Automated regression analysis and its applications such as water analysis
Methods and instrumentation based on the concept of an “Absolute Principle” are relatively rare but, when applicable, offer some very unique advantages. This patent teaches how to automate the use of non-linear curve fitting to widen this concept enabling the determination of multiple properties instead of typically only a single entity as well as dealing systematically with more complex relationships between entities. As a possible application, the invention further deals with analyzing the composition of water regarding inorganic as well as organic species and the use hereto of conductivity measurements as a function of temperature or the mixing ratio with another fluid.
This application is based on PCT/US2003/029961 filing date Sep. 23, 2003, priority date Sep. 23, 2002.
Background1. Field of Invention
This invention relates to “Absolute Principle” based methods and instrumentation, mathematical model descriptions of phenomena, automation and the use of non-linear curve fitting algorithms. In its application it relates to water analysis, organic and inorganic components in aquaous solutions and conductivity measurements relevant to industries in the field of drink and waste water, semiconductor manufacturing, pharmaceutical manufacturing and power generation.
Apart from analysis, wider applications are typically in the field of monitoring chemical or mechanical processes and equipment, machines, engines or vehicles.
2. Description of Prior Art
Absolute Principle:
Analytical instruments based on an “absolute principle” translate the entity to be analyzed into one or more other entities that are directly, without any instrument dependent factors, related to the entity to be analyzed. The relationship between the entities is based on well-established literature values, laws of physics or chemistry, or the like. This means that, in principle, when the measurements of the related entities are correct, automatically the derived value of the entity to be analyzed will be correct.
The “chilled mirror” hygrometer is an example of an instrument based on an absolute principle. It translates the measurement of a concentration of moisture in a gas in the determination of the temperature of the onset of frost formation on the surface of a mirror in the process of being cooled down. Well-established curves from the National Institute of Standards and Technology (NIST) relate dewpoint temperatures to moisture concentrations. Such hygrometers cannot be calibrated by offering it calibrated moisture standards and certainly do not allow tweaking NIST curves making the instrument read the standard. Instead, the hygrometer's temperature sensor is calibrated with a traceable temperature standard. This last calibration is definitely easier to do than the first, specifically at very low concentrations were moisture standards are unavailable.
Calibrating an instrument typically means offering it a number of known calibrated standard concentrations, plotting these concentration values on the x-axis of a calibration curve and plotting the response of the instrument on the y-axis. Then, interpolating a line through the points creates the calibration curve. It is, commonly excepted, bad practice to extrapolate the curve into a concentration range where there are no more calibration points, leaving the instrument to be used only in the interpolated range. Unfortunately, in numerous cases it is very difficult to create standards with very low concentrations. Particularly in such cases, an absolute principle based method can be very valuable.
Non-Linear Curve Fitting:
Non-linear regression or curve fitting is known in the prior art and is one of the more sophisticated, difficult to use and to prepare for parts of the toolbox of a scientist. It enables in principle to mold measured data into a number of parameters used in a mathematical model that describes the phenomenon behind the measured data. It is typically applied as some form of the Levenberg-Marquardt algorithm on a case by case basis as part of a particular study. Its successful application heavily relies on the skills of the operator who needs understanding of the phenomenon as well as of the mathematical intricacies of the method. A number of pitfalls prohibit its use in an automated fashion such as:
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- the difficulty to distinguish sometimes numerous local solutions from the, unique, global solution,
- the requirement to come up with a reasonably good initial value set,
- limited knowledge of the underlying physical or chemical properties and their dependencies,
- the temptation to try to solve more unknowns than the number of equations which is not so easy to recognize as in a linear case,
- the need to define an accurate often typically non-linear mathematical model that fully describes the observable features in terms of a complete set of parameters,
- the non existence of an analytical solution for the mathematical model making that there are no formulas to program into the computer.
- complex interactions making that there is no single mathematical model but many such models depending on the selection of the species under investigation.
- lengthy calculation in terms of the mathematical model can make the use of the curve fitter impractically slow these calculations are down over and over again with slightly variations in the parameters.
Water Analysis:
Large quantities of water are used in industry with quality levels ranging from ultra pure in semiconductor & pharmaceutical manufacturing, to pure water in power generation, down to lesser purity levels in drink and waste water. All applications need some form of analysis capability where the large number of potential contaminants is a major complicating factor. First discussed, section (a), are laboratory type instruments capable of scanning such multitude of species with low detection limits followed by more on line style instruments that typically monitor one specific contaminant only, section (b). TOC analyzers are discussed in section (c) and address contamination by organic species. Section (d) discusses relatively inexpensive, robust, conductivity sensors that are used on line as a non-specific indicator for the overall water quality. Conductivity sensors feature fast response time and high sensitivity. Main disadvantages of conductivity sensors are the lack of information regarding the nature of the contamination and the dependency of the conductivity on the temperature. This last issue is addressed by the use of temperature compensated conductivity readouts. Such readouts however, are still troubled by the lack of knowledge of a sample's composition and the poor accuracy by which some of the chemical properties of the involved species are known, section (e).
(a) Laboratory Instruments:
Laboratory instruments typically rely on grab samples analyzed in a central laboratory. Where semiconductor fabs might have permanent lines going to sample points enabling to perform a number of measurements daily with little sampling contamination, drink and waste water are typically sampled at two week periods. The laboratory type instruments mentioned hereunder have the following common disadvantages of being very labor intensive, requiring high operating costs, high initial investment and frequent calibration.
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- Ion Chromatogrphy: An ion chromatograph is a laboratory instrument. It separates the different ions by their species specific elution time measured after injection of a small amount of analyte into a column. The instruments normally have more columns and need frequent calibration for each species to be analyzed.
- ICP: A nebulizer injects analyte into an argon or helium stream. A plasma is created in this gas stream followed by the observation of the optical emission spectrum (ICP-OES). Alternatively, ions created in the plasma can be routed into a mass spectrometer (ICP-MS).
