A method for measuring the surface temperature of a metal microreactor based on digital twin technology

Through digital twin technology, the internal temperature field of metal microreactors is simulated, which solves the problem of difficulty in accurately testing the surface temperature of metal microreactors in the existing technology, and achieves high-precision, real-time temperature monitoring and standardized testing.

CN119845444BActive Publication Date: 2025-06-13NANJING INST OF MEASUREMENT & TESTING TECH +1
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Patent Information

Application Number
CN202510323532.7
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2025-03-19
Publication Date
2025-06-13
Estimated Expiration
2045-03-19

AI Technical Summary

Technical Problem

The prior art is difficult to accurately test the surface temperature of metal microreactors. The non-removable sensors lead to difficulty in calibration, dynamic temperature changes are difficult to capture in real time, and the lack of unified testing standards leads to poor data comparability.

Method used

Digital twin technology is used to build a digital twin physiological model that combines virtual and real, simulates the temperature changes of the internal microchannels of metal microreactors and surrounding thermal oils, and determines the placement position of the standard sensor by real-time monitoring of the inlet temperature and internal temperature field distribution, calculates the display error of the detected temperature sensor, and obtains accurate temperature values.

Benefits of technology

High-precision measurement of the surface temperature of metal microreactors is achieved, the problems of sensor aging and non-removal are avoided, the accuracy and real-time nature of temperature monitoring are ensured, and the standardization of surface temperature testing of metal microreactors is promoted.

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Abstract

The present invention discloses a method for measuring the surface temperature of a metal microreactor based on digital twin technology, which relates to the technical field of metrological calibration. The method includes: establishing a geometric model according to the actual geometric dimensions and internal structure of the metal microreactor, and performing mesh division; inputting the physical information of the metal microreactor into the divided geometric model, mapping it into a corresponding digital twin physical model, and performing model solution, and inferring the internal temperature field distribution of the metal microreactor according to the solution result to provide a mechanism for surface temperature measurement; by understanding the change mechanism of the internal temperature field, determining the position where the standard sensor is placed on the surface of the microreactor, and then calculating the indication error of the temperature sensor to be tested in the metal microreactor through a comparison method. Through the method of the present invention, high-precision and standardized measurement of the surface temperature of the microreactor can be achieved, effectively extending the service life of the equipment and reducing the maintenance cost.
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Description

Technical Field

[0001] The present invention relates to the technical field of metrological calibration, and particularly to a method for measuring the surface temperature of a metal microreactor based on digital twin technology. Background Art

[0002] As an efficient reaction device, metal microreactors are widely used in the fields of chemical engineering, pharmaceuticals, and energy. Their core advantage lies in achieving high mass and heat transfer efficiency through a miniaturized structure, but the accurate measurement of their surface temperature is relatively complex.

[0003] In the prior art, temperature sensors are usually embedded inside the interlayer of the microreactor to achieve online monitoring. However, due to the requirement of structural compactness, such sensors cannot be disassembled, resulting in the inability to calibrate or maintain them after long-term use. The measurement accuracy decreases significantly with the aging of the sensors, affecting the stability and safety of the reaction. During the operation of the microreactor, due to the rapid flow of reactants and the exothermic / endothermic effect, the surface temperature is always in a dynamic fluctuation state.

[0004] Due to the lack of understanding of the temperature change mechanism inside the microreactor, traditional contact temperature measurement methods directly arrange standard sensors at four corners and the center to measure the temperature. This method has a large measurement error and cannot reflect the true temperature condition of the microreactor. In addition, there is currently no unified standard for measuring the surface temperature of metal microreactors in the industry. Different testing methods are adopted in different research or production environments (such as random point temperature measurement, single-point static monitoring, etc.), resulting in poor data comparability and difficulty in supporting process optimization or horizontal evaluation of equipment performance. Summary of the Invention

[0005] The purpose of the present invention is to provide a method for measuring the surface temperature of a metal microreactor based on digital twin technology. By constructing a virtual-real combined digital twin physical model through digital twin technology, the temperature changes of the microchannels inside the metal microreactor and the surrounding heat transfer oil are simulated, so as to characterize the temperature distribution on the entire surface of the microreactor, aiming to break through the limitations of traditional technologies and provide a standardized and intelligent solution for measuring the surface temperature of metal microreactors.

