Solar heating notes from Norman Saunders (reformatted)

[nick pine]

M1 86601 To Help You Conserve

Copyright Norman B. Saunders, P. E. 15 Ellis Road, M3, Weston MA 02193
may be freely quoted if in context with appropriate credit.

Superinsulation scaled to the climate needs to extend from
the footings over all and be protected by a vapor barrier
on the inside and a wind barrier on the outside. Roughly,
the insulation should have a non-metric R rating close to the latitude
with a bit more on the roof and less on the foundation walls. At half
this R rating, the present value of the future extra fuel required
by the lesser insulation is likely more than the present cost of
installing the extra insulation. The optimum window today is R4+,
as obtained by three glazing surfaces with one low emissivity surface.
To maximize performance, the inner spaces between each pair of sheets
should be more than 1/2 inch.

Berming a house is effective. Burying is not.

The external wind barrier is to keep the insulation working. To
minimize heat loss in the cold of winter and heat gain in the summer,
the inner vapor barrier should be so tight that the total house air
leakage is equivalent to a hole less than a foot in diameter. It is
then necessary to deliberately control air flow the rest of the year.
In such a house, the heat supplied by the occupants and their
activities may be enough to keep it livable.

Glazing's greatest value is in the light admitted. For pleasure in
living, more light and view is desirable. Letting in the sun adds
light and heat. In winter the sun is up only eight hours or so and is
typically covered by clouds half the time, so that solar input
without heat storage can supply no more than 10 to 20 per cent of
the requirements. Conventional Passive Solar houses store heat in
the surfaces of the living spaces. This can supply most of the needed
heat, but you have to live with the necessary swings in living space
temperatures.

The use of hydronic systems, as with water in roof top collectors,
allows use of an isolated heat store. However such systems are
usually too expensive...

M3 86603 The Technical Basis

Comfortable, low-first-cost, very low operating cost houses in many
house styles and in most regions of the country can be built by
carefully attending to details, observing certain restraints, and
following sound engineering principles...

Energy efficiency begins with proper insulation. Rules of thumb for
enough insulation in (hrxft^2xF)/Btu:

R = (winter DD + summer DD)/200 for southern states while
R = (F interior - F mean ambient in coldest month) x 2/3 in the north.

The resulting loss is about 1.5 Btu/(hrxft^2) for typical winter
temperature difference. Non-south windows should have an R-value
at least 1/6 that of the walls and fill no more than 10% of that
wall area. This gives an overall wall heat loss of no more than
2.2 Btu/(hrxft^2).

To get a feel for relative values, consider a 1500 ft^2 house held
at 70F in a 30F ambient. Then if two story, the Wall loss is
8 x 2(25+30+25+30) x 2.2 = 3,872 Btu/hr, and the Roof loss is
25 x 30 x 1.2 = 900 Btu/hr,
total 4,772 Btu/hr.

If one story, (8 x 160 x 2.2) + (1,500 x 1.2) = 4,616 Btu/hr, or
almost the same loss from the exposed skin, whether the 1500 ft^2
is all on one story or on two. Loss of heat to the ground might
bring each up to 5,000 Btu/hr. With one air change every two hours,
replacance is (1,500x8)/120 = 100 cfm and the load resulting from
it 4,000 Btu/hr. An air to air heat exchanger could cut the total
loss for either house to 6,000 Btu/hr.

Were it occupied by four people each giving off 100 W (body heat
fueled by food) and each using 150W for light, etc., the total of
3,412 Btu/hr would supply all the heat needed above 47F, the balance
point in the dark. (The balance point is that external temperature
at which heat available under the stated conditions will hold
the internal house temperature at the desired set point.) With windows
evenly divided between the walls, and little shading, a week-long
average contribution by the sun of 32 W/m^2 (as found in Boston)
would be 1,500 Btu/hr, moving the blaance point down to 37F. By
reducing the non-south glazing, tolerating a slightly lower
temperature, or doing more cooking, the house could be run simply
as superinsulated. However, because of the difference in life styles,
the heat of occupancy could be as high as 6,000 Btu/hr, but it is
about as likely to be only 1,500 Btu/hr.

Using the gains and losses as calculated above, you can get an idea of
how many hundred square feet of glazing you require to be nearly free
of the need to purchase energy for heating. Rigorous calculation of
the need is far more detailed and complicated. With any house at its
particular site, the best one can say is that it can be expected to be
between some temperature A and another temperature B for X fraction of
the time. However, inside temperature and fuel bill are just the start
of what you should consider.

Conventional "Passive Solar" stores excess solar input in room
surfaces for later use. The resulting overheating of living spaces
can be prevented if the store is isolated. The temperature at which
heat is stored can then be higher. Such is my "warm store."

