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Thermal Mass Math

In a message dated 10/20/02 11:35:07
PM, [email-masked] writes:

<<
here's an
example that I found right off using real figures. Simplified a
bit by ignoring air film resistance etc.

I use the symbols
below:

Tr ... for thermal resistance ( defined below )

Tc ... for thermal capacitance (")
Te ... for themral
energy (btus)
Tg .... for thermal gradient = delta temperature
(T_out - T_in) for example
Sh .... specific heat coeficient
for a material

--------------------------
EXAMPLE:
Thermal Mass RC example: (thanks to Nicholas Pine)

Given
that:
* Water has a Sh capacity of 1 btu / lb per deg_F
*
6" fiberglass insulation has a thermal blocking value

(R-Value) of 19. units are: (hr per ft^2 per deg_f )/btu

If you made a big box out of plywood, say a cube, 8' x 8'
x 8' (with a nice low surface to volume ratio), and surrounded it
by the fiberglass insulation mentioned above, and lined it with a
big sheet of rubber roofing material folded up like a Chinese
take-out box, you would have a thermal battery with a time
constant of...

Volume of water = 8' x 8' x 8' = 512 ft^3

Its Thermal Capacitance, Tc = Density EVolume ESh
= 62.4
lb/ft^3 per 512 ft^3 per 1 btu/lb EF ‹
= 31,948.8 btu / F ‹

Its Thermal Mass Resistance, Tr = R-Value / Area
= 19
[hr per ft^2 per deg_F/btu] / (6 sides of (8' * 8')
= 19 [hr
per ft^2 per deg_F/btu] / 384 ft^2
= 0.04948 (hr per deg_F/
btu)

This 8' cube of water would have an RC time constant
of:
RC = Tr * Tc from above we have:
= 0.04948 * 31,948.8

= 1580.83 hours or about 65.88 days

If you put this
thermal energy storage box in a constant temperature 0 degree_F
room with the water at 100 degrees_F , then in a one day period
about:

Te = (Tg) * hrs / Tr which has units of
= ((100
degrees_F - 0 degrees_F) * 24 hrs) / ( 0.04948 hr per deg_F /btu )

= 48,504 btus will be consumed, lost, transferred, sunk....

So about 48,504 buts of heat would leak out of the box, to
heat up the room. This seems like a lot, but, because water has
such a high thermal mass, the water inside the box would only cool
down by, Box_water_delta_deg_F = Te / Tc
= 48,504 btu /
31,948.8 btu / deg_F
= 1.52 deg_F

So after one day the
thermal battery's temperature would be:
100 deg_F - 1.52 deg_F
or 98.48 deg_F.

This looks good, but the storage has to
get charged up some way.

For the earth people we just need
to somehow come to terms with some assumptions about boundaries.
People who use underground umbrellas have
some kind of
R-figure for above but what to do about the sides and
underneath... Some assumptions should be given then tested to get
some kind of usable for other folk modeling information, that
doesn't require corporate expense accounts to give us some kind of
rough answer. It's
always better than picking numbers out of
the air. And we can always modify it to make it more and more
accurate.

Cheers,
Darel Henman

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