BACKGROUND OF THE INVENTION Field of Invention This application is a CIP of pending U. S. application serial no. 08/795,249, entitled"A Heat Engine For The Utilization of Solar Energy". The present invention relates generally to the field of solar energy power generators. More specifically, the present invention is related to collecting and guiding solar-heated air suitable to drive a power generating turbine. The device of this invention has particular application in providing a power source in sunny areas and as a large-scale air exchange system to provide community-wide cooling and cleaning.

Discussion of Prior Art Using heated enclosed air to induce an updraft is known in the art in various configurations as illustrated by US patents: 4,118,636; 4,224,528; 4,275,309; 4,388,533; 4,414,477; 5,096,467; 5,734,202. Typically a surface, capable of absorbing heat, collects energy from impinging sunlight.

A clear glass or plastic material usually encloses the heat-collecting surface to help trap the heat.

The transparent material additionally may have openings which allow heated air to escape out of the greenhouse structure and other openings to allow ambient air to be drawn into the greenhouse structure. More particularly, as air is heated within the greenhouse structure, it becomes less dense than the surrounding air and rises out of any opening in the transparent material. Air in the surrounding environment is drawn into the greenhouse structure to replace the air which escapes.

One notable observation is that the height to which the rising air climbs controls how much air escapes and enters the greenhouse structure. Finally, some type of wind-driven turbine is introduced

into the flowing air to drive an electricity generator. However, such devices, as described in the prior art, have at least two shortcomings: (1) they typically induce only linear air movement within the greenhouse and (2) they require a physical, chimney-like structure to contain any resulting updraft.

A few prior devices mention thermally inducing vortex-like, or rotational, air flow rather than simple linear flow; but appear to do so only within an attached physical chimney. Vortex-like airflow simply means a column of flowing air which has tangential components of velocity of the same order of magnitude as those of the upward motion of the air. The importance of achieving a vortex-like flow relates to the effect such flow has on the height of the generated updraft. The purpose of a chimney or smokestack is to prevent the inward flow of ambient air into the column of hot air rising from the chimney thus allowing the updraft to flow unimpeded. To achieve a higher updraft, a higher chimney is required. However, manufacturing costs and engineering complexity increase as the height of a physical chimney increases. A vortex-like flow, however, inhibits the mixing of the rising air with ambient air without a physical chimney due to the centrifugal forces associated with rotation. The patents to Lucier, 4,275,309, and Valentin, 4,452,046, contemplate rotational air flow but still require a tall, physical chimney-like structure. They differ from the present invention which uses a specific vortex flow instead of a physical smokestack to isolate the resulting rising air. The absence of a physical chimney structure decreases both the engineering complexity and manufacturing costs associated with solar powered wind generators.

Whatever the precise merits, features and advantages of the above cited references, none of them achieve or fulfills the purposes of the present invention. Accordingly, it is an object of the present invention to provide for a solar powered wind generator which is simple to build and inexpensive to manufacture.

Another object of the present invention is to provide wind deflection means within the solar powered wind generator which guides moving air in a rotational direction.

Still another object of the present invention is to provide an appropriately shaped and positioned aperture within a solar powered wind generator which will allow escaping, heated air to flow upwards in a vortex-like manner and, optionally, power a turbine.

Still another object of the present invention is to provide a solar powered wind generator which removes local, heated air in a spiraling updraft and replaces it with a flow of cooler, cleaner air induced in the reverse direction.

A further object of the present invention is to provide a solar powered wind generator which inhibits the mixing of ambient air with the produced updraft by imparting a vortex-like flow on the escaping air rather than using a physical chimney. The virtual chimney created by the vortex-like flow supports updrafts which exceed those supported by feasible physical chimneys. These and other objects are achieved by the detailed description that follows.

SUMMARY OF THE INVENTION These and other problems and disadvantages associated with the prior art are addressed and overcome by the present invention. Deflections within the present solar powered wind generator cause the solar heated air within the entire structure to move in a rotational manner. Additionally, heated air, which escapes the structures as an updraft, continues its rotational movement which allows it to rise high into the atmosphere without mixing with the environment; all without the need for a physical chimney-like structure.

