f5763552 Microsoft Word - Uncertainty Analysis Guideline 2012d

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MEASUREMENT UNCERTAINTY ANALYSIS GUIDELINE REV 2012

For ME 4031 by Scott Dahl

SUMMARY This document presents a discussion of uncertainty in engineering measurements and discusses methods to perform measurement uncertainty analysis.

REFERENCE DOCUMENTS Test Uncertainty, ASME Standard PTC 19.1-2005 Engineering Analysis of Experimental Data, ASHRAE Guideline 2-2005 Theory and Design for Mechanical Measurements, Fourth Edition, Figliola and Beasely Introduction to Engineering Experimentation, Second Edition by Anthony J. Wheeler and Ahmad R. Ganji Measurement Uncertainty-Methods and Applications, Dieck, Fourth Edition, Instrument Society of America Experimentation and Uncertainty Analysis for Engineers, Second Edition by Coleman, ISBN 0-471-12146-0

TABLE OF CONTENTS 1 WHAT IS A MEASUREMENT OR MEASUREMENT SYSTEM? ......................................................................................... 2 2 UNCERTAINTY OF A MEASUREMENT ................................................................................................................................. 2

2.1 WHY IS UNCERTAINTY OF MEASUREMENT IMPORTANT? ................................................................................................... 3 2.2 ERROR VERSUS UNCERTAINTY ......................................................................................................................................... 3 2.3 WHERE DO ERRORS AND UNCERTAINTIES COME FROM? ................................................................................................... 3 2.4 UNCERTAINTY ANALYSIS AND SIGNIFICANT FIGURES ..................................................................................................... 5 2.6 DESIGN-STAGE UNCERTAINTY ANALYSIS ........................................................................................................................ 6 2.7 COMBINING SYSTEMATIC ERRORS FOR A MEASUREMENT ............................................................................................... 6 2.8 COMBINING RANDOM ERRORS FOR A MEASUREMENT ..................................................................................................... 7 2.9 MULTIPLEMEASUREMENT UNCERTAINTY ANALYSIS..................................................................................................... 7

3 UNCERTAINTY OF A RESULT ............................................................................................................................................... 10 3.1 SINGLE MEASUREMENT EXPERIMENTS ...........................................................................................................................14 3.2 UNCERTAINTY IN RESULT FOR MULTIPLE MEASUREMENT EXPERIMENTS ......................................................................15 3.3 SUMMARY OF PROCEDURE FOR UNCERTAINTY ANALYSIS AND ERROR PROPAGATION ...................................................17 3.4 UNCERTAINTY ANALYSIS USING SEQUENTIAL PERTURBATION ......................................................................................20

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1 What is a Measurement or Measurement System? A measurement or measurement system provides us with information on a characteristic or property of something. For example, it might tell us how heavy an object is, or how hot, or how long it is. A measurement gives a number to that property. Measurements are always made using an instrument of some kind (e.g. rulers, thermometers, calipers, pressure gage, and weighing scales) but inevitably the measurement system may also involve people, procedures, and training.

The result of a measurement is normally in two parts: a number and a unit of measurement. For example, the measured ultimate strength of a rubber band tested in a tensile tester using a load cell (e.g. force transducer) is 120 Newtons.

2 Uncertainty of a Measurement Uncertainty of a measurement represents the doubt that exists about the results of any measurement. It provides insight into the quality of the measurement. All reported measurements should consist of an average value, and uncertainty interval (either absolute or relative), a statement of confidence in the specified uncertainty interval, and the units of the measurement. The following form is often used to present these three pieces of information about the measured value.

x=X UX (CI %)

where

X is the average value for the measurement

UX is the uncertainty interval in variable X. The uncertainty interval may be expressed in either absolute terms

or relative terms. Relative uncertainty is defined as UX X

.

CI % is the confidence interval. It is common to use a 95% CI and a 95% CI is used in this course

Alternatively, the uncertainty interval may be expressed in terms of relative uncertainty. The following two presentations of a measured voltage measurement are equivalent; one uses absolute uncertainty to describe the uncertainty interval and the other utilizes relative uncertainty.

(5.0 0.5) VDC (95%)

5 VDC 10% (95%)

For example:

We might say that the length of a certain rod measures 20 mm plus or minus 1 mm, at the 95 percent confidence level. This result could be written:

20 mm 1 mm, at a level of confidence of 95%.

