Enhancing Total Temperature Measurement Accuracy: Calibration Procedures and Novel Two-Wire Probes

To ensure precise efficiency quantification, within 0.5%, in gas turbine engines, a temperature accuracy below 0.5K is required during testing [1]. The design of temperature probes has been extensively studied in the past to improve the assessment of aircraft gas turbine performance [2]. The basic design recommendations for total temperature probes given in the 1960s by Moffat [3] are still valid. The basic understanding of thermocouple behavior together with useful recommendations can be found in Ref. [4]. With the improvement of manufacturing technologies capable of building more complex geometries, and the advancements in computational techniques and their accuracy, new and more precise guidelines to enhance temperature measurements were developed [5]. The advantages of thermocouples over other types of temperature sensors (such as thin-film platinum resistance sensors, platinum resistance thermometers (PRTs) [6]) are the ease of manufacturing and its less expensive price. Additionally, thermocouples can be adapted to high temperature applications [7] and different material combinations have been analyzed in a wide range of standardized conditions [8].

However, thermocouple measurements suffer from a variety of errors due to the heat transfer between the junction, the surrounding gas, and the support [3]. These interactions may show any of the typical mechanisms of conduction, convection, and radiation (see Fig. 1). Efforts were made to provide a correction factor for thermocouples in the past [9–12].

Radiation effects are not significant at low temperatures (<450 K) if the wires are surrounded by a shield that reduces the view factor of the junction to the surroundings [13]. The main source of error results from the conduction and the convection effects [5]. Nevertheless, this is not a practical separation for quantifying the error. Instead, the error caused by the transformation of kinetic energy into thermal energy in the boundary layer around the junction is typically referred to as velocity error. Then, the thermal interactions due to temperature differences between the junction wires, the probe support, and the stem of the probe are considered under the conduction error. Both types of errors are dependent on the shield geometry and the architecture of the probe (i.e., how the wires are located inside this shield). This designation is standard in the thermocouple literature and will be used in the present article. In the actual scenario, the conduction error will also take into account the convection effects not included in the velocity error, like the convection phenomena at the rear of the shield.

Robust calibration processes are required to accurately characterize the performance of total temperature thermocouple probes. This paper outlines each of the stages of the calibration methodology performed at the Purdue Experimental Turbine Aerothermal Laboratory (PETAL). The manufacturing of thermocouples is entirely performed at our facilities, and therefore the full calibration process is carried out in situ. This consists of a static calibration of the thermocouple wire and a velocity error calibration in the open jet test rig. Finally, progress in the design and testing of a two-wire, zero conduction error probe is presented. This probe is used to correct for conduction errors during velocity error calibrations, which will complete the calibration process.

Velocity error will increase with the velocity inside the shield, and conduction error will decrease. A compromise between both effects has been analyzed and values of Mach between 0.1 and 0.15 are optimal to reduce both effects [5].

A conceptually simple solution to separate conduction and velocity errors is to eliminate the conduction error and perform a controlled calibration for velocity error. For the first part, several methods have been studied. Heating of the support to close-to-flow temperatures is an straightforward solution [21], but application requires an elaborate apparatus [22]. Another solution is the use of two-wire thermocouples [23]. These probes record two measurements of the temperature at the same location. Different wire thicknesses provide different conduction errors and therefore there will be a discrepancy in the recording. CFD cases can be run at different conditions to characterize the relations between the measurement difference and the true flow temperature [24]. This calculated temperature is equivalent to the measurement of a junction with zero conduction error.

Once the studied probe measurement is corrected for conduction error, a velocity error calibration can be performed, knowing the recovery error obtained will be solely dependent on the velocity error effects. To achieve the high requirements for measurement accuracy, sophisticated devices have been designed with the purpose of calibrating total temperature probes as a function of the Mach and Reynolds number conditions. These facilities consist of a stagnation chamber where the flow is brought to the desired conditions of upstream total pressure and total temperature, close to stagnation. This flow is then accelerated through a nozzle and discharged into the probe to be calibrated. This concept is borrowed from the well-known method to calibrate multi-hole pressure probes [25–27], but it is suitable for the calibration of total temperature probes. The Oxford Probe Calibration Facility [28], the Loughborough University Probe Aerodynamic Calibration Facility [29], or the Test Facility for Probe Calibration at DLR Göttingen [30], where independent variation of Mach and Reynolds numbers is available, are remarkable examples of probe calibration facilities.

