Comparison of calibration curve fitting methods in the ...

Petroleum Science and Technology

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Comparison of calibration curve fitting methods in the determination of H2S mass fractions in natural gas by GC-SCD Elcio C. Oliveira, Rodrigo F. Calili, Anderson L. S. Ferreira, Alexandre A. Ferreira, Soraya N. Sakalem & Clarisse L. Torres To cite this article: Elcio C. Oliveira, Rodrigo F. Calili, Anderson L. S. Ferreira, Alexandre A. Ferreira, Soraya N. Sakalem & Clarisse L. Torres (2017) Comparison of calibration curve fitting methods in the determination of H2S mass fractions in natural gas by GC-SCD, Petroleum Science and Technology, 35:24, 2241-2248, DOI: 10.1080/10916466.2017.1396608 To link to this article: https://doi.org/10.1080/10916466.2017.1396608

Published online: 27 Nov 2017.

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Date: 06 December 2017, At: 09:20

PETROLEUM SCIENCE AND TECHNOLOGY , VOL. , NO. , – https://doi.org/./..

Comparison of calibration curve fitting methods in the determination of H S mass fractions in natural gas by GC-SCD Elcio C. Oliveira a,b , Rodrigo F. Calilib , Anderson L. S. Ferreirab , Alexandre A. Ferreirac , Soraya N. Sakalemd , and Clarisse L. Torresd Downloaded by [Petroleo Brasileio], [Dr Elcio Oliveira] at 09:20 06 December 2017

a

Technological Development Management and Automation, Services Department, PETROBRAS TRANSPORTE S.A., Rio de Janeiro – RJ, Brazil; b Post-Graduate Program in Metrology, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro – RJ, Brazil; c Division of Geochemistry, PETROBRAS Research and Development Center (CENPES), PETROBRAS S.A., Rio de Janeiro, RJ, Brazil; d Fundação Gorceix, Ouro Preto, MG, Brazil

ABSTRACT

KEYWORDS

This paper discusses metrologically the best practice regarding the calibration curves applied to H2 S mass fractions determination in natural gas by gas chromatography with sulfur chemiluminescence detection (GC-SCD). Three calibration curves were constructed by performing GC-SCD analysis of different H2 S gas standard concentrations (from 3 mg kg−1 up to 500 mg kg−1 ). These experimental curves are better fitted by an unweighted quadratic calibration curve considering ANOVA approach compared to ASTM D5504-12. Despite this, the obtained results show that these two different calibration curve approaches (ASTM and ANOVA) lead to comparable results. Hence, there are no significant statistical differences between these two approaches based on the hypothesis test applied. However, the quadratic calibration curve presents measurement uncertainties of H2 S mass fractions much lower than the ASTM approach.

ANOVA; ASTM D; GC-SCD; H S mass fractions; unweighted linear calibration curve forced through zero; unweighted quadratic calibration curve

1. Introduction Low values, in the order of mg kg−1 of sulfur odorants, are added to natural gas for safety purposes. Some of these odorants are unstable and may react to form compounds with even lower limits (Safadoost et al. 2014; Zhang and Wood 2015). Quantitative analysis of these odorous gases ensures that the odorant injection equipment is meeting the specification. In 2010, a Brazilian group of researchers studied calibration response factors of gas chromatography associated to a sulfur chemiluminescent detector with dual plasma (GC-SCD-DP) for analysis of sulfur-containing petroleum refinery gaseous streams, without, however, focusing in calibration curve fit (Pereira, Schmidt, and Afonso 2010). Periodically, studies are carried out in order to compare the effect of the different calibration equations by means of uncertainty measurement (Forbes and Minh 2013; Kadis and Chunovkina 2016), without however evaluating the statistical significance of these fits. The present study is based on the actual behavior of the experimental data and it is focused on an unweighted second-order regression, better fit. Then, the significance of this regression is metrologically compared to an unweighted linear regression forced through the origin, for the determination of H2 S in natural gas by gas chromatography with chemiluminescence detection, as suggested by ASTM D5504 (ASTM D5504 2012).