The statement says that we are 95 percent sure that the rod length is between 19 mm and 21 mm.

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2.1 Why is uncertainty of measurement important? You may be interested in uncertainty of measurement simply because you wish to make good quality measurements and to understand the results. However, there are other more particular reasons for thinking about measurement uncertainty. You may be making the measurements as part of any of the following:

During Instrument Calibration - where the uncertainty of measurement must be reported on the certificate

During Testing - where the uncertainty of measurement is needed to determine a pass or fail or to meet a certain tolerance - where you need to know the uncertainty before you can decide whether the tolerance is met

During Analysis - you may need to read and understand a calibration certificate or a written specification for a test or measurement.

2.2 Error versus uncertainty It is important not to confuse the terms error and uncertainty. Error is the difference between the measured value and the true value of the thing being measured. Uncertainty is a quantification of the doubt about the measurement result. Whenever possible we try to correct for any known errors: for example, by applying corrections from calibration certificates. But any error whose value we do not know is a source of uncertainty.

2.3 Where do errors and uncertainties come from? Many things can undermine a measurement. Flaws in the measurement may be visible or invisible. Because real measurements are never made under perfect conditions, errors and uncertainties can come from:

The measuring instrument - instruments can suffer from errors including bias, changes due to ageing, wear, or other kinds of drift, poor readability, noise (for electrical instruments) and many other problems.

The item being measured - which may not be stable. For example, the dimensions of an ice cube may be unstable and difficult to measure in a warm room.

The measurement process - the measurement itself may be difficult to make.

Imported uncertainties - calibration of your instrument has an uncertainty, which is then built into the uncertainty of the measurements you make.

Operator skill - some measurements depend on the skill and judgment of the operator. One person may be better than another at the delicate work of setting up a measurement, or at reading fine detail by eye. The use of an instrument such as a stopwatch depends on the reaction time of the operator.

Sampling issues - the measurements you make must be properly representative of the process you are trying to assess. If you are choosing samples from a production line for measurement, dont always take the first ten made on a Monday morning.

The environment - temperature, air pressure, humidity and many other conditions can affect the measuring instrument or the item being measured. Where the size and effect of an error are known (e.g. from a calibration certificate) a correction can be applied to the measurement result. But, in general, uncertainties from each of these sources, and from other sources, would be individual inputs contributing to the overall uncertainty in the measurement.

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Recall the two sources of measurement uncertainty include systematic errors and random errors. The distinction between systematic and random error is shown in Figure 1. Systematic errors are consistent and repeatable errors that are typically characterized though calibrations. Random errors are caused by a lack of repeatability in the measuring system and may arise because of non-repeatability of the measurement system, variability in environmental conditions, data reduction techniques, or measurement methods. Random errors and are typically characterized using statistical methods.

Figure 1 Distinction between systematic and random errors.

Table 1 Systematic and random errors

Error Type Description Sources Examples

Systematic

Errors that are consistent and repeatable Estimate by calculating the Average of measured readings-True Value

Calibration Errors Linearity Accuracy Hysteresis Zero Off-set Spatial Errors Environmental Stability Drift

Random

Errors caused by a lack of repeatability in the output of the measuring system Very simple estimation method is to use using the largest difference between a single reading and the average of all readings

Uncontrolled variables in the measurement process

Measurement system errors

Environment variation

Electrical or Magnetic Noise Resolution Spatial or Temporal Variation Procedural Variation Environmental Stability

Temperature Moisture

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2.4 Uncertainty Analysis and Significant Figures The generally accepted convention or guideline related to the significant figures reported in a result is that the precision level of the reported uncertainty and the nominal or mean value must be the same.

For example:

Incorrect Reporting Correct Reporting (31.25 0.03495) cm (31.25 0.03) cm

To insure consistency, apply the following rules/guidelines when analyzing and presenting your results with an associated uncertainty.

1) Do not apply any rounding or significant figure rules until you have determined the values for the measurement itself and the associated uncertainty.

2) The number of significant figures reported in any uncertainty value should be one. That is, round the calculated uncertainty value to a single significant digit. The only exception to this rule is when the first significant digit has a numerical value of 1. In this case, 2 significant figures may be used in the reported uncertainty value.

3) With the precision of the uncertainty value now determined, adjust the reported nominal/average value significant figures to match the precision level of the uncertainty. The nominal/mean and uncertainty value are to have the same level of precision.

