2. Measurement Process Characterization
2.6. Case studies
2.6.5. Uncertainty analysis for extinguishing fire

2.6.5.1.

Background and data

Background

The data for this case study is taken from a study conducted by Michele Donnelly of the Fire Science Division, Building and Fire Research Laboratory, of the National Institute of Standards and Technology.

The data relates to the volume fraction of nitrogen required to extinguish flame on the surface of a plastic cylinder. Measurements of flux (X) are made on the surface of the object, and corresponding rotameter readings (Y) are converted to flowrate (Y') from a calibration curve for the rotameter. Flowrate is then converted to nitrogen volume fraction.

The case study demonstrates the analysis of uncertainty for values predicted from a calibration curve where there is uncertainty in both the predictors and the predicted values.

Steps in the Analysis

The steps in the analysis are as follows.

1. Create a calibration curve for the rotatmeter.
2. Compute predicted values of nitrogen flowrate from rotameter readings on the twenty-one cylinders in the study.
3. Compute the standard deviations of the predicted values of flowrate.
4. Convert the flowrates to nitrogen volume fractions (%).
5. Calculate the uncertainties due to the calibration curve for the nitrogen volume fractions.
6. Calulate the uncertainties due to flux measurements for the nitrogen volume fractions.
7. Combine the last two components into a total uncertainty.

Data Used to Compute the Calibration Curve

The following is the data used to compute the calibration curve. In this case, we have measurements for both the flux and the rotameter reading.

         Rotameter
   Flux  Readings
    (X)       (Y)
   ====   =======
     40     7.871
     40     7.877
     40     7.863
     40     7.877
     40     7.871
     40     7.877
     40     7.863
     40     7.874
     40     7.871
     40     7.865

     50     9.608
     50     9.591
     50     9.600
     50     9.582
     50     9.582
     50     9.600
     50     9.587
     50     9.587
     50     9.608
     50     9.600
     
     70    12.94
     70    12.93
     70    12.94
     70    12.93
     70    12.94
     70    12.94
     70    12.91
     70    12.92
     70    12.95
     70    12.94
     
     90    15.94
     90    15.84
     90    15.91
     90    16.01
     90    15.98
     90    15.89
     90    15.82
     90    15.89
     90    15.88
     90    15.90
     
     30     5.981
     30     5.980
     30     5.983
     30     5.985
     30     5.985
     30     5.981
     30     5.981
     30     5.978
     30     5.981
     30     5.977
     
     80    14.40
     80    14.39
     80    14.40
     80    14.41
     80    14.42
     80    14.39
     80    14.43
     80    14.37
     80    14.44
     80    14.37
     
    100    17.23
    100    17.17
    100    17.23
    100    17.17
    100    17.15
    100    17.19
    100    17.21
    100    17.24
    100    17.23
    100    17.22

     20     4.012
     20     4.014
     20     4.015
     20     4.011
     20     4.013
     20     4.011
     20     4.020
     20     4.014
     20     4.009
     20     4.011

New Flux Measurements

We also have measurements for 21 new data points. In this case, we only have the flux measurements. We use the computed calibration curve to estimate the nitrogen volume fraction and the associated standard deviation.

  Flux
   X 
======
   26.
   30.
   27.
   44.
   16.
   21.
   14.
   24.
   22.
   43.
   41.
   47.
   45.
   42.
   14.
   14.
   32.
   29.
   47.
   44.
   47.