The paper presents modifications on published demonstration that the counter-current logarithmic-mean-temperature-difference (Delta T-LM) is upper bound compared with the co-current Delta T-LM. In the published demonstration, an approach was proposed but it suffered from wrong assumption. In this paper, corrections to the wrong assumption and a property of a curve derived from the Delta T-LM curve have been used to propose another demonstration that counter-current Delta T-LM is upper bound. Optimization problems have been formulated to verify the proposed developments and demonstrate the results obtained. A class of Delta T-LM approximations including two accurate approximations proposed by the author are discussed. The selected approximations are designated as Underwood's class. This class generates accurate results over the problem temperature-difference-ratio range and has the advantage of direct use of the heat exchanger (HEX) terminal temperature differences. Crown Copyright (c) 2012 Published by Elsevier Ltd. All rights reserved.