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multifamily solar heating
5 sep 2004 drawn, it looks like we have 50x86 = 4300 ft^2 of heated ceiling and 
2(50+86)17 = 4624 ft^2 of walls enclosing the living space, of which about
120 ft^2 is 1st-floor south windows (excluding the stairwell windows) and
148 ft^2 is 2nd-floor south windows and 193 ft^2 is north windows, with
71 ft^2 on the east and west walls. that's about 603 ft^2 of windows, ie
7% of the floorspace. 

with 0.1 air changes per hour (very tight, about 40 ach in a 50 pa blower
door test), it would naturally leak about 0.1x4300x17/60 = 122 cfm, enough 
for 8 full-time occupants at 15 cfm each. an efficient air-air heat exchanger
(eg a bidirectional fan in a partition wall) in series with a humidistat
might provide the rest of the fresh air.

r14 walls and an r34 ceiling (eg 6" of fiberglass under 2" foamboard) and
r2 south windows with 80% solar transmission with r4 and 50% for the rest
would make the thermal conductance to outdoors 268ft^2/r2 = 134 btu/h-f for
the south windows + 335/4 = 84 for the rest of the windows + (4624-603)/14
= 287 for walls + 4300/34 = 126 for the ceiling + 122 for air leakage,
totaling 753 btu/h-f. 

nrel data indicate january is the worst-case month for solar house heating
in nyc, when 610 btu/ft^2 falls on a square foot of ground and 980 btu falls
on a south wall on an average 31.5 f day with an average daily high of 37.6,
for an average daytime temp of 34.6 f. east, west, and north walls get 410,
410, and 190 btu. on a clear vs average day, 690 btu falls on the ground
and 1800 falls on a south wall. the average temp is 76.8 in july, with 1910
btu on the ground and 850 on a south wall.  the yearly average (water) temp
is 54.7, with 1260 and 1510 btu on the ground and on a south wall. 

on an average january day, the south windows would collect 0.8x980x268
= 210.1k btu, if unshaded. the upper south windows might collect more with
a light colored porch roof. the rest would collect 0.5(410(71+71)+190x193)
= 47.4k btu, for a total solar gain of 257.5k btu/day.

an average us family uses about 10,000 kwh per year of electrical energy,
ie about 2500 kwh per year per person. (steve baer and his wife use a total
of 960 kwh/year :-) if 55 energy-frugal people use 1/4 of the us per capita
average amount of electricity indoors, ie 55x2500/4 = 34.4k kwh per year,
that adds 34.4kx3410/365 = 321k btu/day... 55 people sleeping 8 hours would
add 253x55x8 = 111k btu/day.

the house needs about 24h(65-31.5)753 = 605k btu of heat on an average
january day. solar gain and electrical use and people might provide
(257.5+321+111)/605, ie 114% of that. on a cloudy day, electrical use
and people might provide (321+111)/605, ie 71% of the heat. r32 walls
and an r60 ceiling would make it 100%. so would 12" r48 sips.

the hydronic slabs can help spread solar heat around and store it overnight.
three solid 6" x 4300 ft^2 slabs have about 3x4300x6/12x25 = 161k btu/f of
heat capacitance. rc = 161k/842 = 192 hours, which could be raised with a
big water tank. tim ellison spoke on his "house with a 10-day time constant"
at the last portland, me ases conference. a radiant floor and "3 3000 gallon
plastic septic tanks in an outdoor strawbale sculpture." unfortunately,
the tanks contained room temp vs hot water.

with no gain, a 192-hour house would cool from 70 to 30+(70-30)e^(-18/192)
= 66 f over an 18 hour 30 f night. a good passive solar house with a longer
time constant (eg a "barra house" with a hydronic floors over spancrete with
4"x6" holes on 8" centers and a large heat storage tank and thermosyphoning
air heater panels on the south wall) might store heat for 5 cloudy days,
gradually cooling to 60 f, but it wouldn't provide domestic hot water.

a 10 minute 1.25 gpm 110 f shower needs 10x1.25x8.33(110-55) = 5727 btu...
55 showers per day would be 315k btu. an 80% greywater heat exchanger might
preheat fresh water to 55+0.8(105-55) = 95 f with 105 f drainwater from
fully-enclosed showers, reducing the water heating to 55x10x1.25x8.33(110-95)
= 86k btu/day, but that might require lots of money and floorspace. graywater
might flow through 1,000' of 4" pvc pipe in the floorslab, but how would we
keep the pipe full without serous clogging? some septic systems use filters
with nylon stockings surrounding 4" pipe...

