Two layers of greenhouse polyethylene film over a halfcylindrical 8’x16’ x8’tall frame would make the curved greenhouse south wall/roof area 4’Pi16’ = 200 ft^2 with a 200ft^2/R2 = 100 Btu/hF thermal conductance. With R20 insulation, the north wall conductance would be about 200ft^2/R20 = 10 Btu/hF. R20 endwalls would add Pi8’^2/R20 = 10, making the total greenhouse conductance 120 Btu/hF.
Where I live near Philadelphia, 1000 Btu/ft^2 of sun falls on a south wall over an average 30 F January day in about 6 hours. With 80%
solar transmission, the 128 ft^2 south wall projection would transmit 128ft^2x0.8x1000Btu/ft^2/6h = 17067 Btu/h. With no thermal mass, the greenhouse air temp would be about 30+17067/120 = 172 F for 6 hours/day before it suddenly fell back to 30 F, theoretically.
LOTS of thermal mass and mass surface and 102400/24h = 4267 Btu/h on an average January day would make the greenhouse a constant 30 + 4267/120 = 66 F, 24 hours per day.
Each of 2 55 gallon waterfilled drums would have about 450 Btu/F of thermal mass and 25 ft^2 of mass surface and 1.5x25 = 40 Btu/hF of airwater surface conductance with a thermal equivalent circuit which makes more sense if viewed in a fixed font like Courier New after downloading. Tg is the greenhouse air temperature:
17067 Btu/h Tg
  1/120 Fh/Btu
—>*vvv 30 F
 
<
< 1/80
<


 900 Btu/F


with this Thevenin equivalent circuit
https://www.allaboutcircuits.com/textbook/directcurrent/chpt10/theveninstheorem/ :
1/120
www Tg
 
 <
 Vt < 1/80 Vt = 30 + 17067/120 = 172 F.
 <
 
 
  900 Btu/F
 
 
which is equivalent to:
1/48 = 1/(1/120+1/80)
www
 
 172 
  900 Btu/F
 
 
with a 900/48 = 19 hour RC time constant.
137 F  Tg peak air temp, which
 falls quickly at dusk.
 Tg
 Tg
 Tg
84  T T T peak water temp, which does not.
Tg T T
 T Tg T
 T T
Tg Tg T
Tg T
T T
Tg Tg T
T T
54 
6h 24h
If T = Td just before dusk and T = Tn just before dawn, 18 hours later,
Tn = 30 + (Td30)e^(18/19) = 18.4 + 0.388Td, and
Td = 172 + (Tn172)e^(6/19) = 46.6 + 0.729Tn, then
Tn = 18.4 + 0.388(46.6+0.729Tn) = 36.5 + 0.283Tn, so
Tn = 36.5/(10.283) = 50.9 F.
Just before dawn, (50.930)48 = 1004 Btu/h flows out of the drums, warming the greenhouse air to 30 + 1004/120 = 38.4 F.
Just before dusk, Td = 46.6 + 0.729Tn = 83.7 F. The greenhouse air would be hotter: (17283.7)48 = 4238 Btu/h flows into the drums, adding 4238/80 = 53 F to the drum temp to make 137 F air, if I did that right.
We could limit the air temp to about 80 F max by opening a greenhouse vent whenever the indoor air rises to 80 F using an unpowered vent opener, or a thermostat and a 24V 2W motorized damper actuator or a fan. The vent needs to dump about 17067 – (8030)120 = 11067 Btu/h. With an A ft^2 open area for upper and lower vents and an 8’ height difference, A = 11067/(16.6sqrt(
(8030)^1.5) = 0.67 ft^2 min, eg 1 ft^2 vents, which could vent up to 16600 Btu/h at 332 cfm and 80 F.
Ventilation can also reduce wintertime humidity. Air at 80 F (540 R) and 60% RH (an upper limit to avoid mold) has a water vapor pressure Pw = 0.6e^(17.8639621/540) = 0.628 “Hg. The humidity ratio wa = 0.62198/(29.921/Pw1) = 0.0133 pounds of water per pound of dry air. NREL says the average outdoor humidity ratio w = 0.0024 in January in Philadelphia (although the current
online Blue Book is missing Phila data.) A 100 cfm vent airflow would remove 60m/hx100x0.075lb/ft^3(0.01330.0024) = 4.9 lb/h of water vapor from the greenhouse. Some
people say that green plants can evaporate 1 lb of water per day per square foot of greenhouse floor space.
With 4 drums:
1/120
www Ta
 
