Analytical solutions for thermal models of TTHE with adiabatic (TTHE-A) outer wall can be found in the works of Zuritz , Ünal , Batmaz and Sandeep , Peigné et al. , and Bertoli et al. . For this case, Bertoli et al. present a comparison of different solutions and a critical literature review. Regarding the thermal models of TTHEs with heat loss (TTHEs-HL) to the surroundings, the existing analytical solutions can be found in Zuritz -by Laplace transform -, Peigné et al. -by elimination method -, Tribess et al. (Appendix D) -by Laplace transform -, and Bertoli et al. -by self-adjoint operator method -, these last three references dealing with the analogous problem of a MBHE-HL in different flow arrangements: co-current flow , Section 8, expressions for effectiveness are developed. For a better understanding of the underlying physics, a scale analysis is performed in Section 9. Section 10 develops limit solutions of interest. In Section 11, for two case studies, the results of the different analytical solutions are compared with each other and with a numerical solution by the Trapezoidal Rule/Backward Difference Formula (TR-BDF2) composite method. Finally, the main contributions of this work are highlighted in Section 12.