IIT Kharagpur is one of the older establishments of IIT. The dual degree is a 5-year course with BTech in Manufacturing Science and Engineering and MTech in Industrial Engineering and Management. Manufacturing Science is similar to mechanical engineering but also has some differences.
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IIT Bhubaneswar is one of the new IITs. Although it does not have the same reputation as IIT Kharagpur, it offers decent infrastructure and facilities for M.Tech in Mechanical Engineering.
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It is difficult to choose which institute is better as the branches are different. Mechanical Engineering is a core subject, thus choosing it will give you several options in your career. Whereas, IIT Kharagpur is more reputed and will be a better option if you want to study and further research on your subject abroad. The institute has many collaborations with foreign universities like the University of Melbourne, Curtin University, University of Wollongong, University of Alberta, and the University of Auckland. So, choose according to your convenience.