Olivier Collet, Gery Fossaert and validation on spark ignition engines Calibration of TFG sensor for heat flux measurements Academic year 2013-2014 Faculty of Engineering and Architecture Chairman: Prof. dr. ir. Jan Vierendeels Department of Flow, Heat and Combustion Mechanics Master of Science in Electromechanical Engineering Master's dissertation submitted in order to obtain the academic degree of Counsellor: Thomas De Cuyper Supervisors: Prof. dr. ir. Sebastian Verhelst, Prof. dr. ir. Michel De Paepe
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Olivier Collet, Gery Fossaert
and validation on spark ignition enginesCalibration of TFG sensor for heat flux measurements
Academic year 2013-2014Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Jan VierendeelsDepartment of Flow, Heat and Combustion Mechanics
Master of Science in Electromechanical EngineeringMaster's dissertation submitted in order to obtain the academic degree of
Counsellor: Thomas De CuyperSupervisors: Prof. dr. ir. Sebastian Verhelst, Prof. dr. ir. Michel De Paepe
Olivier Collet, Gery Fossaert
and validation on spark ignition enginesCalibration of TFG sensor for heat flux measurements
Academic year 2013-2014Faculty of Engineering and ArchitectureChairman: Prof. dr. ir. Jan VierendeelsDepartment of Flow, Heat and Combustion Mechanics
Master of Science in Electromechanical EngineeringMaster's dissertation submitted in order to obtain the academic degree of
Counsellor: Thomas De CuyperSupervisors: Prof. dr. ir. Sebastian Verhelst, Prof. dr. ir. Michel De Paepe
The authors and promoters give the permission to use this thesis for consultation and to
copy parts of it for personal use. Every other use is subject to the copyright laws, more
specifically the source must be extensively specified when using from this thesis.
Ghent, 2 juni 2012
The authors
Olivier Collet Gery Fossaert
Acknowledgement
This thesis is the final result of one year of hard work and full of challenges. The result of
this thesis would never have been the same without the help of certain people. We would
like to take a moment to thank them.
First and foremost, we would like to thank our supervisors, Prof. dr. ir. S. Verhelst and
Prof. dr. ir. M. De Paepe for their help and their much appreciated advice. We would
especially like to thank them for given us the opportunity to participate in this interesting
research.
Special thanks go to our counselors, ir. S. Broekaert and ir. T. De Cuyper. They were
always available to answer any question we may have and their support during the year
was much appreciated. We do hope that our thesis will help them in their future research
and we wish them the best of luck.
We would also like to thank Prof. K. Chana of Oxford University for his help and useful
insight. His visits were always very inspiring and instructive.
Next, we would like to thank Mr. K. Chielens for his help concerning the CFR setup and
his all round good mood in the laboratory. At the same time, we thank Mr. P. De Pue
for sharing his technical advice to help us with the electronic aspect of our work.
We wish to thank our fellow students for the good times around the setups and in the
class room and specially during these past years.
Last but certainly not least, we wish to thank our parents, for their love, support and for
giving us the opportunity to get an education and prepare us for the future. We thank
our sisters, brothers,family and friends for all the good times we had together.
Finally, we want to thank each other for the wonderful year we had together. It was a
great experience that we will cherish for life.
Olivier Collet and Gery Fossaert
ii
Calibration of TFG sensor for heat fluxmeasurements and validation on spark
ignitions engines
By
Olivier Collet and Gery Fossaert
Supervisors: Prof. dr. ir. Sebastian Verhelst, Prof. dr. ir. Michel De Paepe
Counsellor: ir. Thomas De Cuyper
Master’s dissertation submitted in order to obtain the academic degree of
Master of Science in Electromechanical Engineering
Departement of Flow, Heat and Combustion Mechanics
Chairman: Prof. dr. ir. Jan Vierendeels
Faculty of Engineering and Architecture
Ghent University
Academic year 2013-2014
Summary
Due to the current issues of global warming and decreasing fossil energy resources, inter-
nal combustion engines still are a hot topic for research and development. Fuels, such as
methanol and ethanol, are being researched because they could offer an alternative to the
fossil fuels that are still primarily used today. Multiple techniques have also been intro-
duced over the years, such as charging, exhaust gas recirculation and others, to improve
engine efficiency, fuel consumption and limit the emissions of noxious gasses. However,
further research is still needed to optimize the use of internal combustion engines. This
optimization requires the use of engine simulations. Within the research group Trans-
port Technology of the Department of Flow, Heat and Combustion Mechanics at Ghent
University, a simulation tool is being developed to research the effects of alternative fu-
els and engine enhancements on engine performances. This requires a good knowledge
of multiple processes taking place in the engine, one of which is the heat transfer to the
cylinder walls. Intensive measuring is done to comprehend this process. The researchers
at Ghent University wish to use a Thin Film Gauge sensor to perform heat flux measure-
ments as it offers different advantages compared to previously used sensors. The use of
iii
iv
this sensor requires an adequate calibration. This thesis offers an insight on the function
of the sensor and an overview of the different existing calibration techniques and setups.
Next, the Double Electric Discharge calibration technique and its setup are discussed in
depth. Lastly, heat flux measurements obtained with the calibrated TFG are compared to
results obtained with other sensors to validate the calibration process. Some suggestions
are made to further ameliorate the calibration setup.
Keywords
heat flux measurements, Thin Film Gauge sensor,Double Electric Discharge , spark-ignition
engine,
iv
Calibration of the TFG sensor for heatmeasurements and validation on a SI engine
Olivier Collet and Gery Fossaert
Supervisor(s): Sebastian Verhelst, Michel De Paepe, Stijn Broekaert and Thomas De Cuyper
Abstract—In the development of internal combustion engines, measure-ments of the heat transfer to the cylinder walls play an important role.These measurements are necessary to provide data for building a modelof the heat transfer, which can be used to further develop simulation toolsfor engine optimization. These measurements require an adequate sensor.This research will focus on the Thin Film Gauge (TFG) sensor. To use theTFG sensor, its thermal properties -namely the thermal coefficient and thethermal product- must be correctly calibrated. The Double Electric Dis-charge calibration set-up for the thermal product will be extensively dis-cussed. This paper ends with a comparison between heat transfer measure-ments in a CFR engine done with a non-calibrated TFG sensor, a calibratedTFG sensor and a HFM (Heat Flux Measurement) sensor.
