X-Message-Number: 1979
Date: 17 Mar 93 17:33:19 EST
From: "Steven B. Harris" <>
Subject: CRYONICS Cold Room Calculations
Dear Folks:
Sorry I've been offline during the -135 C room discussion. My
computer has been down, but I'm now up and running on a much
faster system. (I wonder what that statement will mean a couple
of centuries from now...)
Anyway, Mike has asked me to check some figures of Brian's,
and I'm glad to do it on the net so nobody thinks this is all
handwaving. I dug out my old preliminary calculations on a rec-
tangular prism-shaped room-sized enclosure for use at -135 C
(CRYONICS Mag, June 1986, pp 15-17) to see how they compare with
the more sophisticated treatment of Brian Wowk's. I had figured
on a room of 10 x 20 x 8 feet (excuse the English units) holding
18 patients. Surface area would be 82 meters square. I assumed
one foot thick foam, but used a K of ~ 0.01 watt/m*K, which was
the number I got for the best cryogenic foams used in insulating
the non-Dewar tanks of liquid fuel cryogen rockets, like the
space shuttle (NASA reprint #1002). I calculated in 1986 that
(as a simple approximation) heat loss would be:
dQ/dt = KA (deltaT)/thickness = .01*(82)*(160)/.3048
= 37 million (3.7e5) Joules/day
and LN2 use would be 3.7e7/2.1e5 = 176 liters/day (assuming gas
venting at -135 and no recirc of cold gas around the outside),
which worked out to about 10 liters/day/patient (right now our
bigfoots run at about 4 liters/day/patient in best case). Like
Brian, I also came to the conclusion that there isn't enough
"cold" in -135 C gas to be worth a lot of bother to use if for
further cooling.
In Brian's calculation the boiloff rate for this size room
works out by his formula to be:
B = 1.3 * (82 meters square/.3048 meters) = 350 liters/day =
20 liters/day/patient
He's using the same formula I did in 1986. His answer is just
about twice my figure, due to Brian's incorporation of a foam k
value twice as large as mine in his proportionality constant, and
slight differences in his delta H for LN2 vaporization and
temperature difference assumptions.
Brian's analysis of the question of how much foam insulation
to use in a -135 C room is the standard business profit maximum-
ization analysis, and is indeed the way to do this problem, once
we're sure we have all the cost parameters reduced to equations
(what is the labor cost to make insulation thicker, anyway:
surely it isn't a linear function?) Brian's derivation on that
doesn't include all business parameters but is straightforward,
despite the counterintuitive result for the foam thinkness (maybe
it will be better if we can indeed find some K = 0.01 foam).
Let B be boiloff in liters/day
Let C be nitrogen cost in liters/day
Let A be foam area
Let T be foam thickness
Let K be foam cost in dollars per cubic meter
Let r be amortization of foam cost per year
Then nitrogen cost per year is 365 days * BC, or 365*1.3 CA/T
(from Brian's equation above)
Foam cost per year is krAT
The yearly cost Y is nitrogen cost plus foam amortization so:
Y = 475 CA/T + krAT
Differentiating Y with respect to T and setting dY/dT = 0 gives:
krA = 475 CA/ (T^2)
So T^2 = 475 C/kr at minimum cost Y.
Interestingly, in my 1986 letter, I had also included a very
simple formula for the expected effect of external foam in
additively insulate a *dewar* to improve its performance (this is
a separate problem), and after that I had pointed out to me that
John Day and Art Quaife, in CRYONICS October 1981, had earlier
addressed this same problem, but with the same sort of profit
maximizing calculation for foam dewar insulation that Brian just
has for the cold room, albeit in more exhaustive fashion than
either Brian or I had (Art is a professional mathematician).
When we get to the actual design of our cold room, it may be
worth it for Brian (no slouch at math himself, being a physicist)
to re-do his earlier effort with the same level of detail that
John and Art did.
Steve
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