How to increase start cycle time of an induction motor?

In an AC machine that undergoes an inductive type of start - like a squirrel cage induction or synchronous machine started "across-the-line" - the deciding factor on how often a machine can be started in succession is (almost always) the temperature reached by the bars and shorting end rings on the motor rotor. More specifically, it is the brazed joint between the two - because the braze material will "flow" at a lower temperature than the bar or ring material will deform or change properties.

A drive train inertia is composed of all the parts: the motor rotor, coupling(s), gears, and the actual driven equipment - which is a pump, in your case. Note that the pump inertia is comprised of at least two items as well - the actual inertia of the impeller design, and the presence (or absence) of liquid being moved by the pump. Higher inertias require more torque to accelerate - which corresponds to either higher currents, longer acceleration times, or both (and both tend to create more heat in the motor windings).
Induction motor
To make the time to reach operating speed take longer means reducing the gap between the torque required to accelerate the load and the torque produced by the motor over the entire acceleration curve. On the motor side, this may accomplished by increasing the inertia of the motor's rotor. It may also be accomplished by changing the impedance of the rotor winding and thus changing the "shape" of the developed torque curve. A change (i.e. increase) in the number of turns of the stator winding will also reduce the developed torque, leading to a longer time required for acceleration. On the pump side, opening some of the valves so that the casing has more liquid inside will definitely yield a longer acceleration time as well by effectively increasing the driven equipment inertia.

If the idea is to take less time during a single start attempt, the opposite kind of process is necessary. On the motor side - the idea is to reduce the temperatures seen by the rotor bars, rings, and bar/ring joint. Choosing a higher-resistivity bar material can be of some help, as long as it does not go overboard by being too close at the "operating speed" end of the load torque curve. This works because the higher impedance of the path effectively reduces the amount of current that can flow in the circuit - which in turn leads to less heat generation. On the pump side - close off all the valves and have the impeller spinning in a vacuum, or nearly so.

To increase the frequency of start attempts (i.e. less time between starts), not only does the motor need to draw less current, but it also must be more effective at getting rid of the heat produced. We've already talked about how to lower the current draw; to get rid of the heat means moving more coolant (generally air, but sometimes something else in a submersible application) past the "hot" portion of the machine. This can be accomplished by adding separately powered fans (not all that practical for a submersible) that circulate internal coolant over the rotor (increases convective heat removal) - or by circulating more coolant over the exterior of the motor (increases radiated heat removal). Alternatively, the internal components of the motor itself can be adjusted (in the design phase) to give a "low rise" solution.

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