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Packing Heat

BTUser
BTUser Member Posts: 10

I want to install an array of solar thermal collectors on my roof and then dump excess heat into the ground under my house. Hope to accomplish this by way of surface-mounted hydronic radiant panels at slab and ground level. Radiant flooring would be installed on 1st floor rooms, with the intent of modest flywheeling on the thermal mass of the (uninsulated) slab and the ground beneath. By modest, I mean allowing some temperature drift but only within the range of thermal comfort. Perhaps allowing it to dip a bit below that overnight.

Meanwhile, much of the floor, earth, and exposed foundation wall accessible in our crawl space would also be blanketed with hydronic paneling, possibly with insulation on top, and then utilized as a kind of ad hoc thermal energy storage. The temperature of these crawl space surfaces could be allowed to swing through a wider range, since we're not as concerned with maintaining thermal comfort in the crawl space, and insulation on top of the paneling would give us even more wiggle room.

Now, I know an insulated slab or Bob Ramlow sand pit would be better, but it's a retrofit project for a conventional home. I know that volume for volume an insulated water tank would be better still, but I don't think that's going to be the right fit for this project either. Although I haven't come across any examples quite like this so far, I think just plain heating the surface of the ground may suffice. Because I think very low grade thermal energy storage that can hang on to modest temperature levels for short time frames would be enough in my situation.

My situation: Climate averages peak winter lows of 35 and highs of 55. Average annual temp is 62. As I understand it, the ground temperatures under conditioned space should be a little higher than that… somewhere between 62 and room temperature. Meanwhile, I am planning a radiant system for the interior of the house that calls for ultra low temperature differentials between room and water. I believe we can get the required water temp down to 77 at design load. And so, although heat loss to surrounding earth will clearly be significant, the ground and slab surfaces may only need to climb something like 15 degrees to become operational. The heat loss at that dT may be acceptable. A few different ways I've run the numbers, it's looking like adding a single additional solar thermal collector may be enough to offset it. If we pack spare heat from the array into the ground during the fall, the dT should effectively shrink even more.

The hotter the thing gets, the more heat it's going to bleed, but that may be ok. The more topped out it becomes, the less it matters. As an initial ballpark I'm coming up with 1 degree of temperature drop across all exposed slab/foundation/dirt surfaces will contain enough heat for one average winter day. At a dT of 30, presumably it will bleed in the neighborhood of double the heat that it would bleed at a dT of 15. But I could live with that, because at that point the store would contain two weeks of heat.

All I'm looking to do is be able to hang on to a little sunshine during winter storm systems that sometimes last a week or so. A heat pump would supplement it and even extend the range of the storage. For instance, when the storage is at 70 it won't be enough to keep the house warm on its own, but a heat pump could use it as a source and then extract 77 degree water at a very high COP.

Ok fellas. If you've read this far, you've more than earned the right to crucify my plan. Let's hear it…

EdTheHeaterMan

Comments

  • hot_rod
    hot_rod Member Posts: 24,485

    How large is your roof, where are you located, and what do you mean by excess? Are you doing DHW first, then storage?

    The first calculation would be what sort of energy you could harvest throughout the year. That is based on location, array size and orientation.

    There are solar SIM programs that can spit out numbers based on a 30 year database of weather and solar info.

    And you know the ST motto, you get the least when you need it most, when it comes to heating loads.

    If you are considering store bought collectors the SRCC has data like this to show the collector performance under different load conditions and solar radiation. This is a typical flat panel collector 4X6.5 dimension.

    Look at the Categories (Ti- Ta) is collector inlet temperature minus ambient temperature. You will be between C and D most of the year.

    Hopefully you can buy or build this large array for next to nothing, else the numbers will never work out.

    I've been chasing this solar "golden goose" for 40 plus years.

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
    EdTheHeaterMan
  • BTUser
    BTUser Member Posts: 10

    Hi Bob! I've come across some training videos with you and Max over the last year, as I wolfed down all the material I could find on this subject. Thank you for responding to my post!