It would seem from the ensuing discussion that cheap power can be obtained from solar energy by the use of a process which encapsulates the essentials of the world's weather cycle.

The weather cycle represents one of the largest engines operating on this planet. In the weather cycle, as in any heat engine, there is both a source and a sink for heat. Mechanical work is done when the heat is transported between source and sink. In the weather cycle, heat is supplied, mainly at the earth's surface, by radiation from the sun. This heat is converted into mechanical work through convective air currents which transport the heat to the upper levels of the atmosphere, where it is radiated to the heat sink provided by outer space.

Given this appreciation, it is pertinent to enquire as to whether an engine based on this model, could be constructed so as to provide an economic source of power and other benefits. Thus, besides the question of power it is also relevant to bear in mind the economic possibilities of other characteristics of the weather cycle. Specifically, these characteristics are air conditioning on a grand scale and rain making.

The primary task of this enquiry is an examination of the properties of a model system. This system involves three essential elements: a heat collector, a central convection tower which contains a turbine that drives an electric generator and, finally, an air column which forms an extension of the central tower. In turn, the conclusions of this enquiry in respect to these elements are detailed below.

The heat collector will take the form of a large, rather flat truncated cone, circular in plan, highest near its center and consist of light transparent sheets resting on a rigid supporting skeleton.

It should be realized at the outset that commercially significant power is measured in hundreds of megawatts and that, since the sun provides about 1 kilowatt per square meter, we should envisage collectors whose radii, are conveniently measured in kilometers. (A circular collector of 1 kilometer radius would provide an energy input of-3 X 10'watts). Air entering at the periphery with a component of tangential velocity imposed by the presence of vanes will absorb heat from the sun and flow inwards toward the central tower. Conservation of angular momentum will result in an increase in tangential velocity with decreasing radial distance.

The next item is the central tower. This will have circular cross section and enclose an axially disposed turbine which will be caused to rotate by the ascending air. The tower has merely to support itself and the turbine and to be capable of preventing radial air flow. The material can therefore be chosen so as to minimize cost of fabrication.

The final item is the air column. Calculation shows that if the efficiency of the engine is to be such as to allow the off-take of significant power, the height of the column must be such as to be conveniently measured in kilometers. In principle, as in the familiar smoke stack, such a column can be provided by a solid structure. In practice, the cost associated with such an approach would be prohibitive. To recognize how this difficulty can be surmounted it is helpful to understand the role of a smoke stack.

In a smoke stack, the pressure, at the same height, of the air moving inside is lower than that of the still air outside by an amount which increases steadily with distance from the top of the stack.

Thus, the wall of the stack acts as a barrier, that is to say it provides a force which acts to prevent the inward flow of air. Were the wall not there, inward flow and the resultant mixing would continue until the pressure differential and the attendant upward motion of the air were essentially extinguished. Crucial to the operation of the engine is the fact that the barrier to this inward flow need not be material but can be provided through the centrifugal forces associated with rotation.

Indeed, calculation shows that inward motion can be suppressed by imparting tangential components of air velocity whose maximum is comparable in magnitude to the speed of the upward motion in the column. An interesting outcome of these calculations is the prediction of the existence of a

central region in which there is no motion. This is gratifying in view of the observation of well known"eye"in a hurricane and the recent observation using Doppler radar of similar phenomena in tornadoes. It should be remembered that these natural phenomena depend on the same properties of the rotational air column as those invoked in respect of the proposed heat engine.

The phenomena identified above differ from those of conventional heat engine only in the "fuels"involved. In the two cases cited above, as well as in the sea spout, energy is stored in the heat of vaporization of water and liberated when the vapor condenses resulting in a rise in the temperature of the air. In the case of the engine, as in that of the so-called dust-devil, this temperature is raised by the direct action of the sun. In all three cases, energy is transported from ground level to the upper reached of the atmosphere.