Not a feature of the instrumentation itself but of the practice of (grab) sampling is the associated long response time and the potential of sample contamination. Obviously, applying such technologies to scan continuously online would provide a better guarantee of the ongoing water quality but are normally cost prohibitive.
(b) Specific Species Detectors:
For relatively high impurity concentrations there are a number of colorimetric or luminescence based techniques that use specific chemicals to create color change or light emitting reactions. One typically needs one instrument per chemical species to be monitored. Apart from limitations in detection limits, these techniques are typically not rigorously specific for a single species and involve often-delicate controlled flows of costly consumable chemicals. However, these techniques feature fast response times associated with on line applications.
(c) TOC Measurements:
Monitoring water properties such as Total Organic Carbon, TOC, is highly relevant in a number of important industrial processes; in particular, the semiconductor and pharmaceutical industries both use ultrapure water in large quantities.
Most state of the art methodologies for measuring TOC involve oxidizing organic molecules in the analyte water by UV radiation (see U.S. Pat. No. 4,626,413).
The produced CO2 can be measured in situ, typically by measuring the change in conductivity of the analyte water sample itself. One method, the “stopped flow technique”, measures the change in conductivity of a trapped, static, water sample over time, i.e., by monitoring the conductivity as the oxidation reaction proceeds, to determine the end conductivity value at the completion of the reaction (U.S. Pat. Nos. 4,626,413 and 4,666,860). Another method measures the conductivity of a continuously flowing water stream before and after it is exposed to a UV lamp that oxidizes the organics in the stream (See, e.g., Egozy U.S. Pat. No. 5,272,091). Note that this last method is potentially faster than the first, but might suffer from incomplete oxidation giving too low TOC values. The stopped flow technique might suffer from a buildup of contamination in a stagnant analyte from leaching or diffusion of atmospheric carbon dioxide through O-rings, seals and insulators. The continuous flow technique does not have such buildup as it is flushed continuously. These two basic methods use a conversion algorithm to generate a value for the TOC content based on a conductivity value prior and after oxidation. This algorithm assumes that the sample prior to oxidation consists only of pure water and dissolved CO2, referred to as TIC or total inorganic carbon. These methods further assume that after the oxidation only oxidized carbon is added, originating from organic carbon contamination. The algorithm will produce the TOC value strictly based on the conductivity measurements. This means that once the conductivity measurements are calibrated with traceable conductivity standards, no further calibration is required with TOC standards. This so-called “absolute principle” is a particular advantage when measuring very low concentrations for which it is difficult to generate TOC standards. Note that one has to be assured of a hundred percent oxidation of the organic carbon. This assurance can be obtained by a system suitability test that compares the results of a sample of an easy to oxidize component with a sample of a difficult to oxidize component.
The assumptions in the above-discussed algorithm can be too much of a simplification when interfering species are present. Specifically, non-conductive species such as THMs and nitrogen containing species like urea can generate a conductivity change under UV irradiation only partially caused by the presence of carbon atoms thus producing an erroneous TOC reading. The presence of unknown species can also generate errors in the temperature compensation applied in the conductivity measurement, which will impact the TOC reading.
The above interfering species problem is addressed by non in situ methods that remove the produced CO2 from the analyte. The CO2 can then be measured in the gas phase by, e.g., NDIR spectroscopy, although this approach is typically much less sensitive. Alternatively, the CO2 can be separated from the analyte water, e.g., by diffusion across a suitable selective membrane, to generate a conductivity change in a separate UPW (Ultra Pure Water) stream. A drawback of this, more complex, last approach is that the membrane introduces a factor that has to be determined empirically for each individual instrument which requires calibration with TOC standards. This factor can be subject to drift while its behavior at very low concentrations might be compromised.
(d) Conductivity Measurements:
The following paragraphs describing the state of art of conductivity measurements are taken from “Instruments, unique temperature compensation for conductivity and resistivity measurement”, David M. Gray and Anthony C. Bevilacqua, Ph.D., Thornton Associates Inc., Ultrapure Water, January/February 1996, p. 60-62.
Conductivity and resistivity measurements are the most common, reliable, sensitive, accurate, and low-cost means of monitoring water purity for typical mineral contamination. A critical part of this monitoring is to eliminate the temperature dependence. There are three parameters required to make accurate temperature-compensated conductivity measurements: a conductivity measurement, a temperature measurement, and knowledge of the type of impurity. Modern instrumentation is capable of measuring the conductivity and temperature with one sensor device and one meter, but the compensation for a specific impurity depends upon the application or the industry convention. All this presumes that the instrumentation has the capability to compensate for that specific impurity.
Pharmaceutical Industry: With the continuing revision of United States Pharmacopeia 23 (USP 23) water specifications, there is a need to provide limits on a variety of USP-identified contaminating ions. The least conductive of these ions at the allowed levels are chloride (CL−), ammonia (as NH4+), and innocuous ions that are always present (hydrogen [H+], hydroxide [OH−], bicarbonate [HCO3−]), and counterions to chloride [Cl−]. Since a conductivity measurement cannot differentiate among the different ions, it cannot compensate correctly for all possible compositions. In addition, the complex temperature properties of CO2/HCO3− were studied in detail to help with determination of their influence, particularly for offline, atmosphere-equilibrated samples. As temperature rises, the conductivity of HCO3− increases, but the solubility of CO2 decreases and the equilibrium between them shifts. Because of these complexities, it is now proposed to measure uncompensated conductivity and temperature separately, with worst-case conductivity limits established across the temperature range.
Calibration: It is not surprising that in general, the lower the conductivity, the greater the percentage uncertainty in the value. This is due to the potential for contamination in use, the variable concentration of carbon dioxide in the laboratory environment and the volatility of HCl used in some low standards. A very accurate standard can be obtained at high conductivity levels and standards exist at low conductivity levels, but with much greater percentage uncertainty. A “Catch 22” situation has existed where calibration in the range of measurement sacrifies accuracy while a high accuracy standard requires calibration far above the range of measurement. This situation has resulted in fact in the use of UPW as a standard at the very low end of the scale. The uncertainty of pure water as a standard is primarily due to the limitation on temperature measurement and control and on carbon dioxide exclusion in an actual recirculating loop.
