Aside from Stanford University, Rice University, and MIT, what other schools are known for their engineering programs (bioengineering/chemical engineering in particular)?
Cal tech, Michigan, Purdue, Delaware (for chem e), UIUC, UT, JHU(for BME), Cornell, RPI, Case Western, Pitt, etc…
Lots of good programs out there.
https://www.usnews.com/best-colleges/rankings/engineering-doctorate-biological-agricultural
https://www.usnews.com/best-colleges/rankings/engineering-doctorate-biological-biomedical
https://www.usnews.com/best-colleges/rankings/engineering-doctorate-chemical
Firstly, your best choice for the traditional engineering disciplines is usually your state flagship campus. That being said, as regards Chemical Engineering, here are some significant programs know for their substantial ChE academic resources or relatively big ChE departments;
U Wisconsin-Madison,
U Minnesota-Twin Cities
Northwestern U
Iowa State U
Texas A&M
U California-Berkeley
U Pennsylvania
Cornell U
U Delaware
Small schools with very fine programs
Renssalaer Polytechnic Institute
South Dakota School of Mines & Technology
Clarkson University
U Rochester
Case Western Reserve University
Stevens Institute of Technology
New Jersey Institute of Technology