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Data Reconciliation in the Natural Gas Industry: Analytical Applications Elcio C. Oliveira*,† and Paula F. Aguiar‡ Petrobras Transporte S.A., Rio de Janeiro, RJ, Brazil, and Instituto de Quı´mica, UniVersidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil ReceiVed February 18, 2009. ReVised Manuscript ReceiVed May 20, 2009
The composition and physicochemical parameters of natural gas can vary in space and time depending upon the gas source and blending. The key parameters of natural gas are important to know for transportation and efficient consumption. Thus, a natural gas quality measurement is required for technical and economic reasons. The data reconciliation technique has the objective of using measurement redundancies to obtain the best estimate of the conventional true value and, consequently, to minimize its uncertainty. This work discusses this subject in analytical applications related to the natural gas industry. This methodology shows to be more suitable than the traditional ones when it is applied to a characterization study, a dispute between supplier and customer, an operation diagnosis in industrial unit, and an among day homogeneity study.
Natural gas is an attractive fossil carrier of energy. For natural gas production, transportation, trading, and consumption, it is therefore of great economic and technical importance to have knowledge of the physicochemical natural gas properties. The commercial transitions of the natural gas are based on its calorific value, which mainly depends upon composition and density. To guarantee quality in these process measurement analytical applications, a powerful tool is suggested: data reconciliation (DR). Process measurements are subject to errors of either random or systematic nature, causing the conservation laws to be disobeyed.1,2 The procedure known as DR is based on the proposal of adjustments, so that the measurements fit within the conservation laws and the restrictions imposed on the process. DR is a technique that has been developed to improve the accuracy of measurements by reducing the effect of random errors in the data. The principal difference between DR and other filtering techniques is that DR explicitly makes use of process model constraints, degree of statistical dependence between or among variables, and obtains estimates of process variables by adjusting process measurements, so that the estimates satisfy the constraints. The reconciled estimates are expected to be more accurate than the measurements and, more importantly, are also consistent with the known relationships between process variables as defined by the constraints. DR achieves error reduction only by exploiting the redundancy property of measurements. Here, redundancy is the
duplication of critical components of a system with the intention of increasing reliability of the system. Typically, in any process, the variables are related to each other through constraints. Given a set of such system constraints, a minimum number of errorfree measurements is required to calculate all of the system parameters and variables. If there are more measurements than this minimum, then redundancy exists in the measurements that can be exploited. The classical general DR problem deals with a weighted leastsquares minimization of the measurement adjustments subject to the model constraints.3 The reconciliation technique allows that two or more experimental redundant measurements, mei , can be reconciled to a single value, reconciled measurement, mc, that represents the chemical, physical, or physicochemical quantity to be analyzed better, thereby minimizing the measurement uncertainty, umc.4 The DR methodology is based on the estimation of parameters.5 Estimating parameters consists basically of concluding the values of the parameters that cannot be measured nor evaluated a priori from a comparison established between experimental data and an available model for the process. In 1961, Kuehn and Davison6 presented one of the first approaches in relation to DR problem formulation and a method based on Lagrange multipliers to solve the steady-state DR problem. Around 25 years later, in 1980, Knepper and Gorman7 proposed another approach based on successive linearization of the nonlinear equations of the system (constraints). However, Liebman and Edgar8 illustrated that nonlinear programming gives improved reconciliation estimates compared to successive linearization. Recently, Narasimhan and Jordache1 published a
* To whom correspondence should be addressed. Telephone: (5521) 3211 9223. E-mail:
[email protected]. † Petrobras Transporte S.A. ‡ Universidade Federal do Rio de Janeiro. (1) Narasimhan, S.; Jordache, C. Data Reconciliation and Gross Error Detection: An Intelligent Use of Process Data; Gulf Publishing Company: Houston, TX, 2000. ¨ zyurt, D. B.; Pike, R. W. Comput. Chem. Eng. 2004, 28, 381– (2) O 402.