Ultimately, fast response is required to accurately measure the changes of temperature. The thickness of the wire plays a major role in reducing the time response of the thermocouple junction. The dynamic response of bare thermocouple wires of different thicknesses is shown in Ref. [13]. For sufficiently large values of the length-to-diameter ratio in the wire (>40), the thermocouple response is reduced to 0.2 ms. Unshielded junctions do not reduce the velocity of the flow around the junction, and will always yield faster response than the shielded architecture. However, unshielded wires are highly impacted by velocity and radiation errors, which become unacceptable. To correct the thermocouple measurement during transients, the actual temperature can be reconstructed using a discrete model determined with experimental data. The methodology to create this compensation is shown in Ref. [31].

The main source of results in this work comes from tests performed in the PETAL facility [32]. Figure 2 presents a schematic of the facility. This is a blowdown wind tunnel capable of operating at a wide range of Reynolds and Mach numbers. Operating pressure ranges between 0.5 and 6 bar at the inlet of the test section and temperatures vary from 270 to 450 K for continuous operation, but temperatures up to 700 K can be achieved for specific short tests. Two test sections are sourced from a large storage that contains pressurized air at 150 bar. Each section consists of a settling chamber were the flow is homogeneously discharged. To control the mass flow input, a calibrated Venturi measures the flow into the facility.

Figure 3(a) shows every component involved in the setup of the open jet rig to perform a velocity error calibration. A nozzle downstream brings the flow to the desired conditions, based on the mass flow upstream. A robot fixes the probe facing the exit, exposed to the flow. Total pressure and temperature inside the settling chamber, static pressures along the nozzle, and temperatures from the thermocouple rake are acquired for several Mach number conditions. infrared (IR) measurements of the probe stem and nozzle provide with additional information about the probe performance.

The Open Jet Assembly consists of the settling chamber, where flow is discharged close to stagnation conditions; an adapter plate; and a circular convergent nozzle with contraction ratio of 16 (see Fig. 3(b)).

After the area expansion from the upstream 6-in. pipes to the 32-in. settling chamber, the velocity inside the settling chamber would reach 0.035 Mach at maximum mass flow, which translates into maintaining 99.91% of the total pressure at all times. Moreover, the settling chamber contains several layers of honeycomb and mesh screens to reduce the turbulence and secondary flow effects before flowing into the nozzle.

The nozzle internal surface is contoured to maintain a continuous curvature without inflexion points, which promotes a thin boundary layer along the nozzle. This reduces the pressure losses. The length of the nozzle is 1.125 times the inlet area. The outlet area is 3.15 in. diameter (80 mm). This nozzle can hold up to 25 pressure taps divided into 1 array of 10 taps and 3 arrays of 5 taps, distributed at 90 deg from each other. The axial location of the taps is distributed to minimize the errors when interpolating the values between them. The readings from these lines allow for an additional check for the performance of the nozzle. The adapter plate comes with eight inserts to locate pressure and temperature probes and measure the stagnation conditions upstream of the nozzle. The coupling of the nozzle with the settling chamber was evaluated using computational tools to prevent separation at the entrance of the nozzle.

Type-K thermocouples K1X-S304-062-EX-12-MPCX from Evolution Sensors and Controls, LLC. were used to acquire reference temperatures upstream the nozzle. The junction is exposed and flush with a stainless-steel shield. These sensors were calibrated following the static calibration procedure. The data acquisition is performed using the 48-Channel Precision Thermocouple Measurement Instrument EX1048 from VTI Instruments. This device allows for a precise measurement of the voltage in a thermocouple wire. Every four channels come with an independent cold junction reference temperature measured by an resistance temperature detector (RTD). Temperature is calculated afterwards from the calibration coefficients.