I add a "cool store" for the excess heat input from outside or
occupancy. In winter such heat can be used later, even though it is
low-temperature or low-grade heat. In summer it can usually be dumped
later when outside temperatures drop.

In the temperate zones, the ground under and around the building can
be used as a heat sink and heat store. Berming gives better coupling
to the ground and shielding from the air. The excess summer heat can
be dumped into the ground underneath. In one day, a solid surface can
be heated and cooled to a depth of a tenth of a meter and hence store
and release that day's solation striking an equivalent area of
glazing.
Though the useful depth on an annual basis is less than two meters,
carry-over of early fall excess into the typically cloudy early winter
may eliminate the last need for purchased heat. One of the principles
of physics is that the distance that heat penetrates increases as only
the square-root of the time of application.

Conventional air change by leakage in the shell maximizes heat load
from that change. When it is pleasant out, the house is stuffy. As
temperature drops, the flow of air increases. So drafts go up, and
heat load goes up as temperature difference squared. An air tight
house with humidistat controlled air change can partially alleviate
this by keeping air change more nearly constant.

83105 Weather Analysis, Application to Building Design

To get adequate but simple and reliable barriers and entries for wind
and sun, we need to understand the weather quantitatively. Knowledge
of weather variations and the building's heat gain, loss and storage
allows prediction of the variation of internal temperature with time.
The basic design problem then is to find optimum values for the gain
to loss ratio and the storage time constants.

Weather in the Abstract

The physically simple underlying causes of changes in temperature,
solation, and wind are overlaid by an uncountable multitude of chance
events. A proper mathematical description of weather includes both.
(See 81327 Weather Statistical Properties.) The physical basis for
the mathematical abstraction is as follows:

The rotation of the earth around the sun changes the angle between
the equatorial plane and the line to the sun. This gives us our
seasons, with both mean air temperature and possible hours of
sunshine varying sinusoidally. The rotation of the earth about its
axis gives daily changes. The temperature tends to rise with the sun,
but falls more slowly. On any day the sun may be fully visible,
hidden by clouds, or partly getting through. As a rough guide for
New England, there are no clouds 10% of the time, it is overcast 5%
of the time, and the rest of the time all intermediate intensities
of sunshine are equally likely.

The delays, both daily and seasonal, occur because the sun heats
the air by way of the surfaces it strikes. Part of the heat from
the surface is transferred directly to the air touching it. The rate
is between 6 and 60 W/m^2-K for dead still air and howling gale.
These are equally unlikely and the surface is not smooth but has
grass,
trees, buildings, etc. Thus a coefficient of 10 W/m^2-K or so is
representative of heat transfer from earth surface to air in contact.
Air has a thermal capacity of roughly 1000 W-sec/M^3-K. Dividing
the second figure into the first suggests that the depth of air
heated to match the surface temperature might increase at the rate
of 10 mm/sec, or 36 m/hr, or a kilometer a day.

Radiant transfer of heat from the surface is to the general mass
of air and is likely greater than the convective transfer. However,
the reverse flow or transfer of heat from air to ground is essentially
only by contact. The relaxation time for air-mass to ground
temperature is thus on the order of several days. The air masses
themselves move at roughly 10 m/sec in summer and 20 in winter and
are one thousand to several thousand kilometers across Hence the mean
air temperature varies little from one day to the next, but can change
greatly over a few days.

Computing the Expected Performance of a Building

Much of the rest of this paper and of 81327 likely seems remote,
incomprehensible, and from the ivory tower. Even so, it is
a simplified approach in that among other things, it assumes that
the normal, or Gaussian, distribution applies to both temperature and
solation. To show that the approach is real and useful, consider
a building in the Boston area, that described in 82N29. The table on
page one is for the design before adjustments to get 100% solar
heating. This shows that for 20C living space and -5C outside,
the loss rate is 4 kW and the total storage is about 80 kWh/K. From
these figures we get a decay time of 500 hours. (Decay and holding
times are defined in the theory portion of this paper.) The difference
between the first and second examples of the third set on page two,
shows the assumed heat of occupancy to be 1 kW. From the third column,
the normal December solation captured is 3.6 kW.

For the conditions assumed in the table and a five Kelvin degree
permissable temperature swing, the gain to loss ratio is 1.15 and
the holding time is 100 hours.

Referring to 81327, the December temperature in Weston is
9.8 - 12.5 cos(345-29.5), or -1.5C. The standard deviation for 100
hours is 20/sqr(7.5), or 7.3K. The -5C assumed is roughly half a
standard deviation below the mean, and hence it will be at least that
cold out about a third of the time. The variability of the sunshine,
from 81327, is given as 50%/cuberoot(4), or 31% for one standard
deviation. The mean amount of sunshine (Basics p 103) is 52%, so that
no sun is at 1.6 standard deviations. The expectation of four sunless
days in a row (this is seriously stretching the normal law
approximation) is then one in ten. Since no correlation between
temperature and solation has been demonstrated for this area,
the combination of this cold with no sun can be expected to occur
about once in thirty such possible times, so about once in seven
years, purchased heat will be needed in December.