Keywords—SI-engine, thin film gauge, heat flux, calibration, double elec-tric discharge
I. INTRODUCTION
ONE of the key factors in the research of internal combus-tion engines (ICE) is to fully understand the mechanisms
involving the heat transfer in the engine. The heat transfer fromthe combustion gases to the inner cylinder walls has large ef-fects in terms of efficiency, emissions and power output of anICE. Previous research [1] has shown that due to the differentflow conditions during the combustion the heat flux shows a lotof spatial variation. In order to enable a cheap and fast opti-mization of the engine parameters, a simulation model of thecombustion thermodynamics can be used. The development ofsuch a model demands accurate measurements inside the engine.Extensive research [2] has been performed on different kind ofsensors. This research showed that the Thin Film Gauge had themost potential for use in an ICE. They are sturdy and cheaperto manufacture. They have already been used with success inturbo machinery [3]. However, peak temperatures and pressuresare higher in ICE application, this must be taken into accountfor implementing the sensor in the combustion chamber. Also,differences in heat fluxes were observed between the TFG anda very accurate sensor. Therefore, further investigation on TFGsensors is necessary.
II. THIN FILM GAUGE
THE basic thin film gauge - a single layer TFG - consistsof two parts: a thin film of metal which is placed on a
substrate. The film is a resistance temperature detector (RTD).Multiple RTDs are mounted on top of the substrate. As for mostRTDs, the metal used is platinum. This is because platinumhas the most stable resistance-temperature relationship over thelargest temperature range, making it ideal for reliable measure-ments. The substrate is mostly a ceramic that has a low elec-trical and thermal conductivity. A low electrical conductivity isneeded to ensure there will be no short circuiting between thedifferent RTDs. The variation of material properties due to tem-
Before the TFG can actually be used, there are two materialproperties that need to be calibrated. They are the thermal coef-ficient of the RTD and the thermal product of the substrate. Thecalibration of the thermal coefficient αR can be done by usingthe water bath calibration method. By measuring the resistanceof the RTD at different temperatures and by using the linear re-lationship between the temperature and the resistance, αR canbe calculated [1]. The thermal product is not calibrated as easy.That is why previously the thermal product of the bulk materialof the substrate was used in heat flux calculations. However, re-search has showed that the process of placing the thin film uponthe substrate changes the material properties of the substrate [4].It is also worth mentioning that no research has been done so faron the effect of sensor aging and wear on the thermal product.This shows that determining the right thermal product is im-portant. Over the years, two different methods have been usedto calibrate the thermal product: the water droplet method [5]and the hot air gun method [6]. Both methods are based on theone dimensional conduction equation which is solved by usinga step in heat flux or a step in fluid temperature [7]. Billiard [7]has shown that for short flow durations a step in fluid tempera-ture can be considered as a step in heat flux. However, by doingso, an error will be introduced. The water droplet and the hotair gun method are both setups that utilize a step in fluid tem-perature to solve the one dimensional conduction equation. Thehot air gun setup has already been used to calculate the TP whenstep in heat flux is applied. This introduces an error that shouldbe taken into account [6]. When using the water droplet setupto calculate the TP, two dimensional effects have been observedthat introduces errors [5]. Therefore, both methods have beenomitted for determining the TP. A third method exists, using astep in heat flux that is electrically generated.
III. DOUBLE ELECTRIC DISCHARGE
THERE is a third calibration method that can be used to de-termine the thermal product of the substrate: the double
electric discharge method (DED). The difference between this
v
method and the two previous ones, is that the solution of theone dimensional analysis is attained when a step in heat flux isapplied. Therefore, the transient heat flux can be written as afunction of the surface temperature as long as the semi-infiniteprinciple is valid. The heat flux is electrically simulated by thedischarge of a current through the thin film. This dischargecauses ohmic heating of the thin film, therefore, increasing thethin film temperature and its resistance. When a step in heat fluxis considered, the temperature will be proportional with the thesquare root of time as can be seen in figure 1. By controllingthe heat flux and monitoring the surface temperature, the TP canbe achieved. The thin film is placed in a Wheatstone bridge.Once the bridge has been balanced, a voltage pulse is sent to thebridge which causes ohmic heat of the thin film. This voltagepulse functions as the step function. Since the heat flux is gen-erated electrically, only the electrical power or the heat acrossthe thin film will be known. The surface area of the thin filmis necessary to determine the heat flux which is very difficult toobtain accurately. Therefore, the calibration is performed twicein different media to eliminate the knowledge of the thin filmsurface area. The thermal product can then be written as func-tion of the thermal product of the chosen fluid which glycerinand the slopes of the regression of the recorded out of balancevoltage according to equation (1):
√ρck =
√ρckglyc
(∆V/√t)air
(∆V/√t)glyc
− 1(1)
Fig. 1. Out of balance voltage and corresponding regression
Figure 1 represents the recorded out of balance voltages of thecalibrations in air and glycerin together with their regressions.The correlation coefficient of these regressions are higher than99 %. Therefore, the slopes of the regressed data perform a goodrepresentation of the actual ones.
seen in figure 2. In this specific case a voltage pulse of 8 and9 V has been applied to the bridge and pulse time duration of5 and 10 ms has been considered. The voltage pulse level isproportional with the magnitude of the heat flux while the pulsetime duration is related to the time that the heat flux is applied,thus influencing the thin film final temperature.
Fig. 2. TP vs Time variation
Voltage variation results does not represent a specific trendsince the TPs at 8 and 9 V differ from each other for differenttime durations. Time duration variation results in a slight in-crease of TP. However, measurements taken at 9 V do not differa lot from each other. Higher pulse levels resulted in lower in-accuracy. The lowest inaccuracy of 4.5 % has been obtainedwhere from the largest part is due to the inaccuracy of the TP ofthe fluid (4 %).
V. CONCLUSIONS
THE measurements discussed in this paper have led to anumber of conclusions, which will now be summarized.
• A step in heat flux can be perfectly generated with the DEDcalibration.• Higher bridge voltages resulted in the best regression withlowest relative error of 4.5%.• The variation of the amplitude of the voltage pulse does notaffect the thermal product of the substrate much. The mean val-ues and error levels are approximately the same for differentvoltage levels.• Variation of the bridge time duration has also not shown anysignificant changes in the thermal product.• In order to lower the inaccuracies, the accuracy of the thermalproperties of the fluid should be investigated.