(3) Alhaj-Dibo, M. Control Eng. Pract. 2007, doi: 10.1016/j.conengprac.2007.01.2003. (4) Kretsovalis, A.; Mah, R. Chem. Eng. Sci. 1987, 42, 2115–2121. (5) Ramamurthi, Y.; Sistu, P. B.; Bequette, B. W. Comput. Chem. Eng. 1993, 17, 41–59. (6) Kuehn, D. R.; Davison, H. Chem. Eng. Prog. 1961, 57, 44–47. (7) Knepper, J. C.; Gorman, J. W. AIChE J. 1980, 26, 260–264. (8) Liebman, M. J.; Edgar, T. F. Data reconciliation for nonlinear processes. Presented at the AIChE Annual Meeting, Washington, D.C., 1988.
1. Introduction
10.1021/ef9001428 CCC: $40.75 2009 American Chemical Society Published on Web 06/26/2009
DR in the Natural Gas Industry
Energy & Fuels, Vol. 23, 2009 3659
book that provides a systematic and comprehensive treatment of DR and gross error detection techniques. The aim of this work is to make public this subject in analytical applications related to the natural gas industry. In the recent revision of the method that is specified in the International Organization for Standardization (ISO) standard on reference materials,9 it is superficially treated. The 1989 version did not mention this subject, and the last one (2006) does not detail the statistical models used, introducing only a single application. In this work, the methodology is applied and discussed in four different case studies in a natural gas matrix: (i) characterization study of methane as a certified reference material, (ii) dispute between supplier and customer related to ethane content, (iii) operation diagnosis of chromatographs in an industrial unit, and (iv) an among day homogeneity study in terms of density. 2. Methodology 2.1. DR Statistics. The first stage of the theory of DR is the choice of a reference model. The model must be as close as possible from the situation in which the data are generated, process model. Therefore, the evaluation procedure of parameters estimates the existence of an objective function Fob, that is, of a function that measures the distance between the experimental and predicted data for the model. The next step is to optimize the formulated objective function (minimum or maximum, to depend upon the underlying logic of the objective function used). This optimization routine is weighed for the experimental variances, which, in this work, are expressed in terms of measurement uncertainty. The multidimensional model most suitable to describe experimental data fluctuations is the multivariate normal, whose probability density function P(X) with a diagonal covariance matrix V is
P(X) )
1 1 exp - (X - µ)TV-1(X - µ) 1/2 2 ((2π) det V) N
[
maximum likelihood offers a way of tuning the free parameters of the model to provide a good fit. This method is based on the following assumptions: (i) distribution of experimental errors is known; (ii) the hypothesis of the perfect model is valid; and (iii) the hypothesis of the experiment that is well-performed and without systematic errors is valid. The method is described as
P(X) )
1 1 exp - (ε)TV-1(ε) 1/2 2 ((2π) det V)
[
N
]
(4)
P(ε) ) P(Ze - Zc) P(X) )
(5)
1 1 exp - (Ze - Zc)TV-1(Ze - Zc) 1/2 2 ((2π) det V)
[
N
] (6)
P(Ze) must be maximum; that is, P(ε) must be the greatest value. The probability must be maximized because it has been used in the principle of the experiment that is well-performed. According to this principle, the probability of finding the best experimental values is maximum because the experiments are considered to be carefully performed and there is no reason to believe that the residuals are greater than the unavoidable minimum value.
