Lux lucis tepida

In which, energy can be delivered to homes for less than a penny per kilowatt-hour.

The etymology of the word lukewarm comes from old English meaning tepid rather than from the Evangelist whose name means bringer of light is some traditions. The advantages of using real energy to warm water before it is delivered to a home is the subject of this post so we have a Latin title meant to evoke both light and warmth while perhaps spurring a memory that the Romans would take almost any practical step to get a good bath.

The word plumbing comes from the word lead and, despite its unfortunate origins, this is our main theme. Lead, mercury and other heavy metals cause many of our misfortunes because they cause our minds to no longer function properly. As we end the use of coal as we ended the use of lead in gasoline we may well see further reductions in violent crime. Luke, who traveled with Paul, does not mention what Nero did to Paul. Some argue that this is because he wrote his Gospel and Acts before Paul died but others argue that it is because Luke considered Nero an aberration. The behavior of Nero and the other degenerate Emperors could well be explained by the ubiquitous presence of lead in the Roman environment, only some of which was from the plumbing; they used it to flavor food as well.

In our plumbing, water is delivered to homes at the ambient temperature which is just the average temperature of a region over a year (about 55^o F near the Chesapeake Bay). Depending on the depths of the water mains, it may be slightly warmer in the Autumn and cooler in the Spring as the heat from Summer or cooling from Winter reach those depths, but if it is sourced from a well, then these effects are even smaller because the water temperature coming from that low in the earth is constant. The effect of raising the temperature of the delivered water by 20^o F on home energy use is important. When the warmer water is heated in a water heater, less energy is required to bring it up to temperature saving about 25% in energy. Further, when the warmer water is mixed with hot water for bathing, less hot water is needed resulting in a 16% savings on hot water use. Together this comes to a 36% savings. In a home where water heating accounts for 20% of energy use, then the overall savings is about 7%. For a well insulated house where water heating accounts for half the energy use, the fractional savings are even larger. And, there is no need to change anything in the home to take advantage of this.

The water pipes we use to deliver water to homes are close to thermal equilibrium with the ground, but if we are to deliver warmed water they will lose heat to the ground. So, we need an estimate of how much heat they would lose. This depends on the thermal conductivity of the soil and the temperature gradient beside the pipe. We can get an upper limit on the steepness of the temperature gradient if we consider a pipe surrounded by a cylinder of insulating earth with a radius that corresponds to the depth at which the pipe is buried. This is rough upper limit. For dry soil such as under a road the distance down to the water table will often be greater than the distance to the surface implying more insulation in that direction. For soil subject to percolation of rain water, heat transfer could be greater downward if the percolation is rapid enough. For a 20 centimeter diameter pipe buried 2 meters below the surface in soil with a thermal conductivity of 1.5 W/m/K and a temperature difference of 10^o C we can calculate an energy loss of about 31 Watts for every meter of pipe. For a town of 10,000 homes laid out in a square of 5 by 5 kilometers there needs to be about 250 kilometers of water pipes so the energy lost would be 8 MW. If each home uses 350 gallons of water a day that is about 0.018 litres/second and to raise 18 grams of water per second by 10^o C requires 770 watts. So, all 10,000 homes also need 8 MW of power. All told 16 MW of energy input is needed.

For solar energy input we need the collecting area of about 16 house lots. Let's have our town served by four 50 meter high water towers, one at each corner. If we paint the bottom of the tank of each one black, we can arrange mirrors called heliostats around the base of the tower to reflect sunlight on the bottom of the tower. If these are arranged in a circle, then to deliver 2 MW of average power to the bottom of the tower we need to deliver 10 MW at noon so the radius of our circle of mirrors will be about 56 meters. Assuming that the bottom of the tank is 20 meters across, the sunlight will be concentrated in power by a factor of 32. As long as the reflectivity of the bottom of the tank is 10% or less, this poses no danger to vision. The cost of durable (30 year) heliostats is about $126/meter² so the delivered cost of energy, $4.9 million in heliostats for 2 billion kWh over 30 years is 0.23 cents/kWh. Since this mode of delivery displaces electricity or gas use for water heating this looks like a pretty good deal though not all water use is involved with hot water use. In operation, one of our towers would look something like this spanish power plant, but with a water tower instead of a solar furnace. We don't need hot water, just warm.



