Since the heat transfer irreversibility of a heating or cooling process should be measured by the entransy dissipation rate, an equivalent thermal resistance for a heat exchanger was defined based on the entransy dissipation rate of the heat exchanger. The equivalent thermal resistance includes both the overall heat transfer resistance and the extra thermal resistance caused by the non-counter-flow arrangement and the non-equilibrium heat capacity rate ratio of a heat exchanger. The reciprocal of the equivalent thermal resistance was defined as the equivalent thermal conductance. The relationship is established between the heat transfer irreversibility and the effectiveness of a heat exchanger in terms of the equivalent thermal resistance. Finally, a formula describing the relation among the effectiveness, the equivalent thermal resistance and the heat capacity rate ratio is obtained which does not depend on the different flow arrangements. Therefore the effectiveness-thermal resistance （conductance） method is more suitable for the performance comparison of different heat exchangers.