Solar Power: The Turbojet Ploy
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In concept, a turbojet engine is a simple device. A fan at the front draws air into the engine and compresses it; in the burner section, liquid fuel is sprayed into the air and ignited; then the heated, expanding air passes through a turbine, which drives the fan by way of a connecting shaft. If we reconfigure the turbine to produce more mechanical power and less thrust, we can attach a generator to the shaft and produce electricity. And if we heat the air with sunlight instead of burning fuel, we obtain a solar power plant. How effective can such a plant be?
Heating air at a given pressure causes the air to expand its volume in direct proportion to the absolute temperature (temperature measured relative to absolute zero). If we heat air from 300 Kelvin (300 Centigrade degrees above absolute zero or seven Celsius degrees below normal room temperature) to 600 Kelvin, then a given mass of that air will double its volume. To convert degrees Fahrenheit (F) to degrees Kelvin (K), just use the equation
According to the ideal gas law (PV=NkT), if a given mass of gas changes its volume while not changing its absolute temperature, then the pressure in the gas changes in the opposite sense and the energy contained in the gas remains unchanged. If air is compressed or expanded slowly enough through a component that can donate or absorb heat, that isothermal condition will be met to a good approximation. The work that a system must do to compress air thus equals the product of the initial volume of the air and the pressure difference across which the system pushes it. Expansion of the gas is simply the temporal mirror image of compression: that and conservation of energy tell us that the work done by the expansion of air equals the product of the pressure difference and the volume of air coming out of the expander. If that statement did not stand true to Reality, we could connect an expander and a compressor to each other and use the combination to create or destroy energy.
Raw sunlight at Earthís surface carries roughly one kilowatt per square meter onto any flat plane that it strikes at a normal angle. If that plane is perfectly black and loses heat only through radiation, it will achieve a temperature of 364 Kelvin (196 F). If we want to double that temperature to 728 Kelvin (851 F), we must increase the flux of sunlight to sixteen kilowatts per square meter (according to the Stefan-Boltzmann law). We can achieve that increase with Fresnel lenses or mirrors.
For every cubic meter of air that we want to pump into a container against an overpressure of one tenth of an atmosphere (one atmosphere = 14.7 pounds per square inch or 101,325 newtons per square meter), we must apply at least 10.1 kilowatt-seconds of work to the compressor (depending on the efficiency of the compressor). If that cubic meter were to double its volume and then exit the container at the same overpressure, it would do at most 20.2 kilowatt-seconds of work on the turbine (again, depending on the efficiency of the turbine). A system carrying out those compressions and expansions will thus generate as much as 10.1 kilowatt-seconds of net energy through that cubic meter of air, which energy can be fed to an electric generator.
The average American family uses electricity at an average rate of about ten kilowatts, so letís see what a ten-kilowatt solar turbojet system would look like. We can scale it up later, but we must also bear in mind the fact that a large number of small, mass-produced systems is cheaper and more failure resistant than a few large custom-built systems.
According to the above calculation, generating ten kilowatts of power necessitates a system into which we pump air at the rate of one cubic meter per second against an overpressure of one tenth of an atmosphere and which system doubles the volume of that air before the air goes through the turbine. If the fanís inlet has an area of one square meter (a radius of 56 centimeters), the incoming air must move at one meter per second (3.6 kilometers per hour or 2.25 miles per hour). If air enters the system at 300 Kelvin, it only has to achieve 600 Kelvin to double its volume.
For our air heater to maintain a temperature of 600 Kelvin we must put enough sunlight on it to raise its equilibrium blackbody temperature to 728 Kelvin. At that latter temperature the heater re-emits all of the energy that it receives, sending it out as thermal radiation. At 600 Kelvin the heater would radiate half of that energy, so if we put 16 kilowatts of sunlight onto a square meter of heating surface, then the air gets 8 kilowatts of it. If we put a glass cover over the heating plate, to prevent convection from taking away heat, then the thermal radiation will be reflected back onto the plate, enhancing the amount of power that the system can extract from the original sunlight.
