Measurement Uncertainty

The outcome of a measured property depends on the measuring system, the measurement procedure, the skill of the operator, the environment and other effects [207]. As a result, no measurement is exact and measurement uncertainty is stated as a statistical parameter that discusses the possible fluctuations of a measured property [208]. An average value of the dispersion of a number of indication values carries the information of an estimation of the true quantity value and that would not generally be adequate to consider as a final value for a measured property.

Measurement error refers to the deviation of measured quantity from its true value or reference. There are two types of errors involved in measurements known as systematic and random errors. Systematic error is a quantifiable measurement error (offset), which remains constant in replicate measurements. The random error varies in an unpredictable manner in replicate measurements.

The corrections that are made for the measurement errors are often performed prior to the uncertainty analysis as described according to the Guide to the Expression of Uncertainty in Measurements (GUM)[209, 210]. The GUM method provides a different approach rather than express the results of measurement by giving the best estimate of the measurand along with the information about random and systematic errors. The GUM approach enables us to express the results of measurement as the best estimate of measurand together with associated measurement uncertainty.

6.1.1 Uncertainty of Viscosity Measurements

The uncertainty of viscosity measurements is raised due to the uncertainty sources involved in the measuring method. Several uncertainty sources were identified for both aqueous and CO² loaded solutions. Figure 6.1 illustrates the considered uncertainty sources for the viscosity measurement of aqueous amine solutions.

The mathematical model for a double-gap viscometer as shown in Eq (147) provides the relevant uncertainty sources for the uncertainty of viscosity measurements. There are additional uncertainty sources involved in viscosity measurement that are not shown in the model. Such sources are added to the model equation as shown in Eq (148). The uncertainty analysis was done by following the guidelines provided by GUM and QUAM (Quantifying Uncertainty in Analytical Measurement)[210].

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Figure 6.1: Cause and effect diagram for uncertainty analysis of viscosity measurement [211].

𝜂 = ^𝑇

4𝜋𝐿𝜔𝑅²( ^𝑘1 2 𝑘12−1+^𝑘22𝑘3

2 𝑘32−𝑘22)

(147)

Here, T is torque, 𝜂 is dynamic viscosity, L is the liquid height, R is the radius of the inner fixed cylinder, ω is angular velocity, 𝑅1= 𝐾1𝑅 , 𝑅2= 𝐾2𝑅 and 𝑅3= 𝐾3𝑅.

𝜂 = ^𝑇

4𝜋𝐿𝜔𝑅²( ^𝑘12 𝑘12−1+^𝑘22𝑘3

2 𝑘32−𝑘22)

𝑓_𝑝𝑓_𝑡𝑓_𝑤𝑓_𝑟𝑒𝑝 (148)

Where 𝑓𝑝 is purity of MEA, 𝑓𝑡 is temperature, 𝑓𝑤is weight measurement and 𝑓𝑟𝑒𝑝 is repeatability. Those factors are added to the original viscosity expression to consider uncertainty sources, which are not shown in Eq (147).

In the GUM method, the propagation of uncertainty based on the first-order Taylor series approximation is considered in the uncertainty evaluation. A Gaussian distribution is assumed as the probability distribution and that is also used to define confidence intervals.

Consider a measuring system as described in Eq (149).

𝑦 = 𝑓(𝑥₁, 𝑥₂, … , 𝑥_𝑁 ) (149)

Where 𝑦 is the measurand and 𝑥₁, 𝑥₂, … 𝑥_𝑁 are the input quantities. The propagation of uncertainty according to the Taylor series expansion of 𝑦,

𝑢²(𝑦) = ∑ (^𝜕𝑓

𝜕𝑥_𝑖)²𝑢²(𝑥_𝑖) + 2

𝑁𝑖=1 ∑ ∑ ^𝜕𝑓

𝜕𝑥_𝑖

𝜕𝑓

𝜕𝑥_𝑗𝑢(𝑥_𝑖,𝑥_𝑗)

𝑁𝑗=𝑖+1

𝑁−1𝑖=1 (150)

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83 In Eq (150), (𝜕𝑓 𝜕𝑥⁄ _𝑖) gives the partial derivatives (sensitivity coefficients) [212, 213], 𝑢²(𝑦) is the variance of the measuring result, the variance of the input quantity 𝑥_𝑖 is given by 𝑢²(𝑥_𝑖) and the covariance between 𝑥_𝑖 and 𝑥_𝑗 is given by 𝑢(𝑥_𝑖,𝑥_𝑗) [213].

𝑈(𝑦) = 𝑘𝑢(𝑦) (151)

In general, calculated combined uncertainty from Eq (151) is defined as the standard uncertainty with the coverage factor 𝑘=1 which covers 68% of the interval. It is common to represent uncertainty as an expanded uncertainty with 𝑘 =1.96 or 2, which covers approximately 95% of the interval.

Table 6.1 lists the considered uncertainty sources with corresponding distributions and values. The calculated uncertainties for the viscosity of different MEA solutions are given in Table 6.2. The attached Article N and Article O provide more information about the analysis.

Table 6.1: Uncertainty sources with corresponding distributions and values[202].

Input quantity X_i Probability

Distribution Uncertainty U(x_i)

Torque (𝜏) Triangular 0.082 𝜇𝑁𝑚

Level (𝐿) Gaussian 0.45 𝑚𝑚

Angular velocity (𝜔) Triangular 0.01 𝑟𝑎𝑑 · 𝑠⁻¹

Radius (𝑅) Triangular 4.1 𝜇𝑚

Purity Rectangular 2.886x10^-3

Temperature Triangular 2.45 x10^-4

Weight measurement Rectangular 8 x10^-6

CO² loading Gaussian 0.013

Repeatability Gaussian 0.00348

The uncertainty sources in the viscosity measurement was calculated as follows.