The electronic pressure scanning module MPS4264 from manufacturer Scanivalve was used to record the pressure readings from the nozzle pressure taps and adapter plate inserts. This device is specifically designed to be in wind tunnel tests where pressures do not exceed 15 psi (103.421 kPa). The very low-pressure ranges offered make it an ideal fit for applications where precision is critical. This device is designed to accurately correct for any change in the sensor’s behavior due to temperature. The accuracy of the MPS4264 used in the present testing is 0.0621 kPa.

A commissioning phase was carried out with the open jet test rig to validate the use of isentropic relations to compute the downstream Mach number from the total pressure reading in the settling chamber.

During calibration tests, two total pressure measurements are taken at the top and bottom positions in the settling chamber with pneumatic lines. These pressure readings differ by 0.03 kPa in average during testing. The calculated total pressure is the mean of these readings. For the total temperature measurements, the redundancy of this measurement was increased along the test campaign to ensure the best total temperature reading. Up to 6 type-K exposed thermocouples were used to get a reading of the total temperature inside the settling chamber. Figure 3(b) pictures the specific distribution of these probes during the testing.

The probe fixturing setup consists of a KUKA KR 6 R700 six robot and a 3D printed probe mount. Every probe was mounted in the robot with 3D printed fixtures (see Fig. 3(a)). The material used is polyether ether ketone. No significant bending of the fixtures was observed during testing,

The robot is designed for a rated payload of 3 kg in order to optimize its dynamic performance. With reduced load center distances and favorable supplementary loads, a maximum payload of up to 6 kg can be mounted, with a position repeatability of $±0.03mm$⁠. The robot arm provides with sufficient flexibility to mount the probes in any orientation and do fast adjustments during testing if required (e.g., shifting the probe to expose different heads, change the orientation to facilitate IR measurements). Preprogrammed traverse movement is also possible with the robot.

A static calibration is the first step in the procedure to understand the performance of a thermocouple. Every thermocouple sensor will provide with a specific voltage depending on the temperature difference between the junctions. This relation is linear for the range of operation of the thermocouple used. It is possible to expose the junction to a controlled environment at several known temperatures and then perform a linear regression between the voltage readings and the temperature difference between the thermocouple junctions. Then, the temperature of the junction during testing can be calculated with the calculated coefficients and the cold junction temperature, which is measured by an additional sensor.

There exist alternative sources to obtain the calibration coefficients. The International Temperature Scale of 1990 (ITS-90) defines the procedures to perform thermocouple calibrations [33]. Additionally, it provides with an extensive amount of tabulated data to calculate the temperature based on the voltage readings. In the industry, thermocouple manufacturers will follow this standard for the performance of the thermocouples. Then, it is a decision of the end user to perform an additional correction for possible deviations from the standards in the readings.

Since they are defined for the general case, ITS-90 tables will yield inaccurate results when looking at specific sensors. Every thermocouple used in the present work has been calibrated to obtain individual coefficients, which will take manufacturing differences into account. Whenever available, the same data acquisition system that is used during testing was used for this calibration, which will include the environmental conditions from the facility in the calculated coefficients. Performing this procedure in situ also enables access to the raw data directly, which allows to perform a comprehensive uncertainty analysis on the linear fit method and assess the uncertainty of the calibration.

A 9170 Metrology well from FLUKE Calibration is used for the calibration. This device maintains temperature setpoints in a $±0.005K$ range. Calibrations were made for eight setpoints and were ran several times for every thermocouple. Figure 4 shows the results for one of these calibrations.

The aim of a velocity error calibration is to characterize the performance of a total temperature probe when exposed to different upstream flow velocities. This condition is monitored controlling the Mach number and using the thermocouple readings along with the total conditions upstream to compute the recovery factor.

The testing for velocity error calibrations requires a defined procedure which can be consistently reproduced and ensures sufficient repeatability and reliability of the results. This procedure consists of three stages:

• Test: includes a first warm-up stage and the acquisition of data at the calibration points.