Since air is transported upwards there must be a compensating flow in the reverse direction.

Characteristically, this air is cleaner and cooler than that which it replaces. The reduction in temperature can be significant. Its magnitude should approach the temperature increase engendered by local (atmospherically unstable) heating at ground level. Its impact is familiar to all who have experienced the cool breezes which often precede thunderstorms.

It is apparent that the operation of this engine would be environmentally friendly. Again, since fuel employed is free the only apparent ongoing costs are those associated with maintenance. This would suggest that this approach would lead to the production of power at significantly smaller costs than those of more conventional methods.

BRIEF DESCRIPTION OF THE DRAWINGS Figure 1 illustrates a cut away side view of the present invention Figure 2 illustrates an overhead view of the present invention wherein a centrally located aperture is circular in shape and wind deflecting vanes are located at the structure's periphery.

DESCRIPTION OF THE PREFERRED EMBODIMENTS While this invention is illustrated and described in a preferred embodiment, the device may be produced in many different configurations, forms and materials. There is depicted in the drawings, and will herein be described in detail, a preferred embodiment of the invention, with the understanding that the present disclosure is to be considered as a exemplification of the principles of the invention and the associated functional specifications of the materials for its construction and is not intended to limit the invention to the embodiment illustrated. Those skilled in the art will envision many other possible variations within the scope of the present invention.

The primary structure of the present invention, heat collector 110, is illustrated in Figure 1.

Heat collector 110 is used to collect energy radiating from the sun and heat the air located between a supported layer of transparent material 104 and a heat absorbing layer 100. In order to produce commercially significant power, a collector approaching 104-106 m2 in size is contemplated. The sun 116 provides approximately 1 kW/m2 and thus a collector with a radius of 1 km provides an energy input of ~3 x109 W; which is on the magnitude of hundreds of megawatts, a typical measure of

commercial power. However, a structure with as small a radius as 25 m is just as capable of sustaining a vortex-like updraft; but the resulting output would not be in the range of hundreds of megawatts.

Heat absorbing layer 100 acts like a theoretical black body which traps the heat from the sun's rays and, in turn, heats the air located between it and transparent layer 104. Any dark colored material will accomplish this goal; however, the preferred characteristics of any such material are low cost, ease of construction and maintenance, and durability. Asphalt has all those characteristics; but other functionally equivalent dark material like black plastic, coal chips, recycled tires, etc. could also be used. Layer 100 also services a secondary purpose of providing a foundation for braces 108; asphalt also excels at fulfilling this purpose.

Layer 100 has a convex shape near its center which is directly underneath aperture 106. The sloping sides 124 of this convex feature causes air 200, which is rotating in a horizontal plane within structure 110, to transition to vertical motion. This vertically traveling air maintains its vortex-like motion and escapes through aperture 106.

Top layer 104, defining the upper boundary of heated structure 110, is transparent to visible light, opaque to infrared energy, and must contain the air 200 flowing within structure 110. This allows the sunlight to freely enter structure 110 and heat the confined air and thermal absorptive layer 100 in an unimpeded manner. Further, layer 104 prevents the air 200, once it is heated, from mixing with ambient air outside structure 110.

Transparent material 104 is supported above heat absorbing layer 100 by a framework such as wire mesh 109 which rests on the framework of braces 108 which themselves are supported by heat absorbing layer 100. An additional layer of wire mesh 111 rests above material 104. Lower layer 109 provides to material 104 the strength necessary to withstand the forces consequent on the decreased pressure in the disclosure; and layer 111 provides stability against lifting forces which may be occasioned by local winds. Braces 108 are securely attached to layer 100 foundation to withstand the forces of air flow within structure 110 and wind loading of the structure itself.

Because the ultimate goal of the present invention is to induce an airflow within structure 110, the number of braces 108 are the minimum required to safely support upper layer 104 and are designed to minimize any impedance to air movement. Thin, strong metallic braces made of aluminum, steel, or other functionally equivalent alloys provide the needed rigidity while occupying very little space.