REFERENCES
[1] T. De Cuyper and S. Broekaert, “Alcoholen als alternatieve brandstof voorvonkontstekingsmotoren: Experimentele studie naar het klopgedrag en dewarmteafgifte naar de cilinderwanden,” M.S. thesis, Universiteit Gent,2011-2012.
[2] M. Desoete and R. Vyvey, “Evaluatie van warmte uxsensoren voorvonkontstekingsmotoren aan de hand van metingen op kalibratieproefs-tanden en een cfr-motor,” M.S. thesis, Universiteit Gent, 2010-2011.
[3] Schultz. D.L and Jones T.V., “Heat-transfer measurements in short-durationhypersonic facilities,” AGARDograph, 1973.
[4] Lu K. Kinnear K., “design, calibration and testing of transient thin film heattransfer gauges,” Journal of Turbomachinery, 2008.
[5] R. Buttsworth, “Assessment of effective thermal product of suface junc-tion thermocouples on millisecond and microsecon time scales,” Elsevierexperimental thermal and fluid science, 2001.
[6] E. Piccini, S.M. Guo, and Jones T.V., “The development of a nex direct-heat-flux gauge for heat-transfer facilities,” Measurement Science and Tech-nology, 2000.
[7] N. Billiard, F. Illiopoulou, and R. Ferrera, “Data reduction and ther-mal product determination for single and multi-layered substrates thin-filmgauges,” Turbomachinery and Propulsion Department, 2002.
AFR Air to Fuel RatioGUEST Ghent University Engine Simulation ToolIC Internal CombustionICE Internal Combustion EngineATDC After Top Dead CenterBTDC Before Top Dead CenterCFR Cooperative Fuel ResearchCR Compression RatioDAQ Data AcquisitionDED Double Electric DischargeECU Engine Control UnitEGR Exhaust Gas RecirculationFIR Finite Impulse ResponseHFM Heat Flux Micro sensorRTS Resistance temperature sensingHRR Heat Release RateIT Ignition TimingLTI Linear Time InvariantMAP Manifold Absolute PressureNSR Noise to Signal RationPID Proportional Integrating DifferentialPVD Physical Vapor DepositionRPM revolutions per minuteRTD Resistance Temperature DetectorSI Spark IgnitionTFG Thin Film GaugeTP Thermal Product
Designs (des) a filter to convert surface temperature T to heat transfer rate q (T2q) for a two-layer substrate (2l) and gives impulse response (imp) h. Use q = fftfilt(h,T) to convert measured T to q.
[h,shift] = desq2T2limp1(fs,np,rrck1,rrck2,ak1,test) Designs (des) a filter to convert heat transfer rate q to surface temperature T (q2T) for a two-layer substrate (2l) and gives impulse response (imp) h. Use T = fftfilt(h,q) to convert measured q to T.
The basis functions are those for a step in q1(t). In Laplace transformed form, the solution of the heat conduction equations for two layer substrate (Doorly and Oldfield,1987) gives
1
11
1111
2exp1
2exp111
saA
saA
sqskc
sT ,
where 222111
222111
kckc
kckcA and the thermal diffusivity
1
11 c
k .
For a step in q1(t) = u(t), s
sq1
1 , and so
1
123
1111
2exp1
2exp11
saA
saA
skc
sT .
Expanding the denominator as a power series, and taking the inverse Laplace transform,
Semi- infinite layer
T1
222 kc
q1 Thin-film gauge
111 kcInsulating layer
x = 0
x = a
Figure 2 Two layer heat transfer gauge
6
Figure B.2: Model of the TFG double layer [24]
The same manner of the TFG single layer is applied in this case. Equation (B.4) is now
considered for the two layers. The boundary conditions are:
−k1dθ1dx |x=0 = q
−k1dθ1dx |x=a = −k2
dθ2dx |x=a
−k2dθ2dx |x=∞ = 0
θ1(a, t) = θ2(a, t)
(B.15)
In the Laplace domain:
Lq =√k1ρ1c1
√s
[1−A exp(−2a
√sα1
)]
[1 +A exp(−2a
√sα1
)]Lθs (B.16)
With:
A=√ρ1c1k1−
√ρ2c2k2√
ρ1c1k1+√ρ2c2k2
α1= k1ρ1c1
: the thermal diffusivity of the first layer
As with the TFG single layer a step in heat flux is applied at the surface of the sensor.
Equation (B.16) becomes:
Lθs,step =1√
k1ρ1c1s−
32
[1 +A exp(−2a
√sα1
)]
[1−A exp(−2a
√sα1
)] (B.17)
78
Appendix B. Calculations impulse response FIR-method 79
After decomposition into a power series and taking the inverse Laplace transformation,
the obtained set of functions for the TFG double layer are:
pressure signal amplifiers and the PXI-6143 are negligible in comparison with the error of
the pressure sensor itself. The final errors on the pressure signals are listed in table C.4.
Table C.4: Absolute en relative errors for measured pressure signals
Variable X AEX REX [%] Unit
pinlet 0, 03 - bar
poutlet 0, 03 - bar
pcylinder - 1 bar
C.1.4 Temperatures
The inlet, the two outlet temperatures, the oil temperature and the cool water tempera-
ture are all measured with type K thermocouples and read with the PXI-6224-module of
National Instruments. The error on these temperatures are listed in table C.5.
83
Appendix C. Error analysis 84
Table C.5: Absolute errors on the acquired temperatures
Variable X AEX Unit
Ttype K 5 C
C.1.5 Flow rates
The gaseous fuel flow rates are measured with a Bronkhorst F-2010AC mass flow rate
sensor. The liquid fuel flow rate is determined gravimetric by measuring the consumed
mass of fuel over a certain time period. The air flow rate is measured with the Bronkhorst
F-106BZ flow rate sensor. In table C.6, the errors on the volumetric rates are given. The
Table C.6: Absolute errors for volumetric flow rate of gaseous fuels
Variable X AEX Unit
Qlair 0, 2 Nm3/h
Qmethane 0, 036 Nm3/h
Qhydrogen 0, 047 Nm3/h
mass flow rate of liquid fuels is calculated as
mliquid =∆m
∆t(C.1)
The absolute error on the mass flow rate is therefore,
AEmliquid =
√(AE∆m
∆m
)2
+
(AE∆t
∆t
)2 ∆m
∆t(C.2)
The errors on the measured time interal ∆t and the measured fuel mass ∆m are listed
in table C.7. This calculation leads to a relative error on the fuel mass of maximum 2%
when the mass fuel rate is monitored over an interval of 180 s.
