I am currently on the search for an inline conductivity sensor for a cooling circuit project (inline so that I can constantly monitor the conductivity of my medium in the circuit).
The whole circuit is to be controlled by an arduino uno R3 board.

One conductivity sensor which I am favoring has several communication possibilites: RS485 Modbus and analog 4-20mA. The sensor is connected to a transmitter which is what would be connected to the arduino board. I have attached the data sheet of the transmitter.
20276200T10Z000K000.pdf (1.0 MB)

The RS485 Modbus I have read that there are shields that can be attached to the uno board and an exisiting library, which means that the implementation should be fairly do-able.
For analog 4-20mA I have searched in the forum and found that connecting a 250 or 500 ohm resistor between sensor and arduino analog pin will convert the input current into 0-5V. This can then be easily read by the arduino. Is the implementation really this simple? On another website in Google search I read that a special converter would be needed which has a suitable resistor integrated.

I'd certainly start by trying a simple resistor, yes. Ohm's law says that if we want 5V from a 20mA signal we need a resistor of R = V / I = 5 / 0.02 = 250R.
The signal range you can then read is V = I * R = 0.004 * 250 = 1V to 5V. That's within the range of the ADC of the Uno, albeit it with of course 'only' 10 bit resolution, which may or may not be enough for you.
Accuracy might be a problem if you don't feed the Arduino with an exact 5V power supply. In this case, you could offer an external Vref, for instance made with something like a TL431. You could use a 120R resistor in that case to bring the signal within the range of 0.48V to 2.4V and tell the Uno to use the external analog reference.
Another option is to use the internal 1.1V reference, so you'd have to bring down the signal to <1.1V with for instance a 56 Ohm resistor (close enough to 55Ohm probably).

I'd just start with the first approach and a reasonably reliable/precise 5V power supply/reference for the Arduino (so not your computer's USB port).

No. The signal goes directly to the ADC input, the resistor from input to GND. This way the current through the resistor causes the voltage that the ADC can measure.

Thanks for the several approaches!

External power source would mean that I cannot check directly the input data on serial PUTTY on my PC. To check the first approach I could either save the input data on a SD card which I can then open on my PC later on. Or with a display connected to the arduino for live data transmission correct?

Due to my lack of electronics knowledge I don't quite understand sorry. Which input do you mean, to connect the resistor to? Do you mean a resistor that e.g. connects A0 pin with GND on the arduino board itself?

Chose one of A0 to A3 for the signal and its resistor.

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