I tried deriving the LMTD of a parallel flow double pipe heat exchanger by first finding the temperature profiles of the hot and cold fluid and then averaging them over the entire length of the Heat exchanger. Then I subtracted these average values hoping that it would give me the relation for LMTD, but I'm stuck. The relations for temperature variation and average temperature are in the picture.
EDIT: I have come one step closer to make the relation look like the one for LMTD given in books.
The relation would match with the relation of LMTD given in books if we assume the wall surface temperatures to be equal. Do books take that assumption while deriving LMTD?