Of course other materials as are known in the art could also be used as braces 108 but cost may be a factor in excluding exotic elements or composites from consideration.

As for the material of transparent layer 104, glass or clear plastic have the necessary characteristics. They both allow in sunlight; they have the additional benefit of reflecting infrared energy radiating from heat absorbing layer 100, which improves heating of the confined air 200; and they have the strength to withstand harsh environments. Also, they are capable of being securely attached to braces 108 in such a manner as to confine air flowing within structure 110. Because of structure 110 size, layer 104 is preferably constructed from a plurality of panels or sheets of a selected transparent material which are attached to each other and braces 108 to form an air-tight barrier and must conform to size of layer 100.

In a preferred embodiment, to maximize the volume of heated air, transparent layer 104 conforms to the size and shape of heat absorbing layer 100. Also, transparent layer 104 resembles a truncated cone in that its height above layer 100 is greater near the center of structure 100 than at the periphery.

A key feature of transparent layer 104 is aperture 106 which allows heated air to escape structure 110. Allowing air to escape is critical because this causes fresh air to be drawn into openings 114 of structure 110 which subsequently gets heated, becomes a vortex, escapes, and starts the cycle all over again. In a preferred embodiment, aperture 106 is circular in nature, centrally located on structure 110 and approximately one-tenth the size of structure 110.

Another important feature is section 122 which projects above transparent layer 104 for a distance that is on the same order of magnitude as the radius of aperture 106. Section 122 provides a stable transition area where rising, rotating air 119 is protected from ambient air during the critical, early stages of its formation. Once properly formed, rising vortex 119 does not require a physical chimney because mixing with ambient air is prevented by the vortex's angular velocity. An additional purpose of section 122 is to act as a support for a turbine or other power generating device (not shown) which can capture the energy present in the rising vortex. To fulfill both purposes, section 122 is preferably constructed of a sturdy metal and is supported by a plurality of columns 128 which are also made of sturdy metal and attach to layer 100. Columns 128 are numerous enough and strong enough to support section 122 and a turbine but do not significantly impede air flow 200 within structure 110. Alternately, other functionally equivalent materials could be used to construct columns 128 and section 122.

The height of transparent layer 104 above heat absorbing layer 100 and the size of the aperture 106 can be various values in relation to overall structure 110. However, at the periphery, transparent layer 104 must be high enough from layer 100 to allow outside air to be drawn into structure 110. Preferred heights to accomplish this requirement range from 1 to 5 meters. Using this range of periphery heights, all dimensions of structure 110 can then be calculated if the following preferred designs constraints are used: (1) at every point within a circular heat collector 110, the product of radius r and height h is a constant, and (2) the radius r,, of aperture 106 equals the height ho of aperture 106.

Following the dimensional relationships in this preferred embodiment results in optimal flow characteristics for the confined air moving within structure 110 and the air escaping through aperture 106. However, r-h = C is not a necessary characteristic of the present invention, it simply results in a constant radial flux at constant velocity.

The key to starting the rotational movement of air 200 within structure 110 are wind directing vanes 202. Air 118 is drawn into structure 110 through openings 114 in a radial direction. However, vanes 202 are angled, preferably at 45 degrees, so that air entering structure 110 has imposed on it both a radial and tangential velocity component. Vanes 202 are firmly secured to layer 100 in a perpendicular manner and rise substantially to the height of transparent layer 104. In a preferred embodiment, vanes 202 utilize nearby braces 108 to provide strength and rigidity. Further, vanes 202 are made of a strong material such as steel and placed at the periphery of structure 110.

However, any functionally equivalent material can also be used to construct vanes 202 which can just as easily be placed at inner radii as well.

