Counterflow logarithmic mean temperature difference is actually the upper bound: A demonstration

The logarithmic mean temperature difference (ΔTLM or LMTD) is a corner stone concept used in the design and analysis of heat transfer equipment, such as heat exchangers. The LMTD methods are well established in the Chemical Engineering framework and, in general, apply a counterflow LMTD corrected by some factor to assess the “true” mean temperature difference between the fluids exchanging heat. The counterflow LMTD is used because it is undoubtedly believed that its value for two countercurrent streams is higher than when the same streams are arranged in concurrent flow. Although this principle may be supported by direct comparison between the counterflow and parallel flow LMTD’s for any given process conditions, a demonstration of its universal validity is not found in the literature. This article presents a proof founded in mathematical and physical arguments that this belief is indeed a fact.