    I'm near Sacramento. The south-facing side of our roof (slope = 20 degrees; some shade around 9am and again around 4pm) has room for about 300 square feet of thermal collectors; the rest is taken up by our existing 18 PV panels (I believe 3.5' x 5' each). I've done some modeling using the SRCC info, not comprehensive but also not totally simplistic. I'd have to recheck my numbers but I came up with 1000 BTUs/sqft/day as my approximation for winter heat capture. I think I came up with 1650 BTUS/sqft as total solar insolation at winter solstice at my location, so the 1000 per day would come in the form of almost 1500 BTU/sqft some days, 0 some days, etc.

    Working concept is to size the array to capture the average daily BTUs needed for peak winter months. On our existing setup (Forced Air Furnace and AC) we're averaging 3.5 therms/day. Some of that is the low efficiency of the furnace. As part of this process we'll want to make some low-hanging fruit upgrades to the building envelope so hopefully that will come down some, on the other hand I figure we might as well use some of the solar production to preheat our DHW (we have a 50 gallon rheem HPWH). Anyway, I'm ballparking 300K BTUs per day average peak winter heating needs, and 300K BTUs per day production from a 300sqft array.

    By "excess," I meant something like: On sunny winter days the array would produce something like 450K BTUs but the house might only need 250K. I hope to find a way to store the 200K of excess in the ground, literally for a rainy day. Including for runs of say 2 straight weeks of sunny winter days. Basically smooth out the discrepancies between supply and demand in a way that doesn't let any of the array's capacity ever go to waste.

    I also want to invert the process during the summer and use the panels for nocturnal cooling (Peak summer average lows are 58, highs 98, dry summer meditteranean climate). So I would want panels that do a decent job on that front. I didn't think the cost per square foot for thermal collectors would end up being the limiting factor here, but I haven't priced them carefully. Off the top of your head do you have a ballpark range I might expect per square foot of glazed flat panel collectors, installed?

    Whatever incarnation of this I eventually end up going with, I'd be hiring a firm with solid experience with radiant cooling as well as thermal energy storage. DIY on something like this would really be playing with water hehehe

  • DCContrarian
    DCContrarian Member Posts: 1,020

    Are you off-grid? Because if you're connected to the grid, especially if you're in a place with net metering, you're much better off using your money and roof space for electric panels and then using the electricity to heat and cool the house.

    Alan (California Radiant) Forbesethicalpaul
  • BTUser
    BTUser Member Posts: 10

    On grid. Increasing our PV panel count would knock us out of NEM 2.0, so that would be one disadvantage.

    PV & air source heat pump is the way it’s done out here, but for me it’s like nails on the chalkboard of a thermodynamics class. During the season when heat is needed, thermal can capture 4x as much energy as PV. Maybe this isn’t the most sophisticated observation, but it’s also energy in the form that’s needed. Something about going the higher exergy route of PV seems so, well, brutish compared to shuttling around existing heat. If those same panels can do double duty during the summer for cooling, and also provide DHW year round, and if we can even usefully put them to work during the fall & spring as well, it doesn’t look to me like PV will automatically always be the way to go.

    Siegenthaler has made the point that PV complements AC well but as we shift over to heat pumps there’ll be a mismatch between winter night peak demand and solar production. For now there’s no problem with the fact that PV production is spread out all through the year but the kWhs are needed in bulk each winter, so long as your utility lets you true up annually. But with solar rate agreements seemingly always developing loopholes and rates always changing for the worse, I don’t see good reason to trust an arrangement that depends on them.

  • DCContrarian
    DCContrarian Member Posts: 1,020

    "During the season when heat is needed, thermal can capture 4x as much energy as PV."

    But in the season when heat isn't needed, you need to have an elaborate scheme to dump the heat that thermal is capturing, whereas with PV you just return it to the grid. And you need to have an elaborate scheme to keep your system from freezing. And if you size your thermal system to meet your full heating load on the coldest days it will probably be 10x too big in the summer. Even just for domestic hot water thermal systems have trouble aligning availability of sunshine with the load, because most people have their biggest demand for hot water in the morning, when the sun hasn't been out since the previous evening.

    Solar thermal is dead. It doesn't pencil out.

    Alan (California Radiant) Forbes
  • hot_rod
    hot_rod Member Posts: 24,485

    this snap shot shows the data you need. A 23 sq ft collector on a clear day, depending on ambient 25-30,000 BTU/ day, not per hour. A solar day is about 6-8 hours

    Dirt is just not a great conductor or storage medium, but dozens of Ramlows sand bed systems work to some degree.