• Post-process: the calibration points are localized from the raw data and setpoint results are obtained. The recovery factor is calculated.

• Results: once the necessary tests have been performed, all the data are condensed into a single fit. Overall uncertainty values are calculated.

These steps are pictured in Fig. 5. The procedure has been evaluated using a total temperature thermocouple rake of five shielded grounded thermocouples, each of them referred as heads from now on. This device was supplied by the Rolls-Royce Corporation and individual static calibration coefficients were provided. All the results presented in this section belong to the same test campaign performed with this probe.

The probe is located facing the exit of the nozzle, at a distance of 50 mm. This is sufficient to capture downstream conditions before any significant thermal diffusion to the ambient takes place. For the probe studied in this case, only two heads were fitted inside the jet at the same time.

Figure 6 depicts a typical example of how tests are performed, in terms of the total pressure measured in the settling chamber. A first warm-up stage is necessary to condition the temperatures in the settling chamber, the nozzle, and probe being tested. This warm up is made at Mach numbers from 0.4 to 0.6. The target of this stage is to reach a temperature close to that of the upstream flow.

Once the temperatures are stable, the test stage begins. The mass flow through the open jet is tuned to match the desired Mach number set points, based on the total pressure reading in the settling chamber. Specific Mach numbers can consistently be achieved within a $±0.01$ range. Every test consists of 15 points from Mach 0.2 to 0.9 and back to 0.2 in steps of 0.1. Constant velocity and temperature are required at every setpoint. Once a test is finished, another cycle can be performed if the air supply is sufficient. In general, it is possible to run two cycles successively.

For the present work, all tests are performed at ambient temperature and pressure. Values of $γ=1.4$ and $cp=1004J/kgK$ are used.

A last calculation for the recovery ratio is performed. This is defined as the ratio between the probe measurement $TTC$ and the reference temperature $T0$⁠. The definition of recovery factor as a function of the Mach number may bias the result of the calibration, since high recovery factors are always obtained at high Mach numbers. The recovery ratio will provide an additional source of information to analyze the results.

This concludes the first part of the post-processing, which is the calculation of the recovery factor and recovery ratio. However, uncertainties need to be calculated to completely define the performance of the total temperature probe being calibrated.

This definition will be applied to the total temperature and total pressure readings in the settling chamber, and to the total temperature probe readings. Because of the nature of this test, there are no high frequency phenomenon involved and therefore $σs$ will be only dependent on the sensor measurement steadiness. For instance, vibrations of the sensor may cause fluctuations in the reading which do not represent the actual flow conditions. If the sensor is correctly installed, these errors will be negligible if the acquired sample is big enough (⁠$n>1000$⁠).

Furthermore, using Eq. (9), uncertainty propagation is included and calculated for every values just outlined.

The uncertainties for the specific heat parameters were not considered in this analysis. At Mach 0.5, for a change in temperature of 10 K, the recovery factor changes by a 3.5%. If the specific heat ratio and heat capacity are accordingly adjusted to this change, the difference yields a 0.0001% difference. The effect of uncertainty in the specific heat is more than four orders of magnitude lower than the effects of the temperature itself changing with the measurement.

With all these calculations, Fig. 7 can be generated, giving a clear image for a single test results. Uncertainties are calculated to verify if the discrepancies between cycles are within reasonable values. For more clarity in the plot, these are not depicted, but instead they are outlined in Table 1.

After performing several tests with the same probe, condensed results can be plotted to check repeatability and analyze the global performance of the probe. Figure 8 shows those results. From this type of analysis, differences in the dispersion of the results between each head are observed. Variation in the conditions between tests is discarded as the source of these differences, since several heads were tested simultaneously and the dispersion trends are always maintained. Manufacturing errors may be a potential source of these differences. Velocity errors are highly dependent on the probe geometry and therefore different dimensional tolerances impact the recovery factor results.

Additionally, it is observed that, for those tests where several runs were performed successively, the recovery is slightly higher for each of the cycles. Calculating the recovery ratio helps to visualize this effect (see top-right plot in Fig. 8). This shift will be discussed further in this article. Ultimately, the recovery ratio is observed to follow a linear trend with respect to the Mach number, which will be the starting point for the last phase of the velocity error calibration.