Examples showing the application of these rules are presented below in Table 2.

Table 2 Examples of Application of Rules Related to Significant Figures in Reported Uncertainty Before Application of Rules AFTER Application of Rules (Correct Reporting)

(31.25 0.03495) cm (31.25 0.03) cm (9.98238 0.067695) m/s2 (9.98 0.07) m/s2 (23.66789 0.23576) cm (23.7 0.2) cm (23.66789 2.37859) cm (24 2) cm (23.66789 0.005234) cm (23.668 0.005) cm (23.66789 0.1379) cm (23.66 0.14) cm

*Exception of rule since first significant figure equal to 1 (31.25345 0.034953) m/s (31.25 0.03) m/s (1261.2915 200.234) m/s (1300 200) m/s

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2.6 Design-Stage Uncertainty Analysis Design-stage uncertainty analysis is sometimes referred to as zeroth-order uncertainty analysis. It provides a means of estimating the overall uncertainty arising from the instrumentation and method and is typically done during the planning stages of an experimental program. The manufacturers specification sheets are typically the sole source of data for the uncertainty analysis. There are no actual measurements involved in a design-stage uncertainty analysis.

Typically errors are from two sources: (1) known instrument errors and (2) resolution errors. There is not much value in keeping track of individual systematic and random error components in a design-stage uncertainty analysis. Ideally, all errors have the same level of confidence but this may not me known with certainty and it doesnt matter at this stage of the analysis.

The design-stage measurement uncertainty is determined by combining the sources of uncertainty (ek) using the root-sum-squares method (RSS Method). The sources of uncertainty may come from both systematic and/or random errors. That is,

UX,design = e12 + e2

2 + .... + ek2

= ek2

k=1

K

Again, design-stage uncertainty estimate is intended only as a guide for selecting instrumentation before a test, and is never used for reporting results. More detailed analysis is possible once actual measurements are taken.

Example 1: Consider a force-measuring instrument being considered that has the following catalog data.

Range: 0-100N Linearity: within 0.2 N over the range Hysterisis: within 0.3 N over the range Resolution: 0.25 N

In this problem three instrument errors exist. Resolution, Linearity, and Hysteresis. Combine these errors using the RSS method. There is no need to differentiate between systematic and random error during design-stage uncertainty analysis.

Ux,design=(0.1252+0.22+0.32)0.5 = 0.4 N

Note: Interpolation errors are estimated from the given resolution, 0.25N. The interpolation error is the resolution.

2.7 Combining Systematic Errors for a Measurement Systematic error is the portion of the total error that remains constant in repeated measurements throughout the conduct of a test. A systematic error may cause either a high or a low offset. Systematic error is usually derived from calibration information (e.g. accuracy, linearity, hysteresis, offset, and known environmental errors). The total systematic uncertainty is estimated using the root-sum-squares (RSS) method to combine the individual error sources.

The systematic uncertainty, B, is assumed to have 95% CI. These are the values of systematic errors that are often reported in literature or in instrumentation calibrations. The value b is the estimate of the systematic standard uncertainty which has a confidence interval of 1 standard deviation (68%).

MEAUREMENT SYSTEMATIC STANDARD

UNCERTAINTY

/

where K = Total number of systematic error sources bk = Estimate of the systematic error of the kth elemental error.

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It is worth noting that the uncertainty depends on the squares of the individual uncertainties. If the uncertainty from one source is much larger than the others then the larger uncertainty values dominate the values. Many times, the other smaller sources of uncertainty can be neglected.

2.8 Combining Random Errors for a Measurement Estimates for random error are derived using repeated measurements and statistical methods. The random uncertainty is determined from the standard deviation of the mean. The standard deviation of the mean is used with the assumption the errors are normally distributed. The interval has a confidence of one standard deviation, equivalent to a probability of 68% for a population. The random uncertainty at a desired confidence level is defined by the interval, .

RANDOM UNCERTAINTY IN A MEASUREMENT

SX =sx

N

where sx = standard deviation N = number of samples or readings

In the event that you have K random errors for a measurement then Sx is estimated by the RSS method;

SX = (Sx ,k )2k=1

K

1/2

where each elemental source of random error may have its own sample size.

