Something to consider. The below image is of an AC coil solenoid. It could just as easily been a relay coil. The coil is 24 VAC 60 Hz 10 Watts ot at least that is what it was. The removed solenoid was 24 VDC and replaced with this one. My understanding is that it worked real good for about 10 min.
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This was part of my little pile of dead parts I used when explaining to technicians what not to do and always pay attention to detail and work from the drawings and parts listings. This is what can happen when a 24 VAC coil is used in a 24 VDC application.
Ron