Tannin extraction is not instantaneous. Only if you keep the temp over 170 will you be risking noticeable tannin extraction.
After initial vorleuf then drain, your grains will be somewhere in the 150-ish degree area. Your secondary sparge water can go into your mash tun somewhere in the 180-185 degree area depending on water volume and grain volume. Once stirred you should settle in the 165-170 area for temp in the grains.
When I used to batch sparge I usually targetted about 165-168 as my grain bed temp after stirring in the second sparge water. Keep a pitcher of cool water nearby and if you hit it a bit high add in some cool water to bring it down. It is much easier to tweak your mash tun temp down a couple degrees than up a couple.
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