During the starting up period of a centrifugal pump the rotational speed is accelerated rapidly from standstill to its final speed and then established at the rated speed. While flow rate and total dynamic pressure rise are boosted accordingly. In the point of view of a theoretical treatment, pump start-up offers difficulty because both rotational speed and flow rate are variable. A far more difficulty is the linearization of the pump speed which is fundamentally impossible. Since a change in speed of the pump is one of transients cause leading to variation in the pressure and discharge. In order to predict the behaviour of the pump during start-up transient condition a mathematical model is developed. The model is based on a working practical approximation in that the total start-up time is reduced by one third. The influence of the two most important parameters, kinetic energy in the piping system and kinetic energy of the pump, is taken into account in the form of a ratio called hereafter an effective energy ratio. Preliminary test of the model was made on the existing piping systems related to nuclear reactors. As a case of study the safety of the Tehran Research Reactor (TRR) primary piping system, an open pool MTR-type research reactor, is considered. It is demonstrated that, by comparing the curves resulted from the numerical solution of the model in which design data of the TRR are used with the experimental characteristics curves of the TRR, the possibility of primary coolant pump turbining is exceptional.