    A drainback array takes away the stagnation issues or dump requirement. I have owned a number of DB systems, I have one on my house and one on my shop currently.

    The missing link is finding a good use for the summer and highest production months.

    Certainly Im not the one to talk you out of any solar thermal project, no matter how hair brain.

    I saw Mike from Aspen Solar at AHR, he is installing a few dozen ST collectors on a $67 million dollar spec home in Starwood!
    Probably to offset the sin tax they have there :)

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
    BTUser
  • BTUser
    BTUser Member Posts: 10

    The missing link is finding a good use for the summer and highest production months.

    I've come across a few concepts for collectors that would improve night cooling capacity while retaining high daytime heating performance, but I'm unsure if any of them are in production. Has anybody ever heard of a flat panel collector where the cover and rear insulation can be removed? So that during the summer you basically have just the bare innards of a typical glazed flat panel, exposed on both sides to night temperatures and wind and pointed at the sky. This could produce a lot of cooling under the config I described above, and I think 300sqft would still be able to cover DHW needs on summer days. Owner would have to slide a "box" up onto the panel each fall to convert it to heating-only mode.

    Dirt is just not a great conductor or storage medium, but dozens of Ramlows sand bed systems work to some degree.

    One advantage of the ground — even over a Ramlow system — is that it's already there. Every home has a giant cube of dirt underneath it, insulated by a bunch more dirt in every direction. Except for above, but if we're using that cube as thermal storage, we generally want its heat to drift upwards. Even though other materials and arrangements clearly have superior properties, maybe it's possible lowly dirt could make up for it by volume? So far my modeling looks encouraging but I'm probably missing something.

  • hot_rod
    hot_rod Member Posts: 24,485

    A job in Dubai they paint the collectors white in the summer, wash it off in the fall.

    There was a company that made canvas covers for the evac tube arrays, like a mini boat cover

    The Resol controllers have a nightime cooling function. Essentially all the energy you collected during the day could be sent back out to the cold night time sky. But running the system day and night seems silly?

    Cold sky re-radiation. It is used to help protect glycol systems that see little summertime use, vacation homes for example. I data logged that function on the system we had on the Caleffi building in Milwaukee, it only needed about 4 hours to pull the tank temperature down.

    But the DB systems handle the over heat best.

    The new Viessmann collectors have a special coating that changes at a certain temperature to help with over heating.

    Use it or lose it.

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
  • Larry Weingarten
    Larry Weingarten Member Posts: 3,725

    Hi, Inefficient collectors might be something to look at. Inefficient means no serious overheating problems or the need to dump heat. Also means less expensive collectors that you can build from black poly pipe. Stagnation temperature of this type of collector is around 170F. Simple controls and no real need for freeze protection make it easier. Another approach is to go with glazed pool collectors.

    To me, the biggest unknown here is how to effectively put heat into and remove heat from soil. I prefer un-pressurized water for storing and moving BTUs around. There's two cents.

    Yours, Larry

  • BTUser
    BTUser Member Posts: 10

    The white paint thing is hilarious. I took a look at those Viessman collectors. I've come across a variation of that thermochromic function designed for night cooling that works in reverse, increasing IR emissivity at low temperatures and holding the heat in at high ones. That would get me the summer night cooling capacity I'm looking for, but wouldn't solve overheating issues. But I guess I'm unclear on the nature of the overheating problem, because it sounds like plain old drainback works well. Thermosiphon self-cooling fin systems also seem like a good option, I'm curious what the track record looks like on those.

  • BTUser
    BTUser Member Posts: 10

    Huh. Not sure if I follow what you have in mind here. I did take a look at standard pool collectors but couldn't make the numbers work for winter space heating. In my climate during winter they'd have decent efficiency until they get the water up to maybe as high as 70 degrees. But, the TES temp would generally go no lower than that in this arrangement, even when it's fairly depleted.

    I think radiant flooring in the rooms on the slab definitely makes sense and that would give me significant thermal mass to work with. But, I should run a comparison on using a water tank vs. my crawl space concept to supplement that. Got a ballpark of what a 1000 gallon unpressurised tank would cost?

