Yes, you can, but there are reasons why you shouldn't, or at least would be best advised not to. Basically it has to do with the thermal differences involved. The greater the differences in the two bodies in contact, the more efficient will be the heat transfer. So when you run 25 degree tap water straight throught the chiller when the wort is 90+ degrees or so, the rate of temperature change will be quite dramatic. As you have identified in your concern about temperature, the water coming out of the chiller will also be bloody hot. (And by way of energy conservation should be captured by you and used for washing up later, or even in the house somewhere; I once grabbed enough water to give the kids a bath!) Now, if you put that hot water into your icy water bath, you are really going to go through some ice. But without recirculation, it still only takes a few minutes before the water coming out of the chiller is only warm to tepid, even though the temperature of the wort is still perhaps 50 or 60 degrees. This is because the efficiency of heat transfer and rate of temperature change in both bodies is declining all the time. It takes increasingly longer to effect each degree fall in temperature. Somewhere around 40 degrees wort temperature is sort of the optimal time to kick in the ice bath recirculation. The water coming out of the chiller is not going to be so hot as to melt your ice too quickly.
The final word: as another correspondent wrote, if you hook up the pump straight away, you will need the ice and water bath to be large enough that the hot water does not bring down the temperature of the bath. If it became hot, it would not be doing its job. So, like we have been saying, your pump should never be in contact with hot water, at least not very hot water and not for very long.
Does this make sense now?
Regards,
Steve
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