max[P(Ze - Zc)] ) Fob ) max
[
1 × ((2π)N det V)1/2
1 exp - (Ze - Zc)TV-1(Ze - Zc) 2
[
[
Fob ) max ln
]
]]
(7)
]]
1 1 - (Ze - Zc)TV-1(Ze - Zc) ((2π)N det V)1/2 2
[
(8)
1 Fob ) max - (Ze - Zc)TV-1(Ze - Zc) 2
[
]
(9)
(1) Fob ) max[-(Ze - Zc)TV-1(Ze - Zc)]
with the following parameters: σi2 are the variances of the diagonal
(10)
To minimize Fob is the same as to maximize the probability to find experimental measures, provided that the model is perfect and any errors are random, normally distributed and independent.10 This function is usually called the least weighted squares. matrix of variance, V. The reliable region of this curve is where N points have equal density of probability
P(X) ) constant ) (X - µ)TV-1(X - µ)
(2)
From the principle that the mathematical model is perfect, any bias between the experimental data (X or Ze) and the reconciled data (µ or Zc) must only and exclusively be due to the experimental uncertainties
error ) ε ) [Ze - Zc]
(3)
Fob ) min[(Ze - Zc)TV-1(Ze - Zc)]
2.2. Applying the Technique. Detailing eq 11 for the following mathematical model: m1c ) m2c )... ) mNc ; that is, the reconciliation of N results from the same property
Ze )
[] [ ][] me1
mc1 ) mc
me2
)m l c m N ) mc
l meN
Zc )
mc2
A common way to evaluate parameters is to use the maximum likelihood method. Maximum likelihood estimation (MLE) is a popular statistical method used for fitting a mathematical model to some data. The modeling of real-world data using estimation by (9) International Organization for Standardization (ISO). ISO Guide 35. Reference MaterialssGeneral and Statistical Principles for Certification, 3rd ed.; ISO: Geneva, Switzerland, 2006.
(11)
c
mc c ) m l mc
[ ] me1 - mc
Ze - Z c )
me2 - mc l e m N - mc
For quantities not correlated (10) Mansour, M.; Ellis, J. E. Appl. Math. Model. 2008, 32, 170–184.
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OliVeira and Aguiar
umc )
1
(19)
∑ σ1
2 i
2.2.1. For equal variances, σ12 ) σ22 )... ) σN2 , eqs 17 and 19 assume the following forms:
mc )
u mc )
The objective function assumes the general form
Fob ) min
∂Fob c
∂m
[
(me1 - mc)2 σ21
+
(me2 - mc)2
(meN - mc)2
+ ... +
σ22
σ2N
]
(14)
) min Fob )
[
-2(me1 - mc) σ21
2(me2 - mc)
-
σ22 2(meN
-m) c
σ2N me1 mc )
+
me2
+ ... +
- ... -
]
)0
(15)
meN
σ21 σ22 σ2N 1 1 1 + 2 + ... + 2 2 σ1 σ2 σN
m )
2 i
Case Study 1: Characterization Study. Data and their respective uncertainties from 12 laboratories accredited by ISO 1702512 (Table 2) are used in the characterization study of methane in natural gas as a certified reference material. c . Mathematical model: m1c ) m2c )... ) m12 Reconciled value: From eqs 17 and 19, respectively, the reconciled value mc ) 88.71% and the minimized standard uncertainty umc ) 0.02%.
2 i
umc )
In addition, it assumes the following standard uncertainty based on GUM:11
um2 c )
(
1 σ21 1 1 1 + 2 + ... + 2 2 σ1 σ2 σN
(
)
2
um1e
1 σ22
(
( (
) )
88.84 88.71 88.58 88.52 88.29 88.89 + + + + + + 0.222 0.162 0.022 0.202 0.382 0.522 88.58 88.69 88.57 88.50 88.90 88.85 + + + + + 0.222 0.172 0.152 0.342 0.282 0.232 1 1 1 1 1 1 + + + + + + 0.222 0.162 0.022 0.202 0.382 0.522 1 1 1 1 1 1 + + + + + 0.222 0.172 0.152 0.342 0.282 0.232
(
1 1 1 1 1 1 1 + + + + + + 0.222 0.162 0.022 0.202 0.382 0.522 1 1 1 1 1 1 + + + + + 0.222 0.172 0.152 0.342 0.282 0.232
) 88.71%
)
) 0.02%
When the arithmetic average is used instead of DR, its uncertainty by LPU (eq 21) is
+
1 1 1 + 2 + ... + 2 2 σ1 σ2 σN
(21)
√N
3. Results and Discussion
(17)
∑ σ1
umie
2.2.2. For different variances and the most simple model, m1c ) m2c, some situations can be observed in Table 1.
mc )
∑σ
(20)
Specifically, for σ12 ) σ22, mc ) (m1e + m2e)/2, and umc ) 0.71ume1.