ANALYSIS OF MODEL We suppose that the collecting zone is circular and that its outer radius is Ro and that the rate of input of energy is T per unit area per unit time. The rate of input of radiant energy to the system is then <BR> <BR> <BR> <BR> <BR> WnR 2y<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> (1) The immediate consequence of this input of heat is to raise the temperature of the air in the collecting zone and so to reduce it's density. We represent this change as a decrease from a value, <BR> <BR> <BR> <BR> p to p-8p. In its turn, this decrease in density in a column, height H, results in differential pressure, idealized as an amount 5p5pgH and a total force for motion of the air of amount <BR> <BR> <BR> <BR> <BR> FrlbpgHa<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> (2) where a is the radius of the column. In effect, this force acts to steadily increase the length, L, of a column of air which moves at a constant velocity v. Accordingly, we may write <BR> <BR> <BR> dMV<BR> F #a²#V² . <BR> dt

Here, we have written M= vra2pL for the mass of the air in the column. Equating these expressions for F we have <BR> <BR> <BR> <BR> v2 5PgH<BR> <BR> P (3) It may be noted that this result is consistent with the energy changes involved when masses are simultaneously raised and accelerated. It is then seen to be simply a statement that energy is conserved and can be expected to be very general in its applicability.

Now the heat from the sun is expended in two ways: in heating the air and; in doing mechanical work by putting that air into motion.

The rate of heating is given by IIa²VR#T W1 #<BR> where R is the universal gas constant, Q is the molecular volume and 8T the change in the absolute temperature.

The rate of doing mechanical work is W2FV#a²V##gH (4) At steady state W= W, + W2 (5) <BR> <BR> <BR> <BR> <BR> which is when<BR> a²VR#T<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> R02ya²V##gH<BR> <BR> <BR> # Writing |##/#| # #T/T ; #T/T <<1 (6) <BR> <BR> <BR> <BR> <BR> <BR> we have<BR> RT<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> R02ya²##V(gH )<BR> ##<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> Substitutingfrom (3) we have<BR> RT<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> R02ya²#V³(1 )<BR> <BR> <BR> ##gh<BR> (7)

so that <BR> <BR> <BR> <BR> <BR> 3Y QpgH<BR> <BR> <BR> <BR> a 2p QpgHRT Now, as stated, the rate of production of kinetic energy is FV. This, expressed as a fraction of the input radiant energy is, with (4) <BR> <BR> <BR> <BR> <BR> a²gHf5pV<BR> f1 ##V<BR> <BR> <BR> R02y which can, using (3) and (8), also be expressed as <BR> <BR> <BR> <BR> <BR> a²V³# gH##<BR> <BR> <BR> f1<BR> R02y gH##RT (9) while the fraction absorbed in heat is <BR> <BR> <BR> <BR> <BR> ru<BR> <BR> pQgHR T However, these results depend, as mentioned earlier, on an idealization. It has been assume that the density of the atmosphere remains constant with increasing altitude. In fact, the density at 11,000 metres (the height of the troposphere) is only about one third of its value at sea level. We can allow for this variation by taking the height of the column as 7,300 metres. On this basis we

have, taking R = 8.3 joules/degree C, T = 300° C, pQ = 27 X 10-'kg, p = 1.3 kg/m3, that f, = 0.4 and that 0 4 R 2 y a p p <BR> <BR> <BR> <BR> <BR> <BR> Then from (9) we find on substituting the values used earlier that V = 30 m/sec and op/p ~. 01.

Not all the kinetic energy can be converted into the electrical form. Since it is envisaged that energy will be extracted using the type of windmill employed in so-called wind farms, it is appropriate to use an efficiency determined from its study. This number is 47%. The overall efficiency is then 0.4 X. 47 = 0.19.

Clearly, it is impracticable to suppose the use of a solid chimney eleven kilometres in height and this concludes the consideration of the case where the column of air has a solid surround which supports the difference in, pressure op gh'that exists across the solid. Here h'is the vertical distance from the top of the column to the position in question corrected to allow for the variation of density of the atmosphere with altitude. Accordingly, we now turn to the case of real interest in which the difference in pressure is provided by centrifugal forces consequent on rotation in the air column and in which the radial motions which are necessary for the mixing of still and rotating air are precluded by the requirement that in such a process it would be necessary that the several conditions requiring that angular momentum and kinetic energy are conserved be satisfied simultaneously. In particular, it may be noted that the process of mixing which depends

on the formation of vortices at the interface between still and rotating air is essentially precluded by the requirement that angular momentum be conserved. The remarkable stability exhibited by hurricanes, tornadoes, water spouts and smoke rings must be attributed to this feature.

