Estimation and expression of uncertainty for curing parameters of rubber compounds.

Today, most of the decisions related to process problems, quality control, a new development or any process modification are based on the result of quantitative analyses of different kinds. Hence, it is important to have some indication of the quality of the results for the purpose in hand. It is also required to eliminate the replication of effort frequently expended in obtaining them. Confidence in data obtained outside the user's own organization is a prerequisite to meeting this objective. It is now a requirement for laboratories to introduce quality assurance measures such as the use of validated methods of analysis, participation in a proficiency testing program and laboratory accreditation to insure that they are capable of and are providing data of the required quality.

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

General information

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

Mathematical model

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 [3] = 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 [3] = 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])

References

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).