A model of an energy selective electron (ESE) engine with linear heat leakage and Lorentzian transmission probability is established in this paper.The expressions for the main performance parameters of the ESE engine operating as a heat engine or a refrigerator are derived by using the theory of finite time thermodynamics.The optimum performances of the ESE engine are explored and the influences of the heat leakage,the central energy level of the resonance,and the width of the resonance on the performance of the ESE engine are analyzed by using detailed numerical examples.The optimal operation regions of power output and efficiency (or cooling load and coefficient of performance (COP)) are also discussed.Moreover,the performances of the ESE engine with Lorentzian transmission probability are compared with those with rectangular transmission probability.It is shown that the power output versus efficiency (or cooling load versus COP) characteristic curves with and without heat leakage are all closed loop-shaped ones.The efficiency (or COP) of the ESE engine decreases as the heat leakage increases.It is found that as the resonance width increases,the power output and efficiency (or cooling load and COP) increase to a maximum and then decrease due to the finite range of energies which contribute positively to the power generation or refrigeration in the electron system.Especially,when heat leakage is taken into account,the characteristic curves of maximum efficiency (or maximum COP) versus half resonance width are parabolic-like ones,which are quite different from the monotonic decreasing characteristic curves obtained in previous analyses without considering heat leakage.The results obtained in this paper can provide some theoretical guidelines for the design and operation of practical electron energy conversion devices such as solid-state thermionic refrigerators. A model of an energy selective electron (ESE) engine with linear heat leakage and Lorentzian transmission probability is established in this paper. Expressions for the main performance parameters of the ESE engine operating as a heat engine or a refrigerator are derived by using the theory of finite time thermodynamics. the optimum performances of the ESE engine are explored and the influences of the heat leakage, the central energy level of the resonance, and the width of the resonance on the performance of the ESE engine are analyzed by using precise numerical examples The optimal operation regions of power output and efficiency (or cooling load and coefficient of performance (COP)) are also discussed. More over, the performances of the ESE engine with Lorentzian transmission probability are compared with those with rectangular transmission probability. It is shown that the power output versus efficiency (or cooling load versus COP) characteristic curves with and without heat leakage ar e all closed loop-shaped ones. The efficiency (or COP) of the ESE engine decreases as the heat leakage increases. It is found that as the resonance width increases, the power output and efficiency (or cooling load and COP) increase to a maximum and then decrease due to the finite range of energies which contribute positively to the power generation or refrigeration in the electron system. Especially when heat leakage is taken into account, the characteristic curves of maximum efficiency (or maximum COP) versus half resonance width are parabolic-like ones, which are quite different from the monotonic decreasing characteristic curves obtained in previous analyzes without considering heat leakage. The results obtained in this paper can provide some theoretical guidelines for the design and operation of practical electron energy conversion devices such as solid -state thermionic refrigerators.