|
|
re: sample solar problems
17 may 1996
william r stewart wrote:
>nick pine wrote:
>> william r stewart wrote:
>> >nick pine wrote:
>>here's another litmus test for an energy expert: what will the average water
>>temperature be inside a 4' cube full of water surrounded by 5 r20 foam walls,
>>sitting outside in december, when the air has an average 24 hour temperature
>>of 36 f and the sun puts 1,000 btu/ft^2/day of heat into the r1 glazed side
>>of the box? and how will that change over time if we shade the sunny wall?
>>anyone care to answer that question? i'll offer a $10 reward to the first
>>person who answers it correctly.
by the way, i have only seen 3 numerical answers to this simple question
so far. two people got part 1 correct, and one has almost solved part 2 using
simple arithmetic, but there's still a chance for the right person to earn
$10 in 2 minutes with a couple of back of the envelope calculations... this
might also be a rare opportunity for someone else to get paid to learn.
paul erdos used to do this sort of thing with much harder math problems.
he would offer a $100 prize for the solution to some long-standing research
problem, and some grad student who might already be interested in the problem
would get more interested and work for 3 months, day and night, and solve the
problem, and pocket the money, and publish a paper, and feel very happy, and
the science of mathematics would be advanced :-) erdos offered different
rewards for different problems, depending on how difficult they were, from
$10 problems that took a few weeks to $1,000 problems that might take 10 years
to solve, or might take a lifetime...
>> >the water in the cube would rise to a steady state temperature of
>> >36 +80 = 116 of if the energy input were continuous.
>> perhaps, but recall the problem:
>>>>the sun puts 1,000 btu/ft^2/day of heat into the glazed side of the box.
>>>>and how will that change over time if we shade the sunny wall?
>>>when shaded, the water in the cube would fall over time to 36of.
>i showed how it would change over time. you owe me ten bucks, nick :-)
nope. that answer is insufficient and superficial. i'm the judge.
>> >a complete answer with specific times would require calculus, if specific
>> >times was what you were looking for.
>otherwise, simple arithmetic would give the wrong answer, as this is obviously
>not a linear relationship.
>> yes, that was what i was looking for. more than the obvious... finding the
>> form of the answer requires calculus, but newton did that once, a long long
>> time ago. finding the answer just requires plugging some numbers into a
>> simple formula, if you know the formula.
>present the formula and we will see if it produces the correct answer, or even
>a very close answer.
nice try :-) how will "we" know that? the formula is most of the answer...
or an approximate arithmetic equivalent...
>>>4000 lbs of water would require 2.5 hours of 1600 btu/hr energy input, again
>>>without considering the loss through the cube walls.
>so here is an answer with a specific time; what's your beef?
it has nothing to do with the problem.
>> at this point, mr stewart changes the subject:
>> >now, if you;
>> > -moved the water inside to the interior sunspace of a 68of house,
>> > -made the glazing 7'x10 (4 in the entire house on the south side)
>> ie 280 ft^2 of south glazing?
>> > -used a flatter 4"x4'x10' water wall (4 in the entire house), 4" away
>> > from the glazing inside the house,
>> ie about 400 pounds of water? 4" away from the glazing? transparent
>> water? hmmm :-) i guess you won't be able to see very clearly out of
>> those windows, but they will let in some light...
>> > -used an r4 glazing when the sun was shining,
>> with how much solar transmittance, at what cost, and what happens when the
>> argon leaks out? some of these $40/ft^2 high-r windows have low solar
>> transmittance, on the order of 50%. hmmm, 280 ft^2 x $40/ft^2 = $11,200.
>you must be thinking of r8 windows; where do you get your figures?
i don't remember. anderson, pella? popular science? perhaps that's r8.
what brand of windows will you be using and how much do they cost?
>i'm not going with low-e due to insolation losses.
good. i wonder what happened to your uv concerns.