[0006] The present invention realizes the above purpose through the following technical solutions:

[0007] A method for measuring the surface temperature of a metal microreactor based on digital twin technology, the method comprising:

[0008] Arranging a temperature sensor on the feed pipe of the metal microreactor to continuously monitor the temperature change of the feed port and record the indication value of the temperature sensor;

[0009] Establish a geometric model according to the actual geometric dimensions and internal structure of the metal microreactor, and perform mesh division on the geometric model;

[0010] Input the physical information of the metal microreactor as boundary conditions into the divided geometric model, map it into the corresponding digital twin physical model, and perform model solution. Infer the internal temperature field distribution of the metal microreactor according to the solution results, providing a mechanism for surface temperature measurement;

[0011] Determine the placement position of the standard sensor according to the simulation results of the internal temperature field distribution, and calculate the indication error of the temperature sensor to be tested in the metal microreactor by the comparison method, so as to obtain the accurate value of the temperature sensor to be tested.

[0012] As a preferred solution of the present invention, the method for performing mesh division on the geometric model specifically includes:

[0013] Define the size required for mesh division as , then the entire geometric model is divided into tiny models, and the expression is:

[0014] ;

[0015] In the formula, represents the geometric body to be divided;

[0016] Define the global variable for division, that is, the geometric shapes of the tiny models, expressed as:

[0017] ;

[0018] In the formula, is the size of the tiny shape after division, and triangular or quadrilateral division is used by default; tends to 0;

[0019] Create named selections for the edges, faces, and bodies of the divided mesh, and update the named mesh.

[0020] As a preferred solution of the present invention, the physical information of the metal microreactor includes: feed reactant components, density, thermal conductivity, initial temperature and flow rate at the feed inlet, microchannel material, thickness and heat flux, density and heat flux of the heat transfer oil.

[0021] As a preferred solution of the present invention, the method for performing model solution and inferring the internal temperature field distribution of the metal microreactor specifically includes:

[0022] Define boundary conditions, iteration time and number of solution steps, and adopt The turbulence model is used as a solution method to solve the digital twin physical model, and the expression of the turbulent kinetic energy equation is:

[0023] ;

[0024] In the formula, is the fluid density; is the turbulent kinetic energy; represents a small change; is the velocity vector; is the spatial coordinate; is the turbulent viscosity coefficient; is the turbulent kinetic energy generated by the laminar velocity gradient, is the turbulent kinetic energy generated by buoyancy; is the dissipation rate; is the fluctuation generated by diffusion;

[0025] The expression of the diffusion equation is:

[0026] ;

[0027] ;

[0028] In the formula, , , are constants; is the dissipation rate suppression function; is a constant; , ;

[0029] According to the solution results, the temperature field change nephogram inside the metal microreactor is simulated, and the temperature level is judged by different colors. The reasoning principle is:

[0030] ;

[0031] In the formula, represents the temperature change result after model reasoning; is the size of the small shape after division; represents a small change; both represent boundary conditions.

[0032] As a preferred solution of the present invention, the placement position of the standard sensor has the following requirements:

[0033] Avoid the temperature cold spots at the four corners of the metal microreactor;

[0034] Avoid the temperature hot spots between the microchannels in the metal microreactor;

[0035] The standard temperature sensor is arranged around the sensor to be inspected.

[0036] As a preferred embodiment of the present invention, the method for calculating the indication error of the temperature sensor to be tested in the metal microreactor by the comparison method includes:

[0037] Collect the indications of the standard sensor and the sensor to be tested at preset time intervals and calculate the average value. The formula is:

[0038] ;

[0039] ;

[0040] In the formula, is the average value of the standard sensor, represents the th standard sensor, is the maximum number of standard sensors; is the number of measurements; is the maximum number of measurements, is the standard sensor measurement value; is the average value of the temperature sensor to be tested; is the measurement value of the temperature sensor to be tested;

[0041] Then the indication error of the temperature sensor to be tested is: .

