Wright State University, Department of Mechanical and Materials Engineering
ME 495: THERMAL-FLUID SCIENCE LABORATORY
Uncertainty Analysis
Objective: The objective of this analysis is to learn how to estimate the uncertainty of a computed value from measured values.
Reference: Fox, McDonald and Pritchard, Introduction to Fluid Mechanics, 6 th
edn., Appendix F
Preparation: Read the reference material and these instructions. Download the Large Data File, the PAO (Brayco Micronic) property file, the screenshot.jpg and the fleming.pdf file from my webpage: http://www.cs.wright.edu/people/faculty/sthomas/me495.html
Report:
1.
Provide detailed hand calculations showing the computation of one value of heat transfer rate, heat transfer coefficient and thermal resistance, and the associated uncertainties, at steady state using 300 seconds of data. Include a standard unit analysis. Use Excel to obtain averages and standard deviation values for the 300 second interval. Confidence intervals in the data must be computed at a confidence level of 95%.
2.
Create the following figures. Make sure to completely label all axes appropriately, including units. Each figure must have a numbered figure caption, which will be referred to in the discussion. Use a screenshot of each figure in the report instead of inserting it directly from Excel. If this isn’t done, the file size will be so large that attempts to print it out will cause the printers to lock up. a.
Evaporator temperature (TA4) versus time. Use only lines with no data points. b.
The instantaneous time rate of change of the evaporator temperature (TA4) versus time. Use only lines with no data points. c.
Time rate of change of the evaporator temperature (TA4) versus time when averaged over 300 second intervals. Use only lines with no data points.
3.
Answer the following questions by referring to the above figures: a.
When did steady state occur? Provide quantitative proof of your answer by referring to the three figures. b.
Are the magnitudes of the uncertainties of the heat transfer rate, heat transfer coefficient and thermal resistance reasonable? Refer to the Fleming paper on my homepage. c.
What measurement error had the most significant impact on the uncertainties of the calculated heat transfer rate values? Provide quantitative proof of your answer by showing the magnitude of each term under the radical in the root-sum-square equation.
Part 1: Determination of Steady State
In the loop heat pipe experiment, the data was taken where the rate of sampling was approximately 0.5 Hz (one set of data points every two seconds). During experimentation, the heat input at the evaporator section of the loop heat pipe was suddenly increased. At this point, the temperatures across the LHP started to increase. The objective of the experiment was to obtain steady state evaporative heat transfer coefficient and overall thermal resistance data, but

determining when steady state occurred was problematic. A data trace was displayed on the data acquisition computer that showed approximately five minutes of temperature data. Once steady state was approached, it was noted that the temperatures appeared to be steady over this fiveminute window, but was this a sufficient definition of steady state? An experiment was run in which a step heat input to the evaporator was initiated, and the test ran for approximately eight hours in order to determine when steady state occurred. Even with this complete data set, it was not clear when steady state was reached. An alternative method was devised: The time rates of change of the temperatures were tracked. This method proved to be much more sensitive and reliable in terms of tracking when steady state was reached.
Part 2: Uncertainty Analysis
In any experiment, it is of the utmost importance for the experimentalist to attempt to quantify the uncertainties associated with measured values and computed values obtained by using an equation. Errors in measurements arise from several sources, which will be outlined here. In addition, the uncertainty involved in a calculated value based on measurements will be addressed.
Sources of Error
In any measurement, the value obtained by using a device such as a thermocouple is inherently incorrect. It is the experimentalist’s job to estimate the magnitude of that error. This must start with the calibration procedure, where the thermocouple reading is compared to the reading from a NIST-traceable instrument ( http://www.nist.gov/ ) , such as a platinum resistance temperature detector (RTD). To calibrate a thermocouple, the thermocouple and RTD are immersed in a recirculating chiller bath, which holds the temperature of the liquid in the bath at a very nearly constant value. Once steady state has been reached, the temperatures read by the thermocouple and the RTD are recorded. In general, an average and standard deviation of each will be obtained by recording several hundred readings from the thermocouple and the RTD. In this way, a confidence interval of the RTD can be obtained at a specified confidence level. This is important because this will provide a means to account for the variation of the bath temperature, which is called the deadband. The set point temperature of the bath is then increased, steady state is reached, and the temperatures are again recorded. The average thermocouple readings can be compared to the average RTD readings over the range of temperature expected during experimentation, and a best-fit linear curve can be obtained using a statistical package. In order to determine the calibration uncertainty of the thermocouple as a function of temperature, the following sources of error must be summed for each steady state temperature:
1.
Typically, the RTD is provided with a NIST-traceability statement from the manufacturer listing the error associated with the RTD, which may be a function of temperature.
2.
The confidence interval of the RTD readings, which are an indicator of the variation of the bath temperature.
3.
The error associated with the best-fit curve, i.e., the disagreement between the actual data and the prediction provided by the curve.
Once the thermocouple is calibrated, the uncertainty of a reading obtained during use is determined by adding the calibration uncertainty to the confidence interval of the thermocouple reading.
Confidence Interval

