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A zener diode limits a voltage and a solar panel provides a voltage. Then put them together without limiting the current then you have smoke, a burnt zener diode and a burnt solar panel.
Wrong!
Say you have a panel which puts out a no-load Voltage of 7V and a short-circuit Current of 200mA in bright sun, and you shunt it with a 5.1V 1W Zener. The panel will deliver about 150mA @ 5.1V. In that case, the Zener power dissipation in the Zener is 5.1V*0.15A = 0.77W, which is less than the Zener's 1W rating...
A Zener is frequently used as a shunt regulator between a solar panel and its load. If the load consumes some of the panel current (and the load is never disconnected) then the Zener dissipation does not have to absorb the full panel output.
If you need a shunt regulator with a higher power dissipation rating than the Zener alone, you use the following circuit, where a Zener is combined with an NPN Power transistor. In this example, a 12V Zener dissipates <200mW while the NPN dissipates 27W with a panel current of 2A. I have used this simple regulator to float a lead-acid battery with a 22Voc 20W panel. Obviously, the NPN has to be mounted on a large aluminum heatsink...
A solar panel is never damaged by short-circuiting it. In fact, the data sheet always lists the panel's short circuit current rating, which it is is tested for, and which is specified right on the panel's data sheet... A solar panel is just like a current-limited power supply. You cannot hurt it by short-circuiting it...