Estimation and expression of uncertainty for curing parameters of rubber compounds.
To fulfill these requirements, it is essential to demonstrate the quality of results and, in particular, to demonstrate their fitness for the purpose by giving a measure of the confidence that can be placed on the result. This is expected to include the degree to which a result would be expected to agree with other results, normally irrespective of the analytical methods used. One useful measure of this is measurement uncertainty.
The definition of measurement uncertainty is: A parameter associated with the result of a measurement that characterizes the dispersion of the values that could be reasonably attributed to the measurend.
In general, the word uncertainty relates to the general concept of doubt but, on contrary, knowledge and expression of uncertainty of measurement imply increased confidence in the validity of a measurement result.
In this article, we give the calculation of measurement uncertainty for curing parameters of a rubber compound which includes minimum torque, maximum torque, ts2, TC50 and TC90. Minimum torque is a measure of the stiffness of the unvulcanized test specimen, at the specified vulcanizing temperature, taken at the lowest point in the vulcanization curve. Maximum torque is a measure of the stiffness or shear modulus of the vulcanized test specimen at the vulcanizing temperature, measured within a specified period of time. Scorch time is a measure of the time at which a specified small increase in force or torque has occurred and indicates the beginning of the vulcanization. In general, ts2 is being considered as a measure of scorch time. TC50 and TC90 are measures of cure based on the time to develop 50% and 90% of the difference in torque from the minimum to the maximum.
Calculation of measurement uncertainty for curing parameters
The instrument used was a MDR 2000E and the range was 0-200 lb.-in. The sample tested was an internal reference compound. The test temperature was 193[degrees]C and the method used was according to ASTM D 5289
Gx = Ga + e
Gx = estimated value of max.tq./min.tq./ ts2/TC50/TC90
Ga = average value of max.tq./min.tq./ ts2/TC50/TC90
e = uncertainty in max.tq./min.tq./ts2/ TC50/TC90 measurement (error involved in the measurement).
Uncertainty due to random error
Repeatability, which is also known as relative uncertainty or relative standard deviation of mean, is determined by dividing the standard deviation of mean by the mean value. The standard deviation of mean, also known as standard uncertainty for repeatability, is calculated by dividing the standard deviation by the square root of n (number of observations) as described in table 1. For repeatability evaluation, the degree of freedom is obtained from the number of independent repeated observations, i.e., n-1.
Uncertainty due to systematic errors
Sources of uncertainty include: Torque standard calibration uncertainty, [+ or -] 1%; accuracy of torque, [+ or -] 1%; temperature accuracy, [+ or -] 0.3[degrees]C; and accuracy in time measurement, [+ or -] 1%. (Note: Sample weight is not considered as a separate source of uncertainty as the sample weight tolerance is too high [[+ or -] 0.5 g] as compared to the accuracy we have maintained in taking test specimens [[+ or -] 0.1 g]).
The uncertainty of tq. std. calibration is [+ or -] 1% without mentioning confidence level. The divisor for rectangular probability distribution is = [square root of (3)]. For a tq.std, mean value of 20.82 lb.-in., the standard uncertainty for torque standard calibration is 0.01*20.82/[square root of  = 0.1202. Relative uncertainty can be obtained by dividing standard uncertainty by the mean value, thus Uii = 0.1202/20.82 = 0.00577. (Note: We can observe from the above that the mean value is canceled out while calculating relative uncertainty from the standard uncertainty, [(1*20.82)/(100*[square root of (3))]/20.82 = 0.01/[square root of (3). Thus, wherever the calibration uncertainty or measurement accuracy is given as x% without mentioning the confidence level, the relative uncertainty can be directly obtained by x/[100*[square root of (3)]).
As the accuracy of torque is [+ or -] 1% without mentioning the confidence level, the relative uncertainty Uiii = 0.01/[square root of  = 0.00577.
The temperature accuracy is [+ or -] 0.3[degrees]C without mentioning the confidence level. The divisor for rectangular probability distribution is = [square root of (3)]. For a temperature mean value of 193[degrees]C, standard uncertainty for temperature accuracy is 0.3/[square root of (3)] = 0.173 and relative uncertainty can be obtained by dividing standard uncertainty by mean value, thus Uiv = 0.173/193 = 0.0009.