(16)
mei
c
me1 + me2 + ... + meN N
)
uaverage )
2
um2e
+ ... +
1 σ2N
1 1 1 + 2 + ... + 2 2 σ1 σ2 σN
)
2
umNe
(18)
Equation 18 becomes eq 19, when σi2 is assumed to be equal to um2 i for i ) 1, 2,..., N (11) International Organization for Standardization (ISO). Guide for the Expression of Uncertainty in Measurements; ISO: Geneva, Switzerland, 1993.
1 × (0.222 + 0.162 + 0.022 + 0.202 + 0.382 + 122 0.522 + 0.222 + 0.172 + 0.152 + 0.342 + 0.282 + 0.232)
) 0.08%
Figure 1 shows the value found for each laboratory and its respective band of uncertainty. It is possible to observe that the reconciled value is inside of the band of uncertainty of all of the laboratories, which does not happen when the arithmetic average of all laboratories is achieved to represent the best estimate of the conventional true value. In this case, from 12 (12) International Organization for Standardization (ISO). ISO/IEC International Standard 17025. General Requirements for the Competence of Testing and Calibration Laboratories, 1st ed.; ISO: Geneva, Switzerland, 1999.
DR in the Natural Gas Industry
Energy & Fuels, Vol. 23, 2009 3661 Table 1. Reconciled Measure and Uncertainty
relation between variances
reconciled measure
2σ21 ) σ22 4σ21 ) σ22 8σ21 ) σ22 ∞σ21 = σ22 n1σ21 ) n2σ22
mc mc mc mc mc
) ) ) = )
Table 2. Accredited Laboratories and Their Uncertainties laboratory
concentration (%), mie
standard uncertainty (%), umie
1 2 3 4 5 6 7 8 9 10 11 12
88.89 88.84 88.71 88.58 88.52 88.29 88.85 88.58 88.69 88.57 88.50 88.90
0.22 0.16 0.02 0.20 0.38 0.52 0.22 0.17 0.15 0.34 0.28 0.23
Table 3. Dispute between Supplier and Customer
supplier customer
ethane content (%, v/v)
standard uncertainty (%, v/v)
9.65 10.30
0.30 0.15
minimized uncertainty
(2me1 + me2)/3 (4me1 + me2)/5 (8me1 + me2)/9 me1 (n1me1 + n2me2)/(n1 + n2)
specification in out
laboratories, 4 laboratories are outside of the average value and its respective standard uncertainty (laboratories 1, 2, 3, and 12). Case Study 2: Dispute between Supplier and Customer. The maximum content of ethane in commercialized natural gas allowed by law in Brazil is 10% (v/v).13 From real data of a typical study case of the Brazilian industry (Table 3), can the product be commercialized? It is a classical dispute case in the natural gas industry, because on the basis of the supplier results, it is in the specification, but on the basis of the customer results, it is out of specification. Mathematical model: m1c ) m2c. Reconciled value: From eqs 17 and 19, applying the technique of DR, it is observed that the product is clearly out of
umc ) 0.82ume1 umc ) 0.89ume1 umc ) 0.94um1e umc = ume1 umc ) ((n21um2 e1 + n1n2um2 e1)1/2)/(n1 + n2)
specification; therefore, the reconciled value mc ) 10.17% (v/ v) and minimized standard uncertainty umc ) 0.13% (v/v); that is, 10.04-10.30% (v/v) are above the permissible maximum value of 10% (v/v). 9.65 10.30 + 2 0.30 0.152 mc ) ) 10.17% 1 1 + 0.302 0.152 umc )
1 ) 0.13% 1 1 + 0.302 0.152
In this case, it is observed that the great advantage of the use of this technique is a detriment to the arithmetic average and its confidence interval. The average value is 9.98%, with an interval of 0.17%; that is, 9.81-10.15%. For this traditional approach, the product is in the specification; however, the uncertainty band has a partial overlap in relation to the specification limit, placing it outside of specification, generating, once more, a dispute between the parts (Figure 2). Case Study 3: Operation Diagnosis. A typical architecture of the natural gas industry is supposed (Figure 3), where three streams compose the end product to be commercialized in superior calorific value (SCV) terms. The calculation of the superior calorific value is based on the gas composition analyzed by gas chromatography. In each natural gas stream (SCV1e, SCV2e, and SCV3e), there is an on-line chromatograph and the final composition (SCV4e) is also analyzed by a off-line chromatograph. Therefore, redundancy exists in the measurements, and the process model
Figure 1. Uncertainty bands, reconciled value, and averaged value (uncertainty band of each laboratory; s, reconciled value; - - -, arithmetic average).