(e) Difficulties in the Determination of Chemical Properties of Ions:
Scientific literature values describing temperature dependencies of the specific equivalent conductance of species and dissociation constants are typically not very precise. This reflects difficulties in the precise measurement of the properties of species such as weakly ionized acids and bases. One problem is that sample solutions typically contain variable concentrations of carbonic acid due to atmospheric carbon dioxide exposure. Another problem is that scanning properties over a pH range will require samples with multiple components introducing difficulties associated with handling complex chemical interactions.
Objects and Advantages
Accordingly, several objects and advantages of the present invention are:
Absolute Principle:
One object and advantage of the invention is to bring the advantage of an “absolute principle” not only to the determination of a single, relatively straightforward related, component but to the determination of complex related entities or a complete array of properties. Applied to analysis, this allows in principle to determine multiple species at low concentrations without having to perform an individual, low level, calibration for each of the species.
Non-Linear Curve Fitting:
The invention addresses the main obstacles preventing more automated use of non-linear curve fitting including:
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- a wide scan technique capable to automatically finding the global minimum, making it unnecessary to provide a good set of initial values.
- a reversed use of the invention providing it known sample properties in order to let it determine best matching underlying physical or chemical properties and their dependencies. Knowing such physical or chemical properties with a high precision allows to determine a large number of unknown properties under normal use of the invention.
Other obstacles related to the mathematical model are addressed under ‘Further objects and advantages’.
Water Analysis:
(a), b) Ion Analysis:
To obtain knowledge of the nature and concentrations of the ions in a sample that make up the conductivity of an aquaeous analyte with a simple, robust, low maintenance, on-line instrument based on an absolute principle making it unnecessary to calibrate for each individual contaminant, so without calibrated standards or consumable reagents. Such easy to run, low maintenance, calibration and reagents free, on-line analyzer would overcome limitations in terms of number of sampling points and frequency of analysis for applications that now rely on grab sampling and labor intensive, laboratory style equipment.
(c) Analysis of Organic Components and TOC:
To determine the inorganic ion composition resulting from a full oxidation of an organic contaminant in water thus revealing the atomic components that were part of the organic molecules. This information is much more meaningful than just a total organic carbon (TOC) indication.
To use knowledge about the atomic components of organic contaminants to correct chemical interference problems of current TOC analyzers by recognizing the part of a conductivity change that is due to carbon.
To determine the degree of oxidation of an organic contaminant by identifying intermediate ionic species that will fall further apart into stable end product
(d) Conductivity Measurements:
To use knowledge of the sample composition to produce accurate temperature compensated conductivity and pH values. Expanding the practice to use UPW for the calibration of conductivity cells, the invention can make use of information of the conductivity measurement over a continuous range of temperatures. It can determine final contamination such as trace concentrations of carbon dioxide and take their impact into account when using the water as a calibration standard. Finally, the invention can make the offset & span of the involved conductivity & temperature sensor part of the parameters to be determined. Such determination would solely rely on the quality of knowledge of UPW and not on, e.g., the use of secondary temperature standards.
Further Objects and Advantages are Related to the Creation and Use of a Mathematical Model to Describe Complex Phenomena:
State of the art teaches the use of the computer to calculate according to a provided formula. State of the art even teaches to find analytical solutions, or formulas for problems, as is done by symbolical calculators. The invention teaches the use of computing power in a numerical iterative way building up solutions from scratch on a case by case basis as a foundation for in principle limitless complex mathematical models. This ability overcomes major objections against automated use of non-linear curve fitters related to the use of a mathematical model. The numerical iterative way of the invention makes the calculation time go up roughly linear with the complexity of the model. State of the art translation of analytical derived expressions in computer program code makes calculation go up roughly with the square of the complexity of the model.
DESCRIPTION OF THE DRAWINGSThe invention will be better understood with reference to the accompanying drawings, in which:
Wide scan and modified group scan can also be run at startup from block 10 of the device or method according to the invention. It then runs in a kind of warm-up period and will result in a set of startup initial values for the SAMPLE FEATURES. This way, one no longer has to rely on a sophisticated user to perform this task, thus removing another obstacle for automation as discussed in the prior art.
Hereunder, the invention will be further presented mostly narrowed down to a main application which is the analysis of a liquid sample such as water.
Returning to
Block 1, performing sample preparation can trap a quantity of analyte by stopping temporarily a flow of analyte to isolate it from changes in chemical composition during the measurement. Block 1 might also simply do nothing when one just wants to see the components in the sample as it is or when changes in the analyte are much slower than the time to acquire a significant number of data points. Block 3 is a conductivity sensor provision and block 4 contains means to determine the value for the VARIABLE PROPERTY. This last determination might take place by a straight measurement as is likely the best way when dealing with the temperature. This determination might be more involved as is likely to be the case when dealing with the mixing ratio. Determination by block 4 might have to be done in conjunction with block 5 that actively or passively causes the VARIABLE PROPERTY to fluctuate. An example of such conjunction is described later for a preferred embodiment where the mixing ratio fluctuation is setup as an exponential dilution and a value for it can be derived from knowledge of the flowrate, the dilution volume and the time expired since the start of the dilution. Determination can involve repeatable hysteresis data correcting for a lag in the indicated and actual temperature of the analyte. In order to reduce response time of the analysis, the conductivity sensor provision of block 3 might be an array of conductivity sensors each scanning the VARIABLE PROPERTY dependency of the conductivity over only a fraction of the total range. Data acquisition block 6 will gather measured and otherwise derived data regarding the relationship between analyte conductivity and/or between analyte conductivity and mixing ratio value while it can also contain timestamp information associated with these measurements or determinations. Block 7 contains a mathematical model that now calculates the conductivity of an analyte at a certain temperature and/or mixing ratio given a CANDIDATE set of concentrations of the components of the analyte. If applicable the mathematical model might also have to be given a set of concentrations of the components in the fluid used for to mix with the analyte. The functions of the remainder of blocks in
The scheme of
Not shown in
The actual curve fitting routine starts at 40 while block 41 gets the initial concentration settings being: SA concentrations for strong acids, sa10, sa20, . . . ; SB concentrations for strong bases, sb10, sb20, . . . ; WA concentrations for weak acids, wa10, wa20, . . . ; WB concentrations for weak bases, wb10, wb20, . . . and eventually other species concentrations that are involved in more than one reaction.
