Abstract: Abstract: A centrifugal pump can be operated in its reverse rotational direction as a turbine-a prime motor, and its impeller is a kind of centripetal impeller. It can convert the high pressure energy of a fluid into the rotational mechanical energy of the rotor to drive a generator to generate electricity or drive a working machine, realizing an energy recycling of the high pressure fluid. It is well-known that the slip phenomenon exists in turbomachinery, such as hydraulic turbine, steam turbine, roto-dynamic pumps and compressors. This is also true for a centrifugal pump as turbine. The slip velocity may be smaller in a hydraulic turbine than a centrifugal pump. However, this is not the case for a centrifugal pump as turbine because its number of blades is much less compared with the hydraulic turbine. The constraint ability of blade is less for fluid giving rise a smaller slip factor. In addition, a slip factor is the key parameter to estimate the turbine output power based on its existing geometry and designed flow condition to check if its output power has met the requirement by end-users. Unfortunately, this topic is not well documented currently. In this paper, we put forward an analytical method to accurately determine the slip factor at the impeller outlet in a centrifugal pump as turbine in order to predict the turbine performance correctly. Firstly, the relative eddy flow in an impeller passage with closed inlet and outlet was analyzed for a centrifugal pump in its turbine mode. Secondly, three assumptions were made for the induced velocity by the eddy on the blade pressure side, then the slip velocities and the corresponding slip factors at the impeller outlet under these assumptions were derived analytically. Finally, the slip factor formulas were validated by means of the theoretical heads of ten centrifugal pumps as turbines based on their experimental data. Two cases, namely, the slip factors at both impeller inlet and outlet, and the slip factor at the impeller outlet alone, were studied, respectively. It was identified that the slip factor formula based on the second assumption, in which the ratio of the induced velocity on the blade pressure side over the slip velocity on the impeller outlet was equal to the ratio of the areas of two triangles with the pressure side and impeller outlet as one curved edge respectively, was subject to a very good accuracy. Moreover, it was unnecessary to include the slip factor at the impeller inlet when predicting the theoretical head of a pump as turbine. The influence factors of the slip on the centripetal impeller outlet of hydraulic turbine have the entrance and outlet diameter of impeller, entrance and outlet blade angle of blade and the number of blades. The research results improved the accuracy for predicting the performance of hydraulic turbine by the theoretical method, and this kind of method will be convenient and efficient when scholars predict the performance of hydraulic turbine, and may save the cost of research. The results provide a base for more accurately predicting the performance of a pump as turbine by making use of analytical methods.