The breaker protects the wiring not the boiler. 12,000 watts at 240 volts will require 50 amps. So, you will need a 60 amp breaker using AWG# 6 wire on a dedicated circuit.
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A circuit breaker is designed to 'break' in a circuit if a short circuit (or other malfunction) occurs. This prevents overheating (or burn-out) of the circuit wires. In older systems, you would need to find which fuse wire has fused and replace it. In a circuit breaker, once the fault has been found and corrected, the breaker is simply switched back on.
A 30 amp circuit breaker is needed unless the load is a motor circuit, then it has to be sized to 250 percent of the motors full load amps. For 30A circuit is needed breaker 30*1.25=37.5 --> 40A. If load is lamp or heater, then use breaker of group A or B. If load is motor, then use breaker of groupC or D (very hard start) or special safe breaker for motor - with variable amp setpoint. See related link also. By code you are only allowed 80% of the rating of a breaker. So 30x80%=24amps. 24 amps is the MAX allowed on a 30 amp breaker. You would need a 40amp breaker for a 30 amp circuit. 40x80%=32. So you would want a double pole 40amp breaker.
Replace it immediatly, the breaker is no longer within code.
AF refers to the ampere rating of the breaker frame and AT refers to the breaker trip rating in amps. These are not breakers that you would find in your home but industrial molded case breakers. The current trips can be changed out to different values all within the same frame size.
Hard to define... A four year old child has the strength to physically "make" a typical household circuit breaker, while a teen would be able to "make" a typical industrial circuit breaker. (The latter breaker being physically much larger than the former.) Circuit breakers have their "breaking capacity" rated in AMPS. A typical household breaker is rated at 20, 30 or 50 Amps. Industrial breakers might be rated at 500 amps or a thousand or even more. Hope this helps but if it didn't answer your question, please rephrase it and ask again.