PTC INQ. #06-02
TECHNICAL INQUIRY
on
PTC 6.2 – 2004 & PTC 19.5 – 2004 & PTC 19.1 – 1998
Question: How can we be 95% confident that the true value of the discharge coefficient falls within the
uncertainty interval if the curve fit is used for determination of the discharge coefficient?
Answer: Uncertainties within ASME Performance Test Codes are standardized on a 95% confidence
interval. The meaning of the 95% confidence interval is that there is a 95% probability that the
difference between the indicated value and the true value of a measurement or calculation will be less
than or equal to the uncertainty of that measurement or calculation.
PTC 19.5, Flow Measurement, and PTC 19.1, Test Uncertainty, are meant to be used in concert. The
methods provided in ASME PTC 19.1 for the determination of uncertainty should be followed always;
nothing published in either 6.2 or 19.5 was intended to over-rule these principles and methods. Since
the extrapolation method relies both upon an extrapolation equation and calibration data, the uncertainty
of each of these must be factored into the total uncertainty. There is an uncertainty in the intercept, Co,
derived from the scatter of the calibration data; there is a published uncertainty on the ‘slope’
(coefficient ‘d’) in the equation; and finally there is the random variation of the measurements under
test. The consequences of all of these together must be computed per 19.1.
The 95% confidence intervals for the coefficients in the extrapolation equation are given as +/- values in
Mandatory Appendix II (next to the values for the coefficients). These can be used to determine the
combined uncertainty of the curve-fit. Note that the uncertainties of the other coefficients except that of
coefficient “d” are incorporated into the actual calibration data, according to the underlying theory of
this extrapolation method.
The uncertainty of the calibration data is usually provided by the facility that performs the calibration; it
is their estimate of the uncertainty of their calibration, which becomes a systematic uncertainty during
the test.
In the example cited with the question, the ‘Reynolds number dependence of Co’ is insignificant
statistically. As shown in the figure below, the total variation of all the data from the extrapolation
curve is less than +0.12%, which value is quite desirable for repeatability alone.
Coefficient of Discharge
Variation of Calibration Data
0.608
0.6075
0.607
0.6065
0.606
0.6055
0
500000
1000000
1500000
2000000
Pipe Reynolds Num ber
Cd, meas.
19.5 Meth
Since the 1930’s, engineers have been plotting the Coefficient of Discharge against the abscissa of
Reynolds number because they knew it had to be related somehow, and they were conducting
experiments to discover this relationship. It turns out to be a convenient parameter to use to display
calibration results, and so it has remained in use. However, the coefficient of discharge is not a function
of Reynolds number directly; it is a nonlinear function of the boundary layer, which is in turn a
nonlinear function of the Reynolds number.