score:6
Visa-free transfers in China are very limited. In particular, your inbound flight into China must arrive at the same airport as you are the later departing from.
So arriving in Tianjin and departing from Beijing will not be possible. The train trip in between does not make a difference.
For the 72-hour visa-free transit, you can find the rules at regulations at: http://www.travelchinaguide.com/embassy/visa/free-72hour/
For the 24-hour visa-free transit, you can find the rules here: http://www.travelchinaguide.com/embassy/visa/intro2.htm#T - but they are even more restricted, requring you to stay in the international transfer part of the airport. Since you are taking a domestic train, these rules do not apply either.
Upvote:4
This has changed since the question was answered in Dec '15.
At almost all international airports in China, almost all nationalities get 24 hours of transit without any type of visa. There are special restrictions for these passports:
There are also some special restrictions for specific airports (not Tianjin or Beijing).
You can leave the airport during the 24 hours of transit, and can in fact depart from a different airport in China, as long as it is within 24 hours of arriving in China. Your destination must be to a different country than you arrived from. For the purpose of the visa-free transit, the Hong Kong and Macau SARs are treated as different countries.
Source: https://en.wikipedia.org/wiki/Visa_policy_of_China#Visa-free_transit
This link also describes the 72-hour visa-free transit & 144-hour visa-free transit rules applicable to specific nationalities.