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Energy & Fuels, Vol. 23, 2009
Fob )
Figure 2. Reconciled value and averaged value.
∂Fob
[
[
∂SCVc1 Figure 3. Typical architecture of the natural gas industry.
constraints are considered as
SCVc4 )
SCVc1 + 0.99SCVc2 + 1.01SCVc3 3
(SCVe1)2 - 2SCVe1SCVc1 + (SCVc1)2
+
σ21
(SCVe2)2 - 2SCVe2 × 0.99SCVc1 + 0.992(SCVc1)2 σ22 (SCVe3)2 - 2SCVe3 × 1.01SCVc1 + 1.012(SCVc1)2 σ23
+
+
(SCVe4)2 - 2SCVe4SCVc1 + (SCVc1)2 σ24
)
-2SCVe1 + 2SCVc1
-1.98SCVe2 + 1.9602SCVc1
+
σ21
σ22
-2.02SCVe3 + 2.0402SCVc1 σ23 2SCVe1 SCVc1
The constraints can be described as
]
OliVeira and Aguiar
σ21
)
+
+
+
-2SCVe4 + 2SCVc1
1.98SCVe2
σ24 2.02SCVe3
+ + σ22 σ23 2 1.9602 2.0402 2 + + + 2 2 2 2 σ1 σ2 σ3 σ4
]
)0
2SCVe4 σ24
SCVc2 ) 0.99SCVc1
(
)
Applying11 SCVc3 ) 1.01SCVc1 2 uSCV c ) 1
SCVc4 ) SCVc1
σ21 2 1.96 2.04 2 + 2 + 2 + 2 σ21 σ2 σ3 σ4
The general equation (eq 2) assumes these following forms to SCV1c
Fob )
[
(SCVe1 - SCVc1)2 σ21
+
(SCVe3
(SCVe2 - SCVc2)2 σ22 -
SCVc3)2
σ23
+
+
(SCVe4
σ24
[ ] (SCVe1)2 - 2SCVe1SCVc1 + (SCVc1)2 σ21
σ22
(SCVe3)2 - 2SCVe3SCVc3 + (SCVc3)2 σ23
(SCVe4)2
-
2SCVe4SCVc4 σ24
+
]
( (
+
σ22 2 1.96 2.04 2 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
+
) )
2
uSCV2e
2
2.02SCVe3 σ23 2 1.96 2.04 2 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
(
+
uSCV3e
+
)
2
2SCVe4 σ24 2 1.96 2.04 2 + 2 + 2 + 2 σ21 σ2 σ3 σ4
uSCV4e
In the same way, it is described to SCVc2, SCVc3, SCVc4 and their respectively uncertainties 2.02SCVe1
+
(SCVc4)2
uSCV1e
1.98SCVe2
+
(SCVe2)2 - 2SCVe2SCVc2 + (SCVc2)2
Fob )
SCVc4)2
2
2SCVe1
SCVc2 )
σ21
+
2SCVe2 σ22
+
2.04SCVe3
+
σ23 2.04 2 2.04 2.02 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
2.02SCVe4 σ24
(
)
DR in the Natural Gas Industry
( (
2SCVe2
σ21
(
+
2 uSCV c ) 4
+
2SCVe3
+ + σ22 σ23 1.96 1.92 2 1.96 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
)
1.98SCVe4 σ24
2
1.96SCVe2
+
) )
2
σ22 u e 1.96 1.92 2 1.96 SCV2 + + + σ21 σ22 σ23 σ24
+
2
2SCVe3
σ23 u e 1.96 1.92 2 1.96 SCV3 + + + σ21 σ22 σ23 σ24
(
)
2
σ24 u e 2.04 2 2.04 2.02 SCV4 + + + σ21 σ22 σ23 σ24
1.98SCVe1
( (
+
2.0202SCVe4
1.96SCVe2
σ21
)
+
1.98SCVe2
+
2.02SCVe3
