Subject: Ohm's law for heatflow

From: [email-masked] (Nick Pine)
Date: 11 Nov 1996 12:19:23 -0500

My friend in the Canadian Frozen North writes:

>       I figure with 166K Btu insolation and 60K lb of water that the
>       maximum temperature gain by the water is (166KBtu/60Klb = 2.76 F)
>       subtract from that the heat loss day and night and you stabilize
>       at around 115 or 116 F.

You can just pretend the thermal mass is infinite, to find that equilibrium
temperature. (How do you subtract a temperature from a heat loss???)

>       I may be wrong somewhere in my calculations because when I increase
>       the R-value of the hoophouse and decrease the r-value of the pool,
>       (remember, I want this thing to operate at -40 and the 28 degree
>       spread you have in Pennsylvania seems a bit feeble for that) I end up
>       in some sort of a loop where the water temp and air temp fluctuate
>       wildly from positive hundreds to negative 50's and hundreds.

Sounds familiar :-)

"Confusion is like fertilizer.
It feels like shit, but nothing grows without it." :-)

>       I know for sure that the water and air in that greenhouse aren't
>       going to be colder than the outside temp,

That's an excellent insight. One you don't often get with F-charts...
This is more like cooking from scratch than with cake mixes.

>       but I can't seem to figure out the source of the problem.

Keep looking... (I've managed to make those solar air-conditioners too :-) 

>       Thot u mite hev sum ideas, no?

You need to put some limits in your spreadsheets, and make them march slower
in time. You need to tell them things like "if part A is warmer than part B,
let heat flow from A to B, but if part A is cooler, don't let any heat flow
>from  A to B." Your computing knowledge seems to be ahead of your heatflow and
arithmetic knowledge :-) I think you need to slow down, and _simplify_ this
system, and play with it and see what happens, until it makes sense to you.
I know that sounds pretty general, but it's a good way to go...

>       I've gotten some catalogs from greenhouse suppliers... and found them
>       full of good deals, but short on solar and insulating info.

Yes. They don't care much about that, because their customers don't, because
we haven't stopped bombing Iraq yet. Stuppy has a nice $5 catalog at (800)
877-5025, with lots of application notes in the back. More along the lines of
how to build things than their thermal performance.

>       They talk about "high transparency",
>	"the best IR retention","20% warmer","highest insulating
>	value", but they don't give any real numbers for me to crunch.

Yes. I don't think IR greenhouse film is worth buying for a sunspaces that
get cold at night. It costs more, 7 or 8 cents/ft^2 vs 5, and the IR additive
also blocks some of the solar input. Some window screen or shadecloth inside
the sunspace can also reduce reradiation by letting part of the sun pass
through the mesh and heat up the dark wall behind it. Then some of the
reradiation from the wall is blocked by the mesh so it can't get back out
through the poly glazing. Another advantage to using a mesh is that the
sunspace can be filled with cooler air flowing out from the house, while the
air in that thin solar air heater space between the mesh and the house wall
can be warmer, which is thermally more efficient and more comfortable for
people than filling the whole sunspace up with hot air, next to the cool
sunspace glazing. The mesh also increases the sun-air heat transfer surface
area, which lowers the average absorber temperature and reduces reradiation.
If the mesh is hung outside the poly film in the summer, or the poly film
is rolled up and the mesh is left in place, the mesh will reduce solar gain
and the poly film will last a lot longer.

>	I got a greenhouse design book from a prof in Alaska and it
>	gives lotsa numbers, but instead of R-values, he quotes
>	everything in U-value.  Is there a conversion?

Sure R = 1/U.

>       Now I have heard of R-value, U-value, Metric R-value, and R.S.I.

I've never heard of R.S.I. What does that stand for? Royal SI units? :-)

>	Is there a conversion?

US R-values in ft^2-F-hr/Btu are 5.68 times higher than metric ones
in W/m^2-K, for the same material... So metric R1 is like US R5.68,
and a US R1 window would be R0.176 watts/m^2-K or U = 5.68 in metric.

>       I have an old book called _the Theory of Heat_ which has yet
>       another way of quantizing relative thermal conductivity.
>       Beyond me, I'm afraid.

Beyond me too, probably. I do not like calculus.

Perhaps you've seen the book _Structures: Why Things Don't Fall Down_,
by J. E. Gordon, containing this nice little quote...

   When we play tennis or walk downstairs we are actually solving
   whole pages of differential equations, quickly, easily and without
   thinking about it, using the analogue computer which we keep in our
   minds. What we find difficult about mathematics is the formal, symbolic
   presentation of the subject by pedagogues with a taste for dogma,
   sadism and incomprehensible squiggles.

Or this one... 

It's a snap to save energy in this country. As soon as more people become
involved in the basic math of heat transfer and get a gut-level, as well as
intellectual, grasp on how a house works, solution after solution will appear.

					  Tom Smith, 1980
Still waiting...

>       I just want some  basic
>	formulas that I can put numbers into so I can figure out the
>	optimal size and materials for my solar greenhouse experiment.

You might look at your Alaska prof's greenhouse book more carefully, or read
about Ohm's law for heatflow on page 70 of the $35 NRAES-33 Greenhouse
Engineering book (4th rev., 1996? now written by Robert A. Aldrich and John
W. Bartok, Jr., published by the good honest non-profit Northeast Regional
Agricultural Engineering Service at 152 Riley-Robb Hall, Cooperative Extension,
Ithaca, NY, 14853-5701, or (607) 255-7654... (For another $6 they will also
send you a photocopy of their 10 year old biogas booklet that explains how
to make methane from manure and use it in modified gas appliances.) 

