1. Purcell factor(PF) can be calculated as the ratio between power flow from QD with and without metallic nanoparticle nearby.
2. I expect that you represent the QD as a dipole in COMSOL RF/Wave optics module.
3. You can create a small sphere around QD in the geometry domain. It should enclose only emitting dipole. For example radius=5nm.
4. Than you integrate the Poynting vector over the outer surface of that sphere. Make integration both for the case of the free dipole and dipole with the nanoparticle.
5. PF= ratio between two integrals of Poynting vector (with and without the nanoparticle).
P.S.
1. Could you please specify which equations do you need?
2. Could you describe the task? Is QD placed on a surface or not?
Dear Danylo Komisar , It is a great explanation. But I can't understand that how to integrate Poynting vector in it. Can you please explain it more clearly? It will help me a lot.