If we can harness 8 kilowatts from one square meter of black surface, then our solar-power unit needs 1.25 square meters of illuminated surface to produce 10 kilowatts. We need to arrange the area in such a way that it can heat one cubic meter of air per second from 300 to 600 Kelvin. We also need to ensure that the air will pass through the heating element at a speed that allows it to absorb the heat it needs to expand properly.
The energy put into the air through the pressure exerted by the fan can be manifested in one of two ways: it can appear as kinetic energy of the air in motion or it can appear in the compression of the air. If we convert the pressure entirely into kinetic energy, we have
which enables us to calculate the velocity of the airflow. At our one tenth of an atmosphere, P=10,000 newtons per square meter, so, with the density of the air (rho) equal to 1.225 kilogram per cubic meter, we calculate v=128 meters per second (288 miles per hour). If, on the other hand, we use the pressure only for compressing the air, we get air thatís 10% denser than the air coming into the device. The design of the turbine will determine how much of each of those effects will be manifested in the device. We donít want the air to flow at any speed even close to 128 meters per second, so most of the fanís work will be done compressing the air.
The speed at which we want our cubic meter of air to move across the deviceís heating element determines the cross section of the hot box under that element. If we want the air to move at ten meters per second, the cross section of the hot box must be one tenth of a square meter. If we make the heating element ten centimeters wide (about a handsbreadth), we must make it 12.5 meters long and make the hot box one meter deep. Itís unlikely, though, that a solar turbojet of that design would heat the air properly.
We have tacitly been thinking of the heating element as a solid plate with the air rubbing along its surfaces. But it will work so much better if it is in some way porous, so that the air will pass through it to absorb heat (see Appendix). From the direction whence the sunlight approaches it, it must appear to be solid, so that it will absorb all of the light falling on it. Yet it must also allow to blow through it freely enough to absorb heat quickly and efficiently.
The system now needs two hot boxes, one above the heating element and one below it. The top of the upper hot box will be made of glass, so that sunlight can enter the device and heat the heating element. Cool air will enter through the upper hot box and keep the glass relatively cool before going through the heating element, absorbing heat, and passing through the lower hot box and into the turbine.
Ten kilowatts is too small for a commercially viable power plant. So letís imagine building one that generates one thousand kilowatts. Such a machine will require a heating element covering 125 square meters and the main mirrors must intercept 2000 square meters of raw sunlight. To give us the amount of power we want, the machine must draw in one hundred cubic meters of air every second.
If we make the heating element 2.5 meters wide, then it must be 50 meters long. The main mirror, in the form of a segment of a parabolic cylinder, must cover a footprint with a length of 50 meters and with a width equal to 40 meters divided by the cosine of the latitude at which the device sits. That main mirror will focus sunlight onto a convex secondary mirror that will collimate the light and project it onto the heating element. Ducts running alongside the upper hot box will ensure that air comes to the heating element uniformly along its length, so that the heating element is cooled uniformly. At one end of the device we will find the fan, turbine, and generator sharing a short shaft and a cable taking the generated electricity to a switching point to be put onto the distributing grid.
Imagine large numbers of these devices built in the deserts across the world. Vast solar power farms in the Sahara could feed electricity to Europe and Africa. The Gobi Desert and Tibet could power China. The Sonoran Desert could provide electricity for Mexico and Central America, while its northern reaches could power the United States and Canada. Those power sources could bring about a significant reduction in the burning of carbonaceous fuels and mitigate, if not prevent, a full ecological disaster on this planet.
Appendix: The Heating Element
The heating element in this device must be some kind of flat surface that intercepts all of the light projected onto it and yet allows air to pass through it freely. We thus need some kind of structure that is open and yet appears completely opaque from at least one direction.
The best shape for our basic heating element is a helix. Made from blackened metal wire, its structure leaves plenty of room for air to flow through it and if its diameter is small enough, the air will absorb the required amount of heat as it passes through. If the distance between neighboring coils on the helix equals the thickness of the wire, then, seen from the side, the helix will be completely opaque and it will absorb all of the light coming to it from that direction.
Made by machines, the helices will be laid side by side on a grid made of a rigid material. The grid will support the helices, preventing them from sagging and breaking, while not interfering with the air flowing through the heating element. The basic heating element can be mass produced in panels of standard size, say one meter square, that can be fitted together in a larger system.
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