Table C.7: Absolute errors for the calculation of liquid fuel mass rate
Variable X AEX REX [%] Unit
∆m 1 - g
∆t 1 - s
mmethanol - 2 kgs
84
Appendix C. Error analysis 85
C.2 Calculated quantities
To obtain the error on a calculated value, an error analysis must be performed. This
analysis is based on the merit of Taylor. A function f , dependent on variables a, b en c,
the absolute error can be obtained as:
AEf =
√(∂f
∂aAEa
)2
+
(∂f
∂bAEb
)2
+
(∂f
∂cAEc
)2
(C.3)
If no analytical expression is available of a function f , the derivatives in the above equation
may be approximated by an experimental sensitivity analysis. C.2.6 will be dedicated to
this analysis.
The relative error is obtained by taking the ratio of the absolute error to its actual value:
REf =AEff
(C.4)
In the next sections, a representative value of the relative error on methane based measure-
ments, will be given. Details of the operating condition are listed in table reftab:vgl-q-wp.
Wi [J ] Fuel ignition timing [CA BTDC] Throttle position [] λ CR
290 Methane 24 79 1, 3 9
Table C.8: operating condition
C.2.1 Mass in cylinder
The total trapped mass in the cylinder is obtained by taking the sum of the charge that
is sucked into the cylinder and the rest gases that are still present when the exhaust valve
closes.
mmixture = mair +mfuel +mrest (C.5)
Here,
mair =2mair
60 N(C.6)
mfuel =2mfuel
60 N(C.7)
mrest =pcylVcyl
RrestToutlet(C.8)
85
Appendix C. Error analysis 86
mrest is evaluated when the exhaust valve closes.The relative error of these separate com-
ponents are:
REmair =√RE2
N +RE2mair
(C.9)
REmfuel =√RE2
N +RE2mfuel
(C.10)
REmrest =√RE2
pcyl+RE2
Texhaust+RE2
Rrest(C.11)
The relative error of the total mass in the cylinder is
REmmixture =√RE2
mair +RE2mfuel
+RE2mrest (C.12)
These calculations lead to an relative 3, 13% for the mixture mass in the engine.
C.2.2 Air/fuel ratio and air factor
The air/fuel ratio is given by
afr =mair
mfuel(C.13)
The relative error can be calculated as:
REafr =√RE2
mair +RE2mfuel
(C.14)
The air factor λ is calculated as
λ =afr
afrstochiometric(C.15)
The error on the ratio can be calculated as
REλ =√RE2
afr +RE2afrstochiometric
(C.16)
Since afrstochiometric is fixed for a certain fuel, the relative error on λ will be the same as
the relative error on afr:
REλ = REafr (C.17)
This calculation leads to an relative error of 0, 5% on the air/fuel ratio.
C.2.3 Specific gas constant
At fired operation, the specific gas constant Rinlet of the sucked gas mixture is calculated
as:
Rinlet =afr
(afr + 1)Rair +
1
(afr + 1)Rfuel (C.18)
86
Appendix C. Error analysis 87
If the error on the specific heat constant of air an fuel is neglected, the error can be
determined accordingly
AERinlet =
√(Rair −Rfuel)2AEafr (C.19)
Due to remaining rest gases, the value of the specific gas constant of the mixture will differ
from the one of the fresh sucked mixture. The addition on the absolute error is negligible.
Therefore,
AERmixture = AERinlet (C.20)
This results in a relative error of 7, 6% for the specific gas constant of the mixture.
C.2.4 Gas temperature
The gas temperature of the mixture can be calculated by the equation of state:
Tgas =pcylVcyl
Rmixturemmixture(C.21)
The error on the cylinder volume is negligible in comparison with the other errors. The
relative error on the gas temperature is calculated as:
RETgas =√RE2
pcyl+RE2
Rmixture+RE2
mmixture (C.22)
This calculation leads to a relative error of 8, 3% on the gas temperature.
C.2.5 Error analysis calibration TFGs
The calibration has been performed in the linear temperature resistance region, therefore
the resistance can be written as a function of temperature:
R = a T + b (C.23)
The coefficients a and b are calculated according to a least squares method. The absolute
error can be calculated on the coefficients according to:
AEa = AER
√N
∆(C.24)
AEb = AER
√∑(Tj)2
∆(C.25)
N is the amount of data points and ∆ en AER are calculated as:
∆ = N∑
x2 −(∑
x)2
(C.26)
87
Appendix C. Error analysis 88
AER =
√1
N − 2
∑(Rj − b− a Tj)2 (C.27)
The value of α0 can be calculated as:
α0 =a
b+ a T0(C.28)
The absolute error of α0 is then given as:
AEα0 =
√b2(AEa)2 + a2(AEb)2 + a4(AET0)2
(b+ a T0)4(C.29)
C.2.6 surface temperature, flux and convection coefficients
Surface temperature The surface temperature of the TFG single layer is calculated as
Tw = TTFGS =VTFGS
GTFGS α0 V0+ Tatm (C.30)
The absolute error becomes:
AETw =√(
AEVTFGSVTFGS
)2
+
(AEGTFGSGTFGS
)2
+
(AEα0
α0
)2
+
(AEV0
V0
)2
+
(AETamb
GTFGSα0V0
VTFGS
)2
VTFGSGTFGS α0 V0
(C.31)
This calculation leads to relative error of 4, 6% on the surface temperature.
Transient part of heat flux - 1T FIR-method The transient part of the heat flux
is calculated with the 1T FIR-method. This translates itself in Matlab with the function
fftfilt-commando:
qtrans = fftfilt(h, Tw) (C.32)
The flux is dependent of the impulse response h of the 1T FIR-method and the surface
temperature Tw of the sensor. There is no literal function available that relates these
variables with the resulting flux. A sensitivity analysis will be used to determine the
absolute error, which can be written as:
AEqtrans =
√(∂qtrans∂Tw
AETw
)2
+
(∂qtrans∂h
AEh
)2
(C.33)
The impulse response h is only dependent on the thermal properties of the sensor. In case
of the TFG single layer, this is the thermal product of MACOR®. Then:
AEqtrans =
√(∂qtrans∂Tw
AETw
)2
+
(∂qtrans∂TP
AETP
)2
(C.34)
88
Appendix C. Error analysis 89
Since, the partial derivatives in equation (C.34) cannot be determined explicitly, their
values must be estimated on the basis of a measurement. The DED calibration was used
for applying different heat flux levels. It was shown that these had no effect on the change
of TP. Determination on the influence of temperature is explained in the next steps:
1. The flux is calculated on the basis of a measured temperature signal Torig and the
value for the thermal prodcut TPorig of 2050J/m2.K.s1/2.
