The analogous expressions between MBHE-HL and TTHE-HL are listed in Table . These analogies were obtained from Bertoli et al. and expanded in the present work by including the analogy for 𝜁 and an alternative analogy for 𝜂. This alternative analogy is better understood if we consider heat transfer at the solid-fluid interface in the MBHE as occurring across a shell of differential thickness. Thus, the TTHE analogue to ℎ 𝑝 in MBHE is the overall heat transfer coefficient between the fluid in the inner tube and the fluid in the inner annular section, based on the internal area (this work) or external area (Bertoli et al. ) of the inner tube. Thus, in Table , 𝑈 0 is the overall heat transfer coefficient between the fluid in the inner tube of the TTHE and the fluid in the inner annular section, based on the heat exchange area associated with 𝑅 0,𝑖 -analogous to the coefficient ℎ 𝑝 for convective heat transfer between the conveyor fluid and particles in the MBHE -.