The final section of the velocity error calibration focuses on recognizing trends in the results and developing an empirical model. This model will provide the information necessary to characterize the performance of the total temperature probe with respect to the upstream Mach.

The numerical results for each head are shown in Table 2. Notice that $×104$ means the listed value is 4 orders of magnitude bigger than the actual value. As an example, results for the third head for the probe discussed are plotted in Fig. 9.

With the velocity error calibration procedure just described, the performance of the total temperature thermocouple probe can be characterized to correct the measurement during testing. However, the source of the shift in the recovery factor for successive cycles was still not explained. IR thermography will be performed to assess the temperature gradients in the probe stem. The presence of any temperature difference between the thermocouple junction and the probe stem leads to a possible effect of the conduction error in the measurements performed during the velocity error calibration. The purpose of the IR measurements is giving proof of the heat transfer mechanisms driving conduction errors and the impact in the recovery factor calculation.

The TELOPS FAST V1K is a long wave infrared camera capable of recording high resolution surface temperature data at a maximum of 17.2 kHz for reduced acquisition windows and at a maximum of 230 Hz for full $640x512$ pixel resolution. A calibrated 13 mm lens is used for the present testing. The FAST V1K camera was used to perform surface temperature measurements at the heads of the probes. IR measurements are dependent on the emissivity of the material being measured and therefore surfaces need to be treated before taking the measurements to increase the emissivity up to an adequate value, above 0.95 (emissivity for polished metals is typically below 0.3). Dry graphite film spray was used to cover the heads and the stem of the probe. Head inlets and outlets were covered during the application of the spray. This paint has known emissivity of 0.98. The camera is located at a distance of 0.6 m from the probe. Figure 10 presents a measurement instance during testing. Note that the results for this procedure do not belong to the same series of testing introduced during the previous section, but they are performed with the same probe.

Figure 11 depicts the results for the difference between the IR measurements, $TIR$⁠, and the thermocouple junction, $TTC$⁠, against the upstream Mach number conditions. As expected, high temperature differences yield low upstream Mach condition. These differences are monotonically reduced when the upstream Mach increases. It is also observed that these differences are reduced for the second successive cycle performed during the same test.

The results from these measurements demonstrate a difference between the thermocouple reading and the temperature of the probe head. This difference in temperature sets evidence that conduction errors exist during velocity error calibrations, even in steady-state conditions.

The correction for conduction errors in thermocouple measurements requires performing conjugate heat transfer (CHT) calculations to simulate the different temperature response of the wires used in the two-wire, zero conduction error probe. CHT methodology has been used to assess the aerodynamic performance of total temperature probes in Ref. [35]. The following temperature corrections are the result of several steady CHT simulations of the probe head with two wires of 50 and $75μm$ diameter, maintaining the upstream temperature constant and changing the shield temperature to set different temperature differences between the junction and the support. The upstream velocity is also changed to address different Mach number conditions. This procedure is detailed in Ref. [24]. Figure 12 shows the results for the solid domain surface temperatures at one of these simulations.

This correlation is known to be dependent on the specific conditions of the flow and probe geometry.

Guidelines for the design of the Kiel heads of total temperature probes were followed in the design of the head for the temperature probe [5]. It is desired to find a balance between high flow velocities inside the head to increase the heat transfer to the thermocouple, and maintain the velocity error—proportional to the internal flow velocity—low enough. Even though a subsequent open jet calibration can correct for the velocity error, the local conditions during the application are usually unknown and it is always desirable to maintain all errors as low as possible.

However, these guidelines are typically given for a probe which locates the thermocouple wires along the center of the head, parallel to the flow. The proposed probe holds the bare wires perpendicular to the flow, locating both junctions as close as possible to the center of the head. These wires are $50μm$ and $75μm$ thick, sufficiently small to not create an obstruction of the flow. The architecture of the probe with the bare wires exposed to the flow provides a faster response. Finally, the length-to-diameter ratio of the wires is known to yield less conduction error. Altogether, these design directions will reduce the conduction errors for the individual wire measurements.

