Approach to Experimental Design and Analysis: Propagation of Errors and Uncertainties ERR_PROP'ppt

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Approach to Experimental Design and Analysis
Propagation of Errors and UncertaintiesERR_PROP
.ppt
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Contents

• Key concept errors uncertainties are
  inevitable in measurements due to both
  fluctuations in readings basic instrument
  readings they propagate through to final results
• Example mass balance involving addition,
  subtraction, multiplication, division.
• Problem for us to do right now.

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Treatment of Experimental Data

• (read together with the document VARIANCE.doc)
• Primary treatment
• errors and uncertainties associated with
  measurements
• how these errors combine
• Secondary treatment
• analysis of variance
• ANOVA discussed in ANOVA.ppt

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Primary Treatment

• can be done with any/all data
• systematic and random errors (uncertainties)
• few measurements are used on their own need to
  analyse combinations of these errors

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Kinds of Errors and Uncertainties

• Systematic errors
• calibration errors check by standard tests
• e.g. water boils at 100oC temperature
  measurement device should read 100oC

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Kinds of Errors and Uncertainties (cont.)

• Random errors
• from unpredictable (but not unquantifiable)
  fluctuations in experimental situation
• sometimes called uncertainty

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Propagation of Errors and Uncertainties
EXAMPLE

• Cooling Tower
• Suppose that you have done one experiment with
  the Departments cooling tower equipment, giving
  the following results-

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Equipment
Outlet air humidity, 0.0088 kg water / kg dry air
Make-up water, 1.8 x 10-6 kg/s
Inlet air flowrate, 0.018 kg total/s
Inlet air humidity, 0.0080 kg water / kg dry air
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Important Question to Answer

• Do these results show that the water make-up
  flowrate is within experimental uncertainties of
  being equal to the moisture leaving the column in
  the air?
• This question is important because the mass
  balance tells you and the reader of your report
  how reliable your results are.

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Another View

• When you say that your discrepancy is small,
  how small is small? Small means different things
  to different people.
• Here, small means small relative to the
  probable/possible discrepancy resulting from all
  the uncertainties in all the measurements.
• To work out the overall discrepancy, we need to
  propagate through all the individual
  uncertainties into the uncertainty in the final
  result.
• This is what we are doing here.

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Addition/Subtraction Use Absolute Errors
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Multiplication/Division Use Relative Errors
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Fundamentals

• ?A ?B are maxima of
• uncertainty in reading instrument (half of
  minimum scale division or half of instrument
  resolution)
• uncertainty due to fluctuations in readings.
• ?A ?B are like standard deviations cannot be
  added directly.
• (?A)2 (?B)2 are like variances these can be
  added.

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Answer Step 1 Calculate Dry Gas Flowrate (F)

• Required because humidity is expressed on a dry
  basis we have measured total dry flowrate and
  humidity at the inlet
• F FW /(1 Yi)
• (0.018 0.004) kg total s-1 /
• (1 0.0080 0.0001) kg total / kg dry gas
• 0.0179 kg dry gas s-1
• 0.018 kg dry gas s-1

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• Most of the uncertainty is contributed by the gas
  flowrate.

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Step 2 Calculate Rate of Moisture Loss from
Tower in Air

• Rate of loss
• F (Yo - Yi)
• (0.018 0.004) kg dry gas s-1 x
• (0.0088 0.0001 - 0.0080 0.0001) kg water /
  kg dry gas
• (0.018 0.004) kg dry gas s-1 x
• (0.0008 0.00014 ) kg water / kg dry gas
• (1.4 0.4) x 10-5 kg water s-1

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• Most of this uncertainty (78 of it) comes from
  the gas flowrate

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Step 3 Compare

• Compare rate of moisture loss from tower in air
  with water make-up flowrate.
• Rate of loss - make-up flowrate
• (1.4 0.4) x 10-5 kg water s-1
• - (1.8 0.2) x 10-5 kg water s-1
• (-0.4 0.5) x 10-5 kg water s-1

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Significance

• Uncertainty in the discrepancy is greater than
  discrepancy itself, so the discrepancy could
  possibly be zero within the uncertainties in the
  measurements.
• If uncertainty in the discrepancy was smaller
  than the discrepancy, then some other cause
  (apart from random measurement uncertainty) would
  need to be sought for the discrepancy.

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So?

• In the lab, you must not only measure the actual
  values (e.g. a temperature of 55oC), but also
  estimate the uncertainty (e.g. ?2oC).
• You must then use these estimates of uncertainty
  in your propagation of error analysis.

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How to Estimate Uncertainties?

• How much does a measurement fluctuate?
• What is the smallest scale division?
• Uncertainty maximum value of (measurement
  fluctuation, half smallest scale division).
• Uncertainty previous literature experience.

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Exercise

• In the spiral heat exchanger lab, you measure a
  hot water flowrate of 1 kg s-1 a cold water
  flowrate of 2 kg s-1.
• Literature experience suggests that the
  uncertainty in these flowrates is 5.
• Hot water inlet, outlet Ts 95oC, 75oC.
• Cold water inlet, outlet Ts 20oC, 28oC.
• Uncertainties in Ts 1oC.
• Cp (water) 4.2 kJ kg-1 K-1. No uncertainty.

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Equipment
In, 20 C
Cold, 2 kg/s
In, 95 C
Hot, 1 kg/s
Out, 75 C
Out, 28 C
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Questions

• What is the heat balance?
• Do we have a small discrepancy in this heat
  balance?

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Conclusions

• Errors uncertainties are inevitable in
  measurements due to both fluctuations in readings
  basic instrument readings.
• They propagate through to final results in both
  mass energy balances (e.g. heat-transfer
  coefficients) Not just for mass energy
  balances.
• Addition, subtraction, multiplication, division.