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Green Building Bible, Fourth Edition
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General: Help! Help! Is there an Applied Mathematician in the house?!

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  1.  
    • CommentAuthormike7
    • CommentTimeNov 24th 2008 edited
     
    I'm trying to work out a reasonably accurate way of estimating movement of ground-stored heat, but keep banging my head on the limits of my rusty maths. For example the source/sink flow equations that produce this pattern:-
    would be useful if only I knew how to apply them. A bit of coaching needed!:smile:
    • CommentAuthorchuckey
    • CommentTimeNov 24th 2008
     
    Cor, this looks like maths problem that has a useful solution. Looking at your diagram, each path that go through the two nodes, look like a circle, one large loop and an almost straight loop. What I would do is to set up a spread sheet, with the node distance being "1", then calculate the path lengths for the radii .1....9.9,10, then presume each path is actually part a "doughnut" of length as calculated and of width .1, so now you have a table of areas, so you can calculate the individual Rs and so the total R. You could try a few larger values to see if they will change the final figure very much. As you can see, there is an infinite number of paths going up to an infinite radius, so depending on the accuracy required. Still with a spreadsheet you could go up to 6,000 items at least!!!
    Frank
    • CommentAuthormike7
    • CommentTimeNov 24th 2008
     
    Thanks Chuckey - I see what you mean - sort of. I was really hoping to find an algebraic way of dealing with it, to fit in with the algebra of my efforts so far. The advantage of this is that if you're lucky you end up with a neat formula where the importance or otherwise of the basic parameters is clear. Snag is you often end up with a dog's breakfast where nothing is clear! - and its going a bit out of fashion in these digital times, so few people have a clue what I'm on about, it seems.

    I've had a bit of a go spreadsheet-wise at Tony's floor U-value question, but am not very nifty with it. The computer I last had contact with in what I laughingly call my professional life was a PDP8 running Atlas Autocode. I've led a sheltered life since, so I could use a bit of coaching there, too. :sad:
    • CommentAuthorEd Davies
    • CommentTimeNov 24th 2008
     
    Posing questions like this is called nerd sniping:

    http://xkcd.com/356/

    In fact, you've more or less proposed the analog version of the resistor network problem. Hmm...
    • CommentAuthorJackyR
    • CommentTimeNov 24th 2008
     
    Why more points for mathmos? We're three-quarters of the way there already...

    (Conversation between my work colleague and mathmo friend:
    Workie: Jacky, you have to admit you're pretty eccentric.
    Mathmo: Really? We all think she's the most real-world-capable person we know.)
    • CommentAuthorJackyR
    • CommentTimeNov 24th 2008
     
    But I'm a lo-o-o-o-o-t rustier than you, and not v good to start with...
    • CommentAuthorJackyR
    • CommentTimeNov 25th 2008
     
    No, no, leave me alone! I've started looking up http://en.wikipedia.org/wiki/Heat_conduction and http://en.wikipedia.org/wiki/Heat_transfer. Aeeeeiii..... *fwomph*
    • CommentAuthorCWatters
    • CommentTimeNov 25th 2008 edited
     
    Mike7 All I know is the maths won't be easy. The problem look similar to calculating the electric field between two parallel wires. Twenty five years ago when I graduated I might have been able to do that but not now.

    Looks similar eh?
    http://mysite.du.edu/~jcalvert/phys/wav1.gif
    ...lines of equipotential (voltage) correspond to lines of equal temperature and are at 90 degrees to the lines in your diagram.

    What real world problem are you trying to describe? Point sinks ?
    • CommentAuthormike7
    • CommentTimeNov 25th 2008
     
    Thanks CWatters - yes it is quite similar, but spherical rather than cylindrical as per your parallel wires.

    The idea is that the heat loss flow from an underground spherical heatstore into the surrounding field and eventually to the ground surface will be identical to half of the blue diagram above (rotated 90deg).

    The equipotential line along the cut is the uniform average surface temperature, and the store surface will be defined by some equipotential ring of suitable radius - I don't know if it would be exactly circular/sperical - surrounding the point source.

    Just define the two equipotentials/temperatures plus the field conductivity, and it can only be a mere hop and a skip to get the total heat flow rate. I should be so lucky.
    • CommentAuthorwelshboy
    • CommentTimeNov 25th 2008
     
    I am no great mathematician. To my simple mind the diagram is a series of circles with a common chord length.
    Would this help and please don't laugh ! http://www.1728.com/circsect.htm
    • CommentAuthorCWatters
    • CommentTimeNov 25th 2008 edited
     
    Posted By: mike7Thanks CWatters - yes it is quite similar, but spherical rather than cylindrical as per your parallel wires.

