```X-Message-Number: 2011
Subject: cold room approximations; drawers
Date: Tue, 23 Mar 93 11:56:26 EST
From:  var s1 = "Timothy_Freeman"; var s2 = "U.ERGO.CS.CMU.EDU"; var s3 = s1 + "@" + s2; document.write("<a href='mailto:" + s3 + "'>" + s3 + "</a>");

Now that Bob Ettinger has pointed out that the curvature of a room
that is small compared to its insulation causes it to radiate heat
faster, the question arises, how to more accurately approximate the
behavior of a real room?  Brian Wowk used a spherical room.  Here are
two other possibilities:

Option I:  Do the same computation Brian Wowk did, but assume that
cold is lost over the outside surface of the insulation rather than
the inside surface.  Intuitively, it seems that this approach will
give a lower bound to the behavior of the real insulated room, but I
can't prove it.

In this model, the surface area asymptotically increases as the square
of the insulation thickness, and the cold lost per square inch is
inversely proportional to the insulation thickness.  Thus in the limit
the total cold lost is proportional to the insulation thickness, and
it isn't clear that thicker insulation works better.  So this
approximation may be useless because it may give the result that no
amount of insulation helps.

Option II: Assume the insulation covering the room is a slab on each
surface, a quarter cylinder on each edge, and an eighth of a sphere on
each corner.  Treat cold loss on the face with the simple linear
approximation Brian used in his first program.  Approximate cold loss
at each corner as an eighth of the cold that would be lost from the
center of a solid sphere with radius equal to the thickness of our
insulation.  Approximate cold loss along each edge as one fourth of
the cold that would be lost per unit length from a long cylinder with
radius equal to the thickness of the insulation and a cold wire down
the center.

The sphere approximation can use the same formula Brian posted for a
hollow spherical room, except we have an inside radius of zero.  The
cylinder approximation will require repeating the derivation that
Brian used for the hollow sphere, except with one less dimension.

I have no intuition for whether this second option would overestimate
or underestimate the cold lost from the room.  I also have no
intuition about whether people will be able to decipher my description
without a diagram :-).  Here's a top view of the model of the room
which happens to look the same as a side view:

.--------------.
/                \
|  +------------+  |
|  |            |  |
|  |            |  |
|  |            |  |
|  |            |  |
|  +------------+  |
\                /
--------------

The inner square is the usable space, and the outer oval is the outer
boundary of the insulation.  The corners of the insulation should look
like a quarter circle.  (I'm not suggesting that anyone actually build
it in this shape, I'm just suggesting that if we pretend it is built
in this shape we can figure out how it would behave without too much
work.)  The approximation I'm suggesting essentially assumes perfect
planar insulators added where the insulation changes shape:

.--------------.
/ |            | \
|--+------------+--|
|  |            |  |
|  |            |  |
|  |            |  |
|  |            |  |
|--+------------+--|
\ |            | /
--------------

Here there is good thermal contact between the corners of the cold
room and the round-shaped insulation contacting the corner.  Brian's
original approximation simply multiplied the surface area of the inner
room by the conductivity of the insulation, and it was equivalent to
supposing the same perfect insulators are added in the same places,
but there is no thermal contact between the edges and corners of the
cold room and the edges and corners of the insulation.

I don't know if any of this makes sense to the readers.  Maybe email
isn't the right place to describe it.  It seems to me that doing a
detailed finite element approximation before building the thing is a
good idea.  I'm hoping someone (Brian?) has the enthusiasm to modify
Brian's program to implement either option I or option II or some
better idea.  I would do it myself except I have a thesis to write.
:-).

Another idea: Could we put the patients in drawers so we don't have to
walk into a room with no oxygen?  Maybe we could also have two or more
independent LN2 dewars keeping the room cold, and some mechanism that
makes it possible to remove them one at a time for servicing without
walking into the room.  Any practical implementation of this would
have some sort of air lock mechanism for each drawer that would avoid
sucking a lot of room-temperature air into the -130 degree cold room
when you open a drawer.

Tim

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