As the time measurement accuracy is [+ or -] 1% without mentioning the confidence level, the relative uncertainty Uv = 0.01/[square root of (3)] = 0.00577.
The degree of freedom for all systematic errors is infinity, as their uncertainty or accuracy is provided without confidence level.
Calculation of combined uncertainty:
All uncertainty contributors for the cure characteristic measuring parameters being calculated above are relative uncertainties. Now in table 3, specific uncertainty contributors are selected to calculate combined uncertainty. For example, in minimum and maximum torque measurement, the uncertainty contributor from time is excluded, as the uncertainty of time measurement does not affect the value of minimum or maximum torque, but it does affect the measurement of ts2, TC50, TC90.
Combined uncertainty is calculated using the following formula:
Uc(y) = Y * [square root of ([[Ua.sup.2] + [Ub.sup.2] + [Uc.sup.2] + .......]) Where, Uc(y) = combined uncertainty for parameter y;Y = mean value of parameter y; and Ui (i = a, b, c,....) = relative uncertainty contributions.
Calculation of effective degree of freedom
To obtain the effective degree of freedom, the degree of freedom for each standard uncertainty component is required. For components obtained from a repeatability evaluation, the degree of freedom is obtained from the number of independent repeated observations, i.e., n-l. For components obtained from a systematic errors evaluation, the degree of freedom is obtained from the judged reliability of the value of the component. Hence, effective degree of freedom:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Where [U.sub.i], [U.sub.ii], [U.sub.iii ]and [Ui.sub.iv.] are standard uncertainty having [v.sub.i], [V.sub.ii], [V.sub.iii], [V.sub.iv] ... degree of freedom.
Calculation of expanded uncertainty
Coverage factor k = 1.96 (degree of freedom = [infinity]) is used from student t-distribution table for 95% confidence level to calculate expanded uncertainty. Hence, expanded uncertainty:
Ue = k * Uc
Reporting of results
For test parameter minimum torque, the measurement uncertainty U(min. tq.) is [+ or -] 0.019 lb.-in., determined from a combined uncertainty Uc(min. tq.) = 0.010 lb.-in. and a coverage factor k = 1.96 based on student t-distribution table for 95% confidence level. Hence:
Minimum torque = 1.04 [+ or -] 0.02 lb.-in. (at 95% confidence level).
For test parameter maximum torque, the measurement uncertainty U(max. tq.) is [+ or -] 0.116 lb.-in., determined from a combined uncertainty Uc(max. tq.) = 0.059 lb.-in, and a coverage factor k = 1.96 based on student t-distribution table for 95% confidence level. Hence:
Maximum torque = 7.10 [+ or -] 0.12 lb.-in. (at 95% confidence level).
For test parameter ts2, the measurement uncertainty U(ts2) is [+ or -] 0.011 min., determined from a combined uncertainty Uc(ts2) = 0.006 min. and a coverage factor k = 1.96 based on student t-distribution table for 95% confidence level. Hence:
ts2 = 0.52 [+ or -] 0.01 min. (at 95% confidence level).
For test parameter TC50, the measurement uncertainty U(TC50) is [+ or -] 0.014 min., determined from a combined uncertainty Uc(TC50) = 0.007 min. and a coverage factor k = 1.96 based on student t-distribution table for 95% confidence level. Hence:
TC50 = 0.65 [+ or -] 0.01 min. (at 95% confidence level).
For test parameter TC90, the measurement uncertainty U(TC90) is [+ or -] 0.025 min., determined from a combined uncertainty Uc(TC90) = 0.013 min. and a coverage factor k = 1.96 based on student t-distribution table for 95% confidence level. Hence:
TC90 = 1.16 [+ or -] 0.03 min. (at 95% confidence level).
