Entropy generation is recognized as a common measurement of the irreversibility in diverse processes, and entropy generation minimization has thus been used as the criterion for optimizing various heat transfer cases. To examine the validity of such entropy-based irreversibility measurement and its use as the optimization criterion in heat transfer, both the conserved and non-conservative quantities during a heat transfer process are analysed. A couple of irreversibility measurements, including the newly defined concept entransy , in heat transfer process are discussed according to different objectives. It is demonstrated that although thermal energy is conserved, the accompanied system entransy and entropy in heat transfer process are non-conserved quantities. When the objective of a heat transfer is for heating or cooling, the irreversibility should be measured by the entransy dissipation, whereas for heat-work conversion, the irreversibility should be described by the entropy generation. Next, in Fourier’s Law derivation using the principle of minimum entropy production, the thermal conductivity turns out to be inversely proportional to the square of temperature. Whereas, by using the minimum entransy dissipation principle, Fourier’s Law with a constant thermal conductivity as expected is derived, suggesting that the entransy dissipation is a preferable irreversibility measurement for heat transfer.