Torque (𝜏):

The Anton Paar user manual [214] provide the accuracy of torque measurement as, Torque accuracy: max. (0.2 μNm; 0.05%)

Then a triangular distribution was considered to calculate the uncertainty of torque.

Standard uncertainty of torque (τ) measurement 𝑢(𝜏) =^0.2

√6 = 0.082

μNm

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Angular Velocity (𝜔):

All the viscosity measurements were performed under a shear rate of 1000 1/s. for this analysis ± 1 1/s of accuracy was assumed to determine the uncertainty of angular velocity. Using Eq (152), Eq (153) and triangular distribution, the standard uncertainty of angular velocity (ω) was determined as,

𝛾̇ =^𝜋𝑛

30.^1+𝛿2

𝛿²−1 (152)

𝜔 = ^2𝜋

60𝑛 (153)

𝑢(𝜔) = 0.01rad⸳s^-1 Level (𝐿):

The liquid was transferred using a 10 mL syringe with an accuracy of ±0.1 mL. Using a triangular distribution, the uncertainty of volume measurement was determined as ±0.4 mL. Then using the geometry of the double-gap rheometer, the standard uncertainty of level was calculated as 0.45 mm.

Radius (𝑅):

For the analysis, ± 0.01 mm of accuracy was assumed to determine the uncertainty of the radius of the cylindrical geometries. Assuming a triangular distribution for the uncertainty, ± 4.1 μm was obtained for the standard uncertainty of radius.

Purity (𝑃):

The purity of the monoethnaol amine is 99.5%. Considering a rectangular distribution for the standard uncertainty 𝑢(𝑓_𝑝) was calculated as,

𝑢(𝑝) =0.005

√3 = 0.0029 = 𝑢(𝑓_𝑝)

The 𝑢(𝑓_𝑝) vary with the material; here considered only monoethano amine for the aqueous monoethanol amine mixtures.

Temperature (𝑇):

The temperature accuracy of the instrument is ± 0.03 K. A triangular distribution and an average temperature of the measuring range (293.15 K – 393.15 K) were assumed to calculate the uncertainty of temperature 𝑢(𝑓_𝑇).

Standard uncertainty of temperature, 𝑢(𝑇) =0.03

√6 = 0.012 𝐾

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85 𝑢(𝑓_𝑇) =0.01225

50 = 0.000245 Weight measurement (𝑤):

The linearity of the balance is considered as ± 0.2 mg. A rectangular distribution was assumed for the uncertainty of weight measurement. Accordingly, for two weight measurements for a binary mixture:

The standard uncertainty of weight measurement, 𝑢(𝑤) = 2. (^0.2

√3)²

=

0.16mg Considering a sample volume of 20g

𝑢(𝑓_𝑤) =0.16 × 10⁻³

20 = 0.000008 Repeatability

Standard deviation of 0.0076 mPa⸳s was observed for the measured viscosity of 2.179 mPa⸳s. Accordingly,

𝑢(𝑓𝑟𝑒𝑝) =0.0076

2.179 = 0.0035

Table 6.2: Measurement uncertainty of viscosity for different solutions

Property Solution Uncertainty

(95% confidence at k=2) Viscosity

Pure ± 0.0162 mPa·s

Aqueous MEA ± 0.0162 mPa·s

CO² loaded aqueous MEA ± 0.0353 mPa·s

6.1.2 Uncertainty of Density Measurements

A similar approach was adopted as explained in section 6.1.1 to evaluate uncertainty in density measurement. The uncertainty sources of material purity, weight measurement, temperature, CO² loading and repeatability were considered during the uncertainty analysis. The influence of uncertainty in temperature on density measurement was determined by calculating the gradient 𝜕𝜌 𝜕𝑇⁄ of density against temperature. A similar approach was used to calculate the influence of uncertainty in CO² loading on density measurement in which gradient of density against CO² loading, 𝜕𝜌 𝜕𝛼⁄ , was found from measured densities at different CO² loadings. Finally, (𝜕𝜌 𝜕𝑇⁄ )⸳𝑢(𝑇) and (𝜕𝜌 𝜕𝛼⁄ )⸳𝑢(𝛼) were determined to find standard uncertainties of density measurement due to the temperature and CO² loading. For CO² loaded aqueous monoethanol amine mixtures,

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the gradient 𝜕𝜌 𝜕𝑇⁄ of density against temperature was found as 0.73 kg⸳m^-3⸳K^-1 and the corresponding uncertainty in 𝜌 that is (𝜕𝜌 𝜕𝑇⁄ )⸳𝑢(𝑇) was calculated as ±0.009 kg⸳m^-3. The gradient of density against CO² loading, 𝜕𝜌 𝜕𝛼⁄ , was found as 334 kg⸳m^-3 and the corresponding uncertainty in 𝜌, (𝜕𝜌 𝜕𝛼⁄ )⸳𝑢(𝛼) was calculated as ±1.67 kg⸳m-³. The standard combined uncertainties for density measurement 𝑢(𝜌) for different monoethanol amine mixtures are listed in Table 6.3.

Table 6.3: Measurement uncertainty of density for different solutions

Property Solution Uncertainty

(95% confidence at k=2) Density

Pure ± 7.1 kg⸳m^-3

Aqueous MEA ± 7.1 kg⸳m^-3

CO² loaded aqueous MEA ± 7.8 kg⸳m^-3

In document Physicochemical data for amine based CO2 capture process (sider 108-113)