You need to validate your process(ESR by 1ml blood) with standard procedure.Moreover you need to construct a glass tube for that too.You can better to dilute your blood to desire amount with a Solution (Ringed solution) that is compatible with EDTA and Blood cell also.But after all you need to validate it with standard procedure.
I would just say be careful of using a very narrow tube. As the tube narrows, the cells will encounter the tube walls more often and this will effect their settling rate.