Temperature Measurement Error Research Articles

Dual parametric simultaneous measurement of pH and temperature based on optical fiber interference and silanized pH-responsive hydrogel

A no-core and few-mode fiber-based structure coating with polyacrylic acid acrylamide hydrogel is used to realize pH and temperature measurement simultaneously. The proposed structure can be obtained through splicing the no-core fiber, few-mode fiber and no-core fiber (NFN) with a certain length in sequence. Its measurement range can cover 2 to 12 for pH and 20 °C to 90 °C for temperature. Meanwhile, surface silanization of the optical fiber is performed to facilitate the adhesion of the hydrogel and to refine the silanization process. The pH-based transmission-sensitivity is measured to be 0.959 dB/pH and the highest pH-based wavelength-sensitivity is 0.837 nm/pH. The temperature-based wavelength-sensitivities are measured to be 0.01 nm/°C and 0.02 nm/°C, respectively, in two different pH environments of 2 ∼ 5 and 5 ∼ 12, while the temperature-based transmission-sensitivities are measured to be 0.432 dB/°C and 0.356 dB/°C, respectively. The pH response of hydrogel and inherent temperature response of NFN-based Mach-Zehnder interferometer enable the sensor to perform dual parameter measurements for pH and temperature. The dual-parameter matrices are established and validation experiments are carried out. It is demonstrated that the pH and temperature measurement errors can be as low as 0.088 and 1.112°C respectively. This sensor is expected to have broad prospects in environmental monitoring, chemical analysis, and other related fields.

Experimental study on high-precision measurement of surface temperature in friction stir welding aluminium alloy 2219 based on infrared thermometry

In friction stir welding (FSW) of aluminium alloy 2219, the welding temperature critically influences the weld quality. Accurate measurement of the welding temperature is essential for effective welding process control. However, the conventional approach of non-contact infrared thermometry faces significant challenges in FSW due to the inherently low and variable emissivity of aluminium alloys, making it susceptible to environmental radiation interference. This paper proposes a novel, high-precision method for measuring the surface temperature of FSW aluminium alloy 2219 based on infrared thermometry principles. Initially, the study delves into these principles, analysing factors that affect the accuracy of infrared thermometry in the context of FSW in industrial settings. To enhance the surface emissivity of the weldments, a high-emissivity coating is applied to the aluminium alloy surface. Additionally, strategic adjustments in the positioning of the thermal imager and careful selection of the measuring area effectively mitigate interference from the FSW tool’s radiation. Conventional methods report a temperature measurement error greater than 2%, while the proposed method reduces this error to 1.2%. The effectiveness and practical utility of the method are validated through FSW experiments on flat plates and rocket tank rings, corroborated by comparative analyses with surface thermocouple temperature measurements. This confirms the viability of the proposed method for high-precision temperature measurement in aluminium alloy FSW applications.

Multi-spectral radiation thermometry of space point targets based on spectral image pixel binning

The temperature characteristics of space point targets are essential indicators of their operational status and performance. To address the issue of significant temperature measurement errors in space point targets caused by low temperatures and a low imaging signal-to-noise ratio (SNR), we propose a mathematical model for multi-spectral radiation thermometry, derived from the principles of dual-band radiation thermometry. Furthermore, a multi-spectral image pixel binning method is introduced to enhance the SNR and minimize measurement errors. The experimental results indicate that the proposed multi-spectral radiation thermometry outperforms dual-band radiation thermometry. After merging 2 to 20 pixels, multi-spectral radiation thermometry in the 3.75–4.1 and 4.3–4.62 µm bands demonstrates an enhanced SNR and reduced temperature measurement errors. For a 378.15 K blackbody, the relative errors decrease from 1.52% and 2.19% to 0.26% and 0.74%, respectively, after merging six and eight pixels in the two different bands, compared to unmerged images. This method provides a valuable reference for developing techniques to enhance the SNR and improve temperature measurement accuracy for space point targets.