In the case of the heat engine, such rotation can be imparted through the use of deflector plates located at the periphery of the collector or elsewhere. The action of these plates is to deflect the air moving in the collector so that it acquires a velocity component which is normal to the radius of the collector. If the deflectors are at the periphery, the operation of the law of the conservation of angular momentum necessitates that as the air moves towards the center, its tangential velocity increases with the inverse of its distance from the center.

A first consequence of this rotation and the take-off of electrical power, We, is the necessity to reformulate (4) to read W = Wl + W2 + Wr + We where W, refers to the energy associated with rotation.

To evaluate the differential pressure consequent on rotation in the column we recognize that the centripetal force which must be provided on an element of air of thickness 8r, and height oh which subtends an angle a at a radius r can be expressed in the form 2 pbhrCcxbrc r

Then because angular momentum is conserved we require that <BR> <BR> c=r<BR> <BR> (11) where r is constant. On substituting from (11) in (10) we have, by integration, that the total pressure developed, at a distance h'below the top of the column, and at a radius r between the outer (rob,) and any other radius (r,),') within the zone of rotation is

(12) and that the total pressure difference developed between inner and outer radii is <BR> <BR> <BR> PPr21.1z<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> (13) To emulate the effect of a wall it is required, in the first particular, that the pressure difference so developed over the width of the zone equal or exceed that due to the density decrement indeed by solar heating (bp gh'). That is p5pgh'.

(14) As is clear on reflection, the pressure change associated with the operation of a Bernouilli effect, besides being small, is not to be included. It may be remembered that, according to

Bernouilli, the pressure, p, in an incompressible fluid, (including air in this context) stream, whose velocity is a function of position, x, is such that the energy density is constant. That is, so that -p,.) C 2 where C is a constant. Here, the energy of the air entering the collector is increased by the action of the sun by an amount equal to the sum of the kinetic energy acquired and the thermal energy.

The only change in pressure is that due to the decrease in density of the air and this is, p, as given in (14).

Referring to (13), in the region r<rj the pressure decrement is to be constant. For this to be so the rotational velocity must either be zero everywhere in this region or else the rotation is that appropriate for a rigid solid. Radial flow into this region must vanish. If this were true at all values of h, it follows that there would be no upward flow in this region. That is to say, in this region, the value of V, the vertical component of velocity, would be zero.

It should be noted that this somewhat surprising result finds obvious expression in the case of hurricanes. Specifically, in the existence of the so-called eye; a region of calm round the center of rotation. Recently, the existence of such eyes in the case of tornadoes has been confirme through the use of Doppler radar.

Again, if the air in this region were, in fact, static its pressure would equal that of the air outside the column and so would exceed that of its surrounds. Accordingly, one would anticipate

a down draft in this region. We see that the effect of the down-draft would be simply to contribute cold air to that rising in the annulus which surrounds the central column. However, it is apparent that the implicit reduction in efficiency of the engine involved in such flow can be eliminated by using a vane system in the tower of such an arrangement as to make rotational velocity proportional to radius (rigid solid) for all radii less that some value, ri c-In this case and supposing, as seems natural, that the velocity V is continuous as the"cylindrical"surface at r = ri h the differential pressure across this surface vanishes.

To recapitulate, an arrangement is envisaged in which there is a central core within which air moves vertically, with a velocity which is essentially independent of radius and with a rotational velocity which is proportional to radius and that this region is surrounded by one in which the air has a constant vertical component of velocity and in which the rotational component falls inversely with distance from the center.

In response to the forces present in the absence of equilibrium necessary change in pressure is achieved automatically by a radial compression. It is easily shown that the fractional change in pressure at a given radius is simply the negative of the fractional change in that radius.

