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TECHNICAL NOTE Storage rings for thermal energy DAVID FAIMAN Ben-Gurion University of the Negev, Institute for Desert Research, Sede Boqer and Department of Physics, Beer Sheva, Israel (Received 25 January, 1977)

If a continuous supply of energy is to be drawn from an intermittent source, some form of intermediate storage system is necessary. This problem arises for example in all terrestrial attempts to utilize solar energy since the insolation at any fixed position on earth is not constant with time. The particular aspect of storage to which we here address our attention, is that of thermal energy. Typically, a volume V is available and the storage medium is placed in a container whose shape is chosen so as to optimize heat losses and manufacturing costs according to the situation in hand. For example, it is well known that a sphere is the shape that possesses the minimum surface area per unit volume, thus it optimizes both heat losses and material costs. In certain circumstances however, a cylinder might be preferred on account of its lower manufacturing costs. To be more quantitative, consider first a spherical container of volume V. Its surface area is (36'n')1/3V 213. In comparison a right circular cylinder of height h, radius of cross section r and having the same volume, has a total surface area of /h \ zz3 /

which takes its minimum value if h/r= 2. For this situation however, its surface area is (54"lr)x13V213: s o m e 14 per cent larger than the spherical case. It is this 14 per cent deficit in heat losses and material costs that in certain circumstances would be off-set by the greater ease of manufacture of a cylinder. One might well inquire whether some other geometry could perhaps offer an advantage not hitherto considered. In particular, take the torus or anchor ring, with radius of cross section r and having a mean radius R about its axis of symmetry. Its surface area is

R\,/3 16~'2 r ) V2/3 and for a ring with a very 'small hole' i.e. R/r < 27/87r such a container would actually have a smaller surface area than a cylinder of the same volume. At first sight this difference should hardly be worthy of note; after all, if one is prepared to manufacture a torus, why not go straight to a sphere whose surface area is even less. There are however, two important points to notice. First, on account of the toroidal geometry, a not inconsiderable fraction of the heat losses re-enter on the other side of the hole and are consequently not losses at all. Alternatively, in certain applications of such a storage unit, the device that is to utilize the stored heat may be located at the centre of the ring so that part of what would otherwise be lost heat, enters directly into the place where it is to be used. The second point, once we have accepted the hole-in-themiddle concept, is that we need not confine our attention to a regular torus whose dimensions are limited by the physical restriction that R/r>~ 1. For example, if we core out the inner part of our torus, using a cylindrical cutter of radius R, we are left with a ring having a D-shaped cross section. Such a storage ring would have a surface area of 27r[(Ir + 2)(R/r) + 2] V ~ [Ir(~ + ~r(Rlr))] 2j3 where now RIr can be as small as we like. In particular, for RIr = 0 we recover the spherical situation of (36~-)''3 V2'3. For RIr arbitrarily close to zero however, we come as near as we please to the efficiency of a sphere but with the additional advantages of a torus to boot.

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