Distributed optical fiber sensors for real-time tracking of fouling buildup for tubular continuous polymerization reactors

Early online fouling detection is expected to be a real step forward in the operation of continuous tubular reactors. As an online technology, the Rayleigh backscatter based Distributed Optical Fiber Sensor (DOFS) technology was evaluated with respect to temperature measurement resolution, reproducibility and the best online calibration method. Different coatings and terminations were characterized for emulsion polymerization reactors. Commercially available sensors with acrylate primary coatings, dual acrylate coatings, and polyimide coatings were all compared with each other. It is shown that the presence of a secondary coating significantly alters sensor behaviour. Sensors with only a primary coating showed a temperature resolution of 0.1 °C. Online calibration for temperature readout was carried out and validation tested on segments of fiber integrated in a 3D-printed reactor channel (diameter 1.5 mm). Acrylate coated sensors showed a deviation of 20 % for the calibration coefficients when inside the reactor channel compared to the online calibration, leading to errors in temperature measurement. Polyimide coated sensors showed a deviation of just 0.6 % in this validation test, demonstrating good capabilities for online calibration with accurate temperature measurements. The possibility of spatial and temporal monitoring of fouling buildup through a heat exchanging wall equipped with DOFS was evaluated.

Methods to mitigate the impact of work environment on the measurement of building surface temperature by unmanned aerial vehicle onboard thermal imaging system

With the widespread adoption of UAVs and compact infrared cameras, UAV infrared remote sensing technology has extensive applications in qualitative research fields such as finding thermal bridges and air tightness detection in construction projects, but less in quantitative research, mainly due to its large temperature measurement error. When measuring temperature with UAV-mounted thermal imaging cameras, factors such as operational wind conditions and operation methods, in addition to the inherent accuracy of the instrument, can introduce measurement errors. To investigate the impact of operational wind conditions on temperature measurements by UAV-mounted thermal imaging cameras, both field and laboratory experiments were conducted to analyze the error characteristics caused by wind conditions. Devices were designed to mitigate the impact of wind on temperature measurements, and their effectiveness was validated through experiments. Based on these findings, methods were developed for reducing the influence of wind conditions on temperature measurements in UAV operations. The results indicate that wind condition errors primarily affect the stability of temperature data. Under the influence of wind conditions, the temperature data recorded by thermal infrared cameras exhibit periodic and significant fluctuations, with a temperature difference of approximately 4.7 °C between the highest and lowest points within a single cycle, constituting the main source of measurement error. However, by following the prescribed measurement procedures, the temperature measurement error can be reduced by 32 %, from ±5 °C to ±1.6 °C, approaching the accuracy of tripod-mounted thermal imaging cameras. This establishes a foundation for quantitative research in low-altitude infrared remote sensing.

Weak signal ultrafast interrogation of a micro-nano FBG probe sensor based on the hybrid amplified dispersion Fourier-transform method.

To elucidate the thermal transport mechanisms at interfaces in micro- and nanoscale electronic devices, real-time monitoring of temperature variations at the microscopic and nanoscopic levels is crucial. Micro-nano fiber Bragg grating (FBG) sensors have been demonstrated as effective in-situ optical temperature probes for measuring local temperatures. Time-stretch dispersion Fourier transform (TS-DFT) that enables fast, continuous, single-shot measurements in optical sensing has been integrated with a micro-nano FBG probe (FBGP) for local temperature sensing. However, its temperature sensitivity and interrogation resolution are limited by the detection sensitivity. In this paper, we propose a hybrid amplified dispersion Fourier transform (ADFT) method to achieve ultrafast interrogation of FBGP's weak signal. Thanks to the combined effect of TS-DFT and hybrid optical amplification, the reflection signal of the FBGP is amplified, and the wavelength shift of the FBGP sensor is converted to a temporal spacing change between two dispersed pulses through dispersion-induced wavelength-to-time mapping. The proposed method uses a homemade dissipative soliton mode-locked laser as the light source. The hybrid optical amplification technique comprises a L-band erbium-doped fiber amplifier and a distributed Raman amplifier. Their noise figure and net gain for the FBGP are 4.81 dB and 15.93 dB, respectively. In addition, the temperature calibration experiments show that a sampling rate of 51.43 MHz and the maximum temperature measurement error of 1.98°C are achieved within the temperature range of 20.3°C to 97°C. The stability of the net gain provided by the hybrid ADFT system is demonstrated by the coefficient of variation, which ranges from 2.22% to 2.95% in the peak voltage signal of the FBGP. This approach applies to scenarios requiring the handling of weak optical signals, particularly in temperature measurement at the micro-nano scale.