The initial concentrations are passed to the Levenberg routine 42 that runs a loop consisting of block 43 and 44 to determine best fitting concentration values that are passed as the final results to block 49.
Block 43 takes a set of m concentration values (sa1, sa2, . . . , sb1, sb2, . . . , wa1, wa2, . . . , wb1, wb2, . . . ) generated by block 42 and loops through the series of n temperature values (T1, T2, . . . , Tn) from block 47 to calculate a series of n conductivity values σ(sa1, sa2, . . . , sb1, sb2, . . . , wa1, wa2, . . . , wb1, wb2, . . . , Tj) where jε{1, 2, . . . , n). Block 44 now takes, these calculated conductivity values and the measures conductivity values from 48 and presents the array of the differences between these conductivity values to the Levenberg routine 48. Block 48 calculates the NORM, which is the square root out of the sum of the square of the differences, as the quality of the fit which either meets Levenberg algorithm's stopping criteria or results in a modified series of concentration values offered to block 43.
Mathematical Model of Interacting Chemical Species in a Solution:
The math involved in solving the equations that describe the chemistry that takes place in a mixture of interacting species in a solution becomes rapidly unduly complex when increasing the number of species. On the one hand, analytical solutions seize to exist for solutions of the high order polynomial equations involved. On the other hand the expressions for the polynomial coefficients start rapidly to deal with such a large number of parameters that they fall beyond the capacity of symbolic calculators like MathCAD.
Notation & Theory:
Common ions in an aquaous solution are H+ and OH− which are related through the dissociation constant of water leaving only one as an independent parameter, arbitrarily chosen in this case H+. Molar concentrations of species are indicated with square brackets. Atomic species or parts of molecules or ions that remains unchanged in the chemical reactions under consideration are hereinafter called atomic species and notated with small letters. All species that can have a particular atomic species build into their molecular or ionic structure are hereinafter called the associated group of that particular atomic species, and are notated with capital letters where the ions have a + or − sign. The concentrations of the atomic species in an analyte are, in the hereunder presented application, the SAMPLE FEATURES of the SAMPLE searched to be determined. Where the total molar concentration, [carbon] of an atomic species like carbon remains unchanged by the reactions taking place in an aquaous solution, the distribution of carbon among the members of its associated group, {CO2, HCO3−, CO32−}, is determined by these reactions. As an example: the total concentration of carbon from carbonic acid can be written as: [carbon]=[CO2]+[HCO3−]+[CO32−] where carbonic acid is involved in two reactions: [H+].[HCO3−]=ka1.[CO2] & [H+].[CO32−]=ka2.[HCO3−].
The temperature dependent dissociation constants are indicated as kw for water and ka & kb for weak acids & bases with an index for different types.
Summation is noted as:
Σj=0, 1, . . . , NRj.[H+]j=R0+R1.[H+]+R2.[H+]2+ . . . +RN.[H+]N
Mathematical Model of the Chemistry:
Dissociation of Water: [H+].[OH−]=kw (1)
Dissociation of weak acids: [H+].[A−]=ka.[A] (2)
Total concentration [a]of species a, in ionic or dissolves form: [a]=[A]+[A−] (3)
Rewriting equations (2) & (3) gives: [A−]=[a]/(1+[H+]/ka) (4)
Dissociation of weak bases: [B+].[OH−]=kb.[B] (5)
Total concentration [b] of species b, in ionic or dissolves form: [b]=[B]+[B+] (6)
Using (1), rewriting equations (5) & (6) gives: [B+]=[b]/(1+kw/([H+].kb)) (7)
Strong acids and bases can be treated as fully ionized so that [sa]=[sa+] and [sb]=[sb+]. Consider an analyte that contains SA strong acids with concentrations sa1, sa2, . . . , saSA, SB strong bases with concentrations sb1, sb2, . . . , sbSB, WA weak acids with concentrations wa1, wa2, . . . , waWA, and WB weak bases with concentrations wb1, wb2, . . . , WbWB.
Charge conservation dictates:
[H+]+[sb1]+[sb2]+ . . . +[sbSB]+[b1].[H+]/([H+]+kw/kb1)+[b2].[H+]/([H+]+kw
/kb2)+ . . . +[bWB].[H+]/([H+]+kw/kbWB)=(kw/[H+])+[sa1]+ . . . +[sa2]+ . . . +[saSA]+[a1]/(1+[H+]/ka1)+[a2]/(1+[H+]/ka2)+ . . . +[aWA]/(1+[H+]/kaWA) (9)
The product of denumerators of the terms of expression (9) equals:
[H+]*([H+]+kw/kb1)*([H+]+kw/kb2)* . . . *([H+]+kw/kbWB)*(1+[H+]/ka1)*(1+[H+]/ka2)* . . . *(1+[H+]/kaWA) (10)
Multiplying (9) with (10) results in a pth order polynomial expression in [H30 ]: P0+P1.[H+]+P2.[H+]2+ . . . Pp.[H+]p=0.
