Distinguish two types of uncertainty evaluation Do a design stage uncertainty analysis Calculate error of a derived variable Handle uncertainty/error

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Sheila Wilkinson
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1 ME343 Chapter 5

2 Distinguish two types of uncertainty evaluation Do a design stage uncertainty analysis Calculate error of a derived variable Handle uncertainty/error propagation from multiple sources

3 Use available information to estimate the uncertainty range, ux, in the interval estimate of the true value: For current analysis, assume that the true value, x is constant (time independent.

4 Statistical Uncertainty analysis (Type A in NIST terminology based on series of repeated measurements, and calibration tests Judgment based Uncertainty Analysis (Type B in NIST terminology based on previous measurement data, experience with relevant materials and instruments, manufacturer's specifications, data provided in calibration and other reports, and uncertainties assigned to reference data taken from handbooks

6 Design stage Calibration stage Data acquisition stage Data reduction stage

7 We apply design-stage uncertainty analysis when we don t have the system yet, or have not tested it yet. We either rely on the manufacturer s data, or on our own estimates Major facilities may need to be built and equipment ordered with considerable lead time.

8 It is useful for selecting instruments, selecting measurement techniques, and obtaining an appropriate estimate of the uncertainty likely to exist in the measured data. At this point, the measurement system and associated procedures are but a concept. Usually little is known about the instruments, and in many cases they are still pictures in a catalog.

9 Design-Stage Uncertainty is made up of two principle components. 1. Instrument resolution. Instrument uncertainty, uc, (an estimate of the systematic error for the instrument

19 Treat an error as: Precision error if it can be statistically estimated in some manner (data scatter Bias error otherwise (cannot be directly discerned by statistical methods, e.g. By repeating measurements.

20 Calibration Alternate (concomitant Method Interlaboratory comparisons Experience

21 System repeatability and resolution (measurement system Temporal and spatial variations (measured variable properties Operating and environmental conditions (process Measurement repeatability (measurement procedure and Technique

22 A temperature measurement system is composed of a sensor and a readout devices. The readout device has a claimed accuracy of 0.6 C with a resolution of 0.1 C. the sensor has an off-the-shelf accuracy of 0.5 C. Estimate a design-stage uncertainty in temperature indicated by this combination. The design-stage uncertainty is for the combined system is

23 where (ud R is the design-stage uncertainty of the readout device and (uds is that of the sensor. In either case, the individual design-stage uncertainty is found from Sensor

25 The linear displacement of a vehicle due to an applied impact force is measured with a transducer. Transducer specifications are: Input range : 0-5 m Output range: 0-5 V Linearity: ± 0.5% reading Drift: ± 0.05%/ C FSO Sensitivity: ± 0.10% FSO Hysteresis: ± 0.5% reading The transducer output is indicated on a voltmeter (accuracy: within ± 0.1% reading; resolution: 10μV. The expected nominal displacement of the vehicle is to be 4m for an impact force of 000± 100N (95%.

26 The force-displacement relation can be assumed linear. Four replications consisting of 10 measurements are made over the course of one day. The results are given with the ambient temperature for each run. Test run N Mean value [m]

27 At the design-stage, only information known prior to the test are included: For the transducer, we estimate the instrument error from the elemental errors: e L = ± (0.005(4m = 0.01 m e H = ± (0.005(4m = 0.01 m e s = ± (0.001(5m = m e D = ± (0.005/ C(3 C(5m = m so that,

28 The transducer has a sensitivity Kt = 1V/m so that the voltmeter output can be restated in terms of displacement [m]. For the voltmeter: For the transducer-voltmeter system, the designstage uncertainty is:

29 Now we look at the case of a derived quantity that is estimated from the measurement of several primary quantities. The question that needs to be answered is the following: A derived quantity Q is estimated using a formula that involves the primary quantities. a1,a,...an Each one of these is available in terms of the respective best values b1, b,...bn and the respective standard deviations σ1,σ...σn. What is the best estimate for Q and what is the corresponding standard deviation σq?

30 We have, by definition Q =Q(a1,a,...an It is obvious that the best value of Q should correspond to that obtained by using the best values for the a s. Thus, the best estimate for Q given by Q Q( b, b, b3,... b 1 n by definition, we should have:

31 The subscript i indicates the experiment number and the ith estimate of Q is given by Qi = Q(a1i,ai,...ani using a Taylor expansion around the best value as where the partial derivatives are all evaluated at the best values for the a s. If the a s are all independent of one another then the errors in these are unrelated to one another and hence the cross terms.

32 The equation can be written as: Noting that the above equation can be in the form

33 the error propagation formula

34 Two resistors are to be combined to form an equivalent resistance of Readily available are two common resistors rated at and two common resistors rated at 0005%. What combination of resistors (series or parallel would provide the smaller uncertainty in an equivalent 1000 resistance? Case 1 (series resistors R T = R 1 + R If we use the two 500 resistors, then u d 1 The (u 50, d propagation of series = [( = u d R R [(1.u 50 T 1 d1 uncertainty through to u d1 R ( R (1.u d T ] ] 1 71 R T is estimated (95% by

35 Case (paralell resistors If we use the two 000 resistors, then u d 1 The (u 100, d propagation of paralell RT = [( R = [({ u R 35 d uncertainty through to 1 R u R d1 (95% R ( R R1R ( R R 1 T ] 1 }.u d 1 R T is ({ estimated R 1 R 1 R by R1R ( R R 1 }u d ] 1

36 Heat transfer from a rod of diameter D immersed in a fluid can be described by the Nusselt number Nu=hD/k, where h is the heat transfer coefficient and k is the thermal conductivity of the fluid. If h can be measured within 7% (95%, estimate the uncertainty in Nu for the nominal value of h=150 W/m-K. Let D=00.5 mm and k=0.6% W/m-K. u h = (0.07(150 = 10.5 W/m -K u D = m u k = (0.0(0.6 = 0.01 W/m-K We have Nu = f(h,d,k

37 Then, we estimate the Nusselt number here to be Nu = hd/k u Nu = (95% So, Nusselt number can be determined within about 8% in this range of values x0.005 ( ( ( u u k hd ( u k h ( u k D ( u u k N ( u D N ( u h N ( u 1 Nu 1 k D h Nu 1 k u d u h u Nu

Basic RTD Measurements. Basics of Resistance Temperature Detectors

Basic RTD Measurements. Basics of Resistance Temperature Detectors Basic RTD Measurements Basics of Resistance Temperature Detectors Platinum RTD resistances range from about 10 O for a birdcage configuration to 10k O for a film type, but the most common is 100 O at 0