An optical fiber high sensitivity temperature sensor with MZI and FPI parallel connection

Accurate detection of temperature is of great significance in the field of medical research. The focal point of this article centers around an optical fiber sensor that integrates Fabry Perot interferometer (FPI) filled with polydimethylsiloxane (PDMS) and Mach Zehnder interferometer (MZI), leveraging the principle of the vernier effect. By introducing the vernier effect principle, the sensitivity is 31.09 nm/°C in the detection range of 36 ∼ 40°C, and the maximum temperature measurement error is about 0.55 %. In addition, good linear response and longtime stability make it suitable for application in various scenarios where accurate temperature measurement is required.

An ultrafast optical sensor for simultaneous detection of NH3 temperature and concentration based on ultraviolet absorption spectral redshift combined with spectral reconstruction

As a renewable and carbon-free fuel with high energy density, ammonia (NH3) is a promising alternative to fossil fuels. Real-time detection of its temperature and concentration is significant in improving NH3 combustion efficiency. In this paper, we report an optical sensor, which enables simultaneous measurement of temperature and concentration of NH3. The key idea is to acquire the temperature of NH3 through absorption spectral redshift, and determine its concentration by building three-dimensional field map combined with spectral reconstruction. Firstly, we explain the generation mechanism of temperature-induced redshift in ultraviolet differential absorption spectrum based on energy level distribution theory. The relationship between spectral redshift and temperature is then established using spectral autocorrelation algorithm. Secondly, a spectral reconstruction method based on absorbed intensity mapping wavelength is proposed to ensure real-time detection of NH3. Finally, a three-dimensional field map between temperature, reconstruction slope, and concentration is constructed to achieve the detection. Results indicated that the mean relative error in temperature measurement was 1.04 % and the mean absolute error in concentration detection was 0.26 mg/m3 in air. Additionally, the NH3 sensor enabled long-term detection with a detection limit of 15.50 μg/m2, demonstrating its potential to broaden the application of NH3 as a fuel.

Error Analysis and Experimental Research of Temperature/Strain Sensing Based on Few-Mode Fiber Bragg Grating

ABSTRACT To solve the temperature/strain cross-sensitivity problem, simultaneous measurement of temperature and strain can be achieved by using the two spectra of LP01 mode and LP11 mode in few-mode fiber Bragg grating (FM-FBG). However, in the same fiber the temperature/strain sensitivity difference is small, so the sensing accuracy is limited greatly. In this paper, we calculate the relation between sensing accuracy and sensitivity-difference and obtain that the enlarged sensitivity difference could improve the sensing accuracy. FBGs are engraved on two kinds fibers single-mode (SM) and few-mode (FM) with different geometrical dimensions; the sensing coefficients of LP01 mode in SM-FBG and LP11 mode/LP01 mode in FM-FBG are calibrated firstly and then are used to sense the temperature/strain. The experimental results show that parallelly using LP01 mode in SM-FBG and LP11 mode in FM-FBG achieves simultaneous measurement of temperature and strain, and error is reduced greatly compared with the sensing results using LP11 mode and LP01 mode in FM-FBG or using LP01 mode in SM-FBG and LP01 mode in FM-FBG, where the wavelength-temperature/strain coefficient difference is maximum. The temperature measurement error is 1.64°C, and the strain measurement error is 20.04 με, which is consistent with the theoretical analysis. The sensing results provide technical reference for FBG multi-parameter sensing measurement and the practical engineering application.

Distributed vibration and temperature sensing system by multiplexed fiber scattering spectra

A new, to the best of our knowledge, distributed optical fiber vibration and temperature hybrid sensing system is proposed and experimentally demonstrated. The proposed system only employs two signal channels, which is more compact and practical. Based on the structure of the optical time domain reflectometer (OTDR), the Rayleigh scattering light and the Raman anti-Stokes scattering light is extracted for vibration and temperature sensing, respectively. For vibration sensing, a new differential location algorithm based on polarization state analysis of the Rayleigh scattering light is proposed to locate the vibration events. It first rectifies the original OTDR traces by fiber attenuation compensation to make each position in it with the same pulse power level. And then, by difference of adjacent traces and threshold discrimination, the vibration positions are identified and located. For temperature sensing, a temperature calibration unit and algorithm are adopted to dynamically correct the trace data and reduce the temperature measurement error caused by the instability of the pulse laser source. The experiment is conducted with a fiber range of about 12 km and laser pulse width of 60 ns, and the experimental results show that the maximum error range for temperature measurement is −0.7∘C to 1.3°C, with a root mean square (RMS) error of 0.85°C in the entire temperature measurement range. Additionally, the spatial resolution (SR) for both vibration and temperature sensing is 6 m.

