS ele- ment) in CC usually is small compared to the peak width of a component. If so, the PSD of the detector signal will be comparable to the PSD of the compo- nent's peak shape. The PSD of a components peak shape thus determines the maximum frequency that can be monitored. This will be illustrated later. Fast varying concentrations however are smoothed by the averaging effect obtained, calculating the correlo- gram.

Many applications shown in literature applying CC for monitoring purposes show a rather simple separa- tion [2-4,7-9,11]. Often the components involved in the monitored process are known. If so, an alternative approach is suggested in this paper. Chromatographic

pre-information like the retention time and the peak shape of the components is used in the deconvolution process. The deconvolution or correlation procedure in CC is substituted by a fitting procedure. The information about the PRBS injections, and the chro- matographic information of the components is used to estimate the concentration level of a component as a function of time, by appropriately fitting the detector signal.

2. Theory

Fig. 2(a) schematically shows the CC process as described in the previous paragraph with a stationary

A B

PRBS

impulse response

PRBS-response

correlogram

Fig. 2. The CC process schematically for monitoring components with (A) constant and (B) time-varying concentrations. When an impulse response that is constant in time is injected according to a PRBS (A), the PRBS response (detector signal without noise) becomes circular after a presequence. The resulting correlograms are constant in time and equal in shape compared with the impulse response. With an impulse response that is not constant in time (B) the PRBS response does not become circular or constant in time. The correlograms are disturbed by noise and less similar compared to the impulse response.

288 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

impulse response (Eqs. (1) and (2)). In case of time- varying concentrations the impulse response h(t) of Eq. (1) is not constant in time and becomes two- dimensional, h(q, ta). Substitution in Eq. (1) gives

y(tl) ---- X(tl) * h(tl,te) + n(tl). (3)

It follows from Eq. (3) that the impulse response h(t2) changes with the injection time q. Calculating the correlogram on a semi-continuous basis, as described in Section 1, produces a two-dimensional correlogram hl(tl,ta) with the mentioned deficiency. Fig. 2(b) illustrates this process.

If components are known, a situation can be described in which every individual component with a changing concentration has an impulse response. Components with a constant concentration level can be combined in one joint impulse response if neces- sary. When for instance three components: A, B and C, are present in a process, and the concentration levels of all three are varying in time, the two-dimensional impulse response of Eq. (3), h(q,t2), can be divided into three impulse responses. For each component z, a two-dimensional 'component impulse response' hz (q , t2) can be defined.

h(tl~t2) = hA(tl,t2) + hB(tl,t2) + hc(tl,t2). (4)

This two-dimensional component impulse response can be decomposed into a (normalised) one- dimensional component impulse response (peak shape and retention time), h(t), and a function repre- senting the concentration level of the component in time, c(t).

h(tl,t2) = CA(tl) × hA(t2) + CB(tl) × hB(t2)

+ Cc(tl) × hc(t2), (5)

where × is the vector outer product. Fig. 3 illustrates the process. The detector signal of

Eq. (3) then becomes

y(tl) = X(tl) * [CA(tl) × hA(t2)]

-~- X(t l ) * [CB(tl) × hB(t2)]

+ X(tl) * [C¢(tl) × hc(t2)] + n(tl). (6)

Since the time tl has the same meaning for CA, Cn, Cc and x (all runtime of the process) CA(t0, CB(tl) and Cc(tl) can be considered as constants in the convolution integral with respect to X(tl) and Eq. (6) becomes

y(tl) = Ix(t1) • CA(tl)] * hA(t2)

+ [X(tl) • CB(tl)] * hl3(ta)

+ {x(tl) • CB(q)] * hc(t2) + n(q), (7)

where • is the vector inner product. The amount of injected component as a function of

time is obtained by multiplying the PRBS input function x(t) (with value 1 for injection and 0 for no injection) with a value equal to the concentration level of a component, c(t). The result is convoluted with the component impulse response (peak shape and retention time) of the corresponding component. The detector signal consequently is the summation of this process for all components, plus detector noise.

It is impossible to calculate the concentrations, CA(t), Cn(t) and Cc(t), by a straight deconvolution procedure of the detector signal when x(t), hA(t), hB(t) and hc(t), are known. Ca(t), Cn(t) and Cc(t) will give three times more unknown variables than known (measured) detector points. The amount of variables therefore has to be reduced substantially. This only is possible when the signals CA(t), CB(t) and Cc(t) themselves show a correlation in time. C(t) can be described by an nth order polynomial e.g. by a limited number of Fourier terms with increasing frequency. Starting with frequency zero, describing a constant concentration in time, increasing frequencies are added until the fluctuating concentration is described well. When for instance the fluctuations of three components are described, each by a second order polynomial (three variables) nine variables have to be determined. A non-linear fitting procedure can be used to estimate these nine variables.

