Yes there is but it depends on the type of load. The normal alternator equations are based on the assumption that the thing is loaded with a resistive load or one that is not far removed from it.
They come down in the end to deriving the terminal voltage on load from the emf ( open circuit voltage) and the volt drop in the winding due to the load current flowing through its internal resistance. If you look at standard alternator theory they cover this case. It actually becomes quite complex when you start to include all the variables, the resistance is only one factor.
When the load is a battery the whole thing is quite different and I doubt that any standard text book will give you much help as it is not a condition that happens in normal electrical engineering.
In this case the battery voltage is clamped and the current you see depends on the open circuit volts minus battery volts divided by the effective resistance of the alternator. How you determine the effective resistance is a bit of an odd ball but for air gap machines such as the axials the effective resistance is related to the winding resistance but needs a multiplying factor to get sensible results.
Flux
