is there a realistic small or medium scale apparatus that can demonstrate the warming effect of CO₂?
previous attempts by others
Most attempts I have seen attempt to heat identical gas cells with an incandescent lamp. The expectation is that the CO₂ container will absorb more energy and the thermometer inside will register hotter. These are problematic because they are irradiating the cells with shortwave IR, but CO₂ absorbs longwave IR.
Fig 2: Attempts to demonstrate CO₂ forcing using visible and shortwave infrared (won't work).
Fig 3: MODTRAN emission spectrum calculation moving from longwave to shorwave infrared
One can see in Modtran how insignificant the CO₂ absorption effect becomes when heating Earth (the IR emitter) up to lightbulb temperatures. People shining lights on small gas bottles don't have the problem correctly framed in their mind:
CO₂ doesn't cause heating when receiving shortwave IR.
CO₂ prevents cooling when it stands between a warm object emitting longwave IR & a very cold destination (outer space)
new cell design concept
Fig 4: To properly demonstrate only the IR properties of CO₂, the apparatus must have a "hot" emitter operating at room temperatures. This ensures the emitted IR is at wavelengths where CO₂ has significant interaction. Necessarily, we need engineer a cold body to radiate into. Dry ice is the coldest commonly-available substance. Let's imagine a pair of gas cells with and without CO₂ with outer space (T=2.7K) represented by dry ice sitting on a cast-iron plate (T=195K). Colder would be even better, but can't think of an easy way to do it.
Fig 5: The problem with this configuration is convection circulation will establish, which will produce rising warm columns and falling cool columns. How do we measure the avg. temp of the gas now, with it varying widely across the container?
To prevent the convection circulation, we need to invert the cell so that the warmed gas tends to hang around the top and the cold gas sinks and stays at the bottom. Now let's address the walls: we have an interior that is about 100K colder than the room where it sits. Heat will spontaneously flow through the walls near the cold sink and re-start the rising convection. Let's add an insulation layer. The insulation should be the type that has a shiny foil liner facing the interior. Low emissivity surfaces will reflect >95% of the incident IR radiation and emit virtually no IR itself.
How thick should the insulation be? We want internal radiative transfer (RT) of heat to be >10x greater than total exterior-to-interior heat conduction so that RT is the dominating effect. Given our 100K deltaT, if top & bottom plates have ε = 1, we will have q = σ(Ta⁴-Tb⁴) = 308 W/m². How thick of insulation?
We want internal radiative transfer (RT) of heat to be >10x greater than total exterior-to-interior heat conduction so that RT is the dominating effect. Given our 100K deltaT, if top & bottom plates have ε = 1, we will have q = σ(Ta⁴-Tb⁴) = 308 W/m². We need a heat loss calculator to estimate this sort of thing. With a 93 degC temperature difference between the interior & exterior, it will take 25 cm (~1 ft) of polyisocyanurate insulation wrapped around a 1m diameter cylindrical unit to maintain <7.98 W/m².
Fig 8: Estimating heat transmission through wall insulation
We haven't yet considered the aspect ratio (width/height). The taller & skinner it is, the longer the path length for CO₂ to absorb BUT the larger the wall effects (and consequently, more insulation required). Shorter & fatter = less insulation, but lower effect from CO₂. Let's consider a few gas cell sizes:
Fig 9: Mylar ducting "non-conducting, re-radiating walls"
Assuming a 6' segment of 14" mylar ducting, we consider 6 segments of 1' each. The top-most segment is nearest the room-temperature surface (295K), the bottom-most segment is 5K over the temperature of dry ice (195 + 5 = 200K). The four gas segments in between 295K and 200K are assumed to be 19K warmer, each, from bottom to top (219K, 238K, 257K, 276K).
Fig 10: Cell gas transmission schematic
Fig 11: Spectral calculation of a 400 ppm CO₂ gas cell
Fig 12: Spectral calculation of a 800 ppm CO₂ gas cell
The difference in infrared radiation transmission through 0.8m of air between these two cases is 0.291 W/m²/sr.
The design of the apparatus must keep track of three competing effects:
The larger the separation distance, the lower the thermal conduction between the two plates.
The larger the separation distance, the greater the absorption by CO₂ (larger gas cell volume at same CO₂ concentration puts more CO₂ molecules total between the two plates)
The larger the separation distance, the smaller the radiative view factor between the two plates.
Fig 13: Thermal conductivity of gasses as a function of temperature
Fig 14: The thermal conductivity of air is not constant with temperature and must be computed across the range of temperatures that will be present in the apparatus.
Fig 15: The greater the distance between the two plates, the less heat will be conducted through the air from one plate to the other. Assuming 14" diameter plates (0.1m² area, each) the heat flow by conduction is given by the following chart.
Fig 16: The direct radiation exchange also depends on the separation distance. For example, in the case of circular emitter/receiver, the view factor is around 0.18 when the disks are 18" diameter and separated by 63" (1.6m) but the view factor is 0.52 when when separated by 15".
design result v1
The objective is to engineer a gas cell that provides:
Mostly heat transmitted by longwave infrared radiation, not by conduction (mainly by making the diameter larger)
Easier-to-detect high ΔT from radiative exchange (mainly by making the gas cell as long as possible)
Lower cost (mainly by keeping the diameter and length of the gas cell as small as possible)
These objectives contradict each other, so a balance is sought. A calculation of a gas cell with 400 ppm vs 800 ppm CO₂ gives a difference in radiative transfer of less than one watt per square meter (see Fig 11 and 12). Multiple configurations were studied, ranging in diameter from 0.15m (6") to 1.0m (39") and gas cell lengths of 0.1m (4") to 3.0m (9' 10").
Fig 17: The temperature difference (ΔT) produced by gas cells of various diameter (legend) separated by various distance (x-axis). Higher is better. If the ΔT goes below the orange lines, then the ΔT is now the same magnitude as the uncertainty in the temperature measurement.
Fig 18: The ratio of the difference in radiated heat flow to conducted heat flow. Higher is better. If the conduction heat flow dominates, it is unlikely that a temperature difference from the small radiation contribution can be noticed.
We see that larger (= more detectable) ΔT requires a larger diameter cell: preferably 1.0m but no less than 0.25m. The corresponding preferred separation distance is 0.8-1.6m.
However, we also see that maximizing radiative exchange and minimizing thermal conduction prefers the smaller diameters. Having already ruled out 0.15m, this leaves 0.25m (10") and 0.30m (12") as the preferred cell diameters; these diameters of mylar ducting are available.
The optimal separation distance of 1.6m (5' 3"), beyond which the longer gas cell length cannot overcome the smaller radiative view factor.
Sensing the very small temperature change of the gas within the cell relies on extremely sensitive temperature elements. Lakeshore sells cryogenic temperature sensing elements that are "calibrated and matched." A single calibrated PT-102 element is $581 and matched PT-102's are $161 each. A total of four probes for one gas cell would cost $1,064. The range of accurate sensitivity is 30K to 325K (-243 to 52C).
Fig 20: Platinum RTD 30-325±0.023K
CO₂ Gas Demonstration Cell Version 1.0