including dhw, we need 5(605k-432k+315k) = 2440k btu for 5 cloudy days in
a row. with perfect temperature stratification, this might come from two
1500 gallon 5' deep x 8' diameter insulated tanks with copper coil heat
exchangers. over 5 cloudy days, one tank would cool from 170 to 54 f, the
other from 170 to 80 f. stss ( makes standard ul-approved
flexible circular metal tanks and heat exchangers like this for $1-2 per
gallon. they are widely used with tarn woodstoves (although wood is ongoing
work, compared to solar heat.) dhw would flow through a heat exchange coil
in each tank. on cloudy days, the warmer tank would heat the slabs.

a solar attic like those in soldier's grove with water vs warm air heat
collection could heat the tanks with 2 low-power circulating pumps and
heat exchange coils with water from a 6' wide x 72' long greenhouse
polyethylene film heater atop a 6'x60' east-west strip of horizontal pv
panels on the attic floor, with a 6'x12' higher temp heater section with
the poly film pillow covered with 2 additional layers of polycarbonate
glazing, with no pvs beneath it. this polycarbonate glazing costs about 
$1.50/ft^2 in 4'x50'x0.02" rolls, with a 10-year guarantee against loss of
light transmission, outdoors. the 72' heater might be south of an 80'x12'
high reflective wall under the attic ridge, with an epdm liner beneath
for leak protection. the 12/12 south roof could be a 17'x80' single layer
of corrugated polycarbonate dynaglas greenhouse roofing in 20 overlapping
4'x17' sheets.

greenhouse film comes in wide rolls, costs 5 cents/ft^2, and has a 4-year
guarantee outdoors, exposed to wind and uv. water-filled poly film ducts
on the ground store overnight heat in israeli greenhouses. i've boiled this
6 mil uv-inhibited polyethylene film with no apparent change in properties.
its refractive index is close to water and glass, and water absorbs longer
wavelengths than pvs use, and thin layers of water and poly film don't
absorb much, so the heater won't greatly reduce the pv output. when i put
two layers of poly film with an inch of water between them over a pv panel,
the electrical output (isc) only dropped 6%.

a 6'x60' pv array would normally make about 3.6 kw, peak, but the reflective
wall could double that in summertime. this could also work with a smaller pv
array, or none at all, to begin with. the water would drain down at night in
wintertime. it might stay in the heaters all the time with gable attic vents
open in summertime.

the attic would collect about 0.9x12x80(1910+850) = 2385k btu on an average
july day (less, because of the reflective wall shading, and still less, with
white cloth over the south 6' of the floor.) if 0.15x2x6x60x0.9x1910 = 186k
of that becomes electricity and 315k btu heats water, 2385k-186k-315k = 1884k
btu is left, ie 1884k/24h = 79k btu/h. with lots of water inside the attic
and 8 ft^2 gable and floor vents, 79k btu/h = 16.6(16ft^2)sqrt(12)dt^1.5
makes the attic-to-outdoor temp diff dt 19.4 f, with a 16.6(16)sqrt(12x19.4)
= 4050 cfm airflow (which might serve as a whole-house fan at night.) the
average attic temp would be about 76.8+19.4 = 96.2 f in july, or less, with
a thin layer of evaporating water over the heater. cooler pvs produce more

vents through the reflective wall and the north roof could lower the attic
temp more. the north attic roof might extend above the south roof, with 2'
of south-facing windows above it, open in summertime, and a linear lightshaft
below that that leads to some simple skylights in the attic floor. with roof
insulation and heat from the south attic, people might use the space below
the north roof, about 80'x18', down to a 6' headroom. 

this is a sketch. a simple computer simulation would be useful, something
that shows the house would not have required any backup fuel at all over
the last 30 years, using nyc hourly weather data.


appendix: solar attic collection details

with r33 insulation (r14 urethane plus a 6" fiberglass wrap), each tank
would lose about 24h(170-65)150ft^2/r33 = 12.5k btu/day in a 65 f space
we want to collect at least 25k btu/day at 180 f plus 315k btu/day at
120 f or more. 

the tanks and heaters might be plumbed like this (viewed in a fixed font,
eg courier): 

     a 6'x12' wood polycarbonate frame     polycarbonate
     could be laid over the end of         |           | 
     the polyethylene film heater.         polycarbonate
                                           |           |
     polyethylene film.................polyethylene film
   ->|                  1-2" of water                  |<--
  |  polyethylene film.................polyethylene film   |
  |  pv---------------------------------pv | attic floor   |
  |              <--60'-->                 |  <--12'-->    |
  |                                        |               |
  |      initial/final temps               |               |
  |                                        |               |
  | pump 170/54                            |    170/110    | pump
 ---     -------                           |    -------   ---
| ^ |   |       |       120 f              |   |       | | ^ |
| | |   |     * |---<----------------------*->-| *     | | | |
 ---    |   *   |                              |   *   |  ---
  |     | *     |       ground floor           |     * |   |
   -----|       |                              |       |---
         -------      *** heat exchange coil   -------
         120/54                                 170/110

the dhw and radiant floor heat exchange coils are not shown above.

