Uncertainty analysis is a vital part of any experimental program or measurement system design. Common sources of experimental uncertainty were defined in Chapters 2 through 4. In this chapter, methods are provided to combine uncertainties from all sources to estimate the uncertainty of the final results of an experiment.

INTRODUCTION Any experimental measurement will involve some level of uncertainty that may originate from causes such as inaccuracy in measurement equipment, random variation in the quantities measured, and approximations in data-reduction relations. All these uncertainties in individual measurements eventually translate into uncertainty in the final results. This "propagation of uncertainty" is an important aspect of any engineering experiment. Uncertainty analysis is performed during the design stage of an experiment to assist in the selection of measurement techniques and devices. It is also performed after data have been gathered, in order to demonstrate the validity of the results.

we can estimate the maximum uncertainty in R by forcing all terms on the right-hand side of Eq to be positive. In mathematical form, this becomes It is not very probable that all terms in Eq. (7.2) will become positive simultaneously or that errors in the individual x's will all be at the extreme value of the uncertainty interval. Consequently, Eq. (7.3) will produce an unreasonably high estimate for up. A better estimate for the uncertainty is given by (7.3)

Comment: The maximum uncertainty of 40 W is 4% of the power (P = VI = 100 x 10 =1000 W), whereas the uncertainty estimate of 28.3 W is 2.8% of the power. The maximum error estimate is too high in most circumstances.

Solution: The relationship between the pressure and the measured column height for a manometer is: So, The maximum allowable uncertainty in the gage fluid density:

CONSIDERATION OF SYSTEMATIC AND RANDOM COMPONENTS OF UNCERTAINTY In the early phases of the design of an experiment, it is often not practical to separate the effects of systematic and random uncertainties. For example, manufacturer-specified instrument accuracies inherently combine random and systematic uncertainties. The Eq. will then be used with these combined uncertainties in measured variables to estimate the uncertainty of the result. In more detailed uncertainty analyses, it is usually desirable to keep separate track of the systematic uncertainty, denoted B, and the random uncertainty, denoted P, associated with the measurement. Estimating random uncertainty depends on the sample size. A systematic error does not vary during repeated readings and is independent of the sample size. It is this distinction that makes it desirable to handle the terms separately in uncertainty analysis. The random uncertainty is estimated using the t-distribution, introduced in Chapter 6. If the variable x is measured n times, then the standard deviation and mean of the sample can be determined from

In case the mean value is obtained from a different set of tests, the final value of the mean of x, denoted by is determined from M measurements of x and is given by

In this case, M = 1, random uncertainty in this single measurement is given by Usually, S, is determined in tests with a large sample size (n > 30). In addition, a confidence level of 95% is commonly used to determine the random uncertainty interval. For these two conditions, the appropriate value of t is 2.0. A large sample size simplifies an uncertainty analysis considerably, and for this reason, the ASME Standard [ASME, (1998)] suggests the use of large sample sizes in most uncertainty analyses. the systematic uncertainty in the measured variable x, remains constant if the test is repeated under the same conditions. Systematic uncertainties include those errors which are known, but have not been eliminated through calibration, as well as other fixed errors that can be estimated, but not eliminated from the measurement process.

Systematic and random uncertainties are combined to obtain the total uncertainty, using RSS. For the mean of. for a single measurement of x,

Example 7.3 In a chemical-manufacturing plant, load cells are used to measure the mass of a chemical mixture during a batch process. From 10 measurements, the average of the mass is measured to be 750 kg. From a large number of previous measurements, it is known that the standard deviation of the measurements is 15 kg (which implies that t : 2.0 for 95% confidence level). Assuming that the load cells do not introduce any random uncertainty into the measurement, calculate for 95% confidence level (a) the standard deviation and random uncertainty of each measurement. (b) the standard deviation and random uncertainty of the mean value of the ten measurements.

Example 7.4 In estimating the heating value of natural gas from a gas field, 10 samples are taken and the heating value of each sample is measured by a calorimeter. The measured values of the heating value, in kJ/kg, are as follows: 48530, 48980, 50210, 49860, 48560, 49540, 49270, 48850, 49320, Assuming that the calorimeter itself does not introduce any random uncertainty into the measurement, calculate (for 95% confidence level) (a) the random uncertainty of each measurement. (b) the random uncertainty of the mean of the measurements. (c) the random uncertainty of the mean of the measurements, assuming that S was calculated on the basis of a large sample (n > 30); but has the same value as computed inparts (a) and (b).

Example 7.5 The manufacturer's specification for the calorimeter in Example 7.4 states that the device has an accuracy of l.5% full range from 0 to 100,000 kJ/kg. Estimate the total uncertainty of (a) the mean value of the measurement in Example 7.4 and (b) a single measurement of heating value of 49,500 kJ/kg, obtained later than the measurements in Example % confidence assumed) We have assumed that the "accuracy" is entirely a systematic uncertainty. (a) which is 3.1% of the mean value.

Problem

Assuming that the random errors have been determined with samples>30,