    The idea is that the heat loss flow from an underground spherical heatstore into the surrounding field and eventually to the ground surface will be identical to half of the blue diagram above (rotated 90deg).


    Ah I see. Tricky. How about a worse case method.. Lets say the top of the sphere is at depth D. You could model the whole thing as a sphere surrounded by a sphere of earth thickness D.

    If that's not good enough.. If the surface of the ground could be assumed to be at a uniform temperature, I think this would look like a point source near a conducting plane (conducting because temperature is uniform). Google found the electrical equivalent pn page 48 and 49 here..

    http://books.google.co.uk/books?id=VOGZs1G3ZYIC&pg=PA48&lpg=PA49&dq=electric+field+point+source+near+a+conducting+plane&source=web&ots=gdX1mzn6YR&sig=mjglYHTOuVV8Pj5wPJn1zCZstU0&hl=en&sa=X&oi=book_result&resnum=9&ct=result#PPA49,M1

    but still v. difficult.
    • CommentAuthorhowdytom
    • CommentTimeNov 25th 2008
     
    I fear the biggest problem is applying it to a real life situation where the sub-soils, rock, ground water etc will vary through out the site, unless you dig out a 6m cube and back fill with a known media.
    tom
    • CommentAuthormike7
    • CommentTimeNov 25th 2008 edited
     
    Nice try Welshboy - trouble is the blue diagram is an artist's impression, I think, and they probably aren't circles...

    Thanks CWatters. I did a version of your worst case idea in the passive annual heat store thread http://www.greenbuildingforum.co.uk/forum114/comments.php?DiscussionID=921&page=4#Item_4and was looking for something better, as you guessed.

    Your electrical reference looks spot on equivalent, though I haven't had a good look at it to see if I stand a chance of using it. I also found :- .http://web.mit.edu/fluids-modules/www/potential_flows/LecturesHTML/lec1011/node26.html, except we don't want 'a' tending to 0. Oh no.

    And then what. Little wisps of steam coming from the ears now.
    • CommentAuthormike7
    • CommentTimeNov 26th 2008
     
    <blockquote><cite>Posted By: howdytom</cite>I fear the biggest problem is applying it to a real life situation where the sub-soils, rock, ground water etc will vary through out the site, unless you dig out a 6m cube and back fill with a known media.
    tom</blockquote>

    Too right - that is a danger. Yet another pressure toward larger stores for bigger schemes where the cost of careful investigation could be spread.
    •  
      CommentAuthorfostertom
    • CommentTimeNov 26th 2008
     
    Most subsoils, rock etc have reasonably similar densities, conductivities etc, so variations perhaps don't matter. Ground water - if you've got that, forget it - until major study and experimentation proves (perhaps partly) otherwise.
    •  
      CommentAuthordjh
    • CommentTimeNov 29th 2008 edited
     
    Posted By: fostertomMost subsoils, rock etc have reasonably similar densities, conductivities etc

    The values on p57 of the PHPP manual, for example, don't seem too similar?

    silt/clay - lambda 1.5
    rock - lambda 3.5

    I'll grant you that the volume specific heat capacities are more alike, but that's not the prime determinant of heat flow.

    mike, there are TRNSYS models for this kind of thing but those I'm aware of cost money from Tess - http://www.tess-inc.com/trnsys

    There's some more detail of what's available in the files around this location - ftp://ftp.cstb.fr/software/Tess/Doc/src/Type703a.html
    •  
      CommentAuthorfostertom
    • CommentTimeNov 29th 2008
     
    Posted By: djhthe volume specific heat capacities are more alike, but that's not the prime determinant of heat flow
    why not, if it's spacial heat flow i.e. in terms of time, direction and distance? For that, Volume SH would be the one to dial in, not mass SH.

    Is Tess Tas's sis?
    • CommentAuthormike7
    • CommentTimeNov 29th 2008 edited
     
    Thanks for those refs, djh.... but yikes! - I'm not looking to get that involved! What I wanted to do was just extend the idea you also had, ie. a much-simplified mathematical model which would clarify the important parameters, and give some idea of the effect of scale, etc.

    Such a model would not be so accurate a guide for any real design, but it could be a tool for understanding. Otherwise its practical use would be limited to saying 'Watch out - it's going to be at least this difficult' - how much more? Dunno.

    In my heatstore calcs so far, the volumetric heat capacity and the conductivity are both important, perhaps most being the ratio of the two.

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