σ22 σ23 2 1.96 2.04 2 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
+
a
40010 39620 40700 40030
uSCVe1 uSCVe2 uSCVe3 uSCV4e
+
)
2
1.96SCVe4
(
σ21 2 1.96 2.04 2 + 2 + 2 + 2 σ21 σ2 σ3 σ4
( (
)
uSCV1e
1.98SCVe2 σ22 2 1.96 2.04 2 + 2 + 2 + 2 σ21 σ2 σ3 σ4
+
) )
2
uSCV2e
2
2.02SCVe3 σ23 2 1.96 2.04 2 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
(
+
uSCV3e
+
2SCVe4 σ24 2 1.96 2.04 2 + 2 + 2 + 2 σ21 σ2 σ3 σ4
Values are in kJ m-3.
SCVc1 SCVc2 SCVc3 SCVc4
40091 39690 40492 40091
uSCVc1 uSCVc2 uSCVc3 uSCV4c
)
2
uSCV4e
From experimental typical values of superior calorific value and its respective standard uncertainties14 (Table 4), the reconciled values of the calorific value are calculated as well its reconciled standard uncertainties. The superior calorific value analyzed by the off-line chromatograph, SCV4e, presented values of 40 030 ( 56 kJ m-3; that is, 39 974 to 40 086 kJ m-3. This value calculated for the restriction of the process, SCV4e ) (SCV1e + 0.99SCV2e + 1.01SCVe3)/3 ) 40 114 kJ m-3, is outside of this band. Therefore, one or more analyzers had not generated reliable results. It is possible to note that there is a partial overlap between the experimental value and the reconciled one in the analyzers 1, 2, and 4, which does not occur in the analyzer 3 (Figure 4). It indicates a doubt in the results from this equipment. Case Study 4: Homogeneity Study. An among day homogeneity study, in the same natural gas stream based on density is carried out throughout 5 days. Table 5 shows the data of the homogeneity study.
σ24 u e 1.96 1.92 2 1.96 SCV4 + + + σ21 σ22 σ23 σ24
56 55 57 56
σ24
2
2SCVe1
Table 4. Reconciled Calorific Value and Its Standard Uncertaintya SCVe1 SCVe2 SCVe3 SCVe4
2SCVe4
2
σ23 u e 2.04 2 2.04 2.02 SCV3 + + + σ21 σ22 σ23 σ24
)
(
) )
SCVc4
2
2.0404SCVe3
σ21 ) u e 1.96 1.92 2 1.96 SCV1 + + + σ21 σ22 σ23 σ24
2 uSCV c 3
+
σ22 u e 2.04 2 2.04 2.02 SCV2 + + + σ21 σ22 σ23 σ24
1.98SCVe1 SCVc3
2SCVe1
2
2.02SCVe1
σ21 ) u e 2.04 2 2.04 2.02 SCV1 + 2 + 2 + 2 2 σ1 σ2 σ3 σ4
2 uSCV c 2
Energy & Fuels, Vol. 23, 2009 3663
28 28 29 28
Figure 4. Experimental values versus reconciled values.