CONTACT Elcio C. Oliveira [email protected] Technological Development Management and Automation, Services Department, PETROBRAS TRANSPORTE S.A., Rua São Bento, , Centro, -, Rio de Janeiro – RJ, Brazil. ©  Taylor & Francis Group, LLC

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2. Methodology The methodology is divided into five parts. In the first part, the homogeneity of variance is discussed; the second part allows checking the significance of the regression; third and fourth parts are related to the uncertainty measurement associated with the least square fitting and the last one describes a hypothesis testing to compare the different approaches presented in the two previous parts.

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2.1. Homogeneity of variance – Cochran’s test Cochran’s test (Massart et al., 1997) is used for verifying the uniformity (homogeneity) of the regression throughout the calibration curve, comparing the highest variance with the other variances in each point of the calibration curve. If the repeatability of the measurements is independent of the concentration value, this condition of uniform variance is called homoscedastic, and when not uniform, heteroscedastic. 2.2. Regression adjustment evaluation The quantitative verification of the best fit of a calibration curve regression should be evaluated by analysis of variance (ANOVA), although some authors still use the proximity of the coefficient of determination to the unit (Oliveira 2011). The total variability of the answers is decomposed into the sum of the squares due to the regression and the sum of the squares of the residuals; being this latter decomposed in the lack of fit and pure error of the sums of squares. When ANOVA is used to detect lack of fit in a regression adjustment, yˆ = b0 + b1 x + b2 x2 + · · · + bn xn , the total variation of the y values in relation to the mean value, y¯ is described by the total sum of squares, SST , Eq. 1: SST =

ni k   i

(yi j − y¯)2 =

j

ni k  

 

i

j

(yi j − y¯i )2 + 



SSPE

k 



ni (y¯i − yˆi )2 +

i



 SSLOF



k 



i



ni (yˆi − y¯)2 

(1)



SSReg

SSR

k is the number of levels; n is the total number of observations; yi j is the one of the ni replicate measurements at xi and yˆi is the value of y at xi estimated by the regression line. The ANOVA table can be constructed from equations shown in Table 1. k ni k 2 2  SSRes = j (yi j − y¯ j ) ; SSReg = i i ni (yˆi − y¯) ; SST = SSRes + SSReg ;MSReg = SQReg /(k − 1) and MSRes = SQRes /(n − k). When a significant regression is reached, MSReg /MSR is higher than the critical value and MSLOF /MSPE is lower than the critical value. 2.3. Unweighted linear calibration curve forced through zero In cases where the errors present on the x-axis are negligible in relation to the y-axis and the magnitude of the errors in the y-axis is independent of the analyte concentration, an unweighted regression is suitable. Table . ANOVA of simple regression model with replicate observations. Source of variation Regression, Reg Residual, R Lack of fit, LOF Pure error, PE Total

Sum of squares, SS SSReg SSR SSLOF SSPE SST

Degree of freedom, DF 1 n−2 k−2 n−k n−1

Mean squares, MS

F

MSREG MSR MSLOF MSPE

MSReg/MSR MSLOF /MSPE

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The first assumption is that calibration curve takes a straight line, like as y = b0 + b1 x, where b1 is the slope of the line and b0 its intercept on the y-axis. In some situations, the adjusted line may be “forced” to pass through a fixed point (x0 , y0 ), as in the case of ASTM D5504 (ASTM D5504 2012), applied in this study, which recommends that the regression line passes through the origin, i.e., (x0 = 0, y0 = 0). Considering the linear coefficient as zero, yˆ = b1 x , the regression coefficient can be calculated by Eq. (2):  xi yi b1 =  2 xi

(2)

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The standard deviation related to the predicted concentration by the regression sxˆ0 is given by Eq. (3) (Massart et al. 1997): sy/x sxˆ0 = × b1



1 (y0 − y¯)2 1 + + 2 m n b1 i (xi − x) ¯2

(3)

 (yi −ˆy )2 sy/x = (residual standard deviation); yˆi are the predicted values by the regression line; y0 n−2 is the experimental value of y from which the concentration value xˆ0 is to be determined; x¯ and y¯ are mean values; xi and yi are the experimental values; m is the number of measurements to determine x0 and n is the number of measurements for the calibration curve. The expanded uncertainty, U, in relation to xˆ0 can be calculated by Eq. (4): 

