Gas chromatography is a comparative technique, comparing the responses for individual components in an unknown sample with those for the same components in a calibration mixture of known composition. Values for the unknown sample are derived using the composition of the calibration gas and the ratio of responses for the two. The accuracy of the calculated result is determined principally by the accuracy of the calibration gas composition. If the calibration gas composition is in error, then all analyses using that calibration gas will also be in error. The analyzer characteristics include:
(1) Component separation, which allows all relevant components to be measured and avoids interference between them.
(2) Precision uncertainty, which means that the measurements of calibration gas and sample are made with greater or lesser consistency.
(3) Linearity of response, which enables different component quantities to be compared.
Therefore, the analyzer does not in itself offer accuracy of results. However, it can degrade the accuracy derived from the calibration gas by allowing component interference, by failing to compare different quantities in the manner predicted (introducing bias errors) and by operating with poor consistency when repeat measurements are made (precision uncertainty).
In order to optimize accuracy and consistency, we need to consider the calibration gas, the analyzer performance and, less intuitively, the choice of calibration gas composition with respect to the range of sample compositions which may be expected. The combination of these features allows the overall uncertainty and bias error to be calculated.
Calibration gases are the fundamental basis for calculating compositions and hence properties of natural gas. Response to the unknown sample is compared to that of the known calibration gas and hence the composition of the sample is derived. If the stated composition of the calibration gas is wrong, then it will create a bias error in the resulting composition of samples. The uncertainty associated with calibration gas composition will influence the uncertainty of the derived result, whether or not the bias error is effectively zero.
Calibration gases should have a certified composition, traceable to national or international standards and a quoted uncertainty for each component. They should be stable over a period which is stated on the certificate. The composition can be derived from the method of preparation or by comparison with Primary Reference Gas Mixtures issued and certified by a national metrology laboratory. The use of pressurized cylinders is the most practical and convenient way of supplying and using calibration gases.
Gravimetric preparation according to ISO 6142[1] is a well-established and widely used method. Accuracy is ensured through the use of certified mass pieces and starting gases of well defined composition. Although the mass of the cylinder is much greater than the masses of the added components, the accuracy and high discrimination of a well-configured balance system allows smaller preparation uncertainties than with any other method of preparation. A further advantage of gravimetric preparation is that the resulting mass fractions are unequivocally translated into molar fractions.
Comparison of a calibration gas with a number of reference materials according to ISO 6143[2] allows composition to be stated with very low uncertainties, traceable to the mole, and allowing any non-linearity in the measurement system used for comparison to be recognized and allowed for.
The use of comparison gives data on the contents of the calibration gas cylinder. Gravimetric data define what is added to the cylinder, not what comes out of it in use.
EffecTech is accredited by the United Kingdom Accreditation Service (UKAS) to ISO 17025[3] to provide calibration gases with defined uncertainties. Mixtures are prepared gravimetrically, which is the most certain way of achieving target compositions. After treatment to ensure thorough mixing of the components, the mixtures are certified by comparison with reference materials obtained from the National Physical Laboratory (NPL) in the UK or the National Measurement Institute (NMI) in the Netherlands.
Relevant Calibration and Measurement Capabilities (CMCs) under this accreditation are shown in table 1. The uncertainties are expanded values (k=2). These values relate to calibration gases suitable for a typical C6+ analysis, as is widely used for calculation of properties such as Calorific Value, Density and Wobbe Index.
In addition to the data of table 1, EffecTech's accreditation includes hydrocarbons up to C10including aromatics and naphthenes, a range of sulfur-containing gases and refinery gases.
ISO 10723[4] describes the methodology for evaluating analyzer performance. It does not recommend or require performance levels, since that is appropriately decided by regulation or contractual requirements. But it does describe how the tests can be conducted and how quantitative measures of performance are derived.
The analyzer (typically but not exclusively a gas chromatograph) compares the response to an unknown sample with that to a calibration gas. The known content of the calibration gas is then used with the ratio of the responses to derive composition values for the sample. Assuming that the analyzer is properly configured for the application, its performance is characterized by two properties; whether the uncertainty of measurement (precision as repeatability) is low so that the ratio of responses can be well defined, and whether the response to different amounts of components is as assumed.
