I am considering using a KZE series (by United Chemi-Con) electrolytic capacitor, which has a specified lifespan of 5,000 hours at 105°C with rated ripple current. Now, for an example assume I am using it at room temperature (25°C) with half the rated ripple current and half the applied working voltage. How long would the capacitor last? In future, how would I calculate it?
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The general rule used is that usable lifetime doubles for every 10°C reduction in operating temperature. This should be considered an upper bound on expected life.
As the capacitor ages its capacitance will decrease and ESR will increase. The increase in ESR can result in a temperature increase if running high ripple currents which in turn increases the rate of aging.
Also make sure that the increase in ESR / decrease in capacitance over the cap's life span is acceptable. For instance, make sure that the circuit still operates when the cap hits -20% of baseline capacitance, or whatever its end of life rating is.
Other factors can increase the rate of aging as well. For instance, the more air flow over the cap, the shorter its life expectancy due to evaporation effects. On the other hand, a reduction in temperature from the air flow can more than offset this issue if the cap is running warm.