From: mustafa@seas.smu.edu (Mustafa Kocaturk)
Subject: Volume of Cyl. Shell and Thermodynamics
Summary: Weight of tank is ~ wall thickness. Gas may change state
Organization: SMU School Of Engineering and Applied Science
Date: Tue, 17 Nov 1992 04:53:59 GMT

   Hello,

In article <141424@lll-winken.LLNL.GOV> doug@wente.llnl.gov (Douglas S. Miller) calculates the approximate weight of a hollow cylindrical tank:
>           ...  The weight of the tank, however, will go roughly as wall 
>thickness squared (mass = density*pi*(r_outer^2-r_inner^2)*height), so now 

   I will first point out a tiny problem with the qualitative
   statement given above, and then attempt to use laws of thermodynamics
   to test the intuitive results being discussed.

   First, visualize doubling the thickness of the tank walls
   like wrapping the tank by another tank of the same thickness
   and roughly the same diameter, hence the weight would be roughly
   doubled. The weight therefore seems to incease only *linearly*.

   Proof:
   Using elementary algebra to write (ignoring the bases of the cylinder)

       r_outer^2 - r_inner^2 == (r_outer + r_inner)*(r_outer - r_inner)

   and assuming r_outer+r_inner is kept constant (by keeping the average
   diameter constant), it follows that the weight of the tank
   is *proportional* to its thickness.

>we're talking a 40*40*(20 lbs) = 3200 lb tank.  This sucker will also be big; 
>tank walls in scuba are at least an inch thick if memory serves and 
>we're going to increase that by 40. This gives us a tank about seven feet 
>in diameter at a bare minimum.
>
>Now all this assumes that we're scaling up a scuba tank to fit your
>application.  There may be better designs (spheres maybe) that will
>make 100,000 psi practical.  I'm sure there are high pressure

   Secondly, other things may happen before thickness grows
   out of control. The gas inside the tank will probably change state
   (liquefy or solidify) under that pressure, if some of its
   heat energy is taken away by cooling the tank such that
   the gas temperature stays the same (isothermal compression).

   Assuming that the volume and the amount of
   gas stay the same, then the temperature will be given by
       
       T2 = T1 * P2 / P1 (from PV=NRT)

       where

	 T1,T2 are the initial and final temperatures in degrees Kelvin, 
	 P1,P2 are the initial and final pressures of the gas in Pascals,
	 V     is the volume of the enclosure in m^3,
	 N     is the number of moles of gas in the enclosure,
	 R     is a proportionality constant ( ~ 77.8 kJ/((degree K)*mole))

   If the heat is not dissipated, i.e. the compression is
   adiabatic, then one may also use the fact that (PV)^gamma stays
   constant (where 0<gamma<1 -- correct me if I am wrong)
   to discover that T also increases under compression, but more slowly
   than linear. This will result in heat (==energy) loss to the
   surroundings, which will be reflected as a reduction in P * V .

   This means that not only will you need strong walls, but also heat
   insulation to keep the energy in. Note that Q=K*(T_inner-T_outer)
   gives the heat lost, with K being the thermal conductance of the
   cylinder walls (Heat flows from hot to cold).

   In this problem N is not constant, but this can be taken care of by
   treating the volume, temperature, and pressure as variable and then
   integrating.

   The formulas given above summarize the main ideas leading to
   the design of combustion engines and refrigerators, for example.
   They represent laws of thermodynamics, which are so universal
   as to be applicable to chemical reactions, biological systems,
   the life cycles of stars, and superconductivity.

   The idea of using compressed air for storing energy is like the idea
   of using metal springs. Has anybody seen Leonardo da Vinci's
   spring-powered car design ? Its model was on exhibit here at SMU
   recently, along with models of his other ingenious designs like
   the air screw, parachute, log drilling machine, (military) tank, etc..
   I believe he must have also dealt with thermodynamics.

>experimentalist types out there who can give a real answer.
>
   You bet, but I am not one of them.
   
>Doug Miller	doug@retzlaff.llnl.gov
>"But *why* can't I use a gas station air pump on my bike tire?"

   Kind regards,
-- 
Mustafa Kocaturk   mustafa@seas.smu.edu  EE Dept., Room 305A, Caruth Bldg.
Home: 214-706-5954  Office: 214-768-1475  SMU Box 753190, Dallas, TX 75275