  • DCContrarian
    DCContrarian Member Posts: 1,020

    Here's what availability of solar energy looks like where I am, Washington, DC:

    Y-axis is BTU/day for a 128 square foot collector. I did a fairly detailed analysis, factoring the change in efficiency with the changing outdoor temperature, the total insolation and the change in angle of the sun.

    This is what heating demand looks like, BTU/day, heat and hot water:

    So basically, you get the most heat when you need it least, and the least when you need it most*.

    When sizing a system, you have to try and square this circle. At one end, you could size it so that it meets 100% of your needs in January. If you did that, in July it would be approximately 35 times too large, 97% of the heat you capture would have to be discarded somehow. At the other extreme you could size it so that in July it meets 100% of your needs (which is just hot water). If you did that it would meet only 27% of your hot water needs in January and less than 3% of your total heating needs.

    You can pick points between those two extremes but there is no golden point.

    With PV, you get the full benefit of all the solar hitting your collectors, every day of the year.

    *(I probably shouldn't have to point this out, but it's not coincidence that it's hot in summer when there's the most sun and cold in winter when there's the least.)

  • jumper
    jumper Member Posts: 2,415

    You want to store enough heat for the whole year. My experience with radiant heating in Toronto,Canada basements is that over many heating seasons HHW @ ~ 160° stored enough heat that basement stayed habitable during weeks of outage. If you can acquire PV at reasonable cost –doesn't need to be efficient– it is much easier to use resistance heaters to heat earth below slab. If floor becomes too warm then you can repurpose solar electricity for other purposes. Such as DHW. Using PV without interconnecting to utility has to be more economic in long run. Each time your floor is cool enough to use those resistance heaters you will store more heat in ground. And need less the next time.

  • BTUser
    BTUser Member Posts: 10

    Interested to hear more about your basement configs. Was all the heating was done through hydronic tubing embedded in a concrete floor? Were the walls or floor of the basement insulated?

  • Larry Weingarten
    Larry Weingarten Member Posts: 3,725

    Hi, I just checked tank prices and $900 to $1600 seem easy to find for 1000 gallons. These are uninsulated, so that would need to be added. Also I didn't check specs to see maximum temperature, but that does matter 🤠

    Yours, Larry

    BTUser
  • DCContrarian
    DCContrarian Member Posts: 1,020

    @jumper : "You want to store enough heat for the whole year."

    The charts I posted above came from modeling my own house in DC, about 15 years ago. It used 135 million BTU's worth of gas for the whole year. A solar collector of 550 square feet would provide 135 million BTU per year. April through October it would meet all of the heat needs of the house and have a surplus available to store, November through March it wouldn't be enough and you'd have to dip into storage.

    The amount that would have to be stored would be 69 million BTU's. My yield calculations are based on a collector temperature of 120F (yield goes down as the temperature goes up). Let's say you could store all of that heat at 120F and release it at 70F, you'd need 1.38 million pounds of water or 166,000 gallons. That's about 21,000 cubic feet. If you wanted to put that in a tank with a footprint of 1,000 square feet it would have to be 21 feet deep.

    The same 135 million BTU's is about 40,000 kWh. In my climate a solar panel produces about 1,000 hours worth of rated production per year, so you'd need about 40 kW of nominal capacity to net the same amount of energy. That's assuming COP of 1.0, no heat pumps. If you could improve your COP you'd need even less. The installed price of solar is now about $2 per nominal Watt after federal tax credits, so $80K and you're all set. If you got a COP of 2.0 on your heat and hot water it would only be $40K but I won't even go there.

    I'm going to go out on a limb and say there's no way you could build a 166,000 gallon tank in your basement for $80K. I'm not even going to try and count the other pieces that would have to go into making it work.

  • BTUser
    BTUser Member Posts: 10

    Hmmm, that doesn't seem too bad at all. I'll have to weigh the pros and cons carefully.