Table 1--uncertainty evaluation for repeatability Mean SD Uncertainty Parameter Observed value value [[sigma].sub.n-1] Uia Min. tq. 1.06, 1.03, 1.05, 1.04 0.0145 (lb.-in.) 1.02, 1.05, 1.02, 1.06, 1.04, 1.04, 1.04 Uib Max. tq. 7.16, 7.12, 7.11, 7.10 0.0347 (lb.-in.) 7.06, 7.10, 7.07, 7.09, 7.13, 7.05, 7.07 Uic ts2 0.51, 0.51, 0.53, 0.52 0.0074 (min.) 0.52, 0.52, 0.52, 0.52, 0.52, 0.53, 0.53 Uid TC50 O.64, 0.64, 0.66, 0.65 0.0082 (min.) 0.64, 0.64, 0.64, 0.64, 0.65, 0.65, 0.66 Uie TC90 1.15, 1.15, 1.18, 1.16 0.0165 (min.) 1.14, 1.16, 1.15, 1.18, 1.17, 1.17, 1.19 SD of mean Degree of ([[sigma].sub.n-1] Relative freedom Uncertainty Nn) uncertainty (n-1) Uia 0.0046 0.0044 9 Uib 0.0110 0.0015 9 Uic 0.0023 0.0045 9 Uid 0.0026 0.0040 9 Uie 0.0052 0.0045 9 Table 2--uncertainty evaluation due to systematic errors Mean Standard Uncertainty Source of uncertainty value uncertainty Uii Torque standard 20.82 0.1202 calibration uncertainty Uiiia Accuracy of torque 1.04 0.0060 measured at (min. torque) Uiiib Accuracy of torque 7.10 0.0410 measured at (max. torque) Uiiic Accuracy of torque 3.06 0.0177 measured at (ts2) Uiiid Accuracy of torque 4.14 0.0239 measured at (TC50) Uiiie Accuracy of torque 6.60 0.0381 measured at (TC90) Uiv Temperature accuracy 193.0 0.1730 Uvc Accuracy in time 0.52 0.0030 measurement at (ts2) Uvd Accuracy in time 0.65 0.00375 measurement at (TC50) Uve Accuracy in time 1.16 0.0067 measurement at (TC90) Relative Degree of Uncertainty uncertainty freedom Uii 0.00577 [infinity] Uiiia 0.00577 [infinity] Uiiib 0.00577 [infinity] Uiiic 0.00577 [infinity] Uiiid 0.00577 [infinity] Uiiie 0.00577 [infinity] Uiv 0.00090 [infinity] Uvc 0.00577 [infinity] Uvd 0.00577 [infinity] Uve 0.00577 [infinity] Table 3--calculation of combined uncertainty, degree of freedom and expanded uncertainty (uncertainty budget) Mean Uncertainty value Combined Uncertainty contributors (Y) uncertainty Uncertainty in Uia, Uii, 1.04 0.010 min. torque Uiiia, Uiv measurement (lb.-in.) Uncertainty in Uib, Uii, 7.10 0.059 max. torque Uiiib, Uiv measurement (lb.-in.) Uncertainty in Uic, Uii, Uiiic, 0.52 0.006 ts2 measurement Uiv, Uvc (min.) Uncertainty in Uid, Uii, 0.65 0.007 TC50 measurement Uiiid, Uiv, (min.) Uvd Uncertainty in Uie, Uii, 1.16 0.013 TC90 measurement Uiiie, Uiv, (min.) Uve Epanded uncertainty Uncertainty [V.sub.eff] (Ue = k * Uc) Uncertainty in 181 [approximately 0.019 min. torque equal to] [infinity] measurement (lb.-in.) Uncertainty in 7,677 [approximately 0.116 max. torque equal to] [infinity] measurement (lb.-in.) Uncertainty in 327 [approximately 0.011 ts2 measurement equal to] [infinity] (min.) Uncertainty in 470 [approximately 0.014 TC50 measurement equal to] [infinity] (min.) Uncertainty in 328 [approximately 0.025 TC90 measurement equal to] [infinity] (min.) (For degree of freedom > 30, it can be considered as degree of freedom = [infinity])
1. ISO/IEC 17025:1999 General requirements for the competence of calibration and testing laboratories. ISO, Geneva (1999).
2. Guide To the expression of uncertainty in measurement. ISO, Geneva 1993 (corrected and reprinted 1995). This document is also available in American National Standard: ANSI/NCSL Z540-2-1997.
3. EURACHEM, Quantifying uncertainty in analytical measurement. Laboratory of the Government Chemist, London (1995).
4. NABL (National Accreditation Board for Testing and Calibration Laboratories, India) Guide on Estimation and Expression of Uncertainty Measurement (NABL- 141 Guide).
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|Title Annotation:||Tech Service|
|Date:||Dec 1, 2004|
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