3 phase load sizing

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If a 120 volt single phase 24 amp vacuum cleaner is connected to a 3 phase 60 amp panel is the load still 24 amps? Is there an advantage of having a 3 phase panel when all your equipment is single phase? Does three phase mean you can add more load? When does the multiplier 1.732 come in to factor? I two 120 volt 24 amp vacuums that I want to connect to a 3 phase 70 amp panel that is dedicated to a small car wash and all the small car war loads are single phase.....
 
If a 120 volt single phase 24 amp vacuum cleaner is connected to a 3 phase 60 amp panel is the load still 24 amps? Is there an advantage of having a 3 phase panel when all your equipment is single phase? Does three phase mean you can add more load? When does the multiplier 1.732 come in to factor? I two 120 volt 24 amp vacuums that I want to connect to a 3 phase 70 amp panel that is dedicated to a small car wash and all the small car war loads are single phase.....
So lets boil your questions down:

Part 1: is 24 amps single phase still 24 amps single phase? Yes, 24 = 24.

Part 2: Is there an advantage to having a 3 phase panel? Yes, all power is distributed as 3 phase from the utility, but for residential services, they only give us 1 phase. It's irrelevant to the homeowner, but in the grand scheme of this this is a problem for the utilities because they have almost no control over balanced loading on their 3 phase system, which results in their needing to over compensate for it by using bigger more expensive transformers than necessary. Residential customers don't see that cost, it is baked into their higher rates. For a commercial or industrial facility, they actually want to give you 3 phase, it is a better way to ensure loads on the phases are balanced. So if you have 3 phase, you want to use it. Now, if you have 120V, then that means you have a 208Y120V 3 phase system. The 120V part of that is no different than 120V from a residential service. That's why you asking if 24 amps at 120V is the same as 24 amps is kind of a non-issue. Of course it is.

Part 3: Can you add more load with 3 phase? More compared to what? A same size single phase service? Then, more what? More amps or more kW? If you compare a 60A 208Y120V service to a 60A 120/240V single phase service, 60A is still 60A. But that is 60A per POLE, so the same 60A in a 3 phase panel is 3 poles of 60A capacity instead of 2 poles of 60A capacity in the single phase service. So if I were using 15A single pole breakers, I can put in 4 x 15A single pole breakers x 3 poles, so 12 circuits. In the single phase service, I can put in 4 single pole breakers x 2 phases, so 8 circuits. The max amps on each phase is still 60A, but that nets more circuits in a 3 phase panel. So what about kVA? That is where the 1.732 comes in.

Part 4: Where does that 1.732 factor in? If I want to determine my kVA capacity of the above 2 panels, the single phase panel is 60A x 240V = 14.4kVA. On the 3 phase panel, the load is divided between 3 phases, so the load capacity on each phase is shared by a ratio related to the square root of 3, 1.732 and the total kVA = 208V x 60A x 1.732 = 21.6kVA of total circuit capacity.

So can you do what you want? I can't tell you from what you posted, but not likely. You have a small little 70A 3 phase 208Y120 panel, and you are adding 24A to 2 of the poles. That's going to be 1/3rd of the total capacity of each of those phases, so unless they have nearly nothing else on them, it's going to be tough to squeeze that in. Typically on even a small car wash, the pumps and blowers are 5HP 3 phase, so each one is likely around 24A each, and you may have 2 blowers and 2 pumps. Most likely that little 70A panel is already loaded up.
 
Re #4, by the property of the square root of three (its square is three), the other way to get jraef's number is to multiply 120 x 60 x 3. That is what you would get more naturally by looking at balanced 120V line to neutral loads.
Since there is no net current in the neutral, you cannot tell the difference between the two load configurations. So, power factors being equal, the calculated power (and VA) must be the same for both configurations.

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