Some system theoretical aspects have to be taken into consideration when this approach is chosen. Injecting according to a PRBS corresponds to intro- ducing a band limited white noise, noise with a flat PSD in the frequency range of interest, into the (chro- matographic) system. The system can be represented as a filter, only certain frequencies will pass and fre- quency phases will be shifted. In case of a component with a Gaussian impulse response and a time-varying concentration according to the fastness of fluctuation three domains of interest can be distinguished:

(i) A constant C(t) (no or a negligible variation). This is the simplest situation, it is equal to the conventional CC process described in Eq. (1).

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 289

l÷ A -I-

)

Fig. 3. A two-dimensional impulse response of three components each with its own time-varying concentrations. This impulse response can be decomposed in three separate two-dimensional impulse responses, one for each component. Each separate impulse response can be further decomposed in a one-dimensional component impulse response and a time-varying component concentration function.

(ii) The variations in time are slow with respect to the peak width of the component. The convolu- tion process with h(t) will hardly change them and they will be present in the detector signal. (iii) The variations in time are equal or fast with respect to the peak width. The convolution process will level them out and they will hardly or not be present in the detector signal.

Domain (ii) and (iii) can be illustrated with a simple simulation with one component, visualised in Fig. 4. A representation in the Fourier domain of Eq. (7), for one component without noise, is chosen here below.

Y( jw) = [X(]w) * CA(jw)] -HA(]W). (8)

The Fourier transform (FF) of the detector signal, Y(j~), can be described as the convolution of the FT of the PRBS input function, X(j~), with the FT of the concentration of component A as a function of time, CA(jw). The result is multiplied with the F r of the impulse response or the peak shape of component A, HA (j~). When the injection signal (being the convolu- tion of X(j~) and CA(j~3) in the Fourier domain) contains certain high frequencies that are not or hardly present in the peak shape of the component, they respectively will be eliminated or reduced by the multiplication in the Fourier domain. Fig. 4(a) shows a simulation of a PRBS with 1023 periods, modulated with two frequencies, one with 5 and the other with 16

290 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

il/m/ 0 30O 600 90O

~6 5 18 15 20 25

r e F requency

\

periods per sequence. Both modulated frequencies are present in the PSD of this injection signal (Fig. 4(b)). The higher frequency is strongly reduced and the lower frequency is still clearly visible (Fig. 4(d)) after multiplication in the Fourier domain with the simu- lated component peak (Fig. 4(c)). The accuracy in determining fast changes in concentrations which are dampened by the peak shape of the component is strongly reduced, especially when also detector noise is present.

Another important aspect which has to be consid- ered is the situation in which the peak shape of two components are (almost) equal and only their retention time differs. In this case components can only be distinguished on their difference in phase. These phase differences are not influenced by the deconvolution process when according to a PRBS is injected. Using other injection patterns may give problems. When e.g. equidistant injections are used, differences which are once or a multiple times as large as the period between two injections cannot be distinguished. An example will illustrate this. Two components with equal peak shape and a difference in retention time of 40 s are injected every 10 s. A change in concentration of one of the components can be noticed by monitoring the detector signal. It is nevertheless impossible to deter- mine whether this change can be attributed to a changing concentration of the first component at a certain time or to a 40 s earlier changing concentration of the second component. They cannot be distin- guished as the difference in retention time is equal to four times the injection interval. These problems will not occur when a PRBS is used.

r e t F requency

<E \

D 5 10 15

t e l F requency

3. Simulation experiments

To investigate the discussed properties three simu- lation experiments were performed. Experiment I deals with only one component. The limits of the technique and the behaviour of simulated detector noise were investigated. Experiment II deals with

Fig. 4. (a) Modulated PRBS. First 25 points of the PSD of (b) the modulated PRBS, (c) the Gaussian peak shape and (d) the detector signal. The dashed lines in (b) and (d) represent the PSD of the unmodulated PRBS and the resulting detector signal respectively. The offsets or zero frequencies of the PSDs are not included.

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 291

two components, both with equal peak shape but a different retention time so they can only be distin- guished on their difference in retention time (or phase shift). Experiment III deals with three components, each component with its own peak shape and retention time to investigate a more practical situation.