2. On the resulting flux were some recognizable points (eg. peak flux) chosen. The flux
qorig is noted in these points.
3. The variable, temperature is varied 0, 1%, 0, 01% en 0, 001% resulting in Tvar. The
resulting flux qvar is noted again for the previous chosen points.
4. For every variation of the variable, the ratio can be calculatedqorig−qvarTorig−Tvar . Note that
this is an approximation (C.34).
For ∂qtrans∂Topp
we obtain a temperature dependent trace:
∂qtrans∂Topp
∼= 7, 2174e−0,112 Topp (C.35)
Transient heat flux - Fourier method The transient part of the heat flux is given
in equation (A.5):
qtrans = TP ·
∞∑
n=1
√nω
2[(Kn +Gn) · cos(nωt) + (−Kn +Gn) · sin(nωt)]
It is obvious that the partial derivatives are not easily determined of equation (C.34).
Therefore, a sensitivity analysis must be performed. The transient part of the heat flux is
only dependent on the temperature since the TP may be considered constant which has
been proven with the DED calibration. Voor ∂qtrans∂Tw
bekomen we:
∂qtrans∂Tw
= 0, 62 (C.36)
For ∂qtrans∂TP we obtain:
Steady state component flux the steady state part of the flux can be calculated as:
qss =Tw − Tdepth
ak1(C.37)
The absolute error is:
AEqss =
√(AETwak1
)2
+
(AETdepthak1
)2
+
((−Tw + Tdepth)AEak1
ak21
)2
(C.38)
89
Appendix C. Error analysis 90
Total flux The total flux, which is the sum of the transient and the steady state part,
is given by:
qtot = qtrans + qss (C.39)
The absolute error is:
AEqtot =√AE2
qtrans +AE2qss (C.40)
In table C.9, the errors are listed for different calculation methods on the peak heat flux.
Table C.9: Absolute en relative error for flux calculations TFG single layer
Variable X REX [%] AEX Unit
V0 - 10.10−3 V
GTFGS 1 - -
VTFGS - 2, 5.10−3 V
Tdepth - 0, 5 C
TP 4, 2 - J
m2.K.s12
ak1 10 - m2.KW
qtotFIR 1, 2 - Wcm2
qtotFOUR 8, 8 - Wcm2
C.2.7 Convection coefficient
For every sensor, the convection coefficient can be calculated as:
h =q
Tg − Tw(C.41)
The error on the temperature difference ∆T between gas and wall can be written as:
AE∆T =√AE2
Tg+AE2
Tw(C.42)
The error can be calculated as:
REh =√RE2
q +RE2∆T (C.43)
The relative errors are listed in table C.10 for different calculation methods.
Table C.10: Relative errors for convection coefficients
Variable X REX [%] Unit
hTFGFIR 12, 83 Wm2K
hTFGFOUR 20, 29 Wm2K
90
Appendix C. Error analysis 91
C.3 Error analysis on the DED setup
To calculate the error on the thermal product, a proper analysis should be made. Generally
the absolute error of a function f that depends on the variables a,b and c can be calculated
as:
AEf =
√(δs
δaAEa
)2
+
(δs
δbAEb
)2
+
(δs
δcAEc
)2
(C.44)
The relative error can be calculated as the ratio of the absolute error to the function itself:
REf =AEff
(C.45)
To determine the error on the TP, we need to determine the error on the out of balance
voltage. This voltage is given by:
V0 =VB4
[∆R
R+ ∆R2
](C.46)
Where V0 represents the out of balance voltage, VB the bridge supply voltage which is
generated by the data acquisition which has an absolute error of 2µV. R1,R2,R3 and R4
are the resistors of the bridge. R2 is the thin film sensor and functions as the independent
variable in this case. Two resistors have a fixed value and a third is the potentiometer
necessary to balance the bridge. These three resistors have a relative error of 1%, so they
do not influence the error analysis. Their error is the variation of the actual value, supplied
by the data sheet. The bridge can be balanced accurately up to 100 µV . Therefore the
absolute error of the out of balance voltage is 100 µV .
The thermal product is calculated according to:
√ρck =
√ρckglyc(
∆V√t
)air(
∆V√t
)glyc
− 1
(C.47)
And can be written as:
√ρck =
√ρckglyc
(bair)
(bglyc)− 1
(C.48)
Where bair and bglyc are the slopes of the linearized out of balance voltage when regression
is performed. The error on these slopes can be written as:
91
Appendix C. Error analysis 92
AEb =
√√√√1
N−2
∑Ni=1(Tj − aVj − b)2
∑Ni=1 x
2 − (∑Ni=1 x)2
N
(C.49)
The absolute error on the TP can be written as:
AEf =
√(δTP
δbairAEair
)2
+
(δTP
δbglycAEbglyc
)2
+
(δTP
δTPglycAETPglyc
)2
(C.50)
AEf =
√√√√(−TPglycbglycbair − bglyc
AEbair
)2
+
(TPglycbairbair − bglyc
AEbglyc
)2
+
(1
bairbglyc− 1
AETPglyc
)2
(C.51)
Each successful regression will have a correlation coefficient that is 99% or higher. There-
fore, the error on the slopes of the regression are very low. The relative error of glycerin
is 4% [18]. The average relative error of the TP is 4.5%, which is comparable to values
achieved with other setups [14, 27, 28].
92
Appendix D
Double Electric Discharge
calibration appendix
The Double Electric Discharge calibration is a tool for determining the thermal product of
the single layer sensor. This calibration is performed in air and fluid, with known thermal
properties, while a voltage pulse is sent to the RTD which causes ohmic heating. The RTD
which is incorporated in a Wheatstone bridge will cause an out of balance voltage related
to its changing resistance when ohmic heating occurs. A regression will be performed on
the out of balance voltage because the slope of each regression is needed to calculate the
thermal product. This text will explain the setup itself, the calibration process and the
data processing to acquire the thermal product.
D.1 DED setup
The setup consists of a Wheatstone bridge where its input is connected to the DAQ ,
through an electronic circuit as can be seen in figure 1. This electronic circuit functions as
a voltage follower. The voltage follower separates the DAQ from the load to protect the
DAQ from high currents. Secondly, The DAQ can only deliver 5 mA which too low for
the load which makes the voltage follower necessary. The follower supplies the voltage to
bridge which is the same voltage set by the DAQ and sets the current as a function of the
load. The voltage follower consists of the OP amp (AD741) and NPN transistor (2N1711).