[0042] By real-time monitoring of the feed port temperature and combining with the dynamic changes of the internal temperature field, the actual temperature distribution of the reactor under different operating states can be reflected, ensuring temperature monitoring in a high-dynamic reaction environment. Based on the simulation calculation of digital twin technology, it does not rely on traditional hardware sensors to directly measure temperature, avoiding problems such as sensor aging or being unable to be disassembled, effectively extending the service life of the equipment and reducing the maintenance cost. By establishing a digital twin physical model, the temperature field distribution inside the metal microreactor can be accurately simulated, and then the surface temperature of the microreactor can be accurately calculated, overcoming the problem that traditional contact measurement methods cannot effectively capture the dynamic changes of the internal temperature and ensuring the accuracy of the test data. By calibrating the indication error of the temperature sensor to be tested with a standard sensor, the standardization and consistency of temperature measurement can be ensured, which helps to promote the standardization of the surface temperature test of the metal microreactor. Description of the Drawings

[0043] To more clearly illustrate the technical solutions of the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the drawings in the following description are only some embodiments of the present invention. For those of ordinary skill in the art, without creative efforts, other drawings can be obtained based on these drawings. Among them:

[0044] Figure 1 is the method flow chart of the present invention;

[0045] Figure 2 is the schematic diagram of the planar structure of the metal microreactor in the embodiment of the present invention;

[0046] Figure 3 is the schematic diagram of the geometric model established in the embodiment of the present invention;

[0047] Figure 4 is the schematic diagram of mesh division in the embodiment of the present invention;

[0048] Figure 5 is the contour map of the temperature change after stabilization in the embodiment of the present invention;

[0049] Figure 6 is the schematic diagram for comparison of traditional surface temperature measurement methods;

[0050] Figure 7 is the schematic diagram of the surface temperature measurement method in the embodiment of the present invention. Detailed implementation manners

[0051] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the following will clearly and completely describe the technical solutions of the embodiments of the present invention in conjunction with the drawings of the embodiments of the present invention. Obviously, the described embodiments are some, but not all, of the embodiments of the present invention. Based on the described embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art fall within the scope of protection of the present invention.

[0052] As Figure 1 shown, it is an embodiment of the present invention. This embodiment provides a method for measuring the surface temperature of a metal microreactor based on digital twin technology, which is used to solve problems such as difficult calibration due to non-removable sensors in metal microreactors, difficulty in real-time capturing of dynamic temperature changes, and lack of standardized test specifications.

[0053] This method includes the following steps:

[0054] S1: Arrange temperature sensors on the feed inlet pipeline of the metal microreactor to monitor the temperature change of the feed inlet in real time and record the readings of the temperature sensors.

[0055] Due to the high requirements for temperature monitoring of the metal microreactor, the temperature sensor in this embodiment preferably uses an RTD (Resistance Temperature Detector) or a thermocouple. The RTD has high precision and stability and is suitable for applications that require high-precision measurement. The thermocouple performs better in the high-temperature range and is suitable for a wide temperature range. The specific selection can be determined according to the actual temperature range and precision requirements of the feed inlet.

[0056] To accurately capture the temperature changes at the feed inlet, the frequency of temperature data acquisition should be at least once per minute. If the system response is fast, or the reaction process is relatively dynamic, the sampling frequency can be increased to once per second to more precisely capture the instantaneous temperature changes. The accuracy of the temperature sensor should be ±0.5°C or higher.

[0057] S2: Establish a geometric model based on the actual geometric dimensions and internal structure of the metal microreactor, and perform mesh generation.

[0058] Convert the actual geometric features of the metal microreactor into a digital three-dimensional model, and decompose the geometric model into tiny units through mesh generation, which is convenient for numerical simulation and calculation. After mesh generation, each small unit in the model can be calculated according to different physical conditions (such as temperature, flow rate, etc.), so as to obtain an accurate temperature field distribution.

[0059] In one of the embodiments, the method for performing mesh generation on the geometric model specifically includes:

[0060] Define the size required for mesh generation as , then the entire geometric model is divided into tiny models, and the expression is:

[0061] ;

[0062] In the formula, represents the geometric body to be divided;

[0063] Define the global variable for division, that is, the geometric shapes of

[0064] tiny models, which is expressed as:

[0065] In the formula, is the size of the tiny shape after division, and by default, triangular or quadrilateral division is used; tends to 0;

[0066] Create named selections for the edges, faces, and bodies of the meshed grid, and update the named grid.

