I hoped this had been worked out before for an average shower giving a rule of thumb which might have said a quarter (or whatever) of the heat gets passed to the air.
The sort of statistic which might find its way into the newspaper or get quoted, although I hadn't heard it.
As you can see from the formula I provided, the shower/room heat transfer is determined by the shower flow rate and the temperature loss in the shower stream.
The temperature change in the room will, in addition to the power transfer to the air by the shower, be a function of room air volume, the current air temperature, and any additional cooling/air change in the room.
Unfortunatly there will not be an average answer due to the variables involved in each installation.
But I think I answered your original question about "how much heat typically gets transferred from the hot shower water to the room air".
I find the best combination is to run a dehumidifier in the room whilst having a shower. The room gets nice and toasty, and I can shave afterwards as the mirror doesn't fog up.
My dehumidifier is a condenser type that draws about 200w. But the effect in a shower room seems much more than that, so I guess there's quite a bit of heat coming from condensing the water vapour.
Not sure of the maths here, other than to say from experience it would be enough to maintain a comfortable temperature in an already warm room. But pretty useless at warming a cold room.
I did some calculations a while back, when I was thinking about putting a heat exchanger/extractor fan in the bathroom - if I calculated correctly, and recall correctly, most of the energy will be in the water in the air, rather than the air itself. So I stuck with the dehumidifier. I did make some rather off the cuff assumptions about the amount of water in the air, though - something like comparing the heat capacity of a litre or so of water to the heat capacity of a bathroom-full of air.
In spite of various eminent contributions, I still think it is a pretty daft calculation, especially if you have an extractor fan. Hot water comes out of the shower, some of it evaporates (stealing a lot of energy as latent heat). You extract this vapour which then dumps the energy into the plume outside.
Of course you get it back if you run a dehumidifier inside, or if it condenses on the walls and ceiling.
The short answer is that the loss to the air is negligible when comared to the energy that went into heating the water - certainly not a quarter
- or even a tenth of that.
You are using a flow of water (lots of it with a high energy density), to heat small amount of air with a low energy density.
That's because when you do the sums, its not really significant, and ultimately not really that important.
You pay the money to heat the house, you can do that by heating water and pumping it round some radiators, and by squirting it out of a shower head in direct contact with the air. One can argue that the shower heating the room is a bonus from a comfort point of view, even if not from a condensation one.
This seems a good approach but the water will have lost heat to the shower floor before it gets to the plug-hole. As it's a walk-in shower there's quite a bit of shower floor that gets wet.
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