qThe ideal conditions underlying various methods proposed in literature for the thermal design of multi-stream plate-fin heat exchangers are discussed. The basic differential equation for heat transfer from a fin connected to two walls, when the fin bases are at unequal temperatures is solved. Two general situations when T-A > T-B > T-O and T-A > T-O > T-B are investigated. The possible mechanisms, with respect to fin efficiency, of heal exchange in a plate-fin exchanger passage are examined and the relevant heat transfer equations are derived. The significance of the mechanisms is discussed,;It has been shown that the equations for heal transfer across fin bases remain identical for all mechanisms. The relations between various dimensionless factors involved are presented in the form of graphs. The significance of the findings and their use in the development of a general method for rating multi-stream plate-fin heat exchangers are elucidated.