UNCERTAINTY ANALYSIS: 
Mini Flow or Portable Sonic Nozzle-Based Turbine Component Airflow Test System 

ERROR ANALYSIS 

The proposed test stand will have a set of critical flow venturis (sonic nozzles) installed inside a common plenum chamber that is located upstream of the turbine component to be tested. A dried and conditioned compressed air system in conjunction with a pressure regulator bank maintains an adequate pressure ratio across the sonic nozzles to insure that choked or sonic flow exists during operation. This error analysis is provided to demonstrate the flow measurement accuracy that is achieved with tests stands of this style.

The mass flow rate through a sonic nozzle is 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 you use the same area as the laboratory used during flow calibration. Removing these values, the error equation becomes:

The CEESI laboratory will calibrate all of the sonic nozzles located in the test stand. Over the operating range of each sonic nozzle, the laboratory will develop an equation that relates the discharge coefficient (Cd) to the throat Reynolds number. The calibration report will state that the estimated uncertainty in the equation is ±0.50% over the operating mass flow range. Therefore, eCd = ±0.50%.

The inlet absolute pressure at the sonic nozzle inlet will be measured with a combination of Druck model PTX-610 pressure transducers. The atmospheric pressure (barometer) will be measured with a device that has a full-scale value of 15 psia. A second Druck transducer will measure the gage pressure (psig) at the nozzle inlet. This gage pressure transducer has a full scale of 100 psig. The nozzle inlet absolute pressure is the sum of these two transducers. The Druck specifications state that both transducers have an accuracy of ±0.04% of full scale. If we assume that the barometric pressure is approximately 14.7 psia, then the absolute transducer will determine that pressure within ±0.008 psia (0.04% of 15). The gage pressure transducer will determine the line pressure within ±0.04 psig (0.04% of 100). A linear addition of these two error sources results in a measurement error of ±0.048 psia. Now, if we assume the minimum inlet pressure to the sonic nozzles will be 25 psia (10.3 psig) during normal operation of the test stand, then the ±0.046 psia error can be stated as ±0.192% (100*0.048/25). The CB COM Analog Input Module has a minimum accuracy of +/-0.25% if we use a conservative linear addition of the error sources eP = ±0.442%.

The error in temperature (eT) has a sensitivity coefficient of 0.5 applied to it. This is because that in the mass flow equation, the temperature enters as the square root. In the flow bench, the nozzle inlet temperature and part inlet temperature are measured with a T-type thermocouple. The NIST-90 standard for linearization of a T-type thermocouple states a +/-0.9°R accuracy. The flow bench will operate at a temperature of approximately 530°R and at this temperature the error may be expressed as +/-0.170% of reading accuracy. The Omega Thermocouple Transmitter and CB COM Data Acquisition Module will add an additional +0.170% uncertainty. Therefore, by a conservative linear addition of the error sources eT = ±0.340%.

The error in the critical flow function (eC*) depends upon two different parameters. An equation is fit to tabulated data for dry air 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 the tabulated data to within ±0.015%. Therefore, the maximum uncertainty in the dry air data will be ±0.065%. When moist air is used, there will be a slight increase in the uncertainty. Following this dry air analysis is a discussion on the effects of moisture to show the small magnitudes involved. However, for this uncertainty estimate, dry air is assumed. Therefore, eC* = 0.065%.

For dry air, the gas constant (R) is a true constant that does not change with pressure or temperature. Therefore, eR = ±0.00%. However, if the flow stand were to use an air source containing moisture, the gas constant will change due to the amount of moisture that is present. Following this dry air 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 for dry air becomes:

The above value is the uncertainty in the mass flow as determined by the sonic nozzles. Whenever this mass flow rate is used to determine the uncertainty in the "Effective Area" of a test part or component, an additional calculation must be made. The effective area (Ae) of a part is calculated by the following equation:

In the above equation, the proportionality symbol () is used because there are some Mach number terms that are present. These Mach number terms do not add anything of significance to the uncertainty calculation. The uncertainty in Ae is then calculated by:

The error in mass flow has already been determined from the sonic nozzles.  Therefore, eM = ±0.71%. 

The part temperature will be measured with an instrument identical to the one located at the nozzle, and they will have the same level of accuracy.  Therefore, the uncertainty in eTp = ±0.34%. 

The inlet absolute pressure at the test part will be measured with a combination of Druck pressure transducers.  The atmospheric pressure (barometer) will be measured with the same Druck instrument that was described in the nozzle mass flow information shown above.  The turndown of the gage pressure transducer is limited to a 4:1 range.  This is a model PTX-610 that has a full-scale value of 20 psia.  A second Druck transducer will measure the gage pressure (psig) at the part inlet.  The full-scale value of this gage pressure transducer will be selected so that the maximum error in the absolute part pressure will not exceed ±0.200%.  As an example, assume that a 1.04 pressure ratio is to be held across the test part.  If the barometric pressure is 14.7 psia, then the inlet pressure to the part is to be 15.288 psia (1.04 x 14.7) or 0.588 psig.  It will be necessary to measure the absolute inlet pressure to the part within ±0.031 psi to stay within the accuracy band of ±0.200%.  At the barometric pressure of 14.7 psia, the absolute transducer will determine that pressure within ±0.008 psia (0.04% of 20).  Therefore, the selected gage pressure transducer must not add more than ±0.029 psi error to stay within the selected tolerance limit. Therefore we can state that for the worst-case condition,   eP = ±0.200%.  This means that the maximum error in part pressure, including the data acquisition error, is  +/-0.44%.  Therefore, ePp = +/-0.44%. 

To summarize, the three components that contribute to the error in effective area are:

The above analysis shows that with good instrumentation and dry air, the uncertainty in the effective area calculation can be less than ±0.87%.

Humid or Moist Air Consideration

The air supply that is to be used for this test stand has a dew point of -40°F. At this level of moisture, the air can be considered dry and all the information given above applies. However, if a future air source were to be used that has a higher 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 Flow rate of Air through Critical Flow Nozzles". This paper was written by a 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%. However, the flow bench could use a HyCal Model CT-841 Dew Point sensor to measure the amount of moisture in the flow stream. This instrument is capable of determining the dew point of the room air to within 0.7°C with this 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. However, if the flow stand 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, the ratio of wet to dry conditions is related to the mass fraction of the mixture as:

The humidity instrument (HyCal Model CT-841) described 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.

home