Equation (10) can also be expressed as a polynomial expression in [H+]:
Q1.[H+]+Q2.[H+]2+ . . . +QS.[H+]S (11)
The invention's iterative numerical algorithm finds values for the polynomial coefficients P0, P1, . . . , Pp and Q1, Q2, . . . , Qs by adding weakly ionized species one by one. It calculates the polynomial coefficients for a composition with a weakly ionized species from the polynomial coefficient for that composition without that weakly ionized species. If the conservation of charge equation (9) for the composition without the weakly ionized species can be written as:
{P0+P1.[H+]+P2.[H+]2+ . . . +Pp.[H+]p}/{Q1.[H+]+Q2.[H+]2+ . . . +QS.[H+]S}=0
then adding, e.g., a concentration [a]of a weak acid “a” modifies the conservation of charge equation into:
{P0+P1.[H+]+P2.[H+]2+ . . . +Pp.[H+]p}/{Q1.[H+]+Q2.[H+]2+ . . . +QS.[H+]S}=[a]/(1+[H+]/ka)
Notation:
{Σj=0, 1, . . . , NPj.[H+]j}/{Σr=1, 2, . . . , N−1Qr.[H+]r}=[a].ka/(ka+[H+]) or
ka*{Σj=0, 1, . . . , NPj.[H+]j}+{Σj=1, 2, . . . , N+1Pj−1.[H+]j}−[a].ka*{Σr=1, 2, . . . , N−1Qr.[H+]r}=0 (12)
This last expression (12) can be dealt with by a computer program combining old coefficient values to calculate the new coefficient values. Species involved in more than one reaction can be dealt with in a similar way as is illustrated for carbonic acid:
[HCO3−]+2[CO32−]={[H+]+2ka2}*[carbon]/{[H+]**2/ka1+[H+]+ka1} &
{Σj=0, 1, . . . , NPj.[H+]j}/{Σr=1, 2, . . . , N−1Qr.[H+]r}={[H+]+2ka2}*[carbon]/{[H+]**2/ka1+[H+]+ka1}
The iterative numerical method of the invention makes that total concentration value of a each specific atomic species and the values of the reaction constants have to be entered only at one point. State of the art approach would formulate analytical expressions for all coefficients while for each coefficient specific atomic species and the values of the reaction constants have to be entered. The numerical iterative way of the invention makes the calculation time go up roughly linear with the complexity of the model. State of the art translation of analytical derived expressions in computer program code makes calculation go up roughly with the square of the complexity of the model.
Fitting of Offset and Span Values:
Offset and span parameters can also be fitted entering mathematically as follows: {Cspan*σ(a, b, c, . . . , (Tspan*Tj−Toffset))+Coffset−σjmeasured} which is the value of the jth component of the difference vector. Span values should be close to 1.0 and offset values around 0.0.
where σ(a, b, c, . . . , T) is the calculated conductivity for an analyte at temperature T containing concentrations a, b, c, . . . of species.
Fitting with a Sample Undergoing Known Changes:
In the case of a sample undergoing known changes there is more information to incorporate than the discussed collection of pairs of conductiviy and temperature data.
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- exponential dilution. Giving the data pairs σj & Tj a timestamp tj together with the knowledge of the starting moments, composition of the diluting flow (a0, b0, c0, . . . ), volume V and flowrate F of the setup enables to fit with:
- {σ(αj.a+a0,αj.b+b0,αj.c+c0, . . . Tj)−σjmeasured} where αj=exp(−tj*F/V)
Measuring the Temperature Dependency of a Species' Specific Equivalent Conductance or the Temperature Dependency of a Dissociation Constant Involving Several Species:
- {σ(αj.a+a0,αj.b+b0,αj.c+c0, . . . Tj)−σjmeasured} where αj=exp(−tj*F/V)
- exponential dilution. Giving the data pairs σj & Tj a timestamp tj together with the knowledge of the starting moments, composition of the diluting flow (a0, b0, c0, . . . ), volume V and flowrate F of the setup enables to fit with:
In this case we provide data from one or more samples with accurately known compositions. The unknowns to be fitted are the Taylor expansion coefficients of the involved specific equivalent conductance or dissociation constant. The calculated vector now is: {σ(p0, p1, p2, . . . , pn, aj, bj, cj, . . . , Tj)−σjmeasured}
In this case, the p0, p1, p2, . . . , pn parameters are the unknowns to be fitted while the aj, bj, cj, . . . and Tj values are the given known parameters. The non-linear curve fitter works basically the same as in the analysis case.
This makes it possible to, e.g., acidify a sample with a known concentration of carbonic acid with HCl in various concentrations allowing to measure the dissociation constant over a wide pH range. As pointed out in the state of the art description, without using the invention's techniques, complex interaction makes it hard to distinguish temperature effects of the specific conductivity from those of the dissociation constant.
Description of the Invention's Preferred Embodiment Number 1 This embodiment of the invention allows to acquire periodically a set of electric conductivity data from a single sensor over a range of temperatures on a trapped quantity of analyte.
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- stage 1 By opening valve 7, fresh sample enters through inlet 70, flows through the stopped flow cell 72 and exits through outlet 71. Measure initial UPW sample's conductivity & temperature with respectively sensor 3a & 4a and calculate the corresponding total carbon from carbonate, ICC, value assuming all initial conductivity is related to carbonic acid. Note that as the conductivity is strongly dependent on the sample's temperature, both conductivity and temperature has to be measured.
- stage 2 Trap sample water with the stopped flow technique, conductivity and temperature measurements at this point should result identical ICC values as in stage 1
- stage 3 Turn on lamp 1b to apply UV irradiation while determining the end point of chemical oxidizing activity. Oxidation turns non conductive organic species into water and carbon dioxide and possible some other simple molecular species. The carbon dioxide will contribute to the conductivity as carbonic acid. See U.S. Pat. Nos. 4,626,413 and 4,666,860 for possible algorithms for end point detection, where it should be noted that the actual data used for the TOC measurement does not come from this stage but from stage 4. A possible exception might be the case when the curve in stage 3 ends in a linear ramping slope which is, e.g., an indication of a CO2 leak which impact can be taken into account subtracting the leakrate integrated over the expired time from the total carbon.