The probe must maintain the design as similar as possible to the baseline design used for the CHT simulations [24]. This way, the calculated coefficients for the true temperature can be used. The probe was 3D printed with a newly developed ceramic material. All small holes and cavities were checked against the 3D printing capabilities to ensure the geometry prints correctly.

Figure 13(a) presents the final geometry of the probe. The original geometry of the head was not tough enough to prevent failure during manufacturing and operation. The new design maintains the same baseline head geometry except from an increase (17%) of the head walls thickness. The internal geometry of the head remains unchanged (i.e., the inner face diameter was maintained). The head is flush with the tip of the stem, optimizing the space to perform close-to-wall measurements.

An inlet to outlet ratio of 4 is selected. This is the maximum recommended to optimize the recovery factor [2,10]. The body of the stem maintains the baseline geometry, being as thick as the head size. The cavity inside the probe stem was enlarged and the geometry includes some orifices to run the wires internally, out of the main flow.

The manufacturing of the probe follows two phases. First, the probe body is 3D printed and treated to maintain all the features necessary to build a probe. Then, the bare wires are carefully located inside the head. These wires were acquired from OMEGA Engineering and they already include the junction for type-K wires for $50μm$ and $75μm$ thickness. Each of the ends of these wires is subsequently welded to insulated thermocouple wire in the back of the probe to eliminate any exposed bare wire segment. Some design modifications were introduced at this stage, after observing some challenges during the manipulation of the probe and mounting of the wires.

The 3D printing is executed with the XiP Desktop Resin 3D printer by Nexa3D with proprietary material xCERAMIC3280. Holes of 0.7 mm can be printed consistently taking the necessary post-processing measures to prevent the orifices from closing during curing. Some warping was observed during the manipulation of the probe right after printing, and therefore the total length of the stem was reduced from 150 mm in the baseline to 50 mm. Printing the probe in an electrically non-conductive material is an advantage over metal, since now the wires can make contact to the probe body without any risk of short-cutting the thermocouple circuit.

The mounting of the wires requires precise handling. The use of a magnifying system is highly recommended to facilitate this phase. A microscope was used during the whole process. The welding of the wires is problematic due to their thickness and requires painstaking care. Spot welding and soldering were the options evaluated and joining the wires together is possible with both techniques. Soldering was selected to be the most suitable method, because of the lower risk of breaking the wires compared to spot welding, which may break them if the discharge is not precise. This is an inconvenience, since spot welding leaves a clean junction between the wires, while soldering adds more material in a reduced space. Additionally, the solder material does not stick properly to the chromium, which makes the welding more complicated and in many cases the main source of failures during operation.

To solve the welding problems, PETAL has acquired a Laser Welder Orion LZR ECO160 from manufacturer Sunstone Engineering LLC., capable of welding the 50 and 75 $μ$m wires. It is expected that this new equipment will make the manufacturing of the probes less intricate, in addition to improve the performance of the sensor due to the cleaner junction that a laser weld creates, compared to soldering.

The bare wire ends are covered with electrical and thermal insulating epoxy Duraseal Silicone Putty 1532, which remains elastic when cured and holds the wires in position without fixing them, which would provoke breaking when pulling the wires. The segments exposed in the laterals are also fixed to the surface and covered with this epoxy. A second epoxy (8329TCM from MG Chemicals), which hardens once cured and sticks to the ceramic material, is used to close the space at the back of the probe. Kapton tape is used during the curing, but it is recommended to remove it afterwards. Figure 13(b) shows the appearance of the probe with the wires mounted.

Once a dual-wire probe is manufactured successfully, a static calibration is performed to obtain the coefficients for each of the thermocouple junctions. The coefficients shown in Fig. 4 are the result of this calibration, for the $75μm$ wire. The $25μm$ wire yields similar values. A velocity error calibration is performed following the procedures previously outlined.

