Step 1: Read the two reactions as a cycle.
First impure zirconium combines with iodine to make $ZrI_4$, then at high temperature that same $ZrI_4$ breaks back into pure zirconium and iodine. So the metal goes in dirty and comes out clean.
Step 2: Note the key feature, a volatile iodide.
The trick here is making a volatile metal iodide and then decomposing it. This separates the metal from non volatile impurities, which stay behind.
Step 3: Compare with zone refining.
Zone refining uses a moving molten zone and depends on impurities being more soluble in the melt. There is no iodide here, so it is not zone refining.
Step 4: Compare with the Mond process.
The Mond process purifies nickel using carbon monoxide to form nickel carbonyl, not an iodide. So it is not that either.
Step 5: Compare with electrolytic refining.
Electrolytic refining uses an electric current with anode and cathode, again nothing like these gas phase iodide reactions.
Step 6: Identify the method.
Forming a volatile iodide and decomposing it on a hot filament is exactly the van Arkel method, used for Zr and Ti.
\[ \boxed{\text{van Arkel method}} \]