Sx ,k =Sx,kNk

2.9 MultipleMeasurement Uncertainty Analysis This section presents a method for estimating the uncertainty in the value assigned to a variable. The procedure assumes that the errors follow a normal probability distribution and that there are sufficient repetitions to assess random error. The total uncertainty in a measurement is then determined by combining the total systematic error and random error for a measurement in the following manner.

MEASUREMENT UNCERTAINTY

, 2

The degrees of freedom can be calculated exactly using methods described in ASME PTC 19.1 and using the combined degrees of freedom equation shown below. However, a value of 2 is assumed for the t-value in this course to simplify the calculation method. The combined degrees of freedom,, can be calculated using the following equation.

! !#$ #$% %

& '( & '(

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Example 2 Ten repeated measurements of force are obtained from a force transducer that has two sources of systematic error. B1=0.01 N and B2=0.15 N. Estimate the random, systematic, and total measurement uncertainty.

Find the systematic error, random error, and total error, UX, in the measurement. Express the answer in both absolute and relative terms. Recall, B=2b.

bX=(0.052+0.0752)1/2 =0.09 N

SX =sx

N=

3.0410

= 0.96N

, 2 )0.09 0.96 1.92

X=120 2 N at a 95% Confidence

or

X=120 1.7% Relative Uncertainty at a 95% Confidence

Reading Force, N Systematic Error Force, N

1 123.2 b1 0.05

2 115.6 b2 0.075

3 117.1

4 125.7 Total Systematic

5 121.1 Btotal 0.09

6 119.8

7 117.5

8 120.6

9 118.8

10 121.9

Average 120.13

Std Dev 3.04

Count 10

DOF 9

t value 2.2621572

sx_bar 0.96

Measurement Uncertainty 1.9

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Example 3. A calorimeter is used to measure the heating values of samples. The manufacturer of the calorimeter states that the device has an accuracy of 1.5% of the full-scale range where the full-scale range is 0 to 100,000 kJ/kg. The measured heating values for 10 samples are:

Sample Heating Value, kJ/kg 1 48530 2 48980 3 50210 4 49860 5 48560 6 49540 7 49270 8 48850 9 49320 10 48680

Calculate the following:

(a) The systematic and random uncertainty of the mean of the measurements.

a. Systematic error is determined from the accuracy value given in the problem. 1.5% of FS. Meaning, BX=100,000*0.015=1500 kJ/kg.

b. Random error is determined using statistics. Calculate the standard deviation, standard deviation of the means. The standard deviation of the means for the data above is: sx =179 kJ/kg

(b) The total uncertainty of the mean value using a 95% confidence interval.

a. The total uncertainty is determined using the equation:

, 2 )750 179 1542 X=49200 1500 kJ/kg at a 95% Confidence

Reading Heating kJ/kg Systematic Error Heating kJ/kg

1 48530 b1 750.002 489803 502104 49860 Total Systematic5 48560 Btotal 750.006 495407 492708 488509 49320

10 48680

Average 49180

Std Dev 566.31

Count 10

DOF 9

t value 2.262157163

sx_bar 179.08

Measurement Uncertainty 1542.2

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3 Uncertainty of a Result The uncertainty of a result is not obtained directly from measurements. Instead, the individual measurements and associated individual measurement uncertainties are used to determine the uncertainty in the result using propagation of error techniques. In general, the uncertainty of a result, R, is a function of k measured variables, x1, x2, x3,..xk;. or

R=f(x1, x2, x3,..xk)

The root sum of squares (RSS) method is used to propagate the individual measurement uncertainties into the result uncertainty. The partial derivative of the result, with respect to each measured variable, is used along with the uncertainty in the measured variable to determine the overall uncertainty in the result using the following equation. Another common form of the RSS equation utilizes the sensitivity coefficient, , which is defined as the instantaneous rate of change in a result to a change in a parameter.

UR =Rx1

Ux1

2

+Rx2

Ux2

2

+ ... +Rxk

Uxk

2

whereUR = Uncertainty in the result, either systematic or randomUxk = Uncertainty in the measured parameter, xk , either systematic or random

A special case formulation of the above RSS equation exists for cases where the result is dependent only on the product or quotient of the measured variables. This form of the equation is typically easier to use but both methods, the partial derivative method and special case formulation, yield the same result.

