Accuracy of Measurements http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
13.4 Accuracy of Measurements
When a parameter is measured, it is important to assess the accuracy of the measurement. The resulting analysis,
called an uncertainty analysis, provides an estimate of the upper and lower bounds of the parameter. For
example, if Q is a measured value of discharge, uncertainty analysis provides an estimate of the uncertainty UQ
in this measurement. The measurement would then be reported as Q Ä… UQ.
Commonly, a parameter of interest is not directly measured but is calculated from other variables. For example,
discharge for an orifice meter is calculated using Eq. 13.7a. Such an equation is called a data reduction equation.
Consider a data reduction equation of the form
where x is the parameter of interest and y1 through yn are the independent variables. Then, the uncertainty in x,
which is written as Ux, is given by
(13.25)
where is the uncertainty in variable yi. Equation 13.25, known as the uncertainty equation, is very useful for
quantifying the accuracy of an experimental measurement, and for planning experiments. Additional information
about uncertainty analysis is provided by Coleman and Steele 17.
EXAMPLE 13.10 U CERTAI TY ESTIMATE FOR A
ORIFICE METER
For the orifice meter described in Example 13.2, estimate the uncertainty of the calculated discharge.
Assume that uncertainty in K is 0.03, the uncertainty in diameter is 0.15 mm, and the uncertainty in
measured head is 10 mm-Hg.
Problem Definition
Situation:
1. Water flows through an orifice (d = 0.15 m) in a pipe (D = 0.24 m).
2. A mercury-water manometer is used to measure pressure drop.
Find: Uncertainty (in m3/s) for the calculated discharge Q.
Plan
1. Identify the data reduction equation (DRE).
2. Within the DRE, identify each variable that contributes to uncertainty.
3. Develop an equation for uncertainty by applying Eq. 13.25.
4. Calculate uncertainty by using the equation developed in step 3.
Sketch:
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Solution
1. The data reduction equation is the orifice equation, Eq. 13.7a.
2. Variables that cause uncertainty are K, d, g, and h. Neglect the influence of g.
3. Derive an equation for the uncertainty
Evaluate each partial derivative and then divide both sides of this equation by Q2:
4. Substitute values from Example 13.2:
Thus
Review
The primary source of uncertainty in the discharge is due to UK . The term Uh has a small effect, and
Ud has a negligible effect.
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