Thank you for your response. The relays activate the solenoids which turn on the heaters. The relays I'm using are nowhere near the rating they would need to be to operate the heaters themselves (the heaters are operating at 60v and pull approx 34amps each). The thermocouples are physically welded to the metal directly under the center of each heater so that the hottest point is known and control goes from there; this is done so that we don't overheat the material past where it should be. Because it is held at the peak temperature for an extended period of time, the overall mass is all within a few degrees of the same (which we have tested extensively with additional thermocouples in various locations, including the inside of the vessel itself).
We have tried to choose a ramp rate that they could all reach without issue, but there are several problems that arise in doing so. Sometimes the vessels are of differing proportions and we end up choosing wrong for instance because it can be difficult to estimate it. We also have a heater fail sometimes and have to fix it or replace it and need to hold everything else steady while that is done. Also, the vessels have molten salt in them, in various formulations (which is what we do; we are doing research on the best salt formulas for a specific purpose, so we test a different formula every day and eventually plan to move up to testing several at a time). When the salt changes states or flows, it sometimes affects the way the overall system heats up. We have found that it is very simple to resolve these issues manually by simply pausing the temperature of the higher temperature areas while we wait for the ones that are behind to rise, but I am always interested in a better way to do things. If I have them all one Arduino, will it be able to handle a bunch of independent PIDs running? I am sorry for the long response, but while I have a basic or better understanding of the principles of what I need to do, I am relatively new to microprocessor control systems and the Arduino in particular.