[0067] S3: Input the physical information of the metal microreactor as boundary conditions into the divided geometric model, map it into the corresponding digital twin physical model, solve the model, and infer the temperature field distribution inside the metal microreactor based on the solution results, providing a mechanism for surface temperature measurement.

[0068] The physical information of the metal microreactor includes: the components, density, and thermal conductivity of the feed reactants, the initial temperature of the feed inlet (obtained from the indication of the temperature sensor in step S1), the flow rate, the material, thickness, and heat flux of the microchannel, the density and heat flux of the heat transfer oil, etc. A digital twin physical model that combines the virtual and the real is constructed through digital twin technology to simulate the temperature changes of the microchannel inside the metal microreactor and the surrounding heat transfer oil, thereby characterizing the temperature distribution on the surface of the entire microreactor.

[0069] In a specific embodiment, the method for solving the model and inferring the temperature field distribution inside the metal microreactor specifically includes:

[0070] Define the boundary conditions, iteration time, and number of solution steps, and use the turbulence model as the solution method to solve the digital twin physical model. The expression of the turbulent kinetic energy equation is:

[0071] ;

[0072] In the formula, is the fluid density; is the turbulent kinetic energy; represents a small change; is the velocity vector; is the spatial coordinate; is the turbulent viscosity coefficient; is the turbulent kinetic energy generated by the laminar velocity gradient, is the turbulent kinetic energy generated by buoyancy; is the dissipation rate; is the fluctuation generated by diffusion;

[0073] The expression of the diffusion equation is:

[0074] ;

[0075] ;

[0076] In the formula, , , are constants; is the dissipation rate suppression function; is a constant; , ;

[0077] Based on the solution results, a contour map of the temperature field change inside the metal microreactor is simulated. The temperature level is judged by different colors. The reasoning principle is as follows:

[0078] ;

[0079] In the formula, represents the temperature change result after model reasoning; is the size of the small shape after division; represents the small change amount; both represent the boundary conditions.

[0080] The temperature field inside the metal microreactor is calculated through accurate numerical simulation, and the temperature distribution under different time and space conditions is inferred. Using the turbulence model for solution can simulate the complex flow and heat transfer processes of the fluid inside the microreactor, thus more accurately predicting the temperature field change and providing real temperature data support.

[0081] S4: Determine the placement position of the standard sensor according to the simulation results of the internal temperature field distribution, and calculate the indication error of the temperature sensor to be tested in the metal microreactor through a comparison method, so as to obtain the accurate value of the temperature sensor to be tested.

[0082] Specifically, the placement position of the standard sensor has the following requirements:

[0083] Avoid the temperature cold spots at the four corners of the metal microreactor;

[0084] Avoid the temperature hot spots between the microchannels in the metal microreactor;

[0085] The standard temperature sensor is arranged around the sensor to be tested.

[0086] In one of the embodiments, the indication error of the temperature sensor to be tested in the metal microreactor is calculated through a comparison method. The method includes:

[0087] Collect the indications of the standard sensor and the sensor to be tested at preset time intervals and calculate the average value. The formula is:

[0088] ;

[0089] ;

[0090] In the formula, is the average value of the standard sensor, represents the th standard sensor, is the maximum number of standard sensors; is the number of measurements; is the maximum number of measurements, is the standard sensor 's measured value; is the average value of the temperature sensor under test; is the measured value of the temperature sensor under test;

[0091] Then the indication error of the temperature sensor under test is: .

[0092] The error is calculated by comparing the indications of the standard sensor and the sensor under test, thereby realizing the calibration of the temperature sensor under test. Through this method, the deviation of the sensor under test can be identified and its reading can be corrected, thereby ensuring the temperature measurement accuracy of the metal microreactor, improving the reliability and accuracy of the measurement, and ensuring the consistency between the test results and the actual situation.

[0093] Figure 2 Shows a planar structure design of the metal microreactor in an embodiment of the present invention, mainly including elements such as microchannels, temperature sensors in the interlayer, and thermally conductive silicone oil, which can reflect the physical characteristics of the actual device.

[0094] Multiple temperature sensors are arranged in the microchannels of the metal microreactor, and temperature conduction is carried out through the circulating flow of the thermally conductive oil. The temperature sensors monitor the temperature of the oil flow passing through the channels in real time. A temperature sensor is arranged at the inlet of the metal microreactor for real-time monitoring of temperature changes.

