how to iteratively calculate a^emod n with modulus n sized 4096 bits In most sites the exponent of the RSA public key is 24 bits. But the modulus can get to 4096 bits size. I have an accelerator that can get max. 2112 bit size modulus. It calculates a^e mod n. 

Is there a way to separate $n$ (maybe bit wise) and use the accelerator in several iterations to calculate the wanted a^e mod n?
Usually, RSA operations with the public exponent are fast, precisely because the public exponent is short. Hardware accelerators are meant to speed up operations with the private key, which are in much bigger need of it. In particular, hardware accelerators do not need to be "full width" because private key operations use the private key, which contains the factors $p$ and $q$, and allow for doing most of the work modulo $p$ and modulo $q$, both of which being twice shorter than the modulus $n$ (what the CRT is about).

A side note is that some implementation methods of exponentiations are fast but imply some fixed startup and wrapup overheads; e.g. Montgomery multiplication is popular to speed up exponentiations (which consists of many multiplications in a row), but requires converting source values into a special representation, and back at the end, both operations having a non-negligible cost. For a private key operation, the exponentiations used big exponents, and the savings implied by Montgomery multiplications dwarf these overheads, but this might not be true for a short exponent. Therefore, even if your accelerator did support a 4096-bit modulus, it is unclear whether this would have really provided some speedup for your public-key operations.



Anyway, I see no easy way to "split" a 4096-bit exponentiation into several exponentiations modulo smaller values, unless you factor the 4096-bit modulus, which should be infeasible.