I plan on using nichrome wire as the source of heat, and it will be wrapped around an aluminum tube ...
What will you be using for insulation? It will short circuit if wrapped directly around an aluminum tube.
I would like to be able to melt a few different kinds of plastics, so this should be able to at least get up to 250 degrees Celsius.
In an actual single screw plastics extruder, most of the heat is generated by shearing friction between the barrel and polymer. From what I've read in open source link #3, these DIY extruders externally heat feedstock to above their melting point, and use the auger (screw) to convey melt to a single hole die, and generate enough force to extrude it. This must be the case since it has a throughput of only 1 kg/hr - not much shearing there - so nearly all of the heat is generated by the heating element, and it will be the determining factor of how hot the melt gets.
Would you plug the heater into the power outlet? From what I gathered, a PID essentially regulates the temperature by using a thermistor to measure the temperature and compare it to the desired temperature, and keeps said temperature by turning the relay on and off.
I didn't watch the entire video, but yes, the presenter was wiring the output power controller SSR to an AC receptacle.
PID (Proportional-Integral-Derivative) is a type of feedback control loop that compares a desired setpoint value to a measured one (determining the error between them), applies a set of "PID" equations to that error, and manipulates an output to maintain as small an error as possible. It doesn't matter much to the loop, as far as PID equations are concerned, what type of feedback device is used.
For temperature, thermocouple, RTD (Resistance Temperature Device), and thermistors are popular, but if you use a permanent magnet DC motor with an off-the-shelf DC motor controller to turn the screw, chances are it has two "PID" loops - an 'outer' speed loop (matching 0-100% speed potentiometer setting to 0 to 100% armature voltage (or, for the next step in control quality, to tach feedback) and an 'inner' current loop relating current feedback (typically, voltage drop across a small value resistor in series with the armature) to maximum bridge output current.
I've read up some, and have learned that 100K NTC thermistors are popular for 3D printer nozzle control. Keep in mind most of the industrial controllers you'll find are designed for thermocouple and/or RTD sensing. RKC C100s are good, robust controllers (I've used them in hundreds of loops), but at those prices they've either "fallen off of a truck", are good pulls/overstock, or knock-offs. RKC is a Japanese company. If the instructions actually are in Chinese with an English translation as mentioned in the video then the chances are good it's a knock-off. Billy_Joule's 'for a few dollars more' example link to an RKC REX-C100FK02-V*AN that costs only $9.03, hmmm ... typical RKC single unit pricing is more like $200.
Pay attention to controller part numbers! For an RKC C100, 'V' means it has a DC voltage pulse (transistor) output. It could also have been 'M' (relay), 'G' (TRIAC), or a '7' or '8' (0-20 mA, or 4-20 mA analog outputs, respectively). I don't know what the 'FK02' input type is - '02' is the range, but don't know what to make of 'FK'. 'JA3' is standard thermocouple, 'DA1' is RTD, but ''FK' isn't listed in my manual (another clue it is a knock-off?).
Fabricate a guard (1/4" Lexan should do) to enclose your extruder rig. PLA and PET are both extremely hygroscopic, and when heated can easily degrade into a brownish soup with a consistency between motor oil and gear oil. You won't want this stuff flying around - 3rd and 4th degree burns.