Can I connect a normal 12 Volt 150AH C10 battery with a solar 12 volt 150AH C20 battery in series for my 24Volt solar system
Can you - yes. Is it a good idea - no.
A 150AH @ C10 and 150AH @ C20 battery are not the same &quot;size&quot; because one is rated at the 10-hour discharge rate and the other is rated at the 20 hour discharge rate.
Since you have not specified some other chemistry, I'm assuming both are lead-acid 12V batteries, as the majority of 12V batteries are.
If by &quot;normal&quot; you mean &quot;automobile starting&quot; battery, that one may have short life due to construction optimized for a very different service than a deep cycle (or &quot;solar&quot;) battery. On the other hand, by pairing them you are limited to the solar battery's discharge rate if you want to get rated capacity (150AH @ 20hr - 7.5 A - at 24V about 180 Watts before inverter losses, perhaps 150 Watts after inverter losses.)
If that seems like a very low power output, welcome to the ugly realities of solar power, where almost everyone starting out undersizes their battery bank until they have the expensive lesson that it costs a lot to be cheap.