[0095] According to Figure 2 the planar structure in Figure 3 the geometric model shown is established, mapped into the corresponding digital twin physical model to realize scene reproduction, and mesh division is carried out.

[0096] Define the global variable of the division as 0.5 mm, that is , and adopt the global triangular division method to realize the mesh division of the entire model. Define Figure 3 the inlet of the microchannel in Figure 4 as inlet, define the outlet of the microchannel as outlet, the thermally conductive silicone oil in the interlayer of the microreactor as oil-fluid, the boundary of the microreactor channel as re-wall, the outermost edge of the thermally conductive silicone oil as bian-wall, and the result after updating the network is as shown in

[0097] In this embodiment, in order to study the overall heat conduction performance inside the metal microreactor, the reaction substances are simplified, and the component of the inlet is defined as liquid pure water with a density of 1000 kg / m 3 , and the thermal conductivity is 0.6 , the speed at the inlet is 1.2 m / s, and the temperature sensor at the inlet shows 50 °C. Define the edge of the microchannel as Hastelloy metal, and the heat flux is 50 w / m 2 , the thickness of the channel tube is 1 mm. The space between the microchannel and the entire microreactor is filled with heat-conducting silicone oil. Define the density, thermal conductivity, etc. of the heat-conducting silicone oil. Define the material of the outermost part of the metal microreactor as Hastelloy, with a thickness of 2 mm, and also define the density, thermal conductivity, etc.

[0098] Substitute the above relevant physical information into the formula to perform a steady-state analysis of the model. Set the number of iterations to 200 steps and the step size to 1 s. After the temperature stabilizes, the temperature field distribution of the entire metal microreactor is as Figure 5 shown. Figure 5 It is a screenshot of the Fluent software (a commercial CFD software package), where Static Temperature is the static temperature, and [C] represents the temperature unit.

[0099] From Figure 5 , it can be seen that after the temperature stabilizes, the temperature inside the entire microchannel is relatively high, and the overall temperature distribution of the metal microreactor is as follows:

[0100] (1) The temperature range between channels is (47.7 - 50.0) °C;

[0101] (2) The temperature at the inlet is relatively high, and the temperature at the outlet is slightly low;

[0102] (3) The temperatures at the four corners of the metal microreactor are slightly different. The temperature at the inlet is between (45.4 - 48.7) °C, and the temperature at the outlet is relatively low, with a temperature range between (36.1 - 38.4) °C.

[0103] Through the analysis of Figure 5 , the temperature change situation on the surface of the microreactor during the reaction process can be understood, and at the same time, it can be obtained that the temperature inside the metal microreactor is uneven.

[0104] As Figure 6 shown, the four corners (A, B, C, D) and the center (O) of the metal microreactor are the cold and hot points of temperature. The traditional surface temperature measurement method is prone to situations of too high or too low temperature, and the temperature uniformity is poor, so the temperature change situation inside the microreactor cannot be truly understood.

[0105] In the embodiment of the present invention, according to the simulated temperature change cloud map, w standard sensors are placed near the sensors to be inspected and at positions where the temperature field is relatively concentrated, avoiding the temperature cold points at the four corners of the metal microreactor and the temperature hot points between the microchannels, as Figure 7As shown in the figure. Data is collected every 2 minutes, and 10 groups of data are recorded. By comparing the data, the error of the sensor under test can be calculated. The average temperature of the four standard sensors is 44.3 °C, and the average temperature shown by the sensor under test is 47.2 °C. Therefore, the indication error of the sensor under test is 2.9 °C. Using the same method, other temperature sensors on the microreactor can be measured.

[0106] In summary, the present invention provides a high-precision and intelligent temperature measurement solution by establishing a digital twin physical model and combining precise simulation and simulation of the internal and external temperature fields of the metal microreactor, which makes up for the deficiencies of traditional temperature measurement methods in dynamic reaction, temperature change, and standardized testing.

[0107] Using digital twin technology, the present invention can comprehensively and accurately simulate the heat conduction and fluid dynamic changes inside the metal microreactor, breaking through the problem that traditional contact temperature measurement in the prior art cannot reflect temperature fluctuations and changes. By combining temperature sensors with virtual models, higher temperature monitoring accuracy and real-time feedback capabilities are ensured, effectively improving the working stability and safety of the metal microreactor.