    If the crawl space ground thing could work, one advantage is that it could draw down from the earth even below the required source temperature. Call it mini-geothermal. So even as the crawl space ground temps sink down below 70, a heat pump could still get 77f water from that source at a very high COP. The lower the ground temps become, the more self-renewing they'll become, especially if we've been packing heat in all through the fall. If it gets drawn down as low as 65f, I believe it will begin to become entirely self-renewing. And so from a heat pump perspective, the source temperature of the crawl space ground at its worst will be higher than the air temperature at its best (peak winter highs out here basically top out in the low 60s even during warm spells). This is one difference for radiant in my region compared to the cooler climates where it is usually installed, and it makes me think under the right configuration I may be better off coupling the house with the ground than insulating the house from it.

    On another note, I think I understand what you meant before about deliberately seeking out inefficient panels. Originally I was looking for something like a greenhouse enclosure that could be seasonally fitted over pool collectors. The goal was to use them as is for summer nocturnal cooling and then give them some extra oomph for heating during the winter. But maybe I'm underestimating the seriousness of the overheating issue? It sounded like some of the existing solutions discussed above handle it well, but now I'm uncertain.

  • hot_rod
    hot_rod Member Posts: 24,485
    edited 12:34AM

    the only way to accurately model this is with your load info and the solar simulation at your exact location. There are some programs to help with the number crunching. I have used the T-sol program, I think they have a free demo at the site. Www.valentin-software.com/en/

    RETScreen is a free program from Natural Resources Canada, it have an easy to use ST module. It is a huge program the does energy modeling for numerous technologies.

    This graph showed the performance difference between 3 types of collectors. The unglazed pool collectors can reach a high efficiency due to operating at close temperature, ambient/ pool temperature. Taking the glass out of the equation on helps also. But performance drops quickly as ambient drops even 10 degrees.

    Evac tubes really don’t harvest more energy, but the vacuum limits to loss in cold weather condition

    You lose a lot of energy from the place where it strikes the glass to the actual tank of water.

    Here are some of the outputs from the T-Sol program. Model for a 16 collector array on the rec center at the GTMO base in Cuba

    The number worth looking at is the SF, solar fraction. The % of load that the solar will cover. I think in California you had to prove 50% or more to qualify for rebates.

    This is my single collector system in Salt Lake., 53% SF.@ 40% efficiency. The PV array next to the ST runs more like 10% efficiency

    When it comes to heating loads in cold climate a 30-35% SF is what most designers looked for, both from an array that would fit on the roof, and the system cost🤔

    The Weller School near Fairbanks has a Caleffi ST array that warms the ground all summer for the HP loop field. They used to have an online data logger you could view.

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
    BTUser
  • jumper
    jumper Member Posts: 2,415

    I think that radiant floor heating in 1950's basements was steel pipe imbedded in concrete floor. Floors could become too warm for sensitive bare feet. Insulation was gravel under slab. Enough heat must have penetrated gravel over decades that so much heat accumulated. Electric heating can be at higher temperature so heat accumulation can be faster. If you want to control heat output from stored heat then hot water is way to go. I suggest multiple tanks instead of one big one because you can retain high temperatures. Common set up is five tanks, four full plus one empty. Return from first tank goes into empty tank. Subsequently return from second tank goes into now empty first tank. 77° may make your home habitable but I doubt that it will be comfortable.

    BTUser
  • DCContrarian
    DCContrarian Member Posts: 1,020

    I've been interested in "alternative energy" for almost 50 years. Nothing being discussed in this thread is new. If storing energy on site worked it would be commonplace.

    I was excited about phase change materials for a while, see this posting:

    https://www.greenbuildingadvisor.com/question/calcium-chloride-hexahydrate-for-phase-change-heat-storage

    But they just don't pencil out. Like a lot of these ideas, if you just do a bit of analysis you'll see that they're impractical.

    And the landscape is littered with examples of people who built houses that didn't work out, where it would have been obvious that they couldn't work out if just a little bit of analysis had been done.

    A good example is The Sunrise House:

    https://www.greenbuildingadvisor.com/article/the-sunrise-house-in-fairbanks

    They put a 5,000 gallon water tank in the basement to store heat. Only about one thirtieth of the size it needed to be.