The concentration level of the components were independently stochastic fluctuating around an offset. To simulate this, Gaussian distributed noise was pro- duced [15], with a standard deviation (SD) of one and an autocorrelation function (ACF) of ACF(t) = e - ( t / T ) 2 where 7- is the time constant. Sev- eral values of the time constant 7- were chosen to influence the fastness of fluctuation. An offset of ten was introduced to prevent negative concentration values. For each component and for each time constant a record was produced with a length equivalent to ten PRBSs of 127. The detector signal was created accord- ing to Eq. (7); for each component its time varying signal was multiplied with a 10 times repeating PRBS of 127 and then convoluted with its peak. The final detector signal was the summed result. Experiment I was performed with and without additional white noise to simulate detector noise. Fig. 5 shows a chro- matogram of the three components with equal con- centration levels, the peak parameters are listed in the caption of Fig. 5.

The concentrations of the components as functions of time were calculated over a time period of one PRBS. The process was started with a part of the detector signal equal to one PRBS plus the total peak width of the widest peak. This detector signal was fitted with a non-linear fitting procedure minimising the least squares according to the Levenberg-Mar- quardt algorithm [16]. Fourier terms were used to

0 3 0 6 0 9 0 1 2 0 1 5 0

r e l , T I M E

Fig. 5. Chromatogram of the three simulated components with an equal concentration level. The retention time and the sigma of the peaks are 30, 60, 100 and 3, 5, 7, respectively.

describe the varying concentration for every compo- nent. The number of Fourier terms per component was varied. Every Fourier term consists of a sine and a cosine resulting in two fitting parameters per compo- nent. The zeroth Fourier term describing the offset results in only one parameter per component. The time-varying concentrations calculated with the fitted parameters, were multiplied with the PRBS and con- voluted with the peak shape (Eq. (7)) keeping the proper time shifts and retention times into considera- tion. The average squared differences between the resulting signal and the detector signal was minimised by adjusting the parameters with the mentioned algo- rithm. To simulate continuous monitoring this proce- dure was repeated after shifting one PRBS clock period. This resulted in a concentration level as a function of time being the average of 127 calculations (the amount of PRBS clock periods).

4. Simulation results

The results of the experiments are presented in Fig. 6. The average sum of squares from the fits, SSav , and the standard deviation of the difference between the original time-varying concentration and the cal- culated concentration are shown for different time constants, 7-, and for different numbers of fitting parameters.

Fig. 6(a) and (b) show clearly for experiment I without noise, that when 7- decreases from 60 down to 10 time units an increasing number of Fourier terms is needed to describe the signal with the same accu- racy. All calculated signals converge to the 'real' signal. The convergence process is shown in Fig. 7 for T=15 time units in experiment I without added noise. For 7-= 10 time units the convergence however is slow, 18 Fourier terms still are not enough to achieve convergence. The lack of convergence is probably caused by the convolution process of the peak with the original time-varying signal. The peak width dampens fast changes. A second limitation is the detector noise. As the procedure minimises the differ- ences between the original and calculated signal, noise will limit the minimisation. The fitting procedure in this experiment is limited to a level of 0.01, which is the variance of the noise. At this level the signals with 7-=10, 15 and 20 time units do not converge comple-

292 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

i

8

Experiment I

1 + x = I0

0.8 ~ - 4 ~ = 15

~ -°-~= 15+N 0.6" ~ ""t--'~ = 20

• - -b ' -~=20+ N

--X'-X = 00

5 15 25

Number of fit parameters 35

a

3

_.~ 2.5¸

"~ 1.5 ¸

] 1 3= ~ o . 5

Eo

Experiment II

5 15

Number of fit parameters

[ ~ " I,i

25

i

1:

Experiment III 2 .5

2

1.5

1

0.5

0 I "1- .

5 15

Number o f fit parameters

- 4 - - z = 20 - 1 ---o-x= 20_ 2

---e--x = 20_ 3

" O " g = 30 - I

--Jv-~ = 30_ 2 --b--x = 30_ 3 ---e--~ = 00_ 1 -o--,~ = 60_ 2

25

Fig. 6. Resul t s o f s imula t ion exper imen t s . (a), (c) and (e) show the s tandard dev ia t ion o f the d i f fe rence b e t w e e n the or iginal and the

ca l cu la t ed s ignal m e a s u r e d for d i f ferent 7- and c o m p o n e n t s (or noise) . (b), (d) and (f) show the a v e r a g e s u m o f squares f r o m the fits.

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 293

!

!

100

1

0.01

0.0001

0.000001 '

0.00000001

Experiment I

- - m - - ~ = 1 0

" , ~ " ~ ~= . ,~ . .= - '~ -'-~=~

I I I I I I I I

5 15 25 35

Number o f f i t parameters

b

m =-

!