R=C x1a, x2b, x3c,..xkn

URR

=

aUx1x1

2

+bUx2

x2

2

+ .....+nUxk

xk

2

All measurements and the associated error must be independent of each other. That is, an error in one variable must not correlate with an error in another variable. If the variables are not independent, then an alternative approach is required (this alternative approach is not discussed in this course but see the listed references for more information on these alternate approaches).

In either method, the partial derivative or the special case formulation, the uncertainty estimates are typically computed separately for the systematic error and random error components and then combined at the end to yield an estimate of the total uncertainty in the result. However, the effect of combining the systematic and random errors earlier in the calculation does not affect the final total uncertainty in the result.

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The following examples demonstrate the RSS method and are not concerned with keeping track of the systematic and random error components. Other examples will follow that demonstrate the handling of these errors separately.

Example 4. The measurement of power consumption in a simple resistive circuit is determined using the equation, P=V*I. The voltage, V, and current, I, are the measured variables in this problem and the result is the power. The measured quantities and associated uncertainty are:

V=100 3 V (relative uncertainty=3/100=3%) I=10 0.2 A (relative uncertainty=0.2/10=2%)

Confidence levels of both measurements are assumed to be the same (i.e. @95% CI)

Both the partial derivative method and the special form method are shown below. Both methods yield the same results.

In both cases the power measurement is reported as: P= 1000 40 W (95% CI).

Example 5: A pressurized air tank is nominally at ambient temperature (25C). How accurately can the density be determined if the temperature is measured with an uncertainty of 2C and the tank pressure is measured with an accuracy of 1%? The data reduction equation solve for pressure is:

= PRT

UP/P=0.01

UT/T=2/298=0.00671 (Need to covert to the appropriate units firstIn this case degrees Kelvin)

Assume no uncertainty in the gas constant R.

U/=?

Use the relative uncertainty form of the RSS equation to solve this problem.

U

=

UTT

2

+UPP

2

= 0.00671( )2 + 0.01( )2 = 0.012 = 1.2%

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Example 6: The overall heat transfer coefficient U of a system of two fluids separated by a wall of negligible thermal resistance is determined using the relationship:

U = h1h2h1 + h2

where h1 and h2 are the individual heat transfer coefficients for the two fluids.

If h1=15 W/m2 with an error of 5% and h2 =20 W/m2 with an error of 3%, what will the error be in U? Meaning Uh1=0.05*15=0.75 W/m2 and Uh2=0.03*20=0.60 W/m2.

The nominal value of U=8.6 W/m2

This problem requires the long version of the RSS equation. Meaning, the partial derivatives must be calculated.

Iff (x) = h(x)

g(x)Thenddx

f (x) = f (x) = h (x)g(x) h(x) g (x)g(x) 2

therefore

Uh1

=

h2 (h1 + h2 ) h1h2h1 + h2( )2 =

h22

h1 + h2( )2 = 0.32653

Uh2

=

h1(h1 + h2 ) h1h2h1 + h2( )2 =

h12

h1 + h2( )2 = 0.183673

Now insert the partial derivatives and uncertainty of each variable into the RSS equation.

UU =Uh1

Uh1

2

+Uh2

Uh2

2

or

UU = 0.32653 0.75( )2 + 0.183673 0.60( )2[ ] = 0.27W /m2

Report value as (8.6 0.3) W/m2

Converting the absolute uncertainty to relative uncertainty is then 0.27/8.6=3%.

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Example 7: Consider a counter-flow heat exchanger where there is a solution of ethylene glycol on the cold side of the heat exchanger and pure water is on the hot side of the heat exchanger. Thermal losses are assumed to be negligible.

Therefore it is reasonable to expect that the observed energy exchange on both sides of the heat exchanger will be equal. The heat exchange to the cold stream, Qc, should equal that from the hot stream, Qh, within the uncertainty of each quantity. Is this the case?

12 3456 7 6 18 394596& 7 69 A listing of all the measured values and corresponding uncertainty is presented in the following table.

Side Parameter Nominal Value Uncertainty

Cold

cp1 (J/kg-C) 4186 5% &m1(kg/s) 60 10%

T1 (C) 1.1 0.5 C T2 (C) 7.8 0.5 C

Hot

cp3 (J/kg-C) 3768 5% &m3 (kg/s) 56 10%

T3 (C) 12.8 0.5 C T4 (C) 4.4 0.5 C

Hot and cold side energy balances are not identical but they overlap when their respective uncertainties are included. Cold Side Range=1424 to 1942 kW. Hot Side Range=1624 to 2020 kW.

