UNCERTAINTY ANALYSIS: 
Measurement Uncertainty for Fluid Flow in Closed Conduits 

UNCERTAINTY ANALYSIS IN GAS FLOW MEASUREMENT WHEN USING SONIC NOZZLES 

The mass flow rate of a gas passing through a Sonic Nozzle (Critical Flow Venturi) can be calculated from the following equation: 

To calculate the magnitude of the error in the mass flow rate (M), the errors in each of the components of the equation are combined by the root-sum-square method as shown in the following equation:

where eM is the error in the calculated mass flow rate, eP is the error in the pressure measurement, eT is the error in the temperature measurement, etc.

Since gc is a constant, the error (egc = 0) will be zero. Likewise, the error in area (eA = 0) will be zero if the sonic nozzle has been calibrated by a laboratory (CEESI, NIST, etc). The uncertainty for an uncalibrated sonic nozzle is discussed later in this technical note. Removing these zero components, the error equation becomes:

A flow calibration laboratory can perform various tests, which can provide different levels of uncertainty in the discharge coefficient of a sonic nozzle. For the present discussion, assume the discharge coefficient uncertainty to be ±0.50%. A further discussion of the various uncertainty levels in the discharge coefficient will be given later. Therefore, eCd = ±0.500%.

The inlet absolute pressure at the sonic will usually be measured with good quality pressure transducers or gages. If an absolute pressure transducer is used, then the uncertainty in the pressure measurement will be a function of the full scale value of that transducer. Another commonly used method is to use a gage pressure transducer and then add the barometric pressure to determine the absolute pressure at the inlet of the sonic nozzle. For this analysis, we will consider the latter case as it is slightly more complex. 

Assume that both the gage pressure and the barometric pressure are measured with transducers that have an accuracy of ±0.05% of full scale. Also assume that the barometric transducer has a full-scale value of 15 psia, and the gage pressure transducer has a full-scale value of 100 psig. If we assume that the sonic nozzle is to be used at a location near sea level, then the barometric pressure will be near 14.7 psia. In this case, the barometric transducer will determine the pressure within ±0.0075 psia (0.05% of 15). 

Assume that the sonic nozzle will be operating over an inlet pressure range of 25 to 100 psig. The largest uncertainty in the gage pressure will occur at the lowest inlet value of 25 psig. At this condition, the gage pressure transducer will determine the inlet pressure within ±0.05 psig (0.05% of 100). The nozzle inlet absolute pressure is 39.7 psia which the sum of the two transducers readings (25 + 14.7). A conservative linear addition of the two estimated errors results in a measurement error of ±0.0575 psia. At the minimum operating pressure of 39.7 psia, the uncertainty in the pressure measurement is ±0.145% (100*0.0575/39.7). As the sonic nozzle inlet gage pressure is increased, the uncertainty will become less. However, for this example, we will use the worst-case error. Therefore, eP = ±0.145%.

The error in temperature (eT) has a sensitivity coefficient of 0.5 applied to it in the error equation shown above. This is because the temperature enters into the mass flow equation as a square root. For this example, we will assume that the sonic nozzle inlet temperature is measured with a 4-wire platinum RTD and a temperature transmitter. A typical specification for a temperature transmitter is that the measurement accuracy is ±0.40°R. If we assume that the sonic nozzle will be operating at a room temperature of 530°R, then the ±0.40°R error amounts to a ±0.075% of reading accuracy. Therefore, eT = ±0.075%.

The error in the critical flow function (eC*) depends upon two different parameters. For pure gases such as air, nitrogen, oxygen, etc., equations are fit to tabulated data that is published in a document entitled "Real Gas Effects in Critical-Flow-Through Nozzles and Tabulated Thermodynamic Properties", NASA Technical Note D-2565, January 1965. The tabulated data from this publication has an estimated uncertainty of 0.05%. The equation fits to the tabulated data are within ±0.015%. Therefore, for pure gases, the maximum uncertainty in C* will be ±0.065%. When specialty gases or gas mixtures are used, the uncertainty in C* should be evaluated as a special case. In situations where moist air is used instead of dry air, there will also be a slight increase in the uncertainty. 

Attached at the end of this note is a discussion on the effects of moist air and its uncertainty. For this uncertainty calculation example, we will assume a pure gas. Therefore, eC* = 0.065%.

For pure gases, the gas constant (R) is a true constant that does not change with pressure or temperature. Therefore, eR = ±0.00%. However, if the sonic nozzle were to be used with gases that have a changing composition or an air source that contains moisture, the gas constant will change. Following this analysis is a discussion on how moisture affects the mixture gas constant. 

To summarize, the uncertainty in the mass flow rate is the root-sum-square of the component uncertainties. The component uncertainties are:

The uncertainty in mass flow rate becomes:

As can be seen, the uncertainty in the mass flow rate is dominated by the uncertainty in the discharge coefficient (eCd) for calibrated sonic nozzles. For this reason, many flow calibration laboratories can offer different levels of accuracy (also, at different costs). It is possible to obtain calibrations at the following accuracy levels: ±0.50%, ±0.25%, and ±0.10%. The following table shows the uncertainty in mass flow at these accuracy levels and with the same instruments as described above.