>> > and an extra r8 window cover when the sun wasn't shining,
>> how would you do that? movable insulation, at $10/ft^2, installed?
>again, where do you get your prices?
the shelter institute. what will you pay? references, please.
>>will you move it twice a day religiously?
>just like you close your windows when it gets to cool at night.
i hardly ever touch my windows.
>>will it leak any air around the edges?
>probably, but certainly much less than an outdoor thin film sunspace.
i don't think so, if the sunspace is made from a continuous single piece of
plastic film, 16 x 20'. and sunspace leakiness only matters 6 hours a day.
>> >then you would have a direct, passive solar design that retained the sun's
>> >heat energy in the building interior and released it slowly into the
>> >interior at night
>> right. another "direct loss" house :-)
>er, do you have a house that does not lose heat when it is cold outside?
no. but this is a numerical thing, and the less the better at night, and
direct gain houses leak lots of heat to the outdoors at night and during
a week without sun thru their low thermal resistance glazing between the
living area and the outdoors.
>> >care to try the math?
>> sure. i like this arithmetic. let me get a cup of coffee... now i'm back,
>> 200 btu later. i've had this espresso machine for about 3 months now. i
>> guess if some people like me find it fun to wiggle the valves on an espresso
>> machine with all the steam and noise and watch the milk temp rise with a
>> thermometer twice a day, others might find it fun to move window insulation
>> all over a house twice a day, for three months... or, maybe you only move
>> the window insulation on a cloudy winter day--but no, you want to do it at
>> night too... or you might leave some windows covered all winter, like pat
>> hennin or malcolm wells, in semi-hibernation, huddled in cold dark rooms.
>> >assume another 60 ft^2 of window at r4 during the day, adding
>> > another r8 at the other 14 hours.
>you forgot to add r8, you only added r4
oops. would you describe this r8 window insulation, will? who makes it,
how much the material costs, how much installation costs, what the materials
are, how it is operated, and how the edge air leaks affect the r8 rating?
>> >assume 0.25 air changes per hour (optional).
>> ok. that's 0.25 x 18432 ft^3/hr = 4608 ft^3/hr or 77 cfm :-) i'll assume
>> (i almost said "guess") this is leakage thru walls, etc, without an air-air
>> heat exchanger... will the finished house have a blower door testing spec?
>yes, and the top limit will be 0.25 air changes; i expect alot less.
will the house be pressure tested when finished and guaranteed to have
less than 5 ach at 50 pascals, consistent with natural air infiltration of
< 0.25 ach, using the usual rule of dividing the pressure test spec by 20?
the best avis homes informally tested by penn state, as i recall from a
go-around with kurt smith, lyle rawlings and marc rosenbaum, came out at
2.9 ach at 50 pa.
>> may we also assume you use a frugal 500 kwh/month of electricity, ie an
>> average of about 700 watts?
>why does this matter? i will be using a 2kw photovoltaic system.
marc points out that many superinsulated houses are more electrically heated
than solar heated. no, that doesn't matter much. money matters, of course.
eg the yearly backup heating bill, and the non-recurring cost.
>> (steve baer only uses 80 kwh/mo :-)
>how much do you use, coffee and all?
about 600 kwh/month. mostly for my refrigerator, electric water heater and pc.
i just bought a laptop, which should reduce this.
>> i guess we'd want to know the thermal resistance of this house for starters.
>> so let's add up the thermal conductances u = sum(ai/ri) and find
>> the reciprocal 1/u...
>> we have 340 ft^2 of windows,
>4 sets of 10' x 7' windows = 2800 ft^2
let's see. 4 x 10 x 7 = 280. i just did that on my calculator to be sure.
a 2,800 ft^2 window would be 5 stories tall and 50' wide.
>>r4 during a 10 hour day (uday = 340/4 = 85)
>> and r8 during a 14 hour night
>r12 at night; r4 + r8
my mistake.
