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re: home heating efficiency?
14 dec 2004
daestrom wrote:
>> >...the house might have a q = (70-10)400 = 24k btu/h furnace with
>>>th = 256k/(24k+16k) = 6.4 hours and tc = 9.6 hours and tmin = 70-2x9.6
>>>= 50.8 f, or a q = 50k btu/h furnace with th = 3.9 hours and tc = 12.1
>>>hours and tmin = 45.8 f.
on a 30 f day.
>> so it would lose 16h((70+tmin)/2-30)400 btu during the setback,
>> ie 194,560 with q = 24k and 178,560 (9% less) with q = 50k.
>using 'linear' heating/cooling rate for the entire time is pretty ugly.
perhaps i should have used a longer time constant. exponentials are uglier
and more accurate. for instance, the house above with an 8k/400 = 20 hour
time constant would cool from 70 f to tmin = 30+(70-30)e-(tc/20) f (1) in
tc hours on a 30 f night. with q = 24k btu/h, tt = 30+q/400 = 90 f, so it
would warm from tmin to 70 f when 70 = 90+(tmin-90)e-(th/20) (2). combining
(1) and (2) with tc=16-th makes 70 = 90+(30+40e^-(16-th)-90)e^-(th/20), ie
-20 = (40e^-(16-th)/20-60)e^-(th/20), ie 1 = (3-2e^-(16-th)/20))e^-(th/20)
= 3e^-(16-th)/20)-2e^-(16/20), ie 3e^-(16-th)/20 = 1.899, ie e^-(16-th)/20
= 0.6329, ie -(16-th)/20 = -0.4575, ie 16-th = 9.15, ie th = 9.15 and tc
= 6.85 and tmin = 58.4, if i did that right. the numbers are different,
but the conclusions are the same.
>similarly the heating is not very linear.
i suspect the thevenin approximation above (a 90 f battery in series with
a 1/400 fhub resistor and an 8k btu/f cap) is reasonably accurate.
>...unless you have an adaptive thermostat/control that estimates the
>exact time to reset back to 70f such that the temperature just reaches
>70f at the end of the setback period...
or maybe 73 f, with 65 f walls... we might save even more by trying to
dump the heat out of the building's thermal mass into a well-insulated
stratified water tank at the beginning of the setback, letting the
building cool to a lower temp overnight, and then trying to dump that
heat back into the building's mass at the end of the setback. a heat
pump might do ac at the start of a setback and heating at the end.
>such a control might be fun to build and tinker with, but i don't think a
>retail unit is available.
how about an "intelligent building manager" listening to all that winter
degree-day talk on new york city radio? :-) they might get a share of the
energy savings to balance christmas gifts from the tenants.
nick
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