Data Reduction and Uncertainty Analysis

Wright State University, Department of Mechanical and Materials Engineering
Data Reduction and Uncertainty Analysis
Objective: The objective of this analysis is to gain an understanding of the difficulties in dealing
with large data files obtained during experimentation, and to learn how to estimate the
uncertainty of a computed value from measured values.
Reference: Fox, McDonald and Pritchard, Introduction to Fluid Mechanics, 6th 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
Report: Prepare a written report to include the following:
1. Cover sheet
2. Objectives of the analysis
3. List of equations used in determining the uncertainty of each computed value.
Confidence intervals in the data must be computed at a confidence level of 95%.
4. Hand calculations showing the computation of one value of heat transfer rate, heat
transfer coefficient and thermal resistance, and the associated uncertainties. Include a
standard unit analysis.
5. Create the following figures. Make sure to completely label all axes appropriately,
including units.
a. Evaporator temperature (TA4) versus time.
b. Evaporator temperature (TA4) versus time when averaged over 300 second
c. The instantaneous time rate of change of the evaporator temperature (TA4) versus
d. Time rate of change of the evaporator temperature (TA4) versus time when
averaged over 300 second intervals.
e. Heat transfer rate extracted by the calorimeter versus time when averaged over
300 second intervals including error bars.
f. Evaporative heat transfer coefficient versus time when averaged over 300 second
intervals including error bars.
g. Thermal resistance versus time when averaged over 300 second intervals
including error bars.
6. Answer the following questions by referring to the above figures:
a. When did steady state occur?
b. How did you determine when steady state is reached?
c. Is the magnitude of the uncertainty of the heat transfer coefficient reasonable?
Refer to the Fleming paper on my homepage.
d. What measurement error had the most significant impact on the uncertainties of
the calculated heat transfer rate values?
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
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 (, 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
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
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)  
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 Tout  Tin 
Q  m
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,
Q  
 m   
 Tout   
 Tin 
  out
  in
Evaluating the partial derivatives in the above equation gives the root-sum-square uncertainty in
the power output:
 
 
 2 m
 C p Tout 2   m
 C p Tin
Q  C p Tout  Tin   m
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: Q  hAe Te  Tsat  , W
 C p Tout  Tin  , W
Heat Transfer Rate from the Calorimeter: Q  m
Overall Thermal Resistance: Rth  Te  Tc  / Q , K/W
Evaporator Area: Ae  DL , cm2
Evaporator Inner Diameter: D = 2.500 ± 0.05 cm
Evaporator Length: L = 25.00 ± 0.05 cm
Average Evaporator Temperature: Te  (TA4  TA5  TA6  TA7) / 4 , ºC
Saturation Temperature: Tsat  TA8 , ºC
 , kg/sec
Calorimeter Mass Flow Rate: m
Specific Heat of PAO: Evaluate Cp 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: Tout  TA1, ºC
Calorimeter Inlet Temperature: Tin  TA0 , ºC
Average Condenser Temperature: Tc  (TA9  TA10  TA11  TA12) / 4 , ºC
For all measured temperatures, the uncertainty was ±0.09ºC. The uncertainty of the mass flow
rate measurement was ±4.0%. Assume that the uncertainty of the specific heat was ±5.0%.
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