on an average january day, a 12'x12'x80' long attic solar aperture with 90%
transmission might collect 0.9x12x80(610+980) = 1374k btu over 6 hours at a
rate of 229k btu/h. with an attic air temp t, 1358 ft^2 of r1 glazing would
lose (t-35)1358 btu/h. a 6'x60' ew strip near a 90% reflective wall might
receive something like 58.2k btu/h of net heat on an average january day,
after subtracting the electrical output energy. 

if the pv heater glazing film conductance to room air is 1.5, the heaters
might gain (t-120)6x60x1.5 = (t-420)540 btu/h from warmer t (f) attic air.
the polycarbonate heater might gain 10.9k btu/h and lose (180-t)72ft^2/r2.

so we have something like this, viewed in a fixed font:

            120 f     t = attic air temp (f)
     ---    |         |
|---|-->|---*---www---*---www--- 35 outdoor temp (f)
     ---    |  1/540  | 1/1358
  ppn  =    |         |
58.2k btu/h---  i1--> |   (see the basic program below.)
           ---        |
            |         |
            -         |
                      x   opening the circuit at x and replacing the part
                      |   above with its (thevenin) equivalent gives the
            180 f     |   simplified circuit below...
     ---    |         |
|---|-->|---*---www---*   rt = 1/(540+1358) = 1/1898,
     ---    |   2/72  |
10.9k btu/h |         |   1/540+1/1358 = 1/386,
           ---        |
           ---        |   i1 = (120-35)386 = 32841 btu/h, and 
            |         |
            -         |   vt1 = 35+i/1358 = 59.2 f.
     ---              |
 159.9k btu/h       

            180 f     t
     ---    |         |
     ---    |   2/72  | 1/1898  | vt1 = 59.2 f
10.9k btu/h |         |         |
           ---        |        ---   opening the circuit at y and
           ---        |         -    replacing the parts to the right 
            |         |         |    and below with their equivalent...
            -         |         -
     ---              |              vt2 = 59.2+159.9k/1898 = 143.4 f.
 159.9k btu/h       
            180 f     t = 143.4 + (180-143.4)/(1/36+1/1898)/1898 = 144.1 f.
     ---    |         | 
     ---    |   2/72    1/1898  | vt2 = 143.4 f
10.9k btu/h |                   |
           ---                 ---   
           ---                  -
            |                   |
            -                   -

so the 180 f collector gains 6h(10.9k+(144.1-180)72/2) = 57.6k btu/day
and the 120 f pvs gain 6h(58.2k+(144.1-120)540) = 427.2k btu/day.

10 pa=.9*12*80*(610+980)/6'solar power into attic (btu/h)
20 pc=.9*(6*610+.9*1.5*980)/4/6'solar power on collectors (btu/h-ft^2)
30 lpv=60'pv array length (feet)
40 spv=6*lpv'pv array surface (ft^2)
50 scc=6*72-spv'180 f collector surface (ft^2)
60 ppv=spv*pc'total heat power into 120 f pv collector (btu/h)
70 epv=.15'pv efficiency
80 opv=.9*ppv*epv'electrical output (btu/h)
90 ppn=ppv-opv'net heat power into 120 f pv collector (btu/h)
100 pcc=.9*.9*scc*pc'solar power into 180 f collector (btu/h)
110 vt1=35+(120-35)/(1/(1.5*spv)+1/1358)/1358'first thevenin temp (f)
120 rt=1/(1.5*spv+1358)'thevenin resistance
130 vt2=vt1+(pa-pcc-ppv)/1790'second thevenin temp (f)
140 t=vt2+(180-vt2)/(2/scc+rt)*rt'attic temp (f)
150 q120=6*(ppn+(t-120)*1.5*spv)'heat collected from pvs (btu/day)
160 q180=6*(pcc+(t-180)*scc/2)'180 f collector heat (btu/day)
170 print spv,scc,vt1,vt2
180 print t,q120,q180,q120+q180

120 f pv      180 f coll    vt1 (f)       vt2 (f)

360 ft^2      72 ft^2       59.18335      143.4245

air temp (f)  pv heat       coll heat     total heat

144.1053      427235.2      57633.67      484868.8 btu/day

with no venting, the air in my 768 ft^2 attic with a 12/12 south 600 ft^2
single-layer polycarbonate roof and uninsulated stone walls (part of an
1820 stone farmhouse) gets up to 143 f in december. it's a good place
to dry laundry or flowers or escape winter blahs for a few minutes... 

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