We have now to determine v ; h and V,, h. To do so, we first consider the situation at ground level where rO=a, h=H.

We first fix r, the angular momentum density. We suppose that the rate of flow at the

periphery of the collector has a radial component VR so that the rotational component is VR tan a where a is the angle between the direction of the resultant flow and the radius. Taking the radius of the circumference to the Ro we have: (16) r = Ro VR tan a and since roH = a <BR> <BR> <BR> <BR> <BR> # VRR tan α<BR> Voh a a (17) Again, since the matter must be conserved we require that the radial flux of air is given by # = 2#RH0#VR = #a2²(#-##) V, (18) where Ho is the height of the gap around the periphery through which the air enters. Here, we have assumed for convenience and non-critically that, in the column, the vertical velocity, V, is uniform across the section r < r,. Using (18) and neglecting the small quantity, ##, in comparison with, p, we find that <BR> <BR> <BR> <BR> <BR> a²V<BR> <BR> <BR> VR 2R0H0 and from (16) r a2Vtan a 2H 0

Substituting for r in (15) in the case where h = H'we have <BR> <BR> <BR> <BR> <BR> <BR> 1 1 4H02 cot²α<BR> ( )2<BR> <BR> <BR> <BR> rIH a #²a4 where V2 #²<BR> Vr2 Solving we find that <BR> <BR> <BR> <BR> <BR> <BR> #²a²<BR> <BR> <BR> riH<BR> <BR> <BR> #²a2H0 cot α (20) <BR> <BR> <BR> <BR> <BR> Thus, we find that ri H, is monotonic function of p = Ho cot a/a decreasing from the value a with increasing values of p. Apart from the fact that costs of construction are likely minimized when Ho is small there does not at this stage appear to be any grounds for a preferred value for the ratio ri 0/a or for the angle a. This conclusion is in accord with the finding, as we shall now show, that at least in the important range in which r ; H-~ a, the rate of working associated with the rotation is invariant with rush.

To find this energy we recognize first that the total rotational energy may be expressed in the form W = I raz/2, where I is the momentum of inertia of the spinning air mass and X is the

angular velocity. The evaluation of this quantity is complicated by the fact that the air does not <BR> <BR> <BR> <BR> form a solid body so that w varies with radius so that it is necessary to consider the increments in the moment of inertia due to circular elements of thickness 6r and length H', thus oI = 2srH'pr2or Since the conservation of angular momentum requires a radial distribution of speeds specified by vire r where r represents the (constant) angular momentum and r is the radius. The corresponding angular velocity is r r2 so the kinetic energy is The rate at which energy is supplied is then seen to be <BR> <BR> <BR> <BR> <BR> <BR> nVprln--<BR> <BR> <BR> (21) (21) For the important case in which riz'= a, we see from (21) <BR> <BR> <BR> a 2Ho cot a<BR> -1<BR> <BR> <BR> ri A2a

whence we find that <BR> <BR> <BR> <BR> <BR> <BR> a 2H cot a<BR> <BR> a 2H0 cot α<BR> <BR> <BR> ln =<BR> riH #²a Then substituting in (21) for r we have a rate of working <BR> <BR> <BR> <BR> <BR> <BR> 2rtvpT2 Hocot a<BR> <BR> <BR> <BR> <BR> a2a This is just equal to that calculated for uniform flow across the whole section, at speed, V, without rotation. Since the input energy is constant it is apparent that to satisfy energy balance the air speed must be below that specified by the relation (4) namely: pV2opgH.

This is just equal to that calculated for uniform flow across the whole cross section, at speed, V, without rotation.

Since the input energy is fixed, it is apparent that to satisfy energy balance, the air speed must be below that previously specified. We shall refer to this speed as Vu on the basis that the total energy input is fixed and the shape of the fan can be modified so that the total power taken off is unchanged because of the different flow pattern in air flow we see that V'= 2 Vu 3, vu ZO. 8 V.

as stated this change does not reflect a change in the flow of energy since it does not alter the efficiency of the engine.