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Table 5. Measurement Data of an among Day Homogeneity Study of Density in Natural Gasa result
day 1
day 2
day 3
day 4
day 5
1 2 3 4
98.8 98.7 98.9 98.8
99.3 98.7 98.8 99.2
98.3 98.5 98.8 98.8
98.0 97.7 97.4 97.3
98.8 99.1 99.6 99.2
a
ucx )
sr2 sL2 + ) p n×p
sr2 ) MSwithin days ) 0.07 kg m-3
Table 6. Average, Variance, and the Number of Results Per Day (n0)a day day day day day a
count
sum
average
variance
4 4 4 4 4
395.2 396.0 394.4 390.4 396.8
98.8 99.0 98.6 97.6 99.2
0.007 0.087 0.060 0.100 0.109
1 2 3 4 5
Values are in kg m-3.
Table 7. One-Way ANOVA Results from Data from Table 5 source of variation
sum of the degrees of mean F F squares (SS) freedom square (MS) calculated critical
among days within days
6.10 1.09
4 15
1.52 0.07
21.03
3.06
and sL2 )
MSamong days - MSwithin days 1.52 - 0.07 ) ) n0 4 0.37 kg m-3
However, by the technique of reconciliation, whose mathematical model is m1c ) m2c ) m3c ) m4c ) m5c, from eqs 17 and 19, respectively 99.0 98.6 97.6 99.2 98.8 + + + + 0.007 0.087 0.060 0.100 0.109 ) m ) 1 1 1 1 1 + + + + 0.007 0.087 0.060 0.100 0.109 98.75 kg m-3 c
umc )
cx )
p
1
)
p
∑n
ni
∑ ∑x i)1 j)1
ij
)
1 1972.7 ) 98.64 kg m-3 5×4
i
i)1
(13) Ageˆncia Nacional do Petro´leo. Estabelece a especificac¸a˜o do ga´s natural, de origem nacional ou importada, a ser comercializado em todo territo´rio nacional. July 8, 2002; Portaria 104. (14) Oliveira, E. C. Metrologia do ga´s natural: Avaliac¸a˜o de incerteza de suas propriedades fı´sico-quı´micas. Rio Oil and Gas Expo and Conference, 2006. (15) Van der Veen, A. M. H.; Linsinger, T. P. J.; Pauwells, J. Accredit. Qual. Assur. 2001, 6, 26–30.
1 1 1 1 1 1 + + + + 0.007 0.087 0.060 0.100 0.109 umc ) 0.07 kg m-3
Figure 5. Reconciled data of the homogeneity study (s, reconciled value ( minimized uncertainty; - - -, grand mean ( uncertainty).
Traditionally, the technique used to solve this problem is the analysis of variance (ANOVA). The data in Table 5 were treated and are available in Table 6. Table 7 details the ANOVA of the data in Table 5. In this case, because the F calculated is higher than the F critical, it is observed that there is a highly significant effect on one or more days. The parameters to be estimated are the grand mean cx and its uncertainty associated ucx, the between laboratory standard deviation sL, and the repeatability standard deviation sr.15
-3
where p is the number of laboratories in a collaborative study and n is the number of observations. Given that
Values are in kg m-3.
group
0.375 + 50.07× 4 ) 0.28 kg m
When the two approaches (the traditional and the proposal) are compared, it is observed in Figure 5 that the technique of DR is more restrictive. On the basis of the uncertainty, calculated from eq 15, an external pair of horizontal lines (grand mean ( uncertainty ) 98.36-98.91 kg m-3), excludes itself only on day 4 but with the technique of DR, from the minimized standard uncertainty, an internal pair of horizontal lines (reconciled value ( minimized uncertainty ) 98.68-98.82 kg m-3), are added to day 4. Also observed is the existence of a highly significant effect on day 5. 4. Conclusions The technique of DR was provided in more detail in this paper, allowing the reader to use it more easily. It presents different results if compared to the current techniques discussed in the text, being another option available to natural gas industry analytical applications. The case studies presented allowed us to verify that not only is the technique very simple to use with linear systems but that is is also more critical than those currently used in analytical applications in the natural gas industry. The results may serve as a “judge” between the parties involved, besides minimizing the measurement uncertainty. EF9001428
