X-Message-Number: 1938
Date: Thu, 11 Mar 93 01:06:13 CST
From: Brian Wowk <>
Subject: CRYONICS Cold Calculations

Mike Darwin:
 
> Also, I guess we need to come up with some DIMENSIONS for this room.
> Any thoughts out there?
 
        How much money and how much space do you have?  Obviously the 
bigger the better.  There's nothing like a high volume to surface area 
ratio to reduce cooling costs per cubic meter of storage.
 
        I have derived some simple formulas to aid your thinking.  (I 
encourage Steve Harris, Hugh Hixon, or anyone else out there to verify 
them.)  Assume that the thermal conductivity of good foam insulation is 
0.02 watts per meter per degree K.  This is close to the thermal 
conductivity of still air in the temperature range of interest, and good 
foam insulation (such as polyurethane foam or Dow styrofoam) insulates 
just about as well as still air.  Assume that the total temperature 
difference is 150'K (20'C -(-130'C)), and that 200000 joules boils one 
liter of liquid nitrogen (LN2).  The LN2 boiloff rate will then be
 
                        B = 1.3 * A/T
 
where B is liters per day, A is the interior surface area of your room 
in meters squared, and T is the foam wall thickness in meters.  A slight 
variation of this formula allows you to compute the bioloff rate in 
terms of the wall R value.
 
                        B = 300 * A/R
 
where B and A are as defined above, and R is the total room-temperature 
R value of a wall which may be composed of different layers of material.  
(Typical R values per inch are 5 for styrofoam, 3 for fiber glass, and 1 
for wood.) 
 
        You asked how big your room should be.  This I cannot tell you.  
However I *can* tell you exactly how thick your walls should be.  As you 
increase wall thickness, LN2 consumption costs decrease and capital 
costs of insulation increase.  At some point an optimum wall thickness 
is reached that minimizes total operating expenses.  Starting from the 
first equation of this letter, I have solved the optimization problem 
and found a formula which tells you how thick your foam wall should be 
to minimize total costs (derivation available upon request).  Amazingly, 
the answer is independent of room size.  The formula is
 
                        T = sqrt(475*c/(r*k))
 
where T is the optimum thickness in meters, c is the LN2 cost per liter, 
r is the insulation amortization rate, k is the cost of foam per cubic 
meter, and sqrt() denotes taking the square root of the stuff inside the  
parenthesis.  For example, assume LN2 is 10 cents per liter (c=0.1), 
foam is $100 dollars per cubic meter (k=100), and you want to amortize 
insulation expense at 10% per year (r=0.1).  The formula predicts 
operating costs will be minimum when your foam wall is 2 meters thick!
 
        As a practical application, consider a room measuring 5m x 5m x 
3m with a total surface area of A = 110 square meters.  The total annual 
operating costs using the optimum wall thickness of 2 meters will be
 
               (c * 365 * 1.3 * A/T)   +  (r * k * A * T)
 
which is about $5000 dollars a year.  The first term above gives the 
annual LN2 expense, while the second term is the annual insulation 
ammortization expense.  At the point of cost minimization, they are both 
equal at about $2500 each.  I know 2 meters sounds like a lot of foam, 
but anything less will create LN2 expense way out of proportion to 
insulation expense.
 
        By the way, $100 per cubic meter is the cost of Dow styrofoam 
here in Winnipeg.  Expanded polystyrene is a bit cheaper, buts its R 
value is 20% lower than stryrofoam so that it will only add bulk with no 
net savings.  I don't know prices for polyurethane or the Trymer 
cryogenic foam you might want to use for the inner layers.  Of course, 
do you really need Trymer with a wall 2 meters thick?  Thermal gradients 
will be so low that a cheaper foam will probably do fine, especially if 
layered in the way you have desribed. 
 
                                                --- Brian Wowk

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