From (15) we observe that r ; h. and rOh. increase as h'decreases. For this reason the radius of the column would increase with distance from ground level, but as may be verified, only as hwz5. This effect can be expected to be reinforced by the loss of energy consequent on the effects of the action of forces from viscosity.

As we shall see, the existence of these forces whose magnitude is easily estimated, sets a lower limit to the radius of the engine. Two classes of force can be distinguished. Both arise from viscosity and so from changes in the velocity of air flow. The first, and less important, is associated with positionally dependent changes of velocity in the bulk and the second with circumstances in which the velocity falls to zero as at a material surface. The latter is of great importance in the flight of aircraft and in that connection is referred to as skin friction. Von Mises (loc. cit.) gives this force as where I is the length of the surface in a direction parallel to that of motion, b is then the breadth of the surface and V, is the velocity at positions remote from the surface. In our case, we are concerned with a situation in which, notionally, the velocity falls discontinuously from a value V to zero. While there is no material present at this position it would see to be prudent to assume that there is and so make a likely overestimate of the drag associated with this discontinuity.

Focusing our attention on the vertical component of velocity, we have b = 27rr, and I = H. The rate of working compatible with the stress #s is WsFsV1.32#r0V#µ#HV³

This represents a fraction 1.3 2 rOvlppHV3 2 RoY

(23) of the input energy. Taking V = V = 30 m/s, H = 10000, r, = Ro/10, u = 1.6 X 10-'kg/meter sec. and p = 1.29 kg/m3, we find that the rate of loss would reach a tenth of that supplied when Ro = 2.9m. However, it transpires that the length of the column is so great that the flow at the interface becomes turbulent; the value of Reynolds number <BR> <BR> <BR> <BR> <BR> ##ViH<BR> Re µ is so large that (23) is not sufficiently accurate and must be replaced by

. llRe. 66b/pplVi Substituting the numbers previously we find that the critical radius is increased by a factor of 23 to 67 meters. We reiterate, we would expect the efficiency of such engines to be significantly reduced if the radius of the collector were smaller than this.

There is another source of energy loss. This is through the direct conduction of heat from the relatively warm interior of the column. We suppose for simplicity that the temperature gradient within the column is constant and given by 8T roh-rih where 8T is on the basis of the factors employed about 3 °C. The flux of heat through the outer surface radius r,, and length H is <BR> <BR> <BR> <BR> <BR> <BR> <BR> 2#rohHk#T<BR> <BR> <BR> <BR> <BR> rohrih<BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> <BR> where k is coefficient of thermal conductivity. Since rj h/roh iS significantly small than unity this flux approximates to 2 ? tkHoT

The coefficients of viscosity, u, and thermal conductivity, k, in gases, are related. Thus: k = pc,, where c, is the specific heat at constant pressure. For the case of interest here, the gas is diatomic and c,, = 3 R. Using the values for u and R employed above we find that k = 2 X 10-3 watts deg' <BR> <BR> <BR> <BR> <BR> roi'an the rate of heat loss over the whole column is only 105 watts. This again, represents an energy loss, which while larger than due to viscosity, is nevertheless negligible in comparison with the input envisaged.

As a penultimate topic we remark that the assumption that the collection area is circular does not represent an actual constraint. It shall now see that the elliptical form is also permissible.

The essential characteristic of the air flow when the collector is circular is an invariance with respect rotational position ; a position specifiable by an angle 6. Specifically, radial and tangential velocities depend only on radius. If in the circular case, it is arranged that the height of the roof is varied with distance from the center in such a way that the product of radius and height at that radius remains constant, it is found that the radial and tangential velocities are both constant and following Bernouilli that the pressure is also constant. This configuration and its consequences are also applicable to the case where the perimeter of the collector is elliptical.

There remains the possibility of a significant difference in behavior in respect to rotation.

However, since the radial velocity is, with this roof geometry, then independent of radius it is <BR> <BR> <BR> only necessary to adjust the vane angle around the periphery as a function of the angle 6 in order