[0108] By comparing and calibrating the indication values of the sensor under test with standard sensors, the accuracy of temperature measurement can be ensured under different operating environments, solving long-existing problems such as sensor aging and position errors, and providing a strong guarantee for the long-term operation of the metal microreactor equipment. Through simulation calculations and real-time feedback, the changing trend of temperature can be accurately reflected, and adjustments and optimizations can be made based on temperature data at different positions, further improving the thermal efficiency and production efficiency of the metal microreactor.

[0109] The above is only the specific implementation manner of this application, but the protection scope of this application is not limited thereto. Any person skilled in the art within the technical scope disclosed in this application can easily think of various changes or substitutions, which should all be covered within the protection scope of this application. Therefore, the protection scope of this application should be subject to the protection scope of the claims.

Claims

1. A metal microreactor surface temperature testing method based on digital twin technology, characterized in that: The method comprises: A temperature sensor is arranged on the feed port pipeline of the metal microreactor to monitor the temperature change of the feed port in real time and record the indication of the temperature sensor; Establishing a geometric model according to the actual geometric size and internal structure of the metal microreactor, and meshing the geometric model; The physical information of the metal microreactor is used as the boundary condition input into the divided geometric model, mapped into the corresponding digital twin physical model, and the model is solved. The temperature field distribution inside the metal microreactor is inferred based on the solution results, providing a mechanism for surface temperature measurement. The physical information of the metal microreactor includes: feed reactant composition, density, thermal conductivity, feed inlet initial temperature, flow rate, microchannel material, thickness and heat flux, heat transfer oil density and heat flux; The method for solving the model and inferring the temperature field distribution inside the metal microreactor specifically includes: Define boundary conditions, iteration time, and number of solution steps, using The turbulence model is used as a solution method to solve the digital twin physical model. The turbulence kinetic energy equation is expressed as: ; In the formula, is the fluid density; is the turbulent kinetic energy; Indicates a small change; is the velocity vector; is the spatial coordinate; is the turbulent viscosity coefficient; is the turbulent kinetic energy generated by the laminar velocity gradient, The turbulent kinetic energy generated by buoyancy; is the dissipation rate; Fluctuations due to diffusion; The diffusion equation is expressed as: ; ; In the formula, , , is a constant; is the dissipation rate suppression function; is a constant; , ; According to the solution results, a temperature field change cloud diagram inside the metal microreactor is simulated, and the temperature is judged by different colors. The reasoning principle is: ; In the formula, Indicates the temperature change result after model inference; is the size of the tiny shapes after division; Indicates a small change; All represent boundary conditions; Determine the placement position of the standard sensor according to the simulation result of the internal temperature field distribution, calculate the indication error of the temperature sensor under test in the metal microreactor by comparison method, so as to obtain the accurate value of the temperature sensor under test; The placement of the standard sensor has the following requirements: Avoid the cold spots in the corners around the metal microreactor; Avoid high temperature spots between microchannels in metal microreactors; The standard temperature sensors are arranged around the sensor under test.

2. According to claim 1, a metal microreactor surface temperature testing method based on digital twin technology is characterized in that: The method for meshing the geometric model specifically includes: Define the size of the mesh to be divided into , the entire geometric model is divided into A tiny model, the expression is: ; In the formula, Represents the geometry that needs to be divided; Define the global variables of the partition, that is The geometry of a tiny model is expressed as: ; In the formula, To determine the size of the small shapes after division, triangle or quadrilateral division is used by default; tends to 0; Create named selections of meshed edges, faces, and volumes, and update the named mesh.

3. According to claim 1, a metal microreactor surface temperature testing method based on digital twin technology is characterized in that: The method of calculating the indication error of the temperature sensor under test in the metal microreactor by the comparison method comprises: The readings of the standard sensor and the sensor under test are collected once every preset time period and the average value is calculated. The formula is: ; ; In the formula, is the average value of the standard sensor, Indicates Standard sensors, is the maximum number of standard sensors; is the number of measurements; is the maximum number of measurements, For standard sensors The measured value of is the average value of the temperature sensor under test; is the measured value of the temperature sensor under test; Then the indication error of the temperature sensor under test is for: .

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