X-Message-Number: 2002
Date: Sun, 21 Mar 93 18:11:04 CST
From: Brian Wowk <>
Subject: CRYONICS Reply to Ettinger
Robert Ettinger:
> Several people have been talking about room-sized storage with
> conventional (non-vacuum) insulation. Brian Wowk offers
> calculations involving an "optimum" 2 meters of foam insulation,
> with a room 5m x 5m x 3m. However, unless I have misread him
> somehow, he has overlooked a crucial point. He says the thermal
> conductivity of good foam insulation is 0.02 watt/m/degC; this is
> not true. The coefficient (0.02 watt/m/degC in this case) is not a
> constant, but varies with distance. The relation is linear
> (Newton's law of heat transfer) only for fairly thin walls. For
> thick walls, or at least walls that are thick relative to the
> overall dimensions, the temperature gradient is much steeper near
> the inside than near the outside, which means that increasing
> thickness rapidly loses effectiveness. I haven't done the
> calculation for Mr. Wowk's case, but the boiloff will be much
> larger than he says.
Everything here is true except the last sentence. Yes the K
value varies with temperature, and this introduces a non-linearity in
the temperature gradient (as does the room geometry). However, by
using the maximum *room temperature* K value all the way through, I am
setting an upper bound on heat flow. I know this wasn't what I said I
was doing (room-temp K for Stryrofoam is 0.028 not 0.02) but is
effectively what I was doing for Trymer (room-temp K=0.021). My
calculations were only intended to be approximate anyway. (Even so,
the boiloff will not be "much larger" than I calculated.)
Since the push is on for accuracy, I have decided to extend my
model to include important effects heretofore ignored. Most imporantly
I was ignoring the cost of insulation needed to fill in edges and
corners of the room. (I was simply slapping on rectangular slabs of
foam matching the area of walls and ceiling.) When the insulation
thickness is comparable to the room size, this ignored volume can be a
very large fraction of the total insulation volume. For a room with
interior dimensions L x W x H, this previously-ignored volume is given
by
4*T*T*( L + W + H ) + 8*T*T*T
where T is the insulation thickness. Including this volume in the
optization problem and obtaining an analytic solution requires solving
a quartic equation. As is always the case in physics, when the going
gets tough, the tough run home to their computer.
For the benefit of interested hackers, I have included at the
end of this posting a simple BASIC computer program that can be used to
compute the approximate operating costs of a -130'C foam-insulated cold
room. This program includes full floor space costs, and the
previously-ignored insulation volume discussed above. I have adjusted
my former LN2 daily boiloff equation to now include K explicity so that
different foams can be compared. I suggest that room temperature K
values be used to be conservative (K=0.021 for Trymer, K=0.028 for Dow
Styrofoam). The formalu assumes 1 litre of LN2 consumes 234kJ of heat
in the process of vaporizing and warming to -130'C (which differs from
my old approx. value of 200kJ).
So what is the optimum foam thickness now? For our standard 5m x 5m
x 3m room, and Trymer foam at $200 per cubic meter ($6 per cubic foot) the
optimum thickness is 1.4 meters. However as the data below show, savings
increase only moderately as you go from 1 meter to the optimum 1.4 meter
thickness.
INSULATION THICKNESS: 1.00
INSULATION CAPITAL COST: 34000.00
INSULATION AMORTIZATION: 3400.00
LN2 LITERS PER DAY: 128.20
ANNUAL LN2 COST: 14038.45
ANNUAL FLOOR SPACE CHARGE: 2450.00
ANNUAL TOTAL OPERATING COST: 19888.45
INSULATION THICKNESS: 1.40
INSULATION CAPITAL COST: 55574.40
INSULATION AMORTIZATION: 5557.44
LN2 LITERS PER DAY: 91.57
ANNUAL LN2 COST: 10027.46
ANNUAL FLOOR SPACE CHARGE: 3042.00
ANNUAL TOTAL OPERATING COST: 18626.90
To keep capital costs down, and logistics simpler, 1 meter is probably the
way to go. This gives an operating cost of $20,000 per year (neglecting
maintenance and capital costs beyond insulation) for a 20 patient storage
unit.
By the way, I have used this program to study Stryrofoam (at $100 per
cubic meter) and found no savings advantage, even at large wall thickness.
Trymer appears to be the best stuff to use.
7 'ROOM.BAS
9 '
10 DEF FNB= IK*55.5*A/T 'DAILY LN2 BOILOFF (LITERS)
20 DEF FNC= K*( A*T + 4*(L+W+H)*T^2 + 8*T^3) 'INSULATION CAPITAL COST
30 DEF FNF= F*(L+2*T)*(W+2*T) 'ANNUAL FLOOR SPACE CHARGE
49 '
50 IK=.021 'INSULATION K VALUE (Joules/m/degC)
51 L= 5 'INTERIOR LENGTH (METERS)
52 W= 5 'INTERIOR WIDTH
53 H= 3 'INTERIOR HEIGHT
54 A= 2*L*W + 2*H*W + 2*H*L 'INTERIOR SURFACE AREA
55 F= 50 'FLOOR SPACE CHARGE PER SQUARE METER
56 C= .3 'LN2 COST PER LITER
57 K= 200 'COST OF INSULATION PER CUBIC METER
58 R= .1 'AMORTIZATION RATE
99 '
100 FOR T=.5 TO 2 STEP .1
110 PRINT USING"INSULATION THICKNESS: ######.##"; T
120 PRINT USING"INSULATION CAPITAL COST: ######.##"; FNC
130 PRINT USING"INSULATION AMORTIZATION: ######.##"; FNC*R
140 PRINT USING"LN2 LITERS PER DAY: ######.##"; FNB
150 PRINT USING"ANNUAL LN2 COST: ######.##"; 365*FNB*C
160 PRINT USING"ANNUAL FLOOR SPACE CHARGE: ######.##"; FNF
170 PRINT USING"ANNUAL TOTAL OPERATING COST:######.##"; FNC*R+365*FNB*C+FNF
180 PRINT
190 NEXT T
200 STOP
--- Brian Wowk
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