 <
 Vt < 1/160 Vt = 30 + 17067/120 = 172 F.
 <
 
 
  1800 Btu/F
 
 
which is equivalent to:
1/69
www
 
 172 F 
  1800 Btu/F
 
 
with an 1800/69 = 26 hour RC time constant.
If T = Td just before dusk and T = Tn just before dawn,
Tn = 30 + (Td30)e^(18/26) = 15.0 + 0.500Td, and
Td = 172 + (Tn172)e^(6/26) = 35.4 + 0.794Tn, then
Tn = 15.0 + 0.500(35.4+0.794Tn) = 32.7 + 0.397Tn, so
Tn = 32.7/(10.397) = 54.2 F.
Just before dawn, (54.230)69 = 1670 Btu/h flows out of the drums, warming the house air to 30 + 1670/120 = 43.9 F. More drums help. We put 200 55 gallon water drums as plant pallet supports into a 20’x96’ single cover greenhouse in PA, and it never froze in wintertime.
The drum heat would last longer if the greenhouse air were kept at a constant 40 F all night instead of gradually
cooling from the peak to the minimum temp, but that would require some sort of control.
With more controls, 8 drums in an R20 4x4x8’ insulated box with an R1 90% transparent south wall could store more heat, ie 3600(13244) = 317K Btu, vs 8 cooler drums exposed to greenhouse air storing 3600(5844) = 50K Btu. To reduce heat loss at night, a 500 cfm high temp 23 watt fan, eg
https://m.grainger.com/mobile/product/4WT44?cm_mmc=PPC:+Google+PLA&s_kwcid=AL!2966!3!166593068073!!!g!102431315157!&ef_id=WGOu3wAAAH9I1Syw:20180705181606:s could circulate warm air through an R20 partition during the day, with no airflow at night. The 4’x8’ vertical partition could separate the south wall and the vertical drums in a single 2x4 layer under a 4’x8’ bench would have an equivalent circuit like this during the day:
3888 Btu/h 1/500 fan R20/96ft^2 box

—>vvvvvv 80 F
  
R1/32  <
80 F vvv < 1/320
<
 T

 3600 Btu/F


which is equivalent to this:
1/32 1/500 1/4.8
wwwvvvvvv 80 F
 
 <
 Vt < 1/320 Vt = 80 + 3888/32 = 202 F.
 <
 
 
  3600 Btu/F
 
 
and this:
1/34.8
vvv
 
 185 
  3600
 
 
with a daytime time constant RC = 3600Btu/F/34.8Btu/hF = 103 hours.
RC = 3600(20/128+1/320) = 574 hours at night:
R20/128ft^2 box
vvv 40F

<
< 1/320
<
 T

 3600 Btu/F


Tn = 40 + (Td40)e^(18/574) = 1.2 + 0.969Td, and
Td = 185 + (Tn185)e^(6/103) = 10.5 + 0.943Tn, then
Tn = 1.2 + 0.969(10.5+0.943Tn) = 11.4 + 0.914Tn, so
Tn = 11.4/(10.914) = 132.3 F.
Td = 10.5+0.943x132.3 = 135.2 F.
If we insulate the entire greenhouse (including the south wall) with R20
soap bubble foam at night and fill the 6 triangular spaces between 2 drums and a 2’x4’x4’ tall R20 box with an R1 transparent south wall with 90% solar transmission, the 6 spaces (totaling 2’x4’ 2Pi(1’^2) = 1.72 ft^2 could hold 8 4”x3’ vertical PVC perforated pipes (to reduce airflow resistance) surrounded by about 300 pounds of 2” diameter rocks, which would add about 48 Btu/F to the drum thermal capacitance and 108 Btu/hF to the drum thermal conductance, with an equivalent circuit like this, during the day:
1944 Btu/h 1/500 fan R20/48ft^2 box

—>vvvvvv 80 F
  
R1/16  <
80 F vvv < 1/189
<
 T

 948 Btu/F


which is equivalent to this:
1/16 1/500 1/2.4
wwwvvvvvv 80 F
 
 <
 Vt < 1/189 Vt = 80 + 1944/16 = 202 F.
 <
 
 
  948 Btu/F
 
 
and this:
1/16.4
vvv
 
 186 
  948
 
 
with a daytime time constant RC = 948Btu/F/16.4Btu/hF = 58 hours.
RC = 948(20/64+1/189) = 301 hours at night:
R20/64ft^2 box
vvv 40F

<
< 1/189
<
 T

 948 Btu/F


Tn = 40 + (Td40)e^(18/301) = 2.3 + 0.942Td,
Td = 186 + (Tn186)e^(6/58) = 18.3 + 0.902Tn, then
Td = 18.3 + 0.902(2.3+0.942Td) = 20.4 + 0.850Td, so
Td = 20.4/(10.850) = 136 F.
Tn = 2.3+0.942x136 = 130 F.
Foaming the greenhouse at night lowers the heat requirement to (4030)30 = 300 Btu/h. The rocks lower the min usable water temp to 300/189 = 42 F. The 2 130 F drums can keep the greenhouse air 40 F for 948(13042)/(24hx300) = 11 30 F cloudy days in a row.
If cloudy days are like coin flips, a greenhouse that can store enough heat to keep itself 40 F for N 30 F cloudy days in a row would have a Solar Heating Fraction (SHF) of 12^N, eg 0.5 for 1 cloudy day, 0.75 for 2, 12^3 = 0.875 for 3, and so on. For instance, the 8 drums in the box would store 317K Btu. Keeping the greenhouse 40 F for a day requires 24h(4030)120 = 28.8K Btu, so N = 317K/28.8K = 11 days, with an SHF = 12^11 = 0.9995, ie 99.95%, ie solarheated in all but 4 hours per year of 12 months of January weather.
In summary:
min
# of C RC Twp Twm dT usable
drums Btu/F hours peak min F temp SHF
0 0 0 172 30 142  0
2 900 19 87 51 36 55 0
2+foam 948 58/301 136 130 6 42 0.9995
4 1800 26 78 54 24 48 0.23
8 3600 41 75 58 16 44 0.70
8box 3600 103/574 135 132 3 44 0.9995
oo oo oo 66 66 0 40 1
Nick