Thermometer

Board for temperature measurement built around MLX90614 infrared surface temperature sensor.

Platform

Arduino core used is Raspberry Pi Pico / RP2040 by Earle F. Philhower, III v3.7.0.

InterboardSerial custom library is used for communicating with the calibration board.

Libraries

Title	Version
lvgl	8.3.11
TFT_eSPI	2.5.34
Smoothed	1.2.0
SerialTransfer	3.1.3
NeoPixelConnect	1.2.0
JLed	4.13.1
Adafruit MLX90614	2.1.5
Adafruit LC709203F	1.3.4
Adafruit FT6206	1.1.0
Adafruit BusIO	1.15.0
ArduinoJson	7.0.3
InterboardSerial	latest

Theory Highlights

For any object

$$ε + R + 𝜏 = 1$$

$ε$ = Emissivity

$R$ = Reflectivity

$𝜏$ = Transmissivity

For non-transparent object

$$𝜏 = 0; ε + R = 1$$

Stefan-Boltzmann Law

$$M_e = P/A = εσ(T_o^4 - T_a^4)$$

$M_e$ = radiant exitance = defined as $P/A$

$P$ = energy per unit time = power = radiant flux

$A$ = area

$σ$ = Stefan's constant

$T_o$ = temperature of an object

$T_a$ = ambient temperature

Sensor/object arrangement

 sensor (M_s)                                object
+-----------        emitted (M_o)           +------+
|            <==============================|      |
|            <-----------------------------_|      |
+-----------        reflected (M_r)        /|------+
                                          /
                                         /
                                        / ambient 
                                       / to be reflected									   

$$M_s = M_o + M_r = εσ(T_o^4 - T_s^4) + Rσ(T_r^4 - T_s^4)$$

$M_s$ = total sensor exitance

$M_o$ = exitance coming from the object itself

$M_r$ = exitance coming from the surroundings = radiation reflected by the object

$T_o$ = object's temperature

$T_s$ = sensor's temperature (on its die)

$T_r$ = room (ambient, background) temperature

Application

MLX90614 leverages the above to come up with its temperature readings. It has an element monitoring its own temperature, as well as another sensor registering incident energy flux. It uses an emissivity value of 1.00 by default. As a note, in the equation above the sensor's firmware assumes $T_s = T_r$.

A way to adjust the calculations for a different emissivity is to get the temperature reading from the device with its default emissivity, thereby getting the incident flux value. Then use the above formula backward, using an arbitrary emissivity of choice and now taking the temperature for the unknown variable.

Also, when doing calculations manually on MCU, the $T_s = T_r$ assumption can be tweaked, as it has been noted that in real-life scenarios the sensor can have higher temperature (perhaps due to self-heating) than the background.

Calculations

First, we get both the object temperature $T_o$ as well as the sensor temperature $T_s$ with device's emissivity $ε$ set to 1.0. For the sake of this example, we get $T_o = 73.0$ and $T_s = 69.0$. For the sake of simplicity the room temperature is assumed the same as the sensor temperature. Applying the equation above we get:

$$M_s = 1 * σ * ((73.0)^4 - (69.0)^4) + (1 - 1) * σ * ((69.0)^4 - (69.0)^4)$$ $$M_s / σ = 5,731,120$$

Now we calculate the actual object temperature using what we believe is the real emissivity of whatever material we are measuring. Let's say the material is anodized aluminum with emissivity 0.77. Applying the same formula backward we get:

$$M_s = 0.77 * σ * (T_o^4 - (69.0)^4) + 0.23 * σ * ((69.0)^4 - (69.0)^4)$$ $$M_s / σ = 0.77 * T_o^4 - 17,453,683.17$$ $$5,731,120 = 0.77 * T_o^4 - 17,453,683.17$$ $$T_o^4 = 30,110,133.98$$ $$T_o = 74.07$$