Ph.D. study in chemical engineering is often paid for by research funds obtained by your grad-school faculty advisor, and thus it will not cost you anything if you are able to obtain such support. In addition to receiving a tuition award, you will typically receive an offer of financial support for your living expenses.
When you apply to graduate school, you will automatically be considered for support, either in the form of a research assistantship, a teaching assistantship, or a fellowship. These assistantships will usually include full tuition and fees. It’s a great deal for which good students will often qualify, and it is not usually based on need, rather is based on merit. There are also very prestigious graduate research fellowships offered by the National Science Foundation for the best students in the country. If you are able to obtain an NSF graduate research fellowship, you will be highly courted by all the top graduate schools. You must apply directly to NSF for these fellowships.
M.S. study in chemical engineering can be done either in coursework mode (never supported; you are charged tuition and fees) or in thesis mode (may be supported; may include a living stipend). The two-year coursework masters has the advantage of being straightforward to get: you apply, are admitted, take for the necessary courses, get your degree. There is no research project. The M.S. thesis path also starts with you applying and taking courses, but students are working from the beginning on a masters thesis that must be completed before the degree is awarded. There are funded research projects that lead to the M.S., and it is competitive to obtain this funding.
For more on graduate school, please visit the Department’s website on graduate studies.