Table 4-2 on page 64 says poly film has a light (PAR) transmittance of 0.92
and an infra-red transmissivity of 81% (vs glass and polycarbonate at 2% and
1%--polycarbonate is better...) at an 80 F source temperature. Fig 2-8 on
page 31 shows a fairly flat curve of transmittance vs wavelength for poly, so
2 layers should pass about 0.92 x 0.92 = 0.85 of the solar heating spectrum.
But then Table 2-3 on page 34 lists a thermal transmittance of 50% for UV-
stabilized polyethylene, and 20% for IR poly, which seems inconsistent with
page 64. This may be a science like bible interpretation. Table 4-4 on page
66 lists U1.2 for a single layer of plain poly film, with U0.7 for 2 layers.
And table 8-4 on page 146 lists a single layer of clear poly film used as an
internal thermal blanket (no wind) as having U = 0.45 Btu/hr-F-ft^2, ie just
over R2, while black poly film has U = 0.48, a _better_ effective conductor,
as a better IR emitter? If it matters so little whether the poly film blanket
is clear or black, with no wind, how can we lose lots of heat by reradiation?

So... the R-value of poly film with wind outside seems to be "about 1,"
which seems like a nice round number to use for solar designs.

BTW, the thermal conductance of a surface air film is approximately 

U = Cs + V/Cw Btu/hr-F-ft^2, where V is in mph, and 
    Cs = 1.5 and Cw = 5 for smooth surfaces like glass and
    Cs = 2 and Cw = 2 for rough surfaces like stucco (p 22.1, 1993 ASHRAE HOF.)

I suggest you call your greenhouse a design, BTW, not an experiment. Do you
know Ohm's law for electricity? It's the same as Newton's Law of Cooling in
heatflow, with different units.

Newton said that the amount of heat that flows through a wall in an hour
is proportional to the difference in temperatures on each side of the wall
and the size of the wall, and inversely proportional to its thickness.

Ohm's law:      I = (Vh-Vl)/R, eg 2 amps   = (12V-2V)/5 Ohms.
Newton's law:   Q = (Th-Tl)/R, eg 2 Btu/hr = (12F-2F)/5 F-hr/Btu.

I is in the current flow in amps, and Q is the heatflow in Btu/hr
(or watts--all this is easy to do in metric too, easier in fact, but
it seems easier to just talk about one system at a time.)

Vh is a higher voltage and Vl is a lower voltage, while
Th is a higher temperature and Tl is a lower temperature.
(Vh-Vl) is called the potential difference or voltage.
(Th-Tl) is called the temperature difference or "delta T."
You can think of either as the pressure that drives the flow.

R is the electrical resistance in Ohms or 
    the thermal resistance in ft^2-F-hr/Btu
R = rho L/A for an electrical resistor, where rho is the bulk electrical
    resistivity of a material, L is the length of resistor that the
    current flows through, and A is its cross-sectional area of that path.
R = k L/A for a thermal resistor, where k is the bulk thermal resistivity
    of the insulating material, L is the wall thickness (think of a wall
    as a very large flat resistor :-) and A is the area of the wall.

Another way to calculate the value of a thermal resistor is 

R = R-value/A, where "R-value" is the _measured_ US R-value that you see
               stamped on big rolls or sheets of insulation in hardware
	       stores, and A is the area of the insulator in square feet.

A lot of good honest people in the American Society of Heating, Refrigerating
and Air-Conditioning Engineers have measured R-values for individual building
components, so we can easily calculate and reliably predict the values of
thermal resistors composed of combinations of components, using the same rules 
that people use to find the resistance of series and parallel electrical
resistors. This is not very mysterious, altho ASHRAE people seem fond of
footnotes. It works pretty well at low temperatures, eg below 130 F. Above
400 F, heatflow by radiation becomes very important. That depends on the
absolute temperature of a body, ie how warm it is compared to absolute zero,
and what is around it. A "black body" with emissivity 1 at temperature T (F)
with nothing around it radiates 0.174(T+460)^4 Btu/hour/ft^2 of heat. A
selective surface like oxidized galvanized iron or copper can radiate less,
and be hotter in the sun.

It is curious how people don't think of buildings using beams designed with 
arithmetic as structural experiments, but they think of solar structures
designed with Newton (1642-1727)'s Law of Cooling as thermal experiments,
even though that is a older version of Ohm (1787-1854)'s law, which most
people accept, except for new age electricians, perhaps... Is this because
there are so many badly designed solar structures, from a thermal point of
view, designed without much arithmetic at all, or solar common sense? Solar
history books like John Perlin's _Golden Thread_ tell of architects being
_amazed_ over and over, over the years, over decades and decades, one by one,
that a room with a south window gets warm when the sun is shining, a fact that
engineers and physicists have been able to predict very precisely for hundreds
of years. High school students can do this, these days, but architects still
haven't figured out solar basics yet. They just talk as if they know what they
are talking about, and design poorly-performing solar houses, and wave their
hands. They need to use more numbers, as they do when designing simple beams.

>	I have probably invested a week's worth of hours researching
>	this, so you could qualify me as "interested".

I'm just beginning to master the basics of solar arithmetic, after 22 years,
with valuable help from good and honest and generous solar pioneers like Steve
Baer, John Page, PhD, William Beckman, PhD, John Duffie, PhD, Norman Saunders,
PE, Howard Reichmuth, PE, and William Shurcliff, PhD. And special insights
>from  William (Mr Bill) Rosenthal, PhD, who actually likes calculus. And a lot
of distractions and arrogant ignorant talk from architects, and a lot of
expensive and impractical badly performing designs like envelope houses and
direct gain houses (Steve Baer's wife calls them "direct loss houses" :-)
outside of the Southwest. Government programs have not helped my understanding
much, but NREL has helped a lot lately, with some very good solar weather data
for 239 US cities.

Nick