A temperature measurement compensation method for industrial rotary kilns based on infrared multi-feature fusion under dynamic water mist interference

Accurate temperature measurement is vital for assessing the reaction atmosphere of industrial rotary kilns. However, dynamic water mist interference imposes significant challenges to accurate infrared temperature measurement. To this end, this study proposes a temperature measurement compensation method based on infrared multi-feature fusion. First, we developed an industrial system to collect infrared data from the kiln. To address the issue of unclear water mist in infrared images caused by its low resolution and weak texture features, we designed an artificial feature based on infrared images and temperature differences named water mist level (WML). This feature effectively represents the size of the water mist in the image. Additionally, a classification model is established to identify the WML automatically. Subsequently, a novel multi-scale feature fusion network called efficient regression feature pyramid network (ERFPN) is proposed to acquire multi-scale image features. Finally, we propose a grouped feature fusion network (GFFN) to fuse multi-scale image features, WML, and interfered temperature. Experimental results show that the proposed compensation method achieves satisfactory infrared temperature measurement results and significantly reduces the temperature measurement error caused by water mist.

Analysis of cooling effects and measurement errors in water-cooled thermocouples for total temperature measurement in aviation engine combustion chambers

Accurate measurement of the total temperature in the main combustion chamber of an aviation engine is crucial for assessing engine performance. However, a significant challenge exists in balancing the heat transfer errors caused by the necessary cooling of the measuring device with measurement precision. The research employs a combination of numerical simulations and experimental methods to investigate this issue. The numerical simulations utilized the Reynolds-averaged Navier Stokes (RANS) coupled with a conjugate heat transfer approach to study the primary flow characteristics and heat transfer properties within the combustion chamber and the probe body. The experiments were conducted in a heat-calibration wind tunnel, validating the results of the numerical simulations. The results show that by implementing water cooling, the probe's body temperature can be reduced from 1200 °C in high-temperature conditions to 400 °C, with localized temperature imbalances still present in the edge areas of the probe. Tracing the cooling water path reveals that the vertical bends in the internal channels lead to an increase in localized pressure loss of the cooling water. The study quantitatively analyzes measurement errors caused by three heat transfer modes: radiation, convection, and conduction. In low-load conditions of the combustion chamber, heat conduction plays a dominant role, with measurement errors due to conduction reaching up to 70 % of the total measurement error. Radiation further increases the temperature measurement errors of the thermocouple's junctions near the chamber wall. As the combustion chamber temperature increases, entering high-load conditions, the cooling effect of water becomes limited, reducing conduction errors and increasing radiation errors. In these conditions, radiation errors constitute 60–70 % of the total measurement error. It suggests the importance of considering temperature corrections when using water-cooled temperature measurement devices. The research reveals several crucial insights regarding the performance and limitations of water-cooled thermocouples.

Half-wave voltage controllable optical voltage sensor with arbitrary electric field direction modulation*

Abstract An optical voltage sensor with an arbitrary electric field direction modulation (AEFDM) mode is proposed to increase the half-wave voltage. The mode is realized by heterogeneous electrodes arranged in a center-symmetric way, and generates an electric field with a direction at an arbitrary angle to the light propagation direction. The finite element method and coupling wave theory are used to design and optimize the electrodes and field distribution. The experimental results show that heterogeneous electrodes and AEFDM mode are able to regulate the sensor’s half-wave voltage in a range from 1 to 30 times (47–1409 kV), with 0.5 class accuracy and 99.99% linearity. This novel AEFDM method proposes a new electrode shape and position, reducing temperature drift to 50% compared to solid-state voltage divider method.