The probe was calibrated for upstream velocities up to Mach 0.6. The true temperature is calculated from the two-wire measurements, calculating the Reynolds number with the upstream velocity and the characteristic length as the smallest wire diameter (i.e., $50μm$⁠). The coefficient $b$ changes significantly depending on the upstream conditions and therefore it is calculated for every setpoint. Results for this calibration are plotted in Fig. 14. Table 3 outlines the uncertainties for each of the calculated values during testing. Notice that subscript 1 refers to the measurement of the $50μm$ wire, 2 to the $75μm$ one, and true to the calculated corrected true flow temperature.

Recovery factor increases for the calculated true temperature with respect to each of the individual wire measurements, which may point to a reduction of the conduction error. The resulting uncertainty for the true temperature is higher than the individual uncertainties for each wire measurement, due to the formulation in Eq. (34) and the subsequent error propagation that carries the error of variable b with it. Overall uncertainty can be calculated after performing several complete cycles in the probe, in order to address the real accuracy of this measurement.

Next steps in the validation of the two-wire architecture include the installation of the probe in a test scenario. This test will include transients in temperature that will increase the conduction error impacts on the measurement. Then, comparison with CFD results and alternative temperature sensors would be the main source of validation data.

A meticulous analysis of the main sources of error that affect total temperature thermocouples was conducted. This includes the effects of incomplete conversion of kinetic energy into thermal enthalpy, or velocity error, and the remaining heat transfer effects due to temperature differences between the thermocouple junction, the support, and the probe stem, or conduction error. The velocity error is addressed through a rigorous calibration process in an open jet rig. The proposed methodology calculates the recovery factor, accompanied by a thorough assessment of measurement uncertainties and a final empirical formulation of the probe performance with respect to the upstream temperature. IR thermography was used to demonstrate that conduction errors exist during this velocity error calibration procedure, due to temperature differences within the probe body. For correcting the conduction error, a two-wire probe was designed, following directions to reduce conduction and velocity errors. Further iterations in the design were directed to facilitate the manufacturing and a complete description on how the probes were built was given. The two measurements from two thermocouple wires with different dimensions were used to account for the conduction error and calculate the true flow temperature. A velocity error calibration was performed at this new probe to validate both methods, obtaining results of $0.99±0.01$ for the recovery factor at $M=0.6$⁠. The uncertainty of the true flow temperature measurement at that condition is less than 0.2 K.

The authors would like to acknowledge the efforts of Mr. Michael Butzen for assistance during testing and Mrs. Ashlyn Butzen for helping with the manufacturing of the probe. The authors would like to express their gratitude to the Rolls-Royce Corporation for providing with the probes used to commission the velocity error calibration, specially to Mrs. Alanna Crafton and Mr. Roger Bough.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$k$ =

thermal conductivity (W/mK)

$n$ =

number of observations

$p$ =

pressure (kPa)

$r$ =

recovery factor

$s$ =

sample average square of the deviations

$v$ =

velocity (m/s)

$M$ =

Mach number

$N$ =

number of cycles

$R$ =

recovery ratio

$T$ =

temperature (K or $∘C$⁠)

$V$ =

voltage (V)

$cp$ =

specific heat capacity (J/K kg)

$dw$ =

wire diameter (m)

$lw$ =

wire length (m)

$lc$ =

critical wire length (m)

$Rg$ =

gas constant for air (kJ/kg K)

Nu =

Nusselt number

Re =

Reynolds number

$γ$ =

specific heat ratio

$σ$ =

standard deviation

$σs$ =

sample standard deviation

0 =

total

$∞$ =

flow conditions

$ad$ =

adiabatic

$amb$ =

ambient

$cjc$ =

cold junction

$jc$ =

junction

$off$ =

offset

$set$ =

setpoint condition

$s$ =

static

$sp$ =

support

$TC$ =

thermocouple

CFD =

computational fluid dynamics

CHT =

conjugate heat transfer

CI =

confidence interval

IR =

infrared

ITS =

International Temperature Scale

LWIR =

long wave infrared

PETAL =

Purdue Experimental Turbine Aerothermal Laboratory

RTD =

resistance temperature detector

TC =

thermocouple