Chapter VII – Experimental Uncertainty Analysis

CHAPTER VII

Experimental Uncertainty Analysis

Whenever experimental measurements are performed, it is essential to perform uncertainty analysis. This is because the errors usually propagate (propagation of uncertainty). It is sometimes performed during the design of the experiment phase to help the experimentalist to choose the appropriate techniques and devices

7.1. Propagation of uncertainties – General considerations

Let us assume a result R of an experiment. This result will be a function of n measured variables:

Then, a small variation in R will be related to small variations in xi:

This equation is exact if the ’s are infinitesimal; otherwise it is an approximation. The partial derivatives are called: the sensitivity coefficients of result R with respect to variables xi.

Since the right-hand side of the above equation can be + or -, it is better to express it in absolute values:

However, this will tend to maximize (overestimate) the uncertainties, we will express it then in terms of the root of the sum of the squares.

The confidence level in the uncertainty in the result R will be the same as the confidence levels of the uncertainties in the xi’s. It is therefore essential that all uncertainties used have the same confidence level. Another important point is that the above formulation assumes that the measured variables are independent.

Special case:

If the result R is simply the product of the measured variables, then:

This is if:

Example
Orifice meters are used to measure the flow rate of a fluid. In an experiment, the flow coefficient K of an orifice is found by collecting the weighing water flowing through the orifice during a certain interval while the orifice is under a constant head. K is calculated from the following formula:

The values of the parameters have been determined to be as follows, with 95% confidence:
Mass M = 393.00  0.03 kg
Time t = 600.0  1 s
Density  = 1000.0  0.1% kg/m3
Diameter d = 1.270  0.0025 cm
Head h = 366.0  0.3 cm
Find the value of K, its uncertainty (with 95% confidence), and the maximal possible error.

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Instrumentation and Measurements \ LK\ 2009