Thermal analysis of V-pad thermocouple used on once-through steam generator tubes

Once-Through Steam Generators (OTSGs) are used in the oil sands industry to produce steam, which is injected into oil sands reservoirs to produce bitumen. OTSGs consist of a combustion zone where natural gas is combusted to generate thermal energy, which boils water within tubes that pass through the OTSG. In a typical OTSG, the temperature at the external wall of the tubes can reach as high as 500 °C, or even higher. Temperature measurement of the tube wall is critical for safe operation and this can be determined by using thermocouples. However, the placement of the thermocouple and its shield to protect it on the tube outer wall interferes with heat transfer and, thus, the measured temperature from the thermocouple may not be equal to the bare tube (no thermocouple) temperature. Here, we examine how the placement of a thermocouple on a tube wall in an OTSG affects the tube wall temperature by using finite-element-based heat transfer modelling. Depending on the foulant thickness, and the placement of the thermocouple, the local tube wall temperature increases by 45 to 55 °C. The results show that using thermocouples with shields can yield a temperature measurement error of order of 15–30 °C above the actual bare tube temperature. The results provide a basis for correcting the thermocouple/shield derived value to the actual temperature of the tube wall.

Precision Temperature Measurement and Error Analysis for Three-Wire PT100 Resistance Temperature Detector (RTD) using LTSpice

This work focuses on temperature measurement and error extraction for Resistance Temperature Dependence (RTD). RTD is notable for its high accuracy, linearity, and stability. However, obtaining a system error of less than unity in RTD is critical. A platinum RTD is an ideal option if the system requires an accuracy level over a wide temperature range (-200°C to +800°C). Therefore, this work investigated the temperature measurement and extraction of error in RTD by simulating a three-wired PT100 RTD using LTSpice. The analytical calculations were also developed to demonstrate the RTD’s error and were compared with the simulation results for verification purposes. It was discovered that the optimized temperature measurement and percentage errors are 0.01°C and 0.004% respectively. The values of Vc, Sense Resistor (RSENSE), and Reference Resistor (RREF) for the excitation current were found to be significant to maximize the output voltage and mean absolute error (MAE) on the test set, offering insights into the model's overall fit, average deviation, and sensitivity to outliers. Results reveal strong correlations between PV module temperature, irradiance, and AC power generated.

Multi-physical field coupling modeling of microwave heating and reduction behavior of zinc oxide

Microwave heating provides a cleaner pyrometallurgical method for separating zinc from electric arc furnace dust (EAFD, a solid waste generated during steel production with excellent dielectric properties). Despite its undisputed heating efficiency, the impact of microwave heating characteristics on the zinc removal process from EAFD remains unclear. This paper focuses on ZnO, one of the primary components of EAFD, and develops an electromagnetic-thermal-chemical reaction model to analyze the effects of microwave power and graphite addition on heating behavior, reduction reactions, field distributions, and reaction kinetics. The results indicate that the heat generated from the interaction between microwaves and materials exhibits inherent non-uniformity, leading to the localized thermal effect and making the error of general temperature measurement methods. Increasing microwave power and adding graphite enhance heating efficiency and promote the reduction reaction but also result in significant internal-external temperature disparity and worse temperature distribution. At 1000 W, with 1.2 times the stoichiometric amount of graphite addition, the temperature measurement error reaches approximately 200 °C, potentially affecting the kinetic results to a certain degree. The non-isothermal kinetics results from simulations indicate that the localized thermal region exhibits a lower average activation energy of 47.5 kJ/mol compared to experimental results, suggesting that the lower activation energy of the ZnO reduction reaction during microwave heating is primarily caused by the localized thermal effect rather than the non-thermal effect. Additionally, in the energy distribution, the higher proportion of return loss during heating underscores the importance of a well-designed microwave heating system.

Automatic adjustment of the parameters of the temperature measurement algorithm in a wide range

Existing methods and devices for measuring temperature are described, and it is noted that resistance thermometers are most widely used as temperature sensors in systems for collecting and monitoring environmental parameters. It is shown that the main source of temperature measurement errors when connecting sensors remotely is the resistance of the connecting lines, determined by their length. The main methods for reducing the measurement error caused by the influence of the resistance of the vehicle connection line on the temperature measurement results are considered. The advantages of two-wire circuits in comparison with three- and four-wire circuits are formulated. As an analogue, a two-wire method of measuring temperature with an integrating converter is considered in detail, implemented on the basis of a device with a built-in microcontroller, and allowing to obtain such advantages as reducing measurement time, detuning from interference and quantization noise and, as a result, reducing the temperature measurement error in comparison with similar two-wire measurement methods. An alternative two-wire method for measuring temperature with automatic adjustment of the parameters of the measurement algorithm is proposed, aimed at increasing the measurement accuracy by weakening the influence of the time constant. The search for the optimal time of the first integration step was carried out. Experimental studies and evaluation calculations were carried out to confirm the effectiveness of the proposed solution. The coefficient of variation when the resistance of the vehicle changes in the range of 1–4 kOhm lies in the range, and the range of change in the relative error of resistance measurement when using auto-tuning within the specified range has decreased by more than 4 times. The results of the experiment allow us to count on the possibility of using the method in control and regulation systems with remote connection of temperature sensors.