Hot Side

Cold Side

3 4

2 1

Partial Derivative Method

Measurement, x Units Average x Relative Ux Absolute Ux Partial, dR/dx Numerical Partial, dR/dx dR/dx* UxSpecific Heat j/kg-C 4186 5% 209.3 m*(T1-T2) -402 -84138.6Mass Flow Rate kg/s 60 10% 6 cp*(T1-T2) -28046.2 -168277.2T1 C 1.1 0.5 m*cp 251160 125580T2 C 7.8 0.5 m*cp 251160 125580

Calculated ResultQc kW 1683

Total UR 258.7 Watts

Relative Uncertainty 15.4%


Measurement, x Units Average x Relative Ux Absolute Ux Partial, dR/dx Numerical Partial, dR/dx dR/dx* UxSpecific Heat j/kg-C 3768 5% 188.4 m*(T1-T2) 470.4 88623.36Mass Flow Rate kg/s 56 10% 5.6 cp*(T1-T2) 31651.2 177246.72T3 C 12.8 0.5 m*cp 211008 105504T4 C 4.4 0.5 m*cp 211008 105504

Calculated ResultQh kW 1772

Total UR 248.1 Watts

Relative Uncertainty 14.0%

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3.1 Single Measurement Experiments An experiment is considered to be a single measurement experiment when the result is run only once at each test condition. In this case, there are not sufficient test results to calculate the standard deviation of the result. Estimates for the random uncertainty need to be obtained through other sources, such as manufactures data or auxiliary tests.

Estimates for systematic uncertainty are obtained using the equation:

: ;

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3.2 Uncertainty in Result for Multiple Measurement Experiments In experimental situations where a test is repeated multiple times to determine M values of the result, R1, R2, RM a slightly different approach is required to estimate the experimental uncertainty.

The bias error is estimated using the RSS equation or the special function form of the equation.

: ;

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Example 8. The thermal efficiency of a natural gas engine is determined using the equation

= Pm f HV

Where P is the power output in kW, mf is the natural gas flow rate in kg/s, and HV is the heating value of natural gas in kJ/kg. Average values of P, mf, and HV are 50kW, 0.2 kg/minute, and 49180 kJ/kg, respectively. The systematic uncertainties are given as 0.2 kW, 0.003 kg/minute, and 1500 kJ/kg, respectively.

To establish the mean value for the efficiency of engines from a production line, five engines are tested under similar conditions. The results for the calculated resulting efficiency are 31.0, 30.5, 30.8, 30.6 and 30.2 percent. The mean value of the result is 30.6 percent and the standard deviation is 0.303 percent. Meaning, the random portion of the uncertainty in the result is then,

sR =0.00303

5= 0.00136

Bias error is estimated using the special function form of the RSS equation because the result is a product/quotient of the three measured variables.

The value for the efficiency is 0.306 (30.6%).

:= P;0.150?

;0.00150.2 ?

; 75049180?

0.0171

: 0.0171*.306=0.00523

Total Uncertainty in the Efficiency Measurement is then:

: 2 )0.00523 0.00136 0.011 1%UVVWXWUYXZ

Efficiency is reported as: 31 1% (Relative Uncertainty ~3.5%)

(Systematic uncertainty clearly the dominant source of error perhaps instrumentation improvements could be made)

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3.3 Summary of Procedure for Uncertainty Analysis and Error Propagation Uncertainty analysis provides the values on your test results. Performing uncertainty analysis on tests can be complicated at times and requires a consistent approach and systematic bookkeeping. The following steps summarize the activities in performing an uncertainty analysis.

(1) Define the measurement process. Identify all measured values and the relationship between the measured values and the test results.

(2) List all elemental sources. Make a complete list of all possible error sources for each measured parameter. It is sometimes helpful to group uncertainties into categories based on their source (e.g. calibration, data acquisition, etc.)

(3) Estimate elemental errors. Estimate the systematic and random errors. If data is available to estimate the standard deviation of a parameter, or the error is known to be random in nature, then it should be treated as random uncertainty; otherwise, classify it as a systematic uncertainty.

(4) Calculate the systematic and random uncertainty of each measured variable. Total/sum the systematic and random uncertainties for each of the measurement variables.

