# How does an air temperature sensor response time vary with wind speed?

### QUESTION: How does an air temperature sensor response time (or tau 62.3% time constant) vary with air speed?

ANSWER: Simply put, as the air speed past a sensor quadruples, the sensor time constant (and response time) will only halve since the convective heat transfer will only double. The relationship is based on laminar flow across a circular cylinder, which is the typical shape of precision temperature sensors or their filter caps. The following mathematical relationships define the behavior.

### Tau τ63.2% changes inversely with one over the square root of wind speed.

If a temperature sensor time constant (or response time) is given at a particular wind speed, 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.

Multiplication factors to find the equivalent temperature sensor time constant at different wind speeds, if sensor time constant is listed at 4 m/s air flow.
Wind SpeedTime constant τ63.2% multiplication factor
4 m/s1.000
3 m/s1.155
2 m/s1.414
1 m/s2.000
0.5 m/s2.828
Multiplication factors to find the equivalent temperature sensor time constant at different wind speeds, if sensor time constant is listed at 3 m/s air flow.
Wind SpeedTime constant τ63.2% Multiplication factor
4 m/s0.866
3 m/s1.000
2 m/s1.225
1 m/s1.732
0.5 m/s2.449
Multiplication factors to find the equivalent temperature sensor time constant at different wind speeds, if sensor time constant is listed at 2 m/s air flow.
Wind SpeedTime constant τ63.2% Multiplication factor
4 m/s0.707
3 m/s0.816
2 m/s1.000
1 m/s1.414
0.5 m/s2.000
Multiplication factors to find the equivalent temperature sensor time constant at different wind speeds, if sensor time constant is listed at 1 m/s air flow.
Wind SpeedTime constant τ63.2% Multiplication factor
4 m/s0.500
3 m/s0.577
2 m/s0.707
1 m/s1.000
0.5 m/s1.414

### 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 air temperature sensors Ø0.5mm 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, MeteoTemp, E+E Electronic EE08 and others) which use cylindrical porous filter caps have an additional convection effect inside the filter cap to deal with. Their response time may be even slower than the above results for solid cylindrical sensors like PT100, RTD and thermistor probes suggest.

### References

1. Gnielinski, V., "Berechnung mittlerer Warme- und Stoffubergshoeffizienten an laminar und turbulent uberstromten Einzellkorpern mit Hilfe einer einheitlichen Gleichung," Forschung im Ingenierwesen, Vol. 41, pp 145-153, (1975).

2. https://en.wikipedia.org/wiki/Time_constant

3. https://www.sfu.ca/~mbahrami/ENSC%20388/Notes/Forced%20Convection.pdf

4. https://www.brighthubengineering.com/hvac/91056-calculation-of-forced-convection-heat-transfer-coefficients/

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