U = xˆ0 ± t(n−2) × sxˆ0

(4)

2.4. Unweighted quadratic calibration curve However, in some situations, linear fit is not always the one that leaves less residuals. In these cases, quadratic fit may be a good option. The coefficient values bi can be calculated by solving a single matrix equation: b = (X t X )−1 X t Y , as long as the matrix (X t X ) is not singular. For a second-order model, yi = b0 + b1 xi + b2 xi2 , V(b) can be calculated as: ⎡ ⎤ V (b0 ) Cov (b0 , b1 ) Cov (b0 , b2 ) V (b1 ) Cov (b1 , b2 ) ⎦ V (b) = ⎣ Cov (b0 , b1 ) V (b2 ) Cov (b0 , b2 ) Cov (b1 , b2 ) For a second-order model, the combined standard uncertainty, based on first-order Taylor series approximation, is given by Eq. 5 (Medeiros et al. 2016): 2  2  2  2 ∂x ∂x ∂x ∂x uyˆ + = ub + ub + ub ∂ yˆ ∂b0 0 ∂b1 1 ∂b2 2         ∂x ∂x ∂x ∂x +2 × ub × ub × rb0 ,b1 + 2 × ub × ub × rb0 ,b2 ∂b0 0 ∂b1 1 ∂b0 0 ∂b2 2     ∂x ∂x +2 × ub1 × ub × rb1 ,b2 ∂b1 ∂b2 2 

u2c(x)

(5)

√ Where, uyˆ = S/ n; for n numbers of measurements to determine y0 . ub0 , ub1 and ub2 are respectively the standard uncertainties (standard deviations) of b0 , b1 and b2 . Finally, r is the correlation coefficient. The expanded uncertainty, U, in relation to xˆ0 can be calculated by Eq. (6): U = xˆ0 ± t(n−3) × sxˆ0

(6)

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2.5. Hypothesis test In order to evaluate if there is metrological compatibility between the approaches applied, i.e. partial overlap of the results, the following hypothesis test is suggested. Two measurement systems are considered without significant difference if the absolute value of the difference between the measurements is less than or equal to the square root of the sum of the squares of the expanded uncertainties at the same level of confidence, Eq. (7) (Oliveira 2012). |X1 − X2 | ≤ U12 + U22 (7)

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Where X1 ± U1 is the result from the first approach and X2 ± U2 is the result from the second approach.

3. Experimental For the determination of the H2 S concentration in natural gas, an Agilent 7890 gas chromatograph equipped with sulfur chemiluminescence detector (SCD) is used. SCD exhibits high selectivity and sensitivity for sulfur-containing compounds. The employed method is based on the ASTM D5504, which provides guidelines for the determination of volatile sulfur-containing compounds in high methane content gaseous fuels. Experimental data were carried out in three different moments – 2015 May, 2016 February and 2016 June – at the Division of Geochemistry, PETROBRAS Research and Development Center (CENPES), PETROBRAS, Rio de Janeiro, RJ, Brazil. The peak area values of H2 S in bold, Table 2, are considered as outliers by the Grubbs’ test (Grubbs 1969) and were excluded from the data treatment. The operational conditions of 2015 May and 2016 are as follows: Column: DB-Sulfur SCD 70 m × 0.53 mm × 4.3 µm; Retention gap: 30 cm × 0.20 mm (after column); Split ratio: 8:1; Sampling Loop: 250 µL; SCD Attenuation: 100 and Flow rate: 9.6 mL min−1 (constant). 2015 May calibration curve, derived from H2 S cylinders, was built with mass fractions of 98.2 mg kg−1 , 203 mg kg−1 , 398 mg kg−1 and 506 mg kg−1 , whereas 304 mg kg−1 was used as a sample. 2016 February calibration curve, derived from H2 S gas standards obtained by the ACD Cal 2000 electrochemical gas generator, was built with mass fractions of 10 mg kg−1 , 15 mg kg−1 , 25 mg kg−1 and 40 mg kg−1 , whereas 30 mg kg−1 was used as a sample. The operational conditions of 2016 June are as follows: Column: DB-Sulfur SCD 70 m × 0.53 mm × 4.3 µm; Retention gap: 1.5 cm × 0.18 mm (after column); Pre column: 1 m × 0.18 mm; Split ratio: 4:1; Sampling Loop: 1 mL; SCD Attenuation: 1; Flow rate: 9.6 mL min−1 (constant mode). This calibration curve, derived from H2 S gas standards obtained by the ACD Cal 2000 electrochemical gas generator Table . Experimental data. H S peak area ( mV × s)

Mass fraction (mg kg- ) May 2015

February 2016

June 2016

.              