>(unight = 340/8 = 43), with an average daily
>> uwindow = (10xuday+14xunight)/24 = 60.
>> uwalls = (2(24+48)x16-340)/r24 = 82, and uceiling = 24x48/r38 = 30.
>> and uinf = 77 btu/hr-f.
>> so u = 60+30+77 = 167 btu/hr-f and r = 0.006, over 24 hours.
>these numbers have to be revisited.
agreed.
>> at night the u value would be lower, after you devotedly travel around
>> the house and painstakingly put up your night insulation on every window,
>> making sure the edges are ever so carefully sealed, akin to some little
>> twice-daily religious experience... unight = 43+30+77 = 150 btu/hr-f.
>> what will the steady state energy flow be for this house after a long
>> string of average december days, with an average amount of sun?
>> the energy ein that flows into this house in a day might be
>> 280 ft^2 windows :-)
hmm. a new way to measure energy :-)
>that should be 2800 ft^2
i don't think so.
> x 50% transmission
>try 75%
that won't change the picture much.
> x 1000 btu/ft^2/day = 140k
>> + internal energy generation = 72k
>> total ein = 212k.
>> and the energy eout that flows out of the house might be
>> 24 hours x (68-36) x 167 = 128 k btu/day.
>of course, these numbers must be revisited as well.
agreed.
>>now, thermal mass: the house has about 3,500 ft^2 of walls and ceilings with
>>a thermal mass equivalent to 3,500 pounds of water and another 400 pounds
>>of actual water.
>10' x 4' x 4" is ~13 ft^3. water is 62.4 lbs/ft^3. there are
>four water walls.
>the water will weigh close to 3250 lbs.
my mistake again.
>> let's call this 4,000 btu/hr-f.
>you need to revisit this number as well.
somebody should.
>> so if you make the temp of
>> the house, say, 80 f at dusk, at the end of an average day, with some sun,
>> after a long string of december days, each with 1000 btu/ft^2/day of sun,
>> during the night the house will lose approximately 14 hours (80-36) 150 =
>> 100k btu, which might come from the 4,000 btu/f of thermal mass cooling plus
>> 14/24 hours x 72k = 42k of internal energy generation, ie the temperature of
>> the house might drop to 80 - (100k-42k)/4k = 65 f at dawn.
>you have so many corrections to make before you hit this paragraph that i am
>reluctant to mention the ones within.
oh go ahead, mention them.
>you have not yet tried to establish the temperature of the water in the
>water walls.
oh no. i did that.
>your assumption seems to be 1 of, though there is no evidence
>of how you tried to calculate this.
that's pretty cold. that doesn't make any sense to me at all. i don't
understand what you are saying at all here will. and i didn't assume
anything like that. i assumed the water walls were the same temperature
as the air and everything else in the house.
>> how much backup heat will you need in an average year?
>my wife wants a fireplace, so i will likely go with a masonry firestove,
>often referred to as a finnish or russian fireplace, with outside combustion
>air.
how much backup heat will you need in an average year?
>> were you planning to have hot water in this house? how were you planning
>> to do that?
>solar hot water, most likely active.
how much will that cost to buy and operate, and how much backup heat
will you need for the hot water?
>> a better way to compare heating systems
>> might be their yearly heating bills, including the electrical power to run
>> the heating system.
>the model i am planning on building (2250 ft^2) required $168 worth of heat for an entire
>winter. reference "solar today", sept/oct 1995, article on manufactured house.
sounds nice. as steve baer says, the greatest discovery of
solar investigators may have been that lots of insulation is good :-)
>> >some people prefer the passive method, and it is not your perogative
>> >to tell them that they are wrong in preferring that.
>> oh sure it is, and it seems kind to tell people, if their passive houses are
>> such miserable and expensive performers.
>try the numbers again, then we'll talk.
it's your house, will. try the numbers again yourself, if you like.
i'll be happy to look at them.
cheers,
nick
|
|