- stage 4 After the extinction of the UV irradiation by turning off lamp 1b, monitor with sensors 3a & 4a the conductivity and temperature of the analyte as it remains now trapped according to the invention. Heated by the previous UV irradiation the sample will cool down toward ambient temperature. The monitoring during the cool down will provide the empirical temperature profile of the sample in the form of data pairs of a temperature and the corresponding conductivity over a range of temperatures.
As the data from stage 4 comes in, perform curve fitting matching sets of total atomic species such as carbon, chlorine, nitrogen, etc. The match should gain accuracy as more and more data over a growing temperature range is taken into account. The fitting algorithm should provide a quantitative number for the accuracy of the fit.
Compare the accuracy number of the fit with some minimal accuracy requirement. This minimal requirement number can be established in relationship with the specified accuracy of the analyzer. When the minimal accuracy requirements are achieved, step 4 can be halted, the results are reported, and a now measurement cycle can begin by opening the valve that gets a fresh flow of sample into the measuring cell.
Note that a time limit might be set on stage 4, reporting the results, e.g., in combination with a warning regarding any reduced accuracy.
The heater can also be used in conjunction with a control provision based on temperature feedback from sensor 4a. This way, the stage 4 cool down can be controlled by the heater to go at a desirable pace and the above mentioned heat up of an un oxidized sample can take place in a controlled fashion.
Comparing the conductivity temperature dependency of a sample while rising the temperature and while cooling down at various speeds might reveal hysteresis effects. This is caused by a lag between the reading of the temperature sensor and between the actual temperature of the sample. In this preferred embodiment, as well as most of the hereafter presented embodiments, the hysteresis lag should be highly repeatable and can eventually be addressed by a form of lookup table kind of correction.
Description of the Invention's Preferred Embodiment Number 2
This preferred embodiment can analyze species with and without oxidation. A higher concentration of species in analysis cell 85, compared to analysis cell 86, can be considered to result from the breakdown of non conductive, organic species.
Description of the Invention's Preferred Embodiment Number 4
In this mode of operation, contamination emitting from the insulation 104a at the foot of electrode 105 and similarly, the insulation 104b at the foot of electrode 106, will not build up but will be dragged away from the volume of trapped analyte in the measurement area. Closing only one of the valves, say 109, will create a flow from the cavity at the foot of electrode 105 through the measurement area toward the cavity at the foot of electrode 106 continuing toward open valve 110 and the outlet 114. Contaminant buildup in the now small stagnant flow area just upstream of closed valve 109 will have to move counter a small flow in tubing 107 in order to contribute to the measurement. This small flow moves from the cavity around the foot of electrode 105 toward tubing 111 through which it will exit. If valve 110 would have been closed instead of 109, then a flow would have been create from the cavity at the foot of electrode 106 through the measurement area toward the cavity at the foot of electrode 105 continuing toward open valve 109 and the outlet 114. Contaminant buildup in the now small stagnant flow area just upstream of closed valve 110 will have to move counter a small flow in tubing 108 in order to contribute to the measurement. This last small flow moves from the cavity around the foot of electrode 106 to tubing 111 through which it will exit. These on valve open, one valve closed mode of operation allow to replace the analyte in the measurement area by new analyte. The axis of cylindrical cell body 103 might have to be secured in a horizontal position to avoid perturbing effects from gravity.
Varying the temperature of a trapped analyte in this preferred embodiment can be done by taking away from or sending heat into main cell body 103 by means of a cooling or heating provision. This provision can transport heat by direct contact with cell body 103 but also by using the stream of fluid passing through cell body 103 as its means of heat transportation, see preferred embodiment number 8 hereunder.
It should be noted that choosing valves with minimal flow restriction makes that the two discussed cavities around the foot of each electrode are basically interconnected through large diameter tubing downstream of the opened valves. This makes that the cavities will be at the same pressure and the trapped volume will not flow one way or the other.
Description of the Invention's Preferred Embodiment Number 6
To preserve the symmetry in the tee at inlet 100, both legs could be equipped with a quartz spiral around a UV lamp while operating such unit with one lamp on and one lamp off.
Description of the Invention's Preferred Embodiment Number 7
Preferred embodiment number 8 uses the sampling manifold of preferred embodiment number 7 where conductivity cell housing 135 and its sensor 136 are replaced by flow switch based stopped flow cell of preferred embodiment 5 or 6. Varying the temperature of a trapped analyte in the flow switch based stopped flow cell can now be done by varying the temperature of the analyte flow that continues to stream around the trapped volume. The intimate contact of this stream around the trapped volume with cell body 103 and the feet of the electrode bodies 105 and 106 will minimize local temperature gradients when changing the temperature over time. This will ensure a homogeneous temperature distribution in the measuring volume at any point in time.
In order to prevent bubble formation in the trapped volume that might impact the measurement the invention can start flushing the volume with the highest temperature analyte, trapping a volume and then start to measure ramping the temperature down. Bubbles are typically formed when raising the temperature which frees dissolved gases in the fluid.
Description of the Invention's Preferred Embodiment Number 9
C(t)=C(0)*exp(−t*F/V)
where C(0) is the concentration of the component in the analyte, and where the mixing fluid contains zero concentration of the component while the flowrate F is fixed over time.
The mixing fluid can be a strong acid or base which enables to take species properties into account that show more characteristic behavior at extreme pH values.