100

1

0.01

0.0001 '

0.000001 '

0.00000001

Exper iment I I

I I I I I I I I I I

5 15

N u m b e r o f f i t p a r a m e t e r s

a-...------a

I I

25

d

== .=

10(]0'

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0.1

0.001

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0.0000001

Exper iment I I I

!

5 15 25

N u m b e r o f fit p a r a m e t e r s

Fig. 6. (Continued).

tely. Signals with a larger time constant are not affected by this noise level. Estimating the original signal with a certain accuracy is dependant on both ~-

and the amount of detector noise present. With decreasing ~- the acceptable amount of noise also decreases.

294 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

: . . ,, ,... , . . . ,

• .; -:

2 0 0 4 0 0 6 0 0 8 0 0

r e l T I M E

Fig. 7. An example of the convergence process with an increasing number of Fourier terms. From experiment I without noise the convergence of the signal with T=15 time units is shown. From top to bottom 1, 4, 8, and 12 Fourier terms are used for the fit. The dashed line is the original fluctuating signal.

Fig. 6(c) and (d) show the effect of interference when two peaks have equal peak shapes. The signals can only be distinguished on their different retention times (phase differences). Interesting results are obtained when the Fourier terms do not adequately describe the original signal. Going from 1 to 2 and also from 6 to 7 Fourier terms, the Fig. 6(c) and (d) with ~-=20 time units show a decreasing SS,v, the calculated signals compared to the original signals however worsens. Fig. 8 illustrates this in more detail. With 6 Fourier terms the calculated signal fits the original signal only to a certain extent (Fig. 8(a) and (b)), the original signal still contains higher frequencies which are not included in the calculated signal. By adding one Fourier term the sum of the two signals is esti- mated better, but the procedure cannot distinguish properly between the two signals. Where in Fig. 8(c) the signal is overestimated, the signal at the same time in Fig. 8(d) is underestimated.

Fig. 6(e) and (f) show the results of the three components simulation of experiment III. The follow-

Fig. 8. Original and calculated signals from simulation experiment II. The time-varying concentration of two components with an equal peak shape is simultaneously estimated. The dashed lines are the original, the straight lines are the estimated signals. (a) and (c) are the signals of the first component estimated with respectively 5 and 6 Fourier terms, (b) and (d) are the comparable signals for the second component. (c) and (d) show locally under or overestimated signals (marked with ' <== ' ) .

<

®

0 200 400 600 800

tel TIME

m

0 2 0 0 4 0 0 600 800

r e l T I M E

m

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0 2 0 0 400 6 0 0 800

t e l T I M E

0 2 0 0 4 0 0 6 0 0 800

r e l T I M E

a

1000

b

1000

I000

d

1000

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 295

ing can be concluded when experiment I is compared with experiment III:

The time constant of the signals that can be determined with a certain accuracy is higher when more components are involved. The maximum amount of noise that is allowed to achieve a certain accuracy lowers, when the number of components rises for which a time- varying concentration with a certain time constant has to be estimated.

The number of parameters to be estimated in this experiment is linearly related to the number of com- ponents. When the time-varying concentrations of three components are estimated each with for example 5 Fourier term, this requires 27 parameters (3(1+4×2)) to be estimated. Fig. 6(e) clearly shows for T=20 time units that the signals do not converge completely, the quantisation noise of the computer here is the limiting factor (Fig. 6(f)). With a large number of parameters the algorithm also has problems finding the optimum. The response surface in the multi-dimensional parameter space flattens.

5. Experimental set-up

The correlation system used was a slightly modified liquid chromatography (LC) system [17] first

described by Mars [18]. Table 1 lists the equipment, and Fig. 9 shows a schematic diagram of the equip- ment used for the experiments. Chromatographic con- ditions during the experiments are: column, 7.5 cm×4.6 mm i.d. filled with 5 ~tm Hypersil ODS; buffer, 10 mmol 1-1 acetate pH--4.8; eluent, metha- nol/buffer, 40/60 v/v, filtered through 0.2 ~tm; eluent flow rate 0 . 9 m l m i n - 1 ; sample loops, 10~tl; UV detection, 280nm. Sample solutions, sample I: 16 mg 1-1 phenol, sample II: 16 mg 1-1 acetophenon, sample III: 8 mg 1-1 nitrophenol, all sample solutions were made in eluent. The time-varying concentrations were made by using a separate pump for every sample solution. The total flow of the pumps was kept con-

Table 1 List of equipment

Apparatus Type

Valve

Air-control unit

UV detector

Pumps Diode array detector CC-computer

Flow control Data-handling computer

10-port Valco HPLC valve AC 10W, air actuated Valco digital valve interface with speed- up kit Waters model 441 (280 nm, 12.5 ktl flow cell) Kratos Spectroflow 400 Hewlett Packard 1040A Tulip dc386 with DAS-16 DAC/ADC controlled by Assyst Acorn-master microcomputer HP-9000/350

: ' ' "~1 F ' ~

D I . . . . .

