In physics classes, I often catch myself repeating the mantra that heat is a disordered, useless state of energy that is generically the endpoint of an energy flow process. For example, the energy allocated to the fast-spinning wheel of an upside-down bicycle will slowly drain away as the wheel stirs the air, makes sound, and suffers friction at the bearing. Every one of these energy paths results in heat, until 100% of the invested energy is dissipated and the room is a tad warmer as a result. We will never reassemble the lost energy into useful form, once entropy has claimed it. All of this is true enough, but I feel very awkward uttering the words that heat is the graveyard of energy flow, and must place an asterisk on the statement.

The asterisk is that the overwhelming majority of our societal energy consumption makes use of heat—over 90% in the U.S.! So heat does not deserve the bad rap as a worthless waste product. Rather, heat runs our world! Sometimes we just want the heat directly, via: natural gas for furnaces, hot water, and cooking; heating oil for the home; and gas and coal for industrial process heat. This accounts for 20% of our total energy demand, leaving about two-thirds of our total energy consumption in the form of heat that powers heat engines for electricity production, transportation, and machinery. In short, all the energy we get from fossil fuels, nuclear, and biomass derives from heat. That’s hardly useless!

The Sun transmits its energy to Earth across the emptiness of space via radiation. Each square meter of surface at a temperature, T, emits radiation at a rate of σT4, where T is expressed in Kelvin (important!) and σ = 5.67×10−8 W/m²/K4. This constant is easy to remember via the sequence 5-6-7-8. Ignoring for now the subtleties of greenhouse gases, the surface of Earth—typically at 288 K—emits 390 W/m². The Sun, on the other hand, at 5800 K, emits 64 MW per square meter!

Summing over the area of the spherical Sun, at 109 times the radius of Earth, we find the total radiant power of the Sun to be a whopping 3.9×1026 W. Now that’s a light bulb! The Sun’s radiant energy spreads into all directions, creating a sphere of light. At the distance of the Earth, that sphere has an area of 4πr² ≈ 2.8×1023 m², where r is the mean Earth-Sun distance. Dividing these huge figures, we find that the radiant intensity at Earth is 1370 W/m²—which I hope will be a familiar number by now for Do the Math readers.

We can also turn the σT4 relation on its head and say that a patch of full sun (at the ground) receiving 1000 W/m² corresponds to a radiant temperature of 364 K, or a blistering 91°C. This means that a black panel in full sun could get this hot if no paths other than radiation were available for cooling the panel. We would then say that the panel is in radiative equilibrium with the Sun. But air can carry away heat by convection. The self-convection of a hot, flat plate will be about 10 W/m² per degree of difference between the panel and the surrounding air. Requiring the sum of radiative and convective losses to add up to the input power of 1000 W/m² yields a solution of about 55°C (328 K; 131°F) if the surrounding air is at 20°C. This assumes that the plate has no heat paths available through the (insulated) back side. If, on the other hand, it is a thin panel allowing convection on both sides, it will be cooler—although the “heat rises” phenomenon will suppress heat flow on the backside relative to the front, if the plate is indeed level. Just for fun, if we get an additional 5 W/m²/K of convective loss off the back, the equilibrium temperature drops to 47°C (117°F). It all seems reasonable.

The simplest way to replace fossil fuel energy with solar energy is called a window. A single uncoated piece of glass will transmit 92% of visible light (the rest reflected) when light comes straight in (down to 75% at a 20° grazing incidence, 60% at 10° grazing). The glass is opaque to ultraviolet light and mid- to far-infrared (IR) light, but lets over 95% of the unreflected incident solar spectrum pass.

Considering that windows in houses/buildings tend to be vertical, we can evaluate the energy input through windows, taking transmission loss, reflection loss, and angular foreshortening into effect. Because the Sun is higher in the sky in the summer, the window appears foreshortened to the direct sunlight, and also reflects more. So a south-facing window automatically admits more heat in the winter than in the summer, with no adjustment. Putting an overhang over the window—ideally with some vertical space between the window and overhang—can eliminate the summer noon-time contribution entirely. The figure below illustrates the fraction of incident direct-sun energy (think 1000 W/m²) admitted by the window. Vertical reference lines indicate the noon-time elevation of the sun at a latitude of 40° for the winter and summer solstices. The noon-day sun will be somewhere between these values all year. Adjustment to other latitudes involves a simple shift of the dashed lines by the latitudedifference.

Fraction of incident direct energy (perpendicular to rays) making it through a vertical window. The overhang extends an amount that is half the window height, and optionally is spaced 0.2 window-height-units above the top of the window.

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