I have secured 14581 general rank in AIEEE. Can I get admission to Mechanical engineering in NIT? List the NIT's where I can get admission?

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YES!! You can.
NIT Durgapur.
NIT Patna
NIT Nagpur
NIT Kurukshetra

u can get the seat in your state having nit under the name of home state quota. it is not easy to get with all India rank in top 5 nit's i.e., in Warangal , Trichy,Surathkal , Delhi, ... only home state quota can help you to get in any of these nit's if you belong to that particular state having a nit of any of these.

14581 is a bit of a higher rank so you might not get mechanical engineering in most of the top NIT's. mechanical is a very sought after branch nowadays especially after the recession so the demand is very high. there has been a drastic change in the cutoffs for this branch over the last three years. the following link will show you the cut offs for various branches for all the colleges that absorb students based on AIEEE rank. this should give you a clear idea:

http://admissions.aglasem.com/?p=19802