D . . . . . •

component pumps

• v vYT-TYv-7~ TM (0,0,y,y,0,0,0) mixing coil

E3 eJuens pump

. . . . . . .~t... [ '~]. .Ira,- . . . . . .

r3-- ir ec to r column detector

Fig. 9. Schematic diagram of the equipment used for the CC experiments.

Table 2 Flow combinations of sample pumps

gg

"O

a

Pump flow (ml min - I )

Phenol Acetophenon Nitrophenol

1 0.74 0.12 0.04 2 0.42 0.44 0.04 3 0.38 0.12 0.34 4 0.12 0.44 0.34

stant at 0.9 ml min -1, the flow of the individual pumps could be switched between 2 levels for sample solu- tions 2 and 3, and 4 levels for sample solution 1. Table 2 shows for all possible combinations the indi- vidual pump flows. Every 20 s the flow levels are

randomly switched. The outlets of the three pumps were connected with a Valco cross piece to the inlet of a mixing coil. The mixing coil contained 6 ml and dampened the fast changes in component concentra- tion. The outlet of the mixing coil was connected with a diode array detector (DAD), and after that with the CC injection system. The CC experiments were con- trolled by a Tulip dc386, a PRBS of 63 elements was used with period length of 2.4 s, the data were col- lected with a period length of 1.2 s. The sample stream switching process was controlled independently of the CC process with an Acorn master microcomputer. During the CC process the DAD collected a spectrum of the sample mixture every 7.2 s. Spectra of the pure sample mixtures were also collected. The chromato-

0.08

0 .06

0 .04

0.02

0 i 0

Qt~

W

C" w

to

- - a - p h e n o l

- - i , - n i t r o p he n o l

- - O - a c e t o p h e n o n

I I I 3 5 7

N u m b e r o f f i t p a r a m e t e r s

a

296 D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297

• nc01

I I I 3 5 7

r,u.L,=- d f~ Ixrm~W

b

Fig. 10. Results of the experiments. As a function of the number of fit parameters. (A) shows the standard deviation of the difference between the estimated concentration from the spectra and the estimated concentration of the CC process. The standard deviation of the estimated concentration from the spectra itself is plotted at 0 fit parameters. (B) shows the average sum of squares from the fits.

D.J. Louwerse et al./Analytica Chimica Acta 346 (1997) 285-297 297

graphic peak shapes and retention times of the three components were determined by processing the detec- tor signal. The CC detector sequences, after the so called presequence, were added to one sequence. A correlogram was calculated, and the peak parameters were extracted from the correlogram with a non-linear fitting routine [19]. The peak shapes in combination with their retention times were used to estimate the original time-varying concentrations in the same way as the simulation experiments. The original varying concentrations were also estimated by a linear fit of the collected spectra during the CC process with the spectra of the pure sample solutions.

6. Experimental results

.

.

4.

be within range of those of the chromatographic peak shape. The number of components. When the time-vary- ing concentration of more components have to be estimated simultaneously, the maximum number of parameters per component is reduced. Detector noise. The amount of detector noise pre- sent limits the maximum number of parameters that can be used. Equal chromatographic peak shapes. When the peak shape of two or more components is equal, the components only can be distinguished on a difference in phase shift. When the number of parameters is not adequate to describe the fluctu- ating signal, a part of the signal can be assigned to the wrong component.

The results of the experiments are presented in Fig. 10. The average sum of squares from the fits, SSav, and the standard deviation of the difference between the estimated concentration from the spectra and the estimated concentration of the CC process, are shown for different numbers of fitting parameters.

The best results are obtained with 1 Fourier term for nitrophenol and 2 Fourier terms for phenol and acet- ophenon. Adding more Fourier terms did not improve the fit. The SSav hardly reduces and the estimated concentration of the CC process worsens compared to the estimate of the spectra (overfitting).

7. Conclusions

Monitoring time-varying concentrations with mul- tiple input chromatography is possible theoretically as well as in practice. When the time-varying concentra- tions are rapidly changing compared to a PRBS sequence or the length of a chromatogram (f>(sequence) -1) an alternative approach is suggested here. Simulations and practical experiments both prove that this method can be used in situations where the fluctuations are too fast for chromatography or conventional CC.

The fastest signal which can be monitored is limited by:

1. The component peak width. The frequency characteristics of the time-varying signal should

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