Description of the Invention's Preferred Embodiment Number 10The accuracy of the measurement of the conductivity's temperature dependency could be compromised when it involves a ramping up or down of the temperature. The actual water temperature might trail the temperature at the position of the temperature probe or vise versa. This could, e.g., be the case when the temperature probe is attached to the cell's housing and contacts the sample water only indirectly. This preferred embodiment maps such relationship by performing an analysis on a known sample. Looking at the conductivity versus temperature plots show typically monotonously rising curves which enable to define a one on one relationship between a specific conductivity and a specific temperature. Taking a known sample, e.g., containing only a known amount of carbonate, allows to link a conductivity to a temperature. This way, mentioned trailing of the actual analyte temperature with the temperature at the temperature sensor can be determined. A model can now be made, e.g., assigning a certain “inertia” to the “thermal mass” of the cell, to predict the correct analyte temperature from a series of prior temperature measurements made with the temperature sensor. It can be expected that the underlying phenomena leading to the here described effects will be very stable and repeatable. Mentioned correction can also be made using a lookup table or equivalent analytical expression.
Description of the Invention's Preferred Embodiment Number 11 Superimposed on the conductivity signal will be effects from the UV lamp. Effects so far discussed are typically the irreversible breaking down and oxidizing of species. U.S. Pat. No. 4,666,860's
Prolonged, intense, high energetic UV irradiation and the oxidation it provokes will decompose complex species, such as organic species, in a water sample into fragments that will ultimately recombine into molecular and ionic species of the most simple, typically lowest energy, chemical forms. The carbon, nitrogen, chlorine atoms etc. that constituted the original species in the sample prior to the UV irradiation, will end up in the form of only a limited number of mentioned simple species such as CO2, HCO3−, CO32−, NH4+, NH3, NO3−, Cl−, etc.
This embodiment applies UV light for the above purpose as its sample preparation step 1, but does not rely on the completeness of the transformation of complex species into simple molecular or ionic species. It can include intermediate transformation products to the set of sample features used for the curve fitting and takes their atomic composition into account when reporting the analysis result.
Description of the Invention's Preferred Embodiment Number 13This embodiment provides additional means to check the degree of oxidation of the analyte by the UV irradiation. During the oxidation large organic molecules are little by little broken down in smaller pieces. If the oxidation is not complete, larger ions will be present with a corresponding relatively low ion mobility compared to the simplest, smallest form of ions. Matching the temperature dependency of such not fully oxidized analyte with calculated conductivities of an analyte consisting only of simple fully oxidized species will lead to a poor fit. In this embodiment, the curve fitting procedure provides a number for the quality of the fit. A poor number for the fit quality will be an indicator that the sample is not fully oxidized. Detecting such incomplete oxidation could be followed up by slowing down the stream of analyte thus increasing the exposure time to UV.
Summary, Ramifications, and Scope
Accordingly, the reader will see that the invention can create a device or method providing an automated stream of information on hard to determine parameters by scanning easy to measuring parameters while circumventing potentially difficult calibration issues. In its application on water analysis, the measured trace concentrations of ionic contaminants are certainly considered hard to determine while measuring the water conductivity as a function of temperature is much easier. Summarizing this application, the invention measures the temperature and/or dilution ratio dependency of the electrical conductivity of a sample and compares this with calculated conductivity dependencies for various possible sample compositions. The calculation includes the laws of chemistry governing the dependencies including common ion effects that cause non-linear behavior of the conductivity as a function of the concentration of species. The comparison uses a non-linear curve fitting routine such as that of Levenberg-Marquardt to determine the best fit in terms of lowest sum of the squares of the differences between measured and calculated curves. The sample composition with the best fit is then presented as the analysis result.
The invention has various sampling configurations typically trading off speed of response and accuracy of the analysis. Non conductive organic contaminants in water can also be analyzed this way by converting them into basic conductive inorganic species by UV generated oxidation.
The acquired knowledge of the analytical composition of a sample enables also precise calculation of the pH, temperature compensated conductivity values, and calibration of the sensors.
Although the description above contains many specificities, these should not be construed as limiting the scope of the invention but as merely providing illustrations of some of the presently preferred embodiments of this invention. For example, apart from analysis, wider applications of are typically in the field of monitoring chemical or mechanical processes and equipment, machines, engines or even vehicles. To picture the wideness of the scope, based on a mathematical model describing the collaborative functioning of the parts of a car given a number of semi-empirical determined parameters, the invention enables to construct an automated provision that determines the current viscosity of the engine oil by monitoring the engine temperature as a function of the rpm number, speedometer reading and outside temperature.
Thus the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by the examples given.
Claims
1. An instrument, device, method or procedure to determine one or more features, properties or parameters, hereinafter called analyzed feature, of an object, sample or process, hereinafter called object, where the object has another feature, property or parameter, hereinafter called the characteristic feature that is measurable as a function of yet another feature, property, or parameter, of the object hereinafter called the scanned feature, comprising the following
- (a) a mathematical model that given a value of the object's analyzed feature and given a value of the object's scanned feature calculates the corresponding value of the object's characteristic feature, and
- (b) measurement means measuring the object's characteristic feature, and
- (c) variable entity determination means to measure or to control the value of the object's scanned feature, and
- (d) scanning means to actively or passively vary the object's scanned feature, and
- (e) object preparation means to set the scope of the analyzed feature, or to adapt the object to the characteristics of the scanning means, and
- (f) data collection means that, in conjunction with the scanning means, collects data from the measuring means and the variable entity determination means regarding the relationship between the object's characteristic feature and object's scanned feature, and
- (g) data interpretation means determining the most likely object's analyzed feature based on the comparison between the relationship's data from the data collection means and calculated relationships based on the said mathematical model where the calculated relationships are given candidate analyzed feature values by a non-linear regression or curve fitting algorithm such as that of Levenberg-Marquardt to find the candidate analyzed feature value that provides the best match in the comparison.
2. An instrument, device, method or procedure of claim 1 wherein the best matching candidate analyzed feature value is frequently updated using ongoing scanning, data acquisition and interpretation with means that takes older data less into account when doing the comparisons.
3. An instrument, device, method or procedure of claim 2 wherein upon startup or otherwise the data interpretation means can activate a wide scan means generating a grid of initial analyzed feature values with whom a series of runs of a non-linear regression or curve fitting algorithm such as that of Levenberg-Marquardt is started in order to distinguish the overall best fit out of a possible multiple local best fitting analyzed feature values.