If a population of data can be described by a normal distribution, the mean of the population
µ can be determined by finding the mean of a random sample x of n measurements. However, x is only an estimate of
µ
, so the error in x must be provided:
  x
 
. The confidence interval is the range of the estimate for µ : x
     x
 
The error in the estimate of the mean is determined according to the confidence level desired, where typical values are (1 – α) = 0.90, 0.95, and 0.99. The error is given by
  t (

/ 2 , n

1 )
  n where σ is the standard deviation of the population and t (

/ 2 , n

1 ) is the two-tailed Student's tdistribution, which can be calculated using the TINV(
α,n
-1) function in Excel. The confidence interval can be calculated directly using the CONFIDENCE(
α
, σ , n ) function.
Root-Sum-Square Uncertainty
Calculations are often made using measured values. The uncertainty of each measured value impacts the uncertainty of the calculated value. A methodology has been developed in which the uncertainty in calculated values can be estimated, which is called the root-sum-square method.
For instance, the power output of a liquid-cooled heat exchanger is calculated using the first law of thermodynamics, which is used for calorimetry:
  C p

T out

T in

The uncertainty of the heat output due to the measured mass flow rate, inlet temperature and outlet temperature is given in terms of the uncertainties of each measurement. If we assume that the specific heat is a constant with no uncertainty,
  


 
 
  


2



 

T out
 
T out


2



 

T in
 
T in


2
Evaluating the partial derivatives in the above equation gives the root-sum-square uncertainty in the power output:
  
 
C p

T out

T in

     C p

T out
2
 
  C p

T in

2

1 / 2
When using this type of equation, it is important to note that the uncertainties in the measured values include the calibration uncertainty of the device and the confidence interval of the reading. The root-sum-square method can also be used in the experimental design process. When multiple measurements are to be made, where should I spend the most money (in terms of accuracy) to obtain the smallest uncertainty in the calculated values?

The uncertainty in a measured value can be instrumental in determining trends (or the lack of a trend) in data. For instance, the following is an excerpt from an e-mail that I received from a former student:
The question surrounded the volumetric flow rate through the different tubes for both air and water. Merely looking at the average values, one would think that the flow rate of air increases in the gray tubing compared to the clear tubing, but this is not the case when the uncertainties are factored in. The uncertainty literally tells us that the actual value of the flow rate for air in clear tubing lies somewhere between 689 and 869, and the flow rate ranges from 760 to 940 for the gray tubing. Since the ranges overlap, we cannot make any conclusions with regard to the trend in the data. For water, however, it can be seen that the flow rate decreased slightly from the clear tubing to the gray tubing, which was probably due to the increase in the head loss in the smaller diameter tube, as expected.
Equations for Evaporative Heat Transfer Coefficient and Thermal Resistance
Newton’s Law of Cooling:  hA e

T e

T sat

, W
Heat Transfer Rate from the Calorimeter:
  C p

T out

T in

, W
Overall Thermal Resistance: R th


T e

T c

/ , K/W

Evaporator Area: A e
 
DL , cm
2
Evaporator Inner Diameter: D = 2.500 ± 0.025 cm
Evaporator Length: L = 25.00 ± 0.025 cm
Average Evaporator Temperature: T e

( TA 4

TA 5

TA 6

TA 7 ) / 4
, ºC
Saturation Temperature: T sat

TA 8
, ºC
Calorimeter Mass Flow Rate: m , kg/sec
Specific Heat of PAO: Evaluate C p
at the average of the outlet and inlet temperatures of the calorimeter. See Brayco Micronic 889 Spec Sheet for the temperature dependence and units.
Calorimeter Outlet Temperature: T out

TA 1
, ºC
Calorimeter Inlet Temperature: T in

TA 0
, ºC
Average Condenser Temperature: T c

( TA 9

TA 10

TA 11

TA 12 ) / 4
, ºC
For all measured temperatures, the uncertainty was ±0.14ºC. The uncertainty of the mass flow rate measurement was ±3.0%. Assume that the uncertainty of the specific heat was ±1.0%.