But, we can go a bit further. Silicon solar panels are about 20% efficient and currently cost about $5/Watt. They perform within 80% of their initial efficiency for at least 30 years. The main reason they lose their efficiency is owing to cosmic ray induced defects in the crystal lattice which provide places for charge carriers to become trapped. At the bottom of a water tank, they are substantially shielded from cosmic rays and thus should last much longer. Let's assume a lifetime of 60 years. The cosmic rays come mainly from straight up (cos³) and 5 meters of water is about the same as half the atmosphere in terms of mass so that is about 3.8 attenuation lengths for neutrons and panels should last more than 40 times longer mounted under a water tower if they were not also affected by local radioactivity. In practice, we need only be concerned with radioactive contamination of the water in the water tower and the tower itself since the ground is a long way away. Drinking water should not have more than 30 micrograms of uranium per litre according to the EPA so this is a low level of radiation compared to soil. For a steel tower, uranium should have slagged off when the steel was made. So, doubling the life of the panels seems fairly conservative; they might last a couple of centuries or more.

Silicon can be run at much higher light intensity than solar illumination but it loses efficiency when it gets hot. Since we want to heat water in any case, cooling the panels at the bottom of a water tank just requires a bit of plumbing to carry off the 26 kW/m² of power that is not converted to electricity. This is the reason you can boil water in a paper cup over an open flame. The water keeps the paper cool. Keeping the panels within temperature limits also helps with their durability. What is the cost of electricity if we install silicon solar panels with this plumbing? Lets assume that the plumbing costs as much as the solar panels and we employ a company that charges about the same for labor. Then $15/watt over 60 years at 5 hours per day of illumination comes to about 14 cents per kWh with regular illumination, but 0.43 cents per kWh with the concentration we have arranged in any case using the heliostats. This novel method is about ten times less expensive than using coal and twenty times less expensive than buying electricity. The average 1.6 MW of electricity generated this way could be put to use powering public buildings allowing reduced property taxes.

The idea for delivering warmed water to homes appears to be novel and it actually came to me when a friend asked me to attend a Maryland Public Services Commission hearing about a proposed gas fired power plant which is currently dealing with issues of sources of cooling water with the county government. They are asking for flexibility in their licence application to go with either grey water or dry cooling. My friend was concerned about the potable water they intend to draw because each time the water system in Waldorf expands people where I live have to drill new wells and more springs run dry around the county. The issue with the grey water is that it is 15 miles away from the proposed power plant site while the plant itself is much closer to town. The plant will be throwing off about 400 MW of heat so it seems to me that they could ingratiate themselves with the city of Waldorf by offering to warm the potable water using that heat rather than waiting for the water to be used and sent so much farther away. In the end, this idea may be just fiddling while the ghost energy burns but following Luke in attempting to bring more light than heat I put it forward. There may be a number of applications for this approach of delivering power to homes through tepid water. In cases where artificial reservoirs are used instead of water towers and wells, floating absorbing mats might reduce evaporation and warm the water. I like the water tower arrangement best because it shows how to get the most out of silicon solar panels while they are still bit pricey. It seems to me to be the lowest cost for electricity around. We'll be seeing more of that with solar.

The EPA estimates that 147,000 water systems are ground sourced in the US. If 5% of those have suitable water towers then at 400 kW of electricity generation per tower that is about 3 GW of average generation available at less than a penny per kWh. The required nameplate capacity is about 460 MW or 12% of world production of solar cells in 2007. Since the supply of silicon appears to be adequate, getting a home grown heliostat industry going looks as though it might be a good investment opportunity. Because the water tower bottom environment appears to be a good place to avoid soft errors in electronics as well, companies that need very high reliability in data processing might be interested in sponsoring water tower conversions to provide themselves with an extra-reliable server network in a thermally controlled environment using low cost high circuit density components.

Waste heat may also be available from Fischer-Tropsch or Sabatier process plants that produce hydrocarbon fuels from atmospheric carbon dioxide directly. The relatively steady demand for water compared to seasonal heating suggests than towns that don't have suitable water towers may be able to warm their water supply while producing hydrocarbon fuels for other purposes using low cost wind energy as an energy input. The advantage here is that compared to the home scale plants we considered earlier, the need for heat is constant so that the equipment for producing fuel can be used continuously. Conversion of some existing combined heat and power plants might also take advantage of this if their heat production is also configured to produce cooling so that the heat consumption is fairly steady throughout the year.