4. An instrument, device, method or procedure of claim 3 wherein a quality of the fit value is generated by the data interpretation means that is used as a decision criteria to activate said wide scan means.
5. An instrument, device, method or procedure of claim 1 wherein the best matching candidate analyzed feature value is generated as an automated operation on a given set of recently acquired or stored data.
6. An instrument, device, method or procedure of claim 5 wherein upon startup or otherwise the data interpretation means can activate a wide scan means generating a grid of initial analyzed feature values with whom a series of runs of a non-linear regression or curve fitting algorithm such as that of Levenberg-Marquardt is started in order to distinguish the overall best fit out of a possible multiple local best fitting analyzed feature values.
7. An instrument, device, method or procedure of claim 6 wherein a quality of the fit value is generated by the data interpretation means that is used as a decision criteria to activate said wide scan means.
8. An instrument, device, method or procedure of claim 1 where the analyzed feature include offset or span correction parameters for sensors involved in the measurement means or involved in the variable entity determination means.
9. An instrument, device, method or procedure to determine chemical species selected from the group consisting of ions, dissolved gases and neutral species, hereinafter called the composition, present in analytes selected from the group consisting of aquaous solutions, liquids and fluids, hereinafter called the analyte, whereby the analyte features an electrical conductivity property that is measurable as a function of a scannable parameter selected from the group consisting of the analyte's temperature and the degree by which the analyte is mixed with another fluid, hereinafter called the scannable parameter, comprising the following
- (a) model calculation means that given values of a composition and given an analyte temperature produces the corresponding calculated value of an analyte's conductivity, and
- (b) measurement means measuring the analyte's conductivity, and
- (c) scannable parameter determination means for the determination of the value of the analyte's scannable parameter, where determination is selected from the group measurement, derivation and control, and
- (d) scanning means to actively or passively vary the value of the analyte's scannable parameter, and
- (e) analyte preparation means to selectively change the type of chemical species that contribute to the analyte's conductivity or to adapt the analyte to the characteristics of the scanning means, and
- (f) data collection means that in conjunction with the scanning means collects data from the measuring means and the scannable parameter determination means regarding the relationship between the analyte's conductivity and the analyte's scannable parameter, and
- (g) data interpretation means determining the analyte's composition based on the comparison between the relationship's data from the data collection means and relationships calculated by the mathematical model where the calculated relationships are given candidate analyte's composition values by a non-linear regression or curve fitting algorithm such as that of Levenberg-Marquardt to find the candidate analyte's composition value that provides the best match in the comparison.
10. An instrument, device, method or procedure of claim 9 wherein the analyte preparation means selectively changes the type of chemical species that contribute to the analyte's conductivity by irradiating the analyte with UV light converting non conductive species such as organic molecules into other species from which some of them conductive.
11. An instrument, device, method or procedure of claim 9 wherein the analyte preparation means adapt the analyte to the characteristics of the scanning means by isolating, stopping or stabilizing part of the analyte stream so that its analyte's composition is no longer subject to further changes while it is being measured, other than changes resulting from the scanning means or resulting from other analyte preparation means, while its conductivity is measured by the measuring means.
12. An instrument, device, method or procedure of claim 11 where the actual isolating, stopping or stabilizing of a part of the analyte stream takes place by analyte streams redirecting means that flush areas suspect of building up contamination with bypassed, non measured, analyte thus preventing their interfering with the measurement.
13. An instrument, device, method or procedure of claim 10 wherein the analyte preparation means adapt the analyte to the characteristics of the scanning means by isolating, stopping or stabilizing part of the analyte stream so that its analyte's composition is no longer subject to further changes, other than those resulting from the scanning means or resulting from UV light irradiation, while its conductivity is measured by the measuring means.
14. An instrument, device, method or procedure of claim 9 wherein the analyte is flowing while the analyte's conductivity is measured by the measurement means and where the measuring means can consist of more than one conductivity sensor or an array of miniaturized conductivity sensors.
15. An instrument, device, method or procedure of claim 10 wherein the analyte is flowing while the analyte's conductivity is measured by the measurement means and where the measuring means can consist of more than one conductivity sensor or an array of miniaturized conductivity sensors.
16. An method, procedure, instrument or device to generate numerical values for unknown parameters in a mathematical model describing a phenomena involving multiple known or given values for known parameters where known and unknown parameters are linked by a number of mathematically described relationships and where one still has a valid but simpler phenomena by successively leaving unknown parameters out and where an analytical solution is available for an initial simple case featuring none or only few unknown parameters consisting of the following
- (a) the initialization step creating numerical values using formulas derived from the analytical solution for the initial simple case, and
- (b) an iterative step that calculates numerical solutions for a certain phenomena based on the numerical solution for the same but slightly simpler phenomena featuring one unknown parameter less, and
- (c) an loop step that starts with the solution from the initialization step (a) to which it adds successively new unknown parameters while solving each time the next more complex case using the iterative step (b) to end up with a solution for the original phenomena.
17. An method, procedure, instrument or device as claim 16 where the phenomena is the chemistry taking place in aquaous solutions and where the known parameters are the total concentrations of species added to a pure water independent of the ionic or molecular forms that will end up having in the solution and where said ionic or molecular forms are the unknown parameters and where the mathematical model is based on the laws of chemistry associated with aquaous solutions.
18. An method, procedure, instrument or device as claim 17 aking activity coefficients into account build into the mathematical model calculation to reflect effects taking place when dealing with higher concentrations of species.
19. An method, procedure, instrument or device of claim 17 wherein the involved pH range is such that certain strong acids or bases can be considered fully ionized and are treated mathematically that way in the mathematical model.
Type: Application
Filed: Mar 18, 2005
Publication Date: Jul 28, 2005
Inventor: Jacob Mettes (Boulder, CO)
Application Number: 11/083,410