We do not have time periods associated with our ranks, especially for those in the rank of recruit. Usually members are recruits for at minimum 1 week, but depending on the clan and the members schedule this can take much longer. We are looking for quality members, thus we take time to play with the new recruit and get to know them before accepting them into our clans.
We want give our members the opportunity to get to know the recruit to determine if being in a clan is right for them, this also allows your fellow clan members to play with and provide feedback on the new recruit before promotion to private. The time length associated with this process can vary between clans and some of the dependency lies with the recruit (ie. signing up on the website, playing with members, learning the honor code, ect.)
Hope this answers your question, but bottom line is there s not a magic formula that says you have to be a recruit for X amount of days then you are automatically accepted.