  • hot_rod
    hot_rod Member Posts: 24,485

    thus Swiss company specializes in large solar tanks for year around heat and dhw

    Multi thousand gallon tanks that the buildings get built around

    https://jenni.ch/publications-448.html?file=files/jenni/inhalte/pdf/Publikationen/Artikel_CompactThyssen_english.pdf&cid=2910

    Ive used old lp tanks. 500 gallon tanks are common. I use to pick them up for a buck a gallon

    Ill bet they are common in rural Wisconsin

    This tank was installed vertically in my last shop

    Input from 160 sq ft solar, wood gasification boiler

    The top was an expansion space and the air bubble for the solar drainback

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
    BTUser
  • BTUser
    BTUser Member Posts: 10

    Wow, the energy loss at the Cuba installation seems like a lot! Almost half the energy generated at the collectors gone by the time it gets to the DHW? The corresponding energy loss for your single collector system was more like what I would have guessed. Are those kinds of losses par for the course for 16 collector arrays attached to relatively complex systems?

    Really appreciate all these pointers and the time you've taken to put all these photos together. Looks like it's time for me to move on forward to T-Sol or RETscreen. Glanced briefly at them, looks like I'm going to have to figure out how to install some kind of Windows emulator on my MacBook before I can move forward.

    The solar graphs I know well at this point; I somehow spend all my free time these days with my nose buried in either idronics or Siegenthaler's book.

  • DCContrarian
    DCContrarian Member Posts: 1,020

    A thousand gallons weighs 8300 pounds. If you increase the temperature by 50F that's 415,000 BTU, or roughly what you'd net from burning four gallons of fuel oil. At $3.50/gallon that's $14.00 worth of heat. A purchase price of $900 gives a gross payback period of 64 years, assuming:

    1. There's no cost to putting heat into the tank.
    2. There's no cost to getting heat out of the tank.
    3. All of the heat put into the tank comes out.
    4. There's no better source of heat than $3.50/gallon oil

    If you could use the tank once a day rather than once a year it would make sense, tank water heaters are popular. But there's a reason we don't all have big tanks of water in our basements for seasonal energy storage.

  • hot_rod
    hot_rod Member Posts: 24,485

    I keep an old Dell with those windows based programs. RET Screen, F-Chart are what the old timers used.

    T-Sol and PolySun are what the younger generations use:)

    You can define all the components in the piping loop to come up with all the losses along the way, piping, insulation, surrounding air temperature, etc.

    It's really the ambient air temperature around the collector and the single pane of glass that grabs a lot of the energy in the cold winter months. But 120° is still achievable from a basic flat plate collector in most any temperature condition.

    The last version of the Caleffi StarMax collectors had welded frames, spray foam insulation, and a plastic backing, so it did pick up some performance over the riveted, polyiso built collectors. The piping inside sloped to a center outlet for drainback operation.

    This book takes a little deeper dive into the solar design and calculations, piping and wiring concepts.

    Those are Caleffi ST panels going on the MREA building in Custer, Wisconsin

    This was the motivation for that 500 gallon repurposed tank in my shop. The air sep on the hydronic side needs to be piped into that air space so you don't lose the air bubble.

    A pressurized tank like this makes it easy to use "one" water, no heat exchangers needed for the hydronic and solar fluid.

    Bob "hot rod" Rohr
    trainer for Caleffi NA
    Living the hydronic dream
    BTUser
  • Larry Weingarten
    Larry Weingarten Member Posts: 3,725

    Hi @DCContrarian , I agree that seasonal storage which works is rarer than the unicorn 🦄. But 1000 gallons will help with evening out the bumps and dips. I have for about 20 years had a 1000 gallon, solar heated tank in my house for space heating and DHW preheat. It works pretty well, though I do have to burn 1/4 cord of wood yearly.

    Yours, Larry

  • BTUser
    BTUser Member Posts: 10

    So, if I have say 300 sqft of something like those StarMax collectors, installed in a drainback config by someone who knows what they’re doing, do I still have a problem during the summer? During the summer we’d use them for maybe 50-100 gallons of DHW per day, but aside from that they’d just sit there empty and inactive. Outdoor temps out here sometimes get above 110. Is that an issue or can they just sit up there empty in the heat?

    And, are the main advantages of unpressurized storage price and the ability to get a large storage unit into the building?

  • I think I spoke to your wife when she called about Viessmann products. Good luck and keep us up to date on your project.

    8.33 lbs./gal. x 60 min./hr. x 20°ΔT = 10,000 BTU's/hour

    Two btu per sq ft for degree difference for a slab