Consider a double pipe parallel flow heat exchanger, the mean temperature difference in this case turns out to be logarithmic because the temperature profiles are exponential.
If we would have made use of Arithmetic temperature difference, when the profiles were exponential, we would have got erroneous results. However, if the temperature profiles were linear then, the mean temperature difference would've been equal to the arithmetic one.
I started off by considering linear temperature profiles and proceeded just in the fashion we do for calculating the LMTD, however the mean temperature still turns out to be logarithmic, it should come arithmetic isn't it? Can this logarithmic relation be further reduced to arithmetic one?