(5) Propagate the systematic uncertainties and random uncertainties all the way to the result(s). Utilize the RSS equation to propagate the systematic and random uncertainties of the measured variables to the final test results. Keeping the systematic errors and random errors separate allows you to determine which type of error is dominating the measurement uncertainty.

(6) Calculate the total uncertainty of the result(s). Utilize the appropriate RSS equation (for Multiple Measurements or Single Measurements) to combine the systematic and random errors to obtain the total uncertainty in the result.

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Example 9: The efficiency of a pump is determined using the formula

pump. theintoinput power =Wpump theofoutlet andinlet ebetween th aldifferenti pressure=P

rate flow volumetric=Qefficiency pump

=

=

whereW

PQ

The following equipment is used:

Differential Pressure Gage Range 0-1200 kPa Accuracy 0.2% of span (includes linearity, hysteresis, and repeatability) Stability 0.2% of span

Flowmeter Range 1200 LPM (liters per minute) Accuracy 1.5% of reading

Power Accuracy 0.07 kW

The average values and standard deviations of the means of the measured values are as follows.

Measurement Mean Value Standard Deviation Mean N Pressure 702 kPa 10 kPa 20 Flow 340 lpm 5.6 lpm 15 Power 5.1 kW 0.15 kW 10

PART 1: Given the repeated measurements for each measured value to calculate the mean and standard deviation determine the following: (Note: This is a Single Measurement Experiment approach)

(a) Efficiency of the pump (b) Random and systematic uncertainty of the efficiency of the pump (c) Total uncertainty of the efficiency of the pump at a 95% confidence interval.

(a) Pump efficiency is calculated by using the given relationship and converting all measured values to the proper units.

= Q PW

= 0.780

(b) Systematic and random uncertainty is determined using

BR =Rx i

Bi

2

i=1

n

1/ 2

and sR =R xi

sX.i

2

i=1

n

1/2

(c) Total uncertainty determined using RSS method. UR = t , BR

2 + sR2 2 BR

2 + sR2

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Below are the templates of the solution method and results using both the relative uncertainty method and the partial derivative method.

Relative Uncertainty Method


Combining the Systematic and Random Components for Each Measurement Prior to Analysis

In the end, the final answer is the same using any of the above methods. However, only with the first two methods, where the systematic and random components are kept separate until the end, can you determine which source of uncertainty is dominating the uncertainty. That is, are systematic errors dominating or random errors?

Efficiency is presented as: 0.78 0.06 at 95% CI

Measurement, x Units Range Average x Bx bx N DOF t-value Sx_bar

bx/X, Relative

Uncertianty

sxbar/X, Relative

UncertiantyDifferential Pressure kPa 1200 702 3.39 1.70 20 19 2.09 10 0.002417459 0.014245014Volumtric Flow L/minute 1200 340 5.1 2.55 15 14 2.14 5.6Volumtric Flow,Q m 3^/s 0.02 0.0056667 9E-05 0.00 15 14 2.14 9E-05 0.0075 0.016470588Power kW 5.1 0.07 0.04 10 9 2.26 0.15 0.006862745 0.029411765

Calculated ResultEfficiency 0.780 Rel Uncert 0.010449468 Rel Uncert 0.0365958

Total bR 0.008 Total sR 0.029

Total Uncertainty 0.059

Total Rel. Uncertainty 7.6%

Systematic Uncertainty Random Uncertainty

Measurement, x Units Range Average x Bx bx N DOF t-value Sx_barPartial,dR/d

x

Numeric Partial dR/dx*bx

dR/dx*sx_bar

Differential Pressure kPa 1200 702 3.39 1.70 20 19 2.09 10 Q/W 0.001111 0.001886 0.01111111Volumtric Flow L/minute 1200 340 5.1 2.55 15 14 2.14 5.6 0Volumtric Flow,Q m 3^/s 0.02 0.0056667 9E-05 0.00 15 14 2.14 9E-05 DP/W 137.6471 0.00585 0.01284706Power kW 5.1 0.07 0.04 10 9 2.26 0.15 (-Q*DP)/W 2^ -0.15294 -0.00535 -0.0229412

