PhD programs are not necessarily "free". This is very much dependent on the school you are planning to attend and the department in which you plan to study.
In some cases, you can either get funding without doing much other than being a student (for a few years) ... this is generally called a fellowship. In other cases, you can get either a teaching or research assistantship in which you will teach (or TA) a class or you will assist a professor with his or her work.
In either case, you will put in about 20 hours of work a week on your duties (of course the 20 is the "textbook" statement .... in reality you might do 10 .... or, if you have a bastard of a professor you are working for, maybe 40). In addition you will be doing your coursework (about 3 courses per term) ... and coming up with your own research ideas (unless your area overlaps with the professor you are working for).
In chemical engineering (and any science, engineering, or mathematics), you are likely to be able to get an assistantship ... if you are lucky you can get a fellowship. In the arts (philosophy, theater, English, etc. etc.), you will need to be the best and brightest to get any funding. In those disciplines, you may well have to pay out of pocket.
I highly recommend you have some other goals to accomplish during graduate school. If the only thing you have after six years is a PhD, you will have wasted your time. My wife has her PhD and I will have mine in about four months. Trust me on this ... our PhDs are (will be) nice pieces of paper. The other things we've done ... are far more valuable.