Measurement uncertainty is illustrated in figure 1, where the broken line is the response/molar quantity relationship and the Gaussian distributions are the precision uncertainties at different molar quantities. In this instance the uncertainty increases with molar quantity, which is common, but some components show uniform uncertainty over the range.
Error arising from response characteristics (bias) is illustrated in figure 2, where the assumed response is modeled by a straight line through the origin, although the "true" response follows a curve. These two lines coincide at the point of calibration. At all other points an error is introduced which is proportional to the extent to which the two lines differ. Use of a calibration gas of a different composition will change all these errors since it will change the point of coincidence and hence the difference between the lines.
Figure 2 shows an assumed response modeled by a straight line through the origin. This is the default assumption on most analyzers, which is convenient because it implies a constant ratio between response and component quantity, allowing a single calibration gas to be suitable for a wide range of component quantities. In many cases this assumption is sufficiently close to the "true" response, meaning that the resulting bias errors are acceptably small, but for major components, particularly methane, the error can be substantial. Some analyzers have the facility to use more complex response functions and can then be calibrated according to ISO 6974-1, 2[5-6]. Such response functions are commonly polynomials (up to third order) or occasionally exponential functions. Because of the complexity of multi-level calibration, it would be true to say that only a minority of analyzers with this capability does in fact use such response functions. Where multi-level calibration has been used, then the existing functions are the assumed response and the new functions defined by performance evaluation are the "true" ones. The differences between the "assumed" and "true" values are generally much smaller and hence so are the resulting errors.
ISO 10723 requires high quality test gases with which to evaluate analyzer performance. The components to be measured are present in the different test gases at different amount fractions, suitable for the intended range of measurement. Repeat analysis of the test gases enables the measurement uncertainty to be assessed, and whether this is uniform across the range or a function of amount fraction. The mean values from each set of repeats are used to determine the response function, which may be a first-, second- or third-order polynomial. It is assumed that any higher order would indicate a response function which is too complex to be useable.
For each component, seven amount fractions spanning the measurement range are needed to define a third-order polynomial (and hence any lower order if the higher order coefficients prove to be not significant). If a prior knowledge exists that third order is not likely, then five amount fractions enable definition of second-order response, and three enable first-order. Since, however, this information is in most instances not available, it is preferred to use seven levels and to determine the appropriate order of response from the resulting data.
The complete assessment of the errors and uncertainties arising from the use of an instrument could be done by the measurement of an infinite set of well defined reference gas mixtures whose compositions lay within the specified range of operation. However, this is practically impossible. Instead, the principle used is to measure a smaller number of well defined reference gases and to determine a mathematical description of the response functions for each specified component over a pre-defined content range. The performance of the instrument can then be modeled offline using these "true" response functions, the response functions assumed by the instrument's data system and the reference data for the working calibration gas mixture specified for the instrument. The measurement of a large number of gas mixtures can then be simulated offline using iterative techniques to determine performance benchmarks inherent in the measurement system.
Generally, in measurement science it is important to establish any measurement bias through method validation and to correct for any significant systematic errors to give the final result. However, the measurement of natural gas by gas chromatography generally employs the use of a single point working calibration gas in combination with an instrument which assumes a pre-defined relationship between component amount fraction and instrument response. This is termed the "assumed" response. In truth, the instrument often does not respond in this pre-defined way. The principal aim of performance evaluation is to measure the actual or "true" response of the instrument. The difference between the true and assumed response of the instrument gives rise to instrumental errors. When the errors are calculated over a wide range of gas compositions they can become significant. Analysis of the distribution of errors gives rise to a mean error for each measured. As for example in the Network Entry Agreements in the UK, a value for the Maximum Permissible Bias (MPB) applies, which mean bias errors can be compared to the MPB for each component or physical property. Such comparisons may be used as pass/fail criteria for an instrument in its current or proposed application.
It should be noted that any change in the nominal composition of the calibration gas, the calibration gas recipe, will have a profound effect on the distribution of errors and bias arising from the instrument. The distribution of errors from only a single recommended calibration gas recipe is presented here. Recalculations of the redistribution of errors would be required for a change in calibration gas composition.
