re: auto thermostats SOLAR.KnowledgePublications.com

re: auto thermostats
7 nov 2002
>say it's 40 f outdoors, and the house has 500 btu/h-f of thermal
>conductance and 5k btu/f of capacitance, and the heat pump can supply
>2(70-40)500 = 30k btu/h. say it follows this rule: "if the house temp
>drops 0.5 degrees below the setpoint, turn on the heat pump. if it ever
>drops 1.0 degrees below the setpoint, turn on the resistance heat," which
>costs 10 vs 3 cents/kwh. how large can an 8 hour once-a-day setback be
>before it stops saving money? is the breakeven dt only 2 degrees?

if dt is small and the house temp drops quickly to 70-dt, it needs
8h(70-dt-40)500 = 4k(30-dt) btu of heat-pump over 8 hours. this costs
4k/3412(30-dt)0.03 = $0.0352(30-dt). if we reheat to 70 f electrically,
the extra cost is 5000dt/3412(0.10-0.03) = $0.1026dt. the total cost
is $1.056+0.0674dt, minimized when dt = 0. 

>perhaps dt can be larger with a different rule during setback times, like
>"only turn on the resistance heat if the heat pump has been running for
>over 30 minutes without meeting the setpoint."

looks like this could help more, since 30/60x30k = 15k btu can increase
the house temp about 15k/5k = 3 f with no electric resistance heating.

but why 30 minutes, in an 8 hour setback? with a 5000/500 = 10 hour rc
time constant, the house might cool to ts = 40+(70-40)exp(-t/10) over
t hours and warm  back to 70 f with the heat pump in 8-t hours. if it
can supply 30k btu/h, and 30k = (t-40)500+5000dt/dt, 10dt/dt + t = 100,
so t = 100+(ts-100)exp(-(8-t)/10) = 4.78 hours, and ts = 58.6 f. over
8 hours, the average temp is about 64.3, so we use 8h(64.3-40)500 = 97.2k
btu during that time, vs 8h(70-40)500 = 120k btu, a 19% savings worth
20 cents/day. 

we might save more if the house had less thermal mass. in the limit (0), 
we might save 8h(70-40)500 = 120k btu/day worth $1.06, with a 30 f setback.

who sells such thermostats? consult your local hvac criminal...

nick