How to test an air temperature sensor’s response time?
/QUESTION: How to test and determine an air temperature sensor’s response time (tau 63.2 % time constant)?
Vaisala HMP155 mounted in a wind tunnel for sensor response time (sensor time constant) testing.
ANSWER: Since most fluid temperature sensors like PT100, PRT and thermistors are cylindrical in shape, their thermal behavior with respect to flow speed is very predictable and mathematically well documented. Thus, the whole procedure of testing to determine a cylindrical sensor’s response time to temperature changes can be reduced to the following three steps (assuming a laminar flow regime):
Heat or cool the sensor to a temperature significantly different than the fluid medium (air, water, oil, …) for which its time constant has to be determined. The temperature should be different enough so that it will take the sensor 5x - 10x longer than the sensor’s time constant to reach equilibrium with the fluid temperature at the chosen fluid flow speed.
Place the heated sensor in stable uniform fluid flow (air flow, …) and log the temperature changes at a rate at least 10x faster than the time constant. It is important to know the flow speed accurately since the sensor’s time constant changes with the square root of flow speed.
Repeat this test at a second flow speed and compare whether the time constants at both wind speeds are related by the square root of the change in flow speed (if flow speed is doubled, the time constant is divided by the √ 2 = 1.4142 as shown in the table below).
Analysis using a simple Excel spreadsheet
Download an example sensor time constant analysis Microsoft Excel spreadsheet: Sensor Time Constant Analysis Spreadsheet Example.xlsx
Sensor time constant can be easily determined by calculating when the sensor temperature changed by 63.2% between any two points in time as shown in the downloadable excel spreadsheet.
To create your own analysis, follow these steps:
Paste the time series data into an excel spreadsheet. Each row should be made up of 3 columns:
A: time_stamp; B: ambient_fuild_temperature; C: heated_sensor_at_test_temperature;Create column D where for each sensor temperature data point, you calculate the temperature difference to the known ambient fluid temperature flowing past the sensor:
T_diff = T_sensor - T_fluidCreate column E where you calculate the % temperature change in temperature for each T_diff data point in the aforementioned column relative to a single time stamp (starting row time stamp) that you choose as the starting point.
Sensor tau 63.2 % time constant is the time difference between the two rows having a 0 % T_diff (starting row time stamp) and 63.2 % T_diff.
Duplicate column E at least two more times (create columns F & G) using a different time stamp as a starting point to determine the 63.2 % T_diff at different points along the plot. This is used to confirm that the tau 63.2 % time constant is constant and not also dependent on temperature difference T_diff.
IMPORTANT NOTES:
MeteoTemp combined temperature and humidity sensor
The time difference between 0 % T_diff and 63.2 % T_diff should be consistent at different points along the plot of decreasing sensor temperature for the results of measurements to be valid if the sensor design is such that the time constant is not also related to the temperature difference between the sensor and the measurement medium (air, water, …). This is the case with most temperature and humidity sensors in meteorology since, under the filter cap, they rely on natural convection to transfer the exterior temperature changes to the sensor hidden under the filter cap.
After testing and evaluating multiple sensors, you may notice that very few sensors have a stable tau 63.2 % time constant, which is independent of the sensor to fluid temperature difference.
Repeating the above test at a second airspeed is required to validate the test results because sensor thermodynamic behavior at low Reynolds numbers, where most sensors operate, is mathematically predictable. In short, quadrupling the fluid speed will halve the sensor time constant (and response time) since the convective heat transfer doubles with the square of flow speed. Since the typical shape of precision temperature sensors or sensor filter caps is cylindrical, the thermodynamic relationship is based on forced convection equations as developed by Nusselt, Prandlt, and Reynolds for laminar flow across a circular cylinder.
The relationship between fluid speed and temperature sensor time constant
If a temperature sensor time constant (or response time) is given at a single airspeed, such as for many PT100, RTD sensors, thermistors or combined temperature-humidity sensors, the time constant should be increased by the following multiplication factors for lower wind speeds as shown in the tables below.
| TIME CONSTANT CONVERSION MULTIPLICATON FACTORS (Used to find how the temperature sensor's time constant changes at different fluid speeds from the listed time constant on the sensor's datasheet.) | ||||
|---|---|---|---|---|
Fluid speed for which a new time constant is desired| If time constant is listed only for 4 m/s flow speed | If time constant is listed only for 3 m/s flow speed | If time constant is listed only for 2 m/s flow speed | If time constant is listed only for 1 m/s flow speed | |
| 4 m/s | 1.000 | 0.866 | 0.707 | 0.500 |
| 3 m/s | 1.155 | 1.000 | 0.816 | 0.577 |
| 2 m/s | 1.414 | 1.225 | 1.000 | 0.707 |
| 1 m/s | 2.000 | 1.732 | 1.414 | 1.000 |
| 0.5 m/s | 2.828 | 2.449 | 2.000 | 1.414 |
where: τ63.2 = time constant, ρ = fluid density, Cp = specific heat capacity, Vs = sensor volume, h = convective heat transfer coefficient, A = sensor surface area, k = thermal conductivity, Nu = Nusselt number, D = characteristic diameter, Re = Reynolds number, Pr = Prandtl number, U = flow velocity, μ = dynamic viscosity.