Conversely, and quite independently, the uncertainty on measured composition and physical properties arises from the uncertainty on the calibration gas composition and the uncertainty resulting from instrument imprecision. The relationship between instrument precision and amount fraction for each component can be defined in terms of precision functions. Estimates of uncertainty in measured composition and physical properties can then be done by conventional means in accordance with the Guide to the Expression of Uncertainty in Measurement (GUM)[7].
Generally, errors and uncertainties cannot be combined in a satisfactory manner and, as such, they are usually compared. In the past, the point at which the errors in composition exceeded the measurement uncertainty provided a useful benchmark to limit ranges of composition over which the instrument was deemed fit for purpose. However, if the composition ranges of instruments were restricted too frequently, it will be insufficient for practical use. Hence, it is now common practice to accept errors in the measurement of composition provided their effect does not give rise to unacceptable errors in physical properties. As a consequence, we must also make an assessment of the contribution of uncorrected errors to the overall uncertainty of measurement. Uncorrected errors and uncertainties can be combined in accordance with the GUM to give an uncertainty on the mean error for each measured component or physical property. Where a value for the Maximum Permissible Error (MPE) applies, as also quoted in the Network Entry Agreements in the UK, uncertainties on the mean error (uncertainties on the bias) can then be compared to the Maximum Permissible Error (MPE) for each component or physical property. Such comparisons may be used as pass/fail criteria for an instrument in its current or proposed application.
Table 2 shows the component ranges covered by the seven test gases.
For each component, the mean response value at each amount fraction and the measured uncertainty of that response is subjected to regression analysis against the known amount fraction of the component and its uncertainty. The method of Generalized Least Squares is used, as described in ISO 6143 [2]. This allows uncertainties in both parameters (amount fraction and response) to be used in the calculation.
A mathematical description of instrument response (Ri) as a function of amount fraction (xi) is termed the Calibration Function, whereas that describing amount fraction as a function of response is termed the Analysis Function.
The true calibration functions, Fi, true(xi) is determined for each component and caculated by equation (1):
where an are the parameters of the calibration function.
Similarly, the true analysis functions, Gi, true(Ri) is caculated by equation (2):
where bn are the parameters of the analysis function.
Tables 3 and 4 list typical parameters for the calibration and analysis functions respectively.
The assumed instrument response, as an analysis function, is described in terms of factory settings, bi, factory, as equation (3):
where in most cases, b0, factory=b2, factory=b3, factory=0
Following calibration, where the true and assumed functions coincide, the un-normalized measured amount fraction, xi, meas*, is calculated as equation (4):
When this has been calculated for all components in gas, the value is normalized as equation (5):
Properties, Pmeas, such as calorific value, are calculated from the normalized measured composition, and the resulting errors are calculated as equation (6), (7):
The distribution of errors on normalized amount fraction and physical properties can be determined by a Monte-Carlo simulation. A data set of at least 10, 000 random compositions is constructed where each component amount fraction lies within the range of gases possible through the measurement point.
The compositions used in the simulation are not strictly random but are derived using a set of known rules relating the amounts of one component to that of the next hydrocarbon in the homologous series. The natural gas generator also contains rules relating the isomers of butane and pentane to the normal isomer in each case. By this method, non-naturally occurring natural gas compositions, which would not be present in a real sample, are not generated during the simulation.
For example, these ranges may correspond to the maximum possible through the gas metering system and in the UK's transmission system as defined in the Gas Transportation Ten Year Plan[8] issued by National Grid Gas. Public gas transporters are not allowed to convey and deliver gas in the UK with a composition outside of this range.
For each simulated composition, the error on components and properties is tabulated, and from these the mean error for each component and property is derived. Table 5 shows the ranges and mean errors for each component and for calorific value.
The typical Maximum Permissible Bias (MPB) on real superior calorific value for a fiscal /custody transfer metering point is 0.02 MJ/m3. The calculations above show that the measurement of calorific value in this instrument has an acceptably low bias.
The distribution of bias errors for calculated calorific value is shown in figure 3 as a function of methane content for the 10, 000 simulations. Other components also influence the error, but to include them would require a (n+1)-dimensional plot for an n-component mixture. Methane, as the major component, has the most profound effect and is selected to illustrate the variations.
The calibration gas has a methane molar content of 80.5% which is close to the point where the error spread is minimal. As predicted, the error distribution increases the further the methane content is from that of the calibration gas.