24301.2 33163.7 . . 112719.4

. . . . . . . . . . . 35260.3 . 74238.1 110426.3

. . . . . . . . . . . . . . .

. . . .  . . 1078.6 . . . . . 74636.3 .

. . . . . 376.6 601.7 . . . 23570.2 . . . .

. . 11375.7 16170.7 .

22361.9 .

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from H2 S cylinders, was carried out with mass fractions of 3 mg kg−1 , 4 mg kg−1 , 5 mg kg−1 and 10 mg kg−1 , whereas 7 mg kg−1 was used as a sample.

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4. Results and discussion After the outlier’s treatment, the variance behavior is studied. Then, the significance of the linear and quadratic fits of each calibration curve is evaluated by ANOVA. The results showed that the seconddegree fit is the most appropriate. From the recommendation of ASTM D5504, linear regression forced through zero, rather than to the second-order fit (justified by ANOVA), some samples are read in the same calibration curves, with these different regression fits. The estimation of the mass fractions and their respective uncertainties are compared in order to evaluate the equivalence between these two applied approaches. Finally, the samples considered as standard ones, with known true value, are evaluated in curves adjusted by both approaches comparing the consensus value between them. All datasets have uniform variances, i.e., homoscedastic behavior. All calibration curves were fitted by linear regression forced through zero and quadratic regression. The maximum relative standard deviations among the peak areas for each mass fraction are in accordance with the recommended value of 5.0% (ASTM D5504 2012).

4.1. May of 2015 ... Calibration curve fit In both approaches cited previously, the value of the mean square ratio, MSReg /MSR , was statistically significant (F-value > F-crit95% ). However, the value of the mean square ratio, MSLOF /MSPE , was statistically significant (F-value > F-crit95% ), which indicated evidence of lack-of-fit only for the linear through zero model, Table 3. ... Hypothesis test based on uncertainty evaluation H2 S peak areas of the reference material (304 ± 9 mg kg−1 ) described in Table 2 were converted in mass fractions based on the linear through zero model and quadratic model. Their expanded uncertainties were calculated from the equations detailed in the items 2.3 and 2.4, respectively. Mass fractions based on linear through zero model and quadratic one are (303 ± 33) mg kg−1 and (288 ± 14) mg kg−1 , respectively. Table . Analysis of the variance applied to May  data. Source

Sum of squares

DF

Mean square

F-value

F-crit %

.

.

.

.

.

.

.

.

Linear through zero model Regression . Residual . Lack-of-fit . Pure error . Total . Percentage of explained variance = .

    

. . . .

Quadratic model Regression . Residual . Lack-of-fit . Pure error . Total . Percentage of explained variance = .

    

. . . .

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√ From Eq. (7), |303 − 288| < 332 + 142 , i.e., 15 < 36, indicates that there are no systematic errors and these two approaches, linear through zero and quadratic models, are statistically comparable, even without differences observed, the quadratic fit presents lower uncertainty than the other approach.