High-performance Smart Home System based on Optimization Algorithm

Background: With the recent COVID-19 pandemic, people have become increasingly concerned about their physical health. Therefore, the ability to monitor changes in the surrounding environment in real-time and automatically improve the environment has become a current hot topic to improve the overall health level. Objective: This article describes the design of a high-performance intelligent home system that can simultaneously perform monitoring and automatic adjustment functions. Methods: The ESP8266 was used as the core controller, and the DHT11 and G12-04 sensors were used to collect data, such as temperature, humidity, and ambient light intensity. The sampling frequency was increased and the sampled data were processed to improve data accuracy. The sampled data were wirelessly transmitted to a PC or mobile terminal for real-time display. When the sampled data underwent sudden changes, an alert message was sent via the mobile terminal. Based on the real-time changes in ambient light, an improved lighting brightness adjustment algorithm combining bang-bang and single neuron adaptive PID control was used to adjust the lighting brightness. Results: After testing the system designed in this paper and analyzing the errors compared to standard values, the temperature measurement error ranged from 0% to 0.01107%, and the humidity measurement error ranged from 0% to 0.03797%. The improved algorithm was simulated and tested using MATLAB software and compared with traditional PID algorithms and single-neuron adaptive PID algorithms. The improved algorithm did not overshoot during adjustment, and the system reached a steady state much faster than traditional algorithms. Conclusion: The system showed good performance in real-time, stability, and accuracy, fully demonstrating the effectiveness of the devices and algorithms used in the system. This provides ideas for the design and improvement of future smart homes.

Heat transfer error analysis of high-temperature wall temperature measurement using thermocouple

Thermocouples are widely employed as the temperature measuring element for an aero-engine. However, due to the unique characteristics of the aero-engine test environment, there exist significant temperature measurement errors of the thermocouple. The steady-state heat transfer error of the thermocouple can be attributed to convection heat transfer error, wire heat transfer error, and radiation heat transfer error. The flow and heat transfer process of wall temperature measurements using armored K-type thermocouples were simulated numerically by Fluent software at the Mach number of 0.1–0.5 and the temperature of 700∼1200 K. The results indicate that convective error accounts for more than 55 % of the total transfer error. To reduce excessive temperature measurement errors, a novel airflow stagnation cover structure around the thermocouple was presented. When the diameter of the stagnation cover is 2 mm, the error of the thermocouple is reduced to less than 1 %.

Theoretical and numerical investigation of introduced errors in surface temperature measurements of pistons applied in internal combustion engine

DMP (Differences in Material Properties) often exists between temperature sensors and combustion chamber components of ICE (Internal Combustion Engine), resulting in IEs (Introduced Errors) in the temperature measuring process. In this paper, we present a comprehensive theoretical analysis of the correlation between introduced errors on chamber components and the DMP. The analysis process considers the characteristics of the chamber's heat transfer BC (Boundary Condition). Then, simulations are used to investigate the introduced error of hardness plug and thermocouple sensors on the top surface of an ICE aluminum alloy piston. The theoretical analysis results indicate that heat transfer BC can be segmented into the time-averaged and fluctuation parts, and IEs are linearly superimposed under these two parts of BCs. The heat conductivity impacts the time-averaged error and the product of heat conductivity, density, and heat capacity impacts the fluctuation error. The experimental and simulation results indicate that the max time-average error of the hardness plug on piston is 2.4 °C, with a fluctuation introduced error of 3.6 °C and a total introduced error of 6 °C; and the time-average introduced error of thermocouple sensor is 9.4 °C, with a fluctuation error of 5.8 °C and a total introduced error of 15.2 °C. Finally, a fluctuation error correction method is proposed to reduce error. After correction, the error is reduced by 71 % and 66 % for hardness plug and thermocouple, respectively.