As can be seen from the above table, a calibrated sonic nozzle that has good quality instrumentation can provide a mass flow measurement with low uncertainty values.


Uncertainty Considerations for Uncalibrated Sonic Nozzles

When using an uncalibrated sonic nozzle, the error in the discharge coefficient (eCd) must be estimated. The document ASME/ANSI MFC-7M-1987, entitled "Measurement of Gas Flow by Means of Critical Flow Venturi Nozzles", states that a well designed and manufactured sonic nozzle has a discharge coefficient known to about ±0.50%. This estimate has limitations (such as a minimum throat Reynolds number) but the estimate will provide a number that can be used in this example. Therefore, we will assume that eCd = ±0.500%.

The error in throat area (eA) must also be estimated. The following table shows the tolerance values that FLOW SYSTEMS utilizes when manufacturing a sonic nozzle. The important aspect of this table is that as the throat diameter decreases the uncertainty in the throat area increases drastically.

Throat Area Uncertainty
Applicable to Uncalibrated Sonic Nozzles Only

As can be seen, the throat area uncertainty can become very large for small diameter sonic nozzles. This is the major reason to calibrate a sonic nozzle that has a small throat diameter. When a calibration is performed on a sonic nozzle, the calibration laboratory states the throat diameter that they are using and determines the discharge coefficient based on that diameter. Therefore, when the customer uses the sonic nozzle, the error associated with the throat area will become zero. 

The uncertainty in the discharge coefficient a calibrated sonic nozzle is then the only uncertainty that is applied to the mass flow uncertainty calculation. 

For convenience, the mass flow equation and the error equation are repeated below along with the sources of errors that have been previously discussed.

The following table uses the uncertainty estimates shown above for instrumentation and calculates the uncertainty in the mass flow rate for a set of uncalibrated sonic nozzles manufactured to the standard set of tolerances. 

Throat Area Uncertainty
Applicable to Uncalibrated Sonic Nozzles Only

Humid or Moist Air Consideration

If a sonic nozzle is to be used with an air supply that has a dew point of -40°F, then the air can be considered dry and all the information given above for a pure gas applies. However, if an air source were to be used that has a larger moisture content, the moisture will add a small amount to the uncertainty estimate. This discussion on the moisture effects is included so that the magnitude of the effect can be determined.

The amount of moisture that is present in the flow stream will affect the value of C*. The method that is being used to correct the dry air values to the moisture level is described in a paper entitled "The Influence of Humidity on the Flowrate of Air through Critical Flow Nozzles". This paper was written by Mr. A. Aschenbrenner of PTB in Germany. The net effect is that C* will decrease slightly as the moisture content increases. The equation for this decrease is as follows:

The magnitude of the total correction (C*wet/C*dry) is very small. At a room temperature of 70°F and a relative humidity of 100%, the mass fraction (Xv) is approximately 0.016. Using this value, the correction (C*wet/C*dry) is only 0.999. If the correction was totally ignored, the error in C*wet would amount to only a 0.1%. At elevated temperatures, air can hold a greater amount of water vapor. In these situations, a dew point sensor could be utilized. A typical accuracy statement of a dew point sensor is that it is capable of determining the dew point of the air to within 0.7°C. Using this value as the possible error in the dew point, the true mass fraction (Xv) is determined within 5.0%. This 5% error in Xv results in an uncertainty of ±0.005% in the ratio of C*wet to C*dry. By linear addition of the three error components involved, the uncertainty in C* value in the mass flow equation is ±0.07%. The three component errors are 0.05% for the tabular values of dry air, 0.015% for the equation fit of the tabular data, and 0.005% for the wet air correction. Therefore eC* = ±0.070% if a moist air source is used. 


For dry air, the gas constant (R) is a true constant that does not change with pressure or temperature. If the sonic nozzle were to use an air source containing moisture, the gas constant will change due to the amount of moisture that is present. In this context, the term "constant" is a misnomer; however, common practice has kept the term "gas constant". The idea to keep in mind is that the "gas constant" of the air-water mixture does vary with moisture content. In much the same manner as described above for C*, the ratio of wet to dry conditions is related to the mass fraction of the mixture as:

The humidity instrument (dew point sensor) describe above determines the mass fraction (Xv) within 5.0%. Using this 5% variation in Xv, the uncertainty in the ratio of Rwet/Rdry is ±0.04%. Therefore, eR = ±0.04% if a moist air source is used.

Adding the additional uncertainties associated with moist or humid air will increase the uncertainty in the mass flow rate only slightly as shown below. The component uncertainties are:

The uncertainty in mass flow rate becomes:

As can be seen by comparing this value to the value of a pure gas (±0.527%), the effect of moisture can be very small if the moisture is measured accurately.

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