In the measurement of natural gas by chromatography, errors in amount fraction and in physical properties arise due to a difference in the actual and assumed response characteristics of the instrument (see 3.4.1).
Conversely, and quite independently, the uncertainties in measured amount fraction and physical properties arise from the uncertainty on the calibration gas composition and the uncertainty resulting from instrument imprecision.
As there is no uncertainty in the true value of the hypothetical amount fractions, yi, actual, used during the simulation, and the properties calculated from them, Pactual, the uncertainties in the errors, u(E(x)) and u(E(P)), are equal to the uncertainty in the measured amount fraction, u(yi, meas), and the calculated properties, u(Pmeas), respectively.
Estimates of measurement uncertainty in this case can be done by conventional means in accordance with the Guide to the Expression of Uncertainty in Measurement (GUM).
An estimation of uncertainty on measured amount fraction and physical properties is presented below.
(1) Content (Amount Fraction)
From the discussions in the section above we have a measurement equation (8) which defined the unnormalized amount fraction measured by the instrument, xi, meas*.
This measurement equation can be simplified considerably as equation (9) in the case of an instrument which assumes a single point calibration or approximated for components with a non-ideal analysis function defined in the instrument data system.
where
Ri is the measured instrument response (integrated area or peak height wherea ppropriate).
xi, meas* is the unnormalized amount fraction of component i in the measured (meas) sample.
yi, cal is the amount fraction of component i in the calibration (cal) gas.
Therefore, the combined standard uncertainty on the measured unnormalized amount fraction can be obtained by summing the relative uncertainties in quadrature, which is denoted as equation (10):
Hence, the standard uncertainty on the measured unnormalized amount fraction can be calculated from uncertainty sources due to the response of the instrument to the sample and calibration gases and in the composition of the calibration gas.
The standard uncertainties on response, u(R), can be estimated from the precision functions determined in the evaluation described in 3.4 and it can be expressed as equation (11) and (12):
Where Si, true(xi) is the precision function determined for each component, denoted as equation (13):
rn are the parameters of the precision function.
pi is the standard deviation (standard uncertainty) of response of the instrument to component i.
(2) Physical Properties
The uncertainty on physical properties, u(P), is calculated from the uncertainty on the measured component amount fractions, u(xi, meas).
Generally, errors and uncertainties on these errors cannot be combined in a satisfactory manner and, as such, they are usually compared. In the past, the point at which the errors in component amount fraction exceeded its measurement uncertainty provided a useful benchmark to limit ranges of composition over which the instrument was deemed fit for purpose. However, it is now common practice to accept errors in the measurement of component amount fractions provided their effect does not give rise to unacceptable errors in physical properties.
As a consequence, we must also make an assessment of the contribution of such uncorrected but significant errors to the overall uncertainty of measurement. This section uses the principles in the GUM[section F2.4.5] which gives guidance on how to estimate overall uncertainty on mean errors such that they can be compared to a required instrument performance specification.
Content (amount fraction) and physical properties are as follows.
The mean error in amount fraction and properties was calculated by equation (14):
where E(Pt) is the error (in either component amount fraction or property) calculated for the tth of N (usually 10, 000) hypothetical compositions.
The combined standard uncertainty (uc) in the mean error determined above is the positive square root, denoted as equation (15):
is the arithmetic mean of the squared standard uncertainties in error, , calculated for each of the 10, 000 hypothetical compositions.
is the squared standard uncertainty in the mean error, calculated as the variance of the errors determined for each of the 10, 000 hypothetical compositions.
The expanded uncertainty in mean error is then calculated by use of an appropriate coverage factor, k, and denoted as equation (16):
For the present application, a coverage factor of k=2 has been used providing a level of confidence of approximately 95%.
If a condition for Maximum Permissible Error (MPE) exists, then the relation of formula (17)shows compliance with the condition:
The relationship between calorific value data and the MPB is shown graphically in figure 4. The Gaussian curve represents the distribution of the individual errors from the simulation. The maxi mum of this curve lies comfortably within the MPB band. The same data is displayed in figure 5, where the MPE band is also shown. Figures 4 and 5 show compliance with both criteria.
2012, Vol. 41 Issue (1): 1-10
2012, Vol. 41 Issue (1): 1-10