Don Klipstein wrote:

It appears to me that averaged through the heating season, Philadelphia
gets about 120 watts of sunlight per square meter.

NREL data say January is the worst-case month for solar heating in Phila,
where seldom is heard a discouraging word and 1000 Btu/ft^2 falls on
a south wall on an average 30.4 F day with a 37.9 F max. That's 293Wh/24h
= 12.2 W/ft^2 or 131 W/m^2-day.

I don't see this being enough to heat a usual home...

Agreed. A square foot of fairly expensive R4 south window with 50% solar
transmission would gain 500 Btu/day and lose 24h(70-30.4)1ft^2/R4 = 238, for
a net gain of 262, so we could warm a 1 foot cube with a 1 ft^2 south window
if 262 = 24h(70-30)5ft^2/R, ie R = 18 ft^2-F-h/Btu, ie US R18 walls. But to
make it 65 F = 30.4+(70-30.4)e^-(5x24h/RC) after 5 cloudy days, we need RC
= 5x24/ln((65-30.4)/(70-30.4)) = 889 hours. With a thermal conductance G
= 0.25 Btu/h-F for the window + 5ft^2/R18 = 0.28 for the walls, totaling 0.53,
C = RCxG = 889hx0.53Btu/h-F = 471 Btu/F, eg 471 pounds (7.6 ft^3) of water,
so we have what is technically known as "a 5 pound bag problem" :-)

This is exactly analogous to an electronic time constant, but longer :-)

... A home designed to be better insulated and to capture this heat could
get through the winter with little supplemental heat

Sure. More insulation and thermal mass in the house, and a low-thermal-mass
sunspace that collects warm air during the day and gets cold at night, so it
loses little heat to the outdoors at night. A $1 square foot of R1 vertical
polycarbonate sunspace glazing with 90% solar transmission might gain 900
Btu/day and lose 6h(70-34)1ft^2/R1 = 216, for a net gain of 684 Btu/day.
We could keep 1 ft^3 of water with an air heater over 1 insulated wall warm
if G 62.33Btu/F/889h = 0.07 Btu/h-F, with 6ft^2/0.07Btu/h-F = R85 walls :-)
That's doable, but not very practical, with say 2" vacuum aerogel insulation.

Larger cubes are easier, since the available heat storage volume increases
faster than the heat-losing surface, and mass with a higher temp swing can
store more heat, if we don't "live inside the battery." For instance, an 8'
cube with an air heater over one insulated wall can have 130 F mass under
the ceiling on an average January day if 64ft^2x900 = 6h(130-30)64ft^2/R1
for the air heater during the day + 18h(70-30)64ft^2/R for the air heater
at night + 24h(130-30)64ft^2/R for the ceiling + 24h(70-30)4x64/R for the
other 4 walls, which makes R = 23.2 min. With R24, eg 6" SIPs or R19 fiber-
glass + 1" Styrofoam, the cube walls and floor would lose 24h(70-30)5x64/24
= 12.8K Btu on a 30 F cloudy day. If the ceiling mass cools from 130 to 80 F
over 5 days and the average temp is 105, it would lose 24h(105-30)64ft^2/R24
= 4.8K Btu/day (we might add more ceiling insulation.) If the cube loses
5(12.8K+4.8K) = 88K Btu over 5 days, we need can keep it exactly 70 F with
a slow ceiling fan and a room temp thermostat and an occupancy sensor and
88K/(130-80) = 1760 Btu/F of mass, eg 28 ft^3 of water in a 5" layer under
the ceiling. With more insulation, we can use less water...

but summer would be a different matter.

The sunspace needs ventilation and summer shading, eg an overhang that admits
low-angle winter sun (90-40lat-23.5 = 26.5 degrees above the horizon at noon
on 12/21) and blocks high-angle (90-40+23.5 = 73.5 degree) summer sun. With
lots of mass and night ventilation, a house in Phila can get by with no AC
for all but 2 weeks in an average year. July is the warmest month, with an
average daily min temp of 67.2 F. Ceiling fans can make humid humans comfy.

Won't work in places like Seattle though.

Direct gain "mass and glass" doesn't work well there, with 250 Btu/ft^2 on
the ground and 420 on a south wall (not much more) on a 40.5 F December day,
but 180 of that is diffuse, which leaves 240 of direct-beam "gift-wrapped"
sun which can be concentrated. PE Howard Reichmuth's Ecotope greenhouse at
Pragtree Farm near Seattle has been working well for the last 30 years.
A parabolic north wall reflects low-angle winter sun into a water trench
near the base of the wall. A low-thermal mass air heater or sunspace can
also work well, if it only collects solar heat for an hour a day, but it
needs lots of glazing, compared to Phila. A more heroic approach is to store
summer heat for November through February in something like a 20' tall x 16'
diameter well-insulated water tank with a drain-down insulating cover under
a transparent dome.

Nick