The NPN is necessary to deliver the high currents. More details about these components
can be found in the datasheet. The Wheatstone bridge consists of 4 resistors as can be
seen in figure 1. Rx is the RTD of the single layer sensor. R1 is the potentiometer which
is the controllable resistance to balance the bridge and R2 and R3 are fixed resistances as
can be seen in figure 2. Details of the resistors can be found in the datasheets. The DAQ
93
Appendix D. Double Electric Discharge calibration appendix 94
is the PXI-6251 which can deliver voltages from - 10 to 10 V. The DAQ has 8 analogue
input channels and 2 analogue output channels. One of the analogue output channels is
used to generate the signal that is sent to the bridge as can be seen in figure 3.3. Three
analogue input channels are necessary to perform measurements. First, the out of balance
voltage of the bridge will be recorded in order to acquire the data that is necessary for the
digital signal processing as can be seen in figure D.1. Also, the voltage across the RTD is
measured and a shunt resistor is placed in series with the RTD so that the resistance of
the RTD is known. The voltage across the shunt resistor is measured since the DAQ only
can acquire voltages. Details about the DAQ can be found in the datasheet.
Figure D.1: The DED setup
Figure D.2: The potentiometer
Figure D.3: The fixed resistors
94
Appendix D. Double Electric Discharge calibration appendix 95
D.2 DED calibration process
Once the setup is complete, the calibration process can begin. The calibration is performed
with the program ”Labview Signal Express”. This program is compatible with the DAQ
and signals can be generated and acquired on command. First, the bridge needs to be
balanced before sending a voltage pulse to it so that the out of balance voltage remains
zero until the bridge sees the pulse. This process is always performed carefully and after
each calibration, balancing of the bridge must be performed again. The initial voltage
that is sent to the bridge may vary from 0 to 1 V in order to avoid ohmic heating of the
RTD, this voltage range has been derived with the ohmic heating test.
We start by clicking the ”Labiew Signal Express” icon which can be seen in figure D.4.
After that the program is opened, start an empty project to open the workspace.
Figure D.4: Signal express icon
First, we want to initialize the signals that we wish to generate and acquire. We need
to acquire three signals, namely the out of balance voltage, the shunt voltage and the
voltage across the RTD. The shunt voltage and the voltage across the RTD are necessary
to calculate the RTD resistance. To do this, click on the icon Add Signals. Then click
Add Step, Acquire Signals, Analog Input and finally Voltage. Then a screen will
appear which can be seen in figure D.5, here the three input channels must be chosen,
each analog input corresponds with a BNC plug from the DAQ panel. Once the channels
are highlighted, press Ok. A screen appears were we can select the amount of samples and
sample rate as can be seen in figure D.6. The sample rate should be chosen high enough
since the voltage variations occur at small time intervals. Choose a sample rate higher
than 100 000 Hz. The amount of samples can be set as desired, when the program is run
95
Appendix D. Double Electric Discharge calibration appendix 96
continuously, this has no meaning. When the program is run once, this will determine the
amount of samples. When this procedure is done, return to Data View, then add two
more displays via Add Display and drag the three input signals to a separate display.
Figure D.5: choose analog input channel
96
Appendix D. Double Electric Discharge calibration appendix 97
Figure D.6: choose amount of samples and sample rate
Now that the inputs are defined, the outputs can be defined. First we need to calibrate
the Wheatstone bridge. Therefore, a constant DC signal must be generated to balance the
bridge. Two steps need to be done here, first we need to create the signal, then we need to
generate this signal to a desired output channel. We start by clicking Add Step followed
by Create Analog Signal, then a screen appears which can be seen in figure D.7. In the
box Signal type the waveform needs to switched to DC signal. The amplitude can be
modified in box Offset, here the amplitude may be set from 0 to 1 V as already explained,
in this example 300 mV has been chosen. The other parameters may stay default.
97
Appendix D. Double Electric Discharge calibration appendix 98
Figure D.7: Signal express output signal window
Then the output signal needs to be generated, this is done by clicking Add Step followed
by Generate Signals, DAQmx Generate, Analog Output, Voltage. A similar screen
appears as when signals are acquired, choose the appropriate output channel. Note that
the appropriate PXI slot is connected with the BNC panel, otherwise, Signal Express will
not be able to generate output signals.
Every signal is now initialized, the final step is to run to program. This done by clicking on
the arrow besides the Run icon as can be seen in figure D.8. Click on Run Continuously,
otherwise, the program will run once which is the time corresponding with the amount
of samples. In this case, we want to calibrate the bridge properly, therefore, the program
needs to run the whole time. The out of balance voltage can be monitored in one of the
displays. By varying the resistance of the potentiometer, the bridge can be balanced. The
limits of the vertical axis can be set to proper value, close to zero. The resolution of this
calibration can be set 100 µV, so that the bridge can be balanced carefully.
98
Appendix D. Double Electric Discharge calibration appendix 99
Figure D.8: run continuously
Once the bridge is balanced, the program may be stopped by clicking Stop. Now the
voltage pulse needs to be generated to perform the double discharge calibration. This can
be simply done by clicking the Create Signal on the left tab. The screen of the DC
signal appears again. The Signal type needs to be changed to Square Wave now as
can be seen in figure D.9. It is very important to set every parameter to its correct value.
First, the Frequency must be chosen, in this example, the value is set 100 Hz which
corresponds to pulse duration of 5 ms during one period. The Phase is chosen to 180 °so
that the waveform starts from its low value. The pulse voltage level is set by adjusting
two boxes namely Amplitude and Offset. For example, in this case we have chosen a
pulse of 4 V, therefore the Amplitude is set to 2 V and Offset is set to 2.3 V. Note that
the offset corresponds with 2 V offset to start from 0 V, however 2.3 V is necessary since
the bridge is balanced to 300 mV and this functions as zero level. The final adjustment is
Sample rate, this is the amount of samples that is desired to create the function. This
is chosen to 67 kS/s so that one pulse is sent to the bridge instead of periodic signal. See
to it that the pulse needs to drop to 300 mV, otherwise the output of the DAQ remains
high. This could cause sensor burnout and must be avoided. The other parameters may
be set to their default values.
