I have never seen any particular benefit to a permanently-installed thermocouple in a forge for typical knifemaking use. To tell you anything useful, it would seem logical that the thermocouple needs to measure the temperature at the workpiece location. Very few forges I have seen provide an even temperature profile throughout the entire forge volume, so measuring the temperature in the working zone will require putting the thermocouple in the working zone, which is typically quite small. Sticking a fragile thermocouple in the sweet-spot seems like it will inevitably reduce the space available for the workpiece, result in lots of broken thermocouples and prove frustrating.
I tend to use a handheld Mineral Insulated TypeK thermocouple 6mm diameter and 600mm long below the handle (1/4" x 24"). This is rigid enough to poke about in the forge, hand-held, and establish the temperature profile within the forge interior. IF this shows that the forge is at constant temperature throughout, I see no problem with putting in a fixed thermocouple somewhere that reads the "correct" temperature whilst being out of the precious working zone.
I tend to use burners based on a commercial Venturi mixer (the Amal" atmospheric injector) that allow very precise adjustment of the air:fuel ratio and therefore the flame temperature. Putting the thermocouple in and adjusting the temperature, then removing the thermocouple in the confident expectation that the temperature will remain constant, is no problem with this system. This is a significant factor in my thinking about permanently-installed thermocouples. There are homebuilt blown burner systems that provide similarly fine mixture adjustment, but I have not encountered a well-documented online Naturally-Aspirated "build-a burner" that comes anywhere close to the Amal in terms of precise temperature control. The commercial Australian "Gameco" burners look like they should be a credible commercially-made alternative.
The "set-and-forget" adjustment is manual and takes some time, which is not normally a problem for most hobby smiths. I can see the advantages of a PID-controlled forge for the pro, who can't afford the 10-20 minutes of hands-on adjustment time and would prefer to go and do something that pays while the controller does the adjusting. The pro probably has the experience and equipment to ensure that the PID control is not having an adverse effect on his/her process. I am not convinced that the hobby smith will be able to apply similar QC.
PID control effectively switches between 2 states, which we can regard as "too cool" and "too hot", to control the temperature. This gives a saw-tooth graph of temperature against time. The size of the saw-teeth depends on the output cycle time. With a short output cycle, the saw-teeth are small. With a long output cycle, the saw-teeth are large. For electric HT ovens, I set a 2-second output cycle time and use an SSR to allow fast switching, so the saw-teeth are small enough to be negligible. With a forge, the switching to the "too hot" stage is usually with a solenoid valve, which cannot switch very fast and needs a relatively long output cycle. For gas-fired ceramics kilns, the output cycle time is typically 30 seconds. I would not see this as a particular problem for a forging forge (the temperature cycles widely when forging anyway), but it would probably make it a pretty poor way to Heat-Treat. The Pro will usually have a proper HT setup separate from the forge, so this is irrelevant, but the amateur on a budget is much more likely to use their temperature-controlled forge for HT and I feel that adjusting manually to a steady temperature is the better option.
The reason I use typeK for adjusting forges is that they are relatively cheap and have a maximum range of 1372 degC, 2500 degF. This is high enough to read welding temperatures for those who have little or no previous experience of welding.
I should throw in a couple of caveats here: I work with gas burners and temperature control systems as part of my job and I'd like to think I'm reasonably good at it, but my actual forging and welding experience is minimal compared to many on the forum. I have built a few forges and electric HT ovens for smiths and knifemakers. To the best of my knowledge these work pretty well, but I can't absolutely discount the possibility that the guys are just too polite to tell me my stuff sucks.
When I first wondered about welding temperatures, I had no real idea what constituted a "good" welding temperature. I spoke to several very good smiths and was still none the wiser. I went to a hammerin a few years ago and there was a vertical Propane forge set up next to a little power hammer (a 15 kg, 33 lb Anyang) that was being used for pattern welding by many people, but mainly by beginners. The forge had been adjusted by eye and experience by the guy who built it and it just worked. After seeing a lot of Damascus made with very few failures, I stuck in a type S thermocouple (about $500-worth) and took a reading. It was running at 1300 degC, 2372 degF. I could fish around in the working zone and get readings from 1285 to 1315 degC, but it was so close to 1300 degC that, if I'd set up a PID control system to run at 1300 degC and got the same result, I'd have been entirely happy with it.
I therefore tend to regard 1300 degC as a good welding temperature: If a beginner has a forge running at 1300 degC and makes a bad weld with steel of around 0.8% Carbon, they can be pretty sure it is not the fault of the forge temperature and can look elsewhere for the problem. This is not to say that 1300 degC is "the best" welding temperature: the guys who pattern weld a lot tend to do it at lower temperatures because the contrast tends to be better, but as far as I can tell, they tend to get the technique down at "around" 1300 degC and then play with reducing the temperature once their skill level is no longer a major variable.
Type N thermocouples were developed as an "improved" type K for temperatures above about 1000 degC, 1832 degF, with greatly reduced "drift". They were developed as a system of thermocouple alloy and sheath material for Mineral Insulated assemblies. I use them when I build Heat-Treat ovens because I did some investigation into drift rates with types K, N and S at work a few years back and type N came out as a no-brainer. The biggest limitation of type N for the forge user is that they top out at 1300 degC, 2372 degF. Using type K gives the ability to read and adjust to 1300 degC plus-or-minus a few degrees.
For a forging-only forge, I'd use type N every time. Typically, forging temperatures seem (to me: see caveat above) to range from about 1000 degC, 1832 degF, up to about 1150 degC, 2102 degF; well within the range of typeN. Nicrobell-sheathed Mineral Insulated type N thermocouples are readily available for use to 1250 degC, 2282 degF.
If low initial cost is not a major consideration, I'd probably use type S, or maybe type R if that is the regional preference. Both tend to be expensive (Platinum for one leg and Platinum/Rhodium alloy for the other leg; Pt/10%Rh for type S, Pt/13%Rh for type R, IIRC), and fragile, but they last a long time and keep their accuracy. I have a few type S industrial thermocouples removed from decommissioned equipment and use them when testing forges and burners, but they were around $500 apiece when new and I wouldn't buy new ones if I broke them. They are 500 mm (20") long, the wire diameter is 0.45mm (.018") and the sheaths are Recrystallized Alumina. They are good to at least 1600 degC, 2912 degF. I did buy a couple of cheap ones off ebay, but the wires are very thin (.012mm, .0005") and the sheaths are probably Aluminous Porcelain, good to at least 1400 degC, 2552 degF. I've not used them yet.
Pt100 Resistance sensors are very accurate, but the tables only go to 850 degC, 1562 degF. max. Most of the PT100 assemblies I see are only rated to 400 degC, 752 degF, max. I'd use them in a dedicated tempering oven, but that's about it.
I think type K is probably the "best" choice for a smith learning to forge, so long as the limitations are understood.
