Electrical – How to calculate heat in wire

edit: changed calculations to the diode's package

I would use SMT diodes (I'll explain in a minute why). 45 °C is not very hot, they can easily handle that without derating. You can use the BAS16 in SC79 package.

Thermal resistance for the SC79 package is 120 K/W. That means that without thermal losses supplying 200 mW will already heat the package up from room temperature (20 °C) to 45 °C.

You'll have to experiment a bit with power, because at equilibrium you supply as much power as is lost to the environment, and that last thing is not easy to calculate theoretically. Keep in mind that the diodes will lose more heat through conduction (to the PCB) than through convection (to the air).

Arduino can supply 100mW per pin, so you can't multiplex directly from the I/O. Three problems.

• Not only is the 100mW insufficient, if you want to multiplex 25 columns and have 200mW for each, that means you have to supply 5 A at a 4 % duty cycle. Arduino can't source nor sink that, so you'll need transistors for controlling both rows and columns.
• And that's where the diodes come in, you can't just multiplex an array of resistors, nor Nichrome wire. The resistors in adjacent colums/rows are parallel to them and they will also take a share in the load. You can't control them individually. With diodes you can do that, LED display multiplexing is done all the time.
• 25 diodes \$\times\$ 200 mA average = 5 A. Arduino can't supply that. Using an external 5 V supply is a Bad Idea™: the diodes drop only about 1 V, so the rest of the 5 V would cause 20 W dissipation in transistors and current limiting resistors. You want a power supply at around 2 V.

Further reading
Thermal Management in Surface-Mounted Resistor Applications

Two subjects are covered here. First, reality. Second, the questions posed.

First, Reality : About the most you can get from a U.S. 120 volt plug in is 1500 watts (generally speaking). So your heater element will draw more current (because of lower resistance) until the nichrome element heats up. During this transition time, one depends upon the circuit breaker characteristics to not shut off (trip off). Then, counting upon your forced air cooling (fan), the nichrome element should stabilize (temperature wise) at 9.6 ohms. That is as good as it gets for a constant applied voltage.

Second, your question : "How do I figure out the cold resistance". You need to get the manufacturer to supply you with the expected operating temperature where the resistance is 9.6 ohm ( or rather, at what temperature will the nichrome be at 1500 watts). From there, you can use the nichrome's temperature coefficient to calculate the resistance at room temperature.