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4.2. February of 2016 ... Calibration curve fit The value of the mean square ratio, MSReg /MSR , was statistically significant (F-value > F-crit95% ). However, the value of the mean square ratio, MSLOF /MSPE , was statistically significant (F-value > F-crit95% ), which indicated evidence of lack-of-fit for the linear through zero model, Table 4. However, for the quadratic model, the value of the mean square ratio, MSReg /MSR , was statistically significant (F-value > F-crit95% ). However, the value of the mean square ratio, MSLOF /MSPE , was statistically significant (F-value < F-crit95% ), which evidenced no of lack-of-fit for this approach, Table 4. ... Hypothesis test based on uncertainty evaluation H2 S peak areas of the reference material (30 mg kg−1 ) described in Table 2 were converted in mass fractions based on the linear through zero model and quadratic model. Their expanded uncertainties were calculated from the equations detailed in 2.3 and 2.4, respectively. Mass fractions based on linear through −1 −1 zero model and quadratic one are (29.95 √ ± 1.94) mg kg and (30.25 ± 0.31) mg kg , respectively. 2 2 From Eq. (7), |29.95 − 30.25| < 1.94 + 0.31 , i.e., 0.30 < 1.97, which indicates that there are no systematic errors and these two approaches, linear through zero and quadratic models, are statistically comparable, even without differences observed, the quadratic fit presents lower uncertainty than the other approach. 4.3. June of 2016 ... Calibration curve fit The value of the mean square ratio, MSReg /MSR , was statistically significant (F-value > F-crit95% ). However, the value of the mean square ratio, MSLOF /MSPE , was statistically significant (F-value > F-crit95% ), which indicated evidence of lack-of-fit for the linear through zero model, Table 5. The value of the mean square ratio, MSReg /MSR , was statistically significant (F-value > F-crit95% ). However, the value of the mean square ratio, MSLOF /MSPE , was statistically significant (F-value < F-crit95% ), which indicated no evidence of lack-of-fit for the quadratic model, Table 5. Table . Analysis of variance applied to February  data. Source

Sum of squares

DF

Mean square

F-value

F-crit %

.

.

.

.

Linear through zero model Regression . Residual . Lack-of-fit . Pure error . Total . Percentage of explained variance = .

    

. . . .

Quadratic model Regression . Residual . Lack-of-fit . Pure error . Total . Percentage of explained variance = .

    

. . . .

.

.

.

.

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Table . Analysis of variance applied to  June data. Source

Sum of squares

DF

Mean square

F-value

F-crit %

.

.

.

.

.

.

.

.

Linear through zero model Regression  Residual . Lack-of-fit . Pure error . Total  Percentage of explained variance = .

    

 . . .

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Quadratic model Regression  Residual . Lack-of-fit . Pure error . Total  Percentage of explained variance = .

    

 . . .

... Hypothesis test based on uncertainty evaluation H2 S peak area of the reference material 7 mg kg−1 described in Table 2 were converted in mass fractions based on the linear through zero model and quadratic model. Their expanded uncertainties were calculated from the equations detailed in 2.3 and 2.4, respectively. Mass fractions based on linear through −1 −1 zero model and quadratic one are (7.18 √ ± 0.89) mg kg and (6.91 ± 0.03) mg kg , respectively. 2 2 From Eq. (7), |7.18 − 6.91| < 0.89 + 0.03 , i.e., 0.27 < 0.89, which indicates that there are no systematic errors and the two approaches, linear through zero and quadratic models, are statistically comparable, even without differences observed, the quadratic fit presents lower uncertainty than the other approach.

5. Conclusions This study discusses a comparison of a proper fit of calibration curves in the determination of H2 S mass fractions in natural gas standards by GC-SCD. The results based on the ASTM D5504-12 approach are perfectly satisfactory; since the hypothesis test used shows that there are no significant differences between the results coming from the quadratic calibration curve and from the linear calibration curve forced through zero. Thus, despite the noticeable difference among the uncertainties obtained by the two approaches, it is recommended to use the ASTM D5504-12 calibration method for geochemical issues because of its simplicity for a routine work as the calculated uncertainties using this approach will not interfere in the interpretation of the geochemical data. However, in situations where more rigorous measurement uncertainties are required, quadratic calibration curve is recommended. As future work, this research group intends to study the dilution of samples in gas phase, with high contents of H2 S, using calibrated syringes, as natural gas samples may contain higher amounts of H2 S than the upper limit of the established calibration curve, as well as to perform all calibration curves with certified reference materials (CRM) in order to obtain more accurate analytical results.

ORCID Elcio C. Oliveira

http://orcid.org/0000-0001-8979-4131

References ASTM International. 2012. Standard test method for determination of sulfur compounds in natural gas and gaseous fuels by gas chromatography and chemiluminescence. ASTM D5504, West Conshohocken, PA: ASTM.

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