Physics and math behind the results
Governing equations of laminar heat transfer for cylindrical temperature sensors and their response to temperature changes. (Ref 1: Gnielinski) Valid for all naturally ocuring wind speeds in nature and all temperature sensors with a diameter of Ø 0.5 mm and larger.
Sensor time constant (tau 63.2 %): τ = ρ*Cp*Volume/(h*Area)
Convective heat transfer coefficient: h = Nu*2k/(π*D)
Nusselt number: Nu = 0.664*Re^(1/2)*Pr^(1/3),
Reynolds number: Re = (π*D/2)*V*ρ/μ
h = convective heat transfer coefficient, V = free stream air velocity, D = cylinder diameter, k = thermal conductivity of the air, ρ = density of the air, μ = viscosity of the air, Cp = heat capacity of air.
Combined temperature and humidity sensors
Combined temperature and humidity sensors like the Vaisala HMP155, Rotronic HC2A & Hygromet4, BARANI DESIGN MeteoTemp, E+E Electronic EE08, and others that use cylindrical porous filter caps have an additional convection effect inside the filter cap to deal with. Their response time will be dependent on the speed and magnitude of temperature changes.
τ63.2 Temperature Sensor Time-Constant Calculator
Paste a temperature step-response test into the table below. The calculator finds local time constants as the starting sensor-to-air temperature difference changes during the test.
Correct first-order target: completed response = 1 − e−1 = 0.6321205588; residual ratio = e−1 = 0.3678794412. This is τ63.2, not τ62.3.
1. Input data
Required columns: time or elapsed seconds, ambient/reference/fluid temperature, sensor temperature.
With headers, column names may include words such as time, ambient,
reference, fluid, and sensor.
Rejects late-stage calculations where the remaining signal is too small.
Use 10 to reproduce the Barani Excel example start points; use 1 for every row.
Documentation
Calculation method
For each selected start row i, the calculator computes
ΔT_i = T_sensor,i − T_ambient,i. It then searches later rows for the first point where
ΔT_j / ΔT_i ≤ e^-1. The crossing time is linearly interpolated between the sample before
and the sample after the threshold. The local time constant is
τ63.2 = t_cross − t_i.
Why 63.2% is used
A first-order sensor response follows ΔT(t) = ΔT_0 e^(−t/τ). At one time constant,
ΔT(t)/ΔT_0 = e^-1 = 0.3678794412. Therefore the completed response is
1 − e^-1 = 0.6321205588, or 63.212%. A 62.3% threshold would give
−ln(1 − 0.623) = 0.9755τ, about 2.45% low before any sampling-row error.
Experimental requirements and limitations
The method assumes a step-like temperature change, stable known airflow, stable ambient/reference temperature, and a sampling interval much faster than the expected time constant. Near equilibrium, small temperature differences are easily dominated by sensor resolution, noise, and ambient drift; use the minimum |ΔT| setting to exclude those rows. If τ changes with starting |ΔT|, the sensor/filter-cap system is not behaving as a single fixed first-order time constant over the whole test.
Python tool that complements the web calculator
This Python tool can be used independently from the command line or imported into other Python workflows. It calculates local \(\tau_{63.2}\) values from temperature step-response data, writes result tables, and generates plots for \(\tau_{63.2}\) versus starting temperature difference.
Files:
What the Python tool does
For each selected start point, the tool calculates the residual sensor-to-reference temperature difference:
A first-order temperature time constant is reached when the remaining temperature difference has fallen to e−1 of its starting value:
This is equivalent to a completed response of:
The crossing time is linearly interpolated between the samples surrounding the threshold, and the local time constant is:
Repeating the calculation from multiple start points produces τ63.2 versus |ΔTstart|, showing whether the apparent time constant changes as the sensor approaches the reference temperature.
References
How to does an air temperature sensor response time vary with wind speed? by Jan Barani, BARANI DESIGN Technologies s. r. o., 2019/08/09
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