99
Appendix D. Double Electric Discharge calibration appendix 100
Figure D.9: Signal express pulse waveform
Once the signal is created, we can go back to the Data View. Then the program can be
run. This is done by clicking on the arrow besides run as can be seen in figure D.10. See
to it that the program is run once. If the program is continuously run, too much heat is
generated in short time which can destroy the sensor.
Figure D.10: run the program once
When the calibration is performed, a function which is proportional with the square root
100
Appendix D. Double Electric Discharge calibration appendix 101
time should be monitored at the out of balance voltage display. Click right mouse button
and store the data to clipboard. Then open an empty text file and paste the data in it.
This text file will then be used for data processing which is explained in the next section.
Repeat the same procedure for storing the RTD voltage and shunt voltage.
When this procedure is performed, repeat the same steps with the sensor immersed in
glycerin. The thermal product can then be determined.
D.3 DED data processing
In order to achieve the calibrated value of the thermal product, the recorded data must
be processed to achieve the actual thermal product. The routine is explained in the m-file
itself. The structure of the file will be explained here as well.
The first part of the algorithm reads the three recorded signals which are the voltage
across the shunt in series with the RTD, the voltage across the RTD and the out of
balance voltage for measurements taken in air and glycerin. With this data, the power
across the RTD can be calculated.
Once the data is read, the data that used for regression needs to be determined. This
determination relies on the power function that has been calculated in the previous step.
The power function is proportional with the heat flux and therefore, this function will also
be a step function. However, the power function still represents a transient part that does
not follow the step function well. Therefore, an algorithm to determine the data points
that follow the step function well, will be conducted to achieve a reliable set of data points
to perform the regression. First, the end point of the step function will be located since
the step function is constant there. This is performed with a while loop that runs from
the end of the data points until a value higher than the noise is achieved. This value will
be the end point of the step function. Next, the point of origin of the step function needs
to be determined, this point lies a certain amount of steps earlier. This amount of steps
can be calculated when the pulse time duration and sample rate are known. In this case, a
step of 5 ms time duration sampled at 100 000 Hz requires 500 steps. Then the amount of
regression points needs to be determined, this can be performed by a loop that determines
the point that the step function varies 2 % from its end point, determined earlier. The
interval between this point and end point will then be used for regression. However, this
loop struggles with noise, therefore, the amount of regression points has been chosen hard
coded in order to achieve a reliable set of regression points.
101
Appendix D. Double Electric Discharge calibration appendix 102
After that the regression points are determined, the regression process on the out of balance
voltage can begin. First, the useful data must be derived before actual regression can be
performed. When the DAQ is triggered, it sends a pulse of certain time duration and data
is recorded at the same moment. The amount of samples that is read can be adjusted with
”Signal Express”. In this case, the amount of samples has been chosen twice as high than
pulse time duration. Every time a new set of samples are taken, the trigger has some delay
resulting in another start of rise time for each recorded voltage. Therefore, an algorithm
must be defined that takes the useful data out of this. This is done by taken the part
of the out of balance voltage that is defined from the origin and end point as discussed
in previous paragraph. When the step function originates the out of balance voltage will
start to rise, therefore, these points are taken for defining the subset of the out of balance
voltage that will be used for regression. Before regression can be performed the this subset
of out of balance voltage needs to be shifted to the origin. The out of balance voltage
must originate from zero in order to perform a regression.
Once, the out of balance voltage is shifted to the origin, the regression can be performed.
An non linear model is used where the only unknown is the slope which is required. This
slope will be calculated with a least squares method. The slopes of the measurements
taken in air and glycerin will be used to determine the thermal product.
D.4 Linearity error
It has been mentioned that the bridge implies a linearity error when the bridge consists
of a single varying element. The out of balance voltage will not be proportional with the
change of resistance due to this error. However, for small resistance variations the error
will be small. When the resistors of the bridge are chosen equal at ambient conditions,
the linearity error will be 0.5 % per % variation of thin film resistance.
There is a solution to this problem. Figure D.11 represents a Wheatstone bridge with
an OP amp configuration that might be applied in order to avoid the non linearity of
the bridge. Here, the out of balance voltage functions as the input of the operational
amplifier. The OP amp produces a forced null, by adding the a voltage in series with
the variable arm. This voltage is equal in magnitude and opposite in polarity to the
incremental voltage across the varying element and is linear with ∆R. The output of the
OP amp can then be connected to the DAQ. This active bridge has a gain of two over the
standard single-element varying bridge, and the output is linear, even for large values of
∆R. The amplifier used in this circuit requires dual supplies because its output must go
102
Appendix D. Double Electric Discharge calibration appendix 103
negative. Note that the resistances are chosen equal when no change in the single-varying
element is monitored. Therefore, each resistance should be an accurate potentiometer
that needs to be set to appropriate value when a calibration is performed since the thin
film resistance varies with the temperature. Also, each thin film has another resistance at
ambient conditions which makes the potentiometers necessary.
Figure D.11: The optimized DED setup
Another benefit of this setup is that the change in resistance can be calculated easily since
the out of balance voltage is proportional to the change in resistance. The error on the
voltage pulse can also be omitted by applying the radiometric principle. It can be seen
that the output of the bridge or OP amp is proportional with the bridge supply voltage.
By dividing the output, by hard or software, with the bridge supply voltage, the error or
drift on voltage will be omitted, so reducing the error.