Calculated ResultEfficiency 0.780 Total BR Total sR

0.008 0.029



Measurement, x Units Range Average x Bx bx N DOF t-value Sx_bar UxPartial,dR/d

x

Numeric Partial dR/dx*Ux

Differential Pressure kPa 1200 702 3.39 1.70 20 19 2.09 10 20.286 Q/W 0.001111 0.02253995Volumtric Flow L/minute 1200 340 5.1 2.55 15 14 2.14 5.6Volumtric Flow,Q m 3^/s 0.02 0.0056667 9E-05 0.00 15 14 2.14 9E-05 0.0002 DP/W 137.6471 0.02823256Power kW 5.1 0.07 0.04 10 9 2.26 0.15 0.3081 (-Q*DP)/W 2^ -0.15294 -0.0471148

Calculated ResultEfficiency 0.780 Total UR

0.059



20

3.4 Uncertainty Analysis Using Sequential Perturbation A numerical approach can also be used to estimate the uncertainty of a result. The method known as sequential perturbation is often used or preferred when direct differentiation is too cumbersome/complex and/or the number of variables involved in the calculation of uncertainty is large.

The method utilizes a finite difference method to approximate the derivatives in the RSS equation and can be easily implemented in a spreadsheet software program or using a hand calculator. The steps in the method are as follows:

1. Calculate the result R0=f(x1, x2, x3.xi) using the nominal values for all the independent measured variables.

2. Next, increase each independent variable by their respective uncertainties and recalculate the result. Only increase one variable at a time and leave the other independent variables set to their respective nominal values. These values are called Ri+. That is,

R1+=f(x1+U1, x2, x3, xi)

R2+=f(x1, x2+U2, x3, xi)

R3+=f(x1, x2, x3+U3, xi)

Ri+=f(x1, x2, x3, xi+Ui)

3. Next, decrease each independent variable by their respective uncertainties and recalculate the result. Again, decrease one variable at a time and leave the other independent variables set to their respective nominal values. These values are called Ri-. That is,

R1-=f(x1-U1, x2, x3, xi)

R2-=f(x1, x2-U2, x3, xi)

R3-=f(x1, x2, x3-U3, xi)

Ri-=f(x1, x2, x3, xi-Ui)

4. Now calculate the difference Ri+ and Ri- for each independent measurement. Ri+= Ri+-R0 and Ri-= Ri--R0

5. Evaluate the approximated uncertainty for each variable.

Ri = Ri+ Ri

2

Rxi

Ui

6. The uncertainty in the result is then

UR = Ri( )2i =1

k

1/2

21

Example 10. The measurement of power consumption in a simple resistive circuit is determined using the equation, P=V*I. The voltage, V, and current, I, are the measured variables in this problem and the result is the power. The measured quantities and associated uncertainty are:

V=100 3 V (relative uncertainty=3/100=3%) I=10 0.2 A (relative uncertainty=0.2/10=2%)

1. Calculate the result R0=f(x1, x2, x3.xi) using the nominal values for all the independent measured variables. R1=V, R2=I in this case.

R0=V*1=1000

2. Next, increase each independent variable by their respective uncertainties and recalculate the result.

R1+=f(100+3, 10) = f(103, 10) = 1030 R2+=f(100, 10+0.2) = f(100, 10.2) = 1020

3. Next, decrease each independent variable by their respective uncertainties and recalculate the result.

R1- =f(100-3, 10) = f(97, 10) = 970 R2- =f(100, 10-0.2) = f(100, 9.8) = 980

4. Now calculate the difference Ri+ and Ri- for each independent measurement. R1+ = 1030-1000 = 30 R2+ = 1020-1000 = 20

R1- = 970-1000 = -30 R2- = 980-1000 = -20

5. Evaluate the approximated uncertainty for each variable.

R1 = R1+ R1

2=

30 (30)2

= 30 RV

UV

R2 = R2+ R2

2=

20 (20)2

= 20 RI

U I

6. The uncertainty in the result is then

UR = 30( )2 + 20( )2 0.5

= 36.06

A copy of a spreadsheet layout for this problem is shown below.

In both cases, Example 4 and Example 10, the power measurement is reported as: P= (1000 40) W (95% CI).

P=V*I

Variable Nominal U_I U+ U- R+ R- dR+ dR- dR

V 100 3 103 97 1030 970 30 -30 30

I 10 0.2 10.2 9.8 1020 980 20 -20 20

P 1000 UP 36.06

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f5763552 Microsoft Word - Uncertainty Analysis Guideline 2012d