The ASHRAE 1954 Handbook gives the 1% and 2 1/2% temperatures for
Boston as 0 and 8 F (-17.8 and -13.4C.) For -15C, the holding time
for our houses is three days, giving a standard deviation of 7.8
Kelvin degrees. In December, the 1% temperature is 2.1 standard
deviations down (again a serious stretching of the normal law) and
so can be expected to occur about once every 4.5 years. This suggests
the need to purchase heat in December once in 35 years.

Designing a building

Obviously, to have solar heat at night, there must be heat storage.
The question is how much. For the house to remain comfortable when it
is very cold out is as much a matter of heat storage as it is of
insulation. Heating system design should consider heat storage
quantitatively.

Every heating system is designed to be inadequate for some small part
of the time. System size has been made simply proportional to
the difference between the inside and the design temperature. This
design value had been that temperature which the observed outside air
was below for 1% of the time.

With better insulated and tighter houses, the temperature bounding
2 1/2% of the time has been used. For both values, the resulting
heating systems have been adequate for a greater fraction of the time
than that for which the design was supposedly adequate. Heat storage
carries the house over part of the coldest times. Deliberate inclusion
of heat storage in the calculations and quantitative consideration of
the variability of the weather allow prediction of the extent of
inadequacy of the heating system.

The Building Parameters

The building values can be normalized by using the ratios of heat
gain to heat loss and heat storage to heat loss. The gain and loss
are expressed in watts or Btu per hour. The amount of heat stored is
given in mega-joules, kilowatt hours, or Btus. The gain and loss
should each be averaged over time before taking their ratio. The decay
time constant is the ratio of heat storage to heat loss, each value
being for one Kelvin degree. The holding time is the product of
the decay time by the ratio of (permissable temperature drop from
the set point) to (temperature difference between the living space
and the outside air.) To a first approximation, the decay time is
actually constant, while the holding time varies inversely with
the outside temperature. The holding time is roughly the period
between heating system on times.

Design for "100%" Solar Heating

To make the analysis simple, start with the worst month of the year
and find for it the mean gain and loss from the weather data averaged
over many years. Select a time interval and the fraction of the time
it is permissable to be outside of the set temperature range. For
each set of such selections, compute for both ambient temperature and
solation the expectation of departure from the mean. Use the lower
temperature and the smaller amount of solation to calculate the gain
to loss ratio. For "100%" solar heating, the holding time must be
great enough to bridge the expected periods of little or no solar
input. For periods greater than the holding time, the gain must be
greater than the loss.

As the permissable swing around the set point is reduced, so is
the holding time. So for a tightly thermostated house, the primary
heat store must be separate from the living space. With a well-
insulated and nearly airtight house having south windows, there can
be overheating anytime in the winter. So I use two heat stores.
The warm, primary, solar heated store runs above house temperatures,
and it is typically a water store overhead. The tempering or cooling
store is just below house temperature and is typically a rock store.

On sunny days, the cooling fan holds the house temperature down to
the upper set point by moving heat from the living spaces to the rock
store. In the night or on cold cloudy days, when the house reaches
the lower set point, the heating fan pulls heat from the warm store
into the living spaces. There is also a lowest temperature set point
which, if ever reached, will turn on the purchased power. The holding
time now breaks up into two parts: the set point holding time and
the free energy holding time. The first is time during which heating
is from the warm store, and the second is the subsequent time until
the temperature reaches the point at which power is purchased.

Since the warm store is isolated from the living spaces, it can
cycle through a large temperature swing. The set point holding time
is given by the thermal capacity of this warm store multiplied by
the difference between its peak possible temperature and the lower
set point. The free energy holding time is given by the product of
the difference between the lower and lowest set points and the overall
heat storage within the house. The heat removed in cooling the house
is thus subsequently available as backup.

Solar backup for a solar system.

Modification of the 82N27 design for 100% Solar Heating

The modification of the design to transfer most of the heat from
the sun into the warm store is described elsewhere. The warm store
has a capacity of 8.6 kWh/K and its temperature swing is limited to
a 70C upper limit set by the water heating use and the 20C living
space lower set point. Thus the set point holding time is 8.8(70-20)/4
= 110 hours. With a purchased power set point of 8C, the free energy
holding time is 40 hours, giving a total holding time of 150 hours or
two days more than would be obtained with a simple direct gain system
as calculated above. The expectation of being outside of the set point
limits is less than that derived for the direct gain system, and
the expectation of ever having to purchase energy is very small.