103
Appendix E
MATLAB code
104
% DED processing tool% Created by Gery Fossaert & Olivier Collet% This tool reads recorded out of balance voltages wherefrom a regression% is perfomed in order to acquire the thermal product
% First, the recorded data must be read
% Data from air measurementsR_shunt = 1.45; %shunt resistanceV_I1 = dlmread('shunt_8V_5ms_air2.prn'); %voltage across shuntV1 = dlmread('RTD_8V_5ms_air2.prn'); %voltage across RTDOB1 = dlmread('8V_5ms_air2.prn'); %out of balance voltageI1 = V_I1(:,2)/R_shunt; %calculation of actual currentR1 = V1(:,2)./I1; %calculation of RTD resistanceI_square1 = I1.^2;Q1 = R1.*I_square1; %calculate power across RTD
% Data from glycerin measurementsV_I2 = dlmread('shunt_8V_5ms_glyc1.prn'); %voltage across shuntV2 = dlmread('RTD_8V_5ms_glyc1.prn'); %voltage across RTDOB2 = dlmread('8V_5ms_glyc1.prn'); %out of balance voltageI2 = V_I2(:,2)/R_shunt; %calculation of actual currentR2 = V2(:,2)./I2; %calculation of RTD resistanceI_square2 = I2.^2;Q2 = R2.*I_square2; %calculate power across RTD
% The first step is to acquire the indeces where the heat flux is constant% this is done by the following while loops that determines the% indices where end_i1 is the index where the heat flux is constant at the% end of the step. Then start_i1 is the index where the heat flux lies in% a high confidence interval. With these indices, actual regression data is% determined
i1 = length(V1);while( Q1(i1) < 0.08) %determination of end point step power function,
i1 = i1 - 1; %the threshold of the condition is hard coded sinceend %it varies for different voltage rangesend_i1 = i1-1 ;
zero1 = end_i1-498; %origin of square root of time defined as zero1
% j1 = end_i1-1;% while( Q1(j1) > 0.99*Q1(end_i1) && Q((j1) < 1.01*Q1(end_i1))% j1 = j1 -1;% j1 = end_i1 - 1;%% end% determination of the instant when step function start to vary
start_i1 = end_i1-400; %the instant where the power is constant, hard coded
%In order to perform regression, the resistance and voltage trace needs%to be shifted to the origin. First, the subset of useful data is taken,%this is when the out of balance voltage starts to rise. Then a new time%array is created that corresponds with the shifted traces
delta_x1 = end_i1 - zero1; %defining the subset of dataR11 = R1(zero1:end_i1); %subset of resistanceR11 = R11 - R11(1); %subset originates from zero now
1
OB11 = OB1(zero1:end_i1,2); %subset of out of balance voltageOB11 = OB11 - OB11(1); %subset originates from zero nowx1 = [0:0.00001:(delta_x1*0.00001)]'; %defining new time array
%startv_reg1 and endv_reg1 are the shifted values from where the heat%power step function is constant. It is between these values that the%regression is performed.
%signal to noise ratio%calculation of signal powerpow_ER1 = (sum(abs(ERv1).^2))/length(ERv1);pow_ER2 = (sum(abs(ERv2).^2))/length(ERv2);pow_v1 = (sum(abs(v1).^2))/length(v1);pow_v2 = (sum(abs(v2).^2))/length(v2);NSR_air = pow_ER1/pow_v1;NSR_glyc = pow_ER2/pow_v2;
%plot RTD resistancefigure(6)subplot(2,1,1)plot(V1(zero1:end_i1,1),R1(zero1:end_i1));title('RTD resistance in air')xlabel('Time (s)')ylabel('Resistance (Ohm)')subplot(2,1,2);plot(V2(zero2:end_i2,1),R2(zero2:end_i2));title('RTD resistance in glycerin')xlabel('Time (s)')ylabel('Current (R)')
%plot RTD powerfigure(7)subplot(2,1,1)plot(V1(1:zero1,1),Q1(1:zero1),V1(zero1:start_i1,1),...Q1(zero1:start_i1),V1(start_i1:end_i1),Q1(start_i1:end_i1),...V1((end_i1):(length(V1))),Q1((end_i1):(length(V1))));title('RTD power in air')xlabel('Time (s)')ylabel('Power(W)')legend('start points','origin points','regression points','end points')subplot(2,1,2);plot(V2(1:zero2,1),Q2(1:zero2),V2(zero2:start_i2,1),Q2(zero2:start_i2),...V2(start_i2:end_i2),Q2(start_i2:end_i2),V2((end_i2):(length(V2))),...Q2((end_i2):(length(V2))));title('RTD power in glycerin')xlabel('Time (s)')ylabel('Power (W)')legend('start points','origin points','regression points','end points')
%plots of the curves and the regressionfigure(8)subplot(2,1,1);plot(x1(1:(startv_reg1-1)),R11(1:(startv_reg1-1)),...
3
'g',x1(startv_reg1:endv_reg1),R11(startv_reg1:endv_reg1),...'r',x1,y1,'b');title('Regression of resistance in air')xlabel('Time (s)')ylabel('Resistance (Ohm)')legend('data points','regression points','regression')subplot(2,1,2);plot(x2(1:(startv_reg2-1)),R22(1:(startv_reg2-1)),...
title('Regression of resistance in glycerin')xlabel('Time (s)')ylabel('Resistance (Ohm)')legend('data points','regression points','regression')
%plots of out of balance voltage with regressionfigure(9)subplot(2,1,1);plot(x1(1:(startv_reg1-1)),OB11(1:(startv_reg1-1)),'g',...
x1(startv_reg1:endv_reg1),OB11(startv_reg1:endv_reg1),'r',x1,v1,'b');title('Out of balance voltage in air')xlabel('Time (s)')ylabel('Out of balance voltage (V)')legend('data points','regression points','regression')subplot(2,1,2);plot(x2(1:(startv_reg2-1)),OB22(1:(startv_reg2-1)),'g',...
x2(startv_reg2:endv_reg2),OB22(startv_reg2:endv_reg2),'r',x2,v2,'b');title('Out of balance voltage in glycerin')xlabel('Time (s)')ylabel('Out of balance voltage (V)')legend('data points','regression points','regression')
% Out of balance voltagesfigure(10)subplot(2,1,1);plot(OB1(1:zero1,1),OB1(1:zero1,2),OB1(zero1:start_i1,1),...
title('Out of balance voltage in air')xlabel('Time (s)')ylabel('Out of balance voltage (V)')legend('data points','origin points','regression points','end points')subplot(2,1,2);plot(OB2(1:zero2,1),OB2(1:zero2,2),OB1(zero2:start_i2,1),...
title('Out of balance voltage in glycerin')xlabel('Time (s)')ylabel('Out of balance voltage (V)')legend('data points','origin points','regression points','end points')
title('Out of balance voltage in air')xlabel('Time (s)')ylabel('Out of balance voltage (V)')legend('data points','regression points','regression')subplot(2,1,2);plot(x1(1:(startv_reg1-1)),R11(1:(startv_reg1-1)),...
rck_gly = 0.0925; %TP taken from AGARD documentrck_sub_R = rck_gly/((beta1/beta2)-1); %TP from resistance tracerck_sub_OB = rck_gly/((betav1/betav2)-1); %TP from out of balance voltag
%error on slopes of regressionxlin1 = x1.^(1/2);xlin2 = x2.^(1/2);[b1, bint1] = polyfit(xlin1(startv_reg1:endv_reg1),...