This month’s open thread.
Seed topics: The genealogy of climate models, how to compare different greenhouse gases, whether a 2 deg C temperature target makes sense (Stoat has already weighed in), or reflections on the Nenana Ice classic (which has just concluded for this year). But you decide.
Jim Bullis,
Please, not another thread on the EPA fuel-economy stickers. We’ve been all over that: https://www.realclimate.org/?comments_popup=5443
Please, no more hyperbole about repealing the laws of thermodynamics.
By all means remind us that EVs are only (roughly) as clean as the electric power mix. But your “marginal response” assumption about EVs being 100% coal-powered isn’t reasonable, and repetition will not make it so. (More than half the 164 results Google gives for the search string “marginal response” “electric vehicles” coal seem to be comments by yourself — filtering with -bullis leaves 74 hits, and the top four of those were also yours…).
#329–Hunt, the case of the Carteret Islanders is apparently the only one so far. Perhaps you’re aware of it already, but in case not, the total evacuation of about a thousand residents of that island group was funded by the Papua New Guineau government a few years back; it’s estimated that the islands will be completely submerged by 2015.
The cause is unclear, but may be down largely to tectonic subsidence, abetted by SLR and–according to denialists who want to blame the victims–the dynamiting of reefs which allegedly worsened erosion. (I personally find this last a very dubious allegation–unlikely on the face of it, and contemptible if it is indeed a canard. But it is a claim that exists, and I don’t have definitive knowledge about it.)
There is a documentary about the situation.
http://www.global-greenhouse-warming.com/Carteret-Atoll.html
@patrick: Hoping to get back to you before this UV is closed.
Re 348 Rod B
I don’t think I liked the cross section factor in your #340. Molecular cross section has very little effect on spontaneous emission, though some effect on absorption. But I had a difficult time following this post and might be misinterpreting what you said.
Emission cross section = absorption cross section at LTE.
PS
cross section per unit volume csv * an increment of distance dx (small enough to avoid cross section overlaps) = effective fractional area that intercepts radiation. Optical thickness per unit distance = cross section per unit volume; optical thickness is 1 when the cross section contained in a volume is equal to the area of the volume facing the direction being considered – but overlaps in cross sections make the effective fractional area not 1 but exp(-1).
Cross sections and the optical thicknesses attributed to them add linearly by type and source (absorption + scattering = extinction) and optical thickness adds linearly by distance.
Of course a molecule at any one time probably doesn’t look like a sphere (well it wouldn’t necessarily look like anything at all, would it? You’d need to have a photon interaction to see it… – we’re talking about photon interactions so electron microscopy images might not apply directly)
But in the average of a molecule over time or the average of a sufficient population of molecules over a shorter time, if there is not a prefered orientation, the same cross section area, an effective area that acts like a blackbody, will be the same in all directions. Of course such an area, relative to molecular center of mass, would not be hard-edged presumably but rather a fuzzball/’cloud’ – partially transparent and covering a larger area, but the integrated fractional area would be equivalent to an opaque circle, and a sphere of the same size would present such a cross section in all directions. A blackbody sphere would emit the same area-integrated intensity in all directions (flux per unit solid angle), which would be the flux per unit solid angle emitted in a direction from a molecule (time average) – and you get the intensity by taking the cross sections and dividing by volume and multiply by distance (for an infinitesimal distance, that’s it – otherwise take the 1 – exp[-that value] because of overlaps – and then multiply by the Planck function you get a spectral intensity (flux per unit solid angle per unit area, per unit of the spectrum). And such a blackbody sphere must emit in total a flux from an area 4*pi*r^2, and a fuzzball that emits the same amount in each direction must emit the same as the sphere in total.
If the LW emission equals absorption in CO2 you would not have any atmospheric heating and have only surface warming by only that portion of the downward directed emission (nominally half of total emission) that makes it back to the surface. (Though I admit to not having gone through multiple mirror effect of the up/down emission now joined with up/down absorption.)
I was not refering to the effect of greenhouse gases (or clouds) on temperature as in climate change (adding or taking them away) – but rather the ongoing process at equilibrium climate. CO2, etc, maintains the temperature (surface and troposphere) warmer than otherwise but the CO2+H2O+etc. molecules and clouds are generally emitting more LW photons then they are absorbing (with some exceptions – base of a cloud layer if there isn’t an inversion below it, gases near temperature minima or points of curvature in the Planck function of the temperature profile, as graphed over an atmospheric column measured in optical thickness), depending on sign and spatial scale and opaqueness) because of convective heating from the surface and SW (solar heating) of the air itself – particularly H2O, clouds, and ozone. Consider that if there were no greenhouse effect, adding greenhouse gases would initially cool the atmosphere…
350 CM
It is not hyperbole if the laws of thermodynamics are ignored as is the case with the EPA MPGE formula. It is not hyperbole to be outraged that a Nissan Leaf would get a 99 MPGE rating when it should get more like a 33 MPGE rating. This kind of deceit is very damaging to future hopes for reducing CO2 and also damaging to the financial structure of the USA.
It is not an assumption that the reserve capacity of the electric system determines the response to new loads, not the ‘mix’ that so many like to assume. Neither is it an assumption that coal is a cheaper source of heat than natural gas.
It is a foolish assumption that renewables will soon take a place as part of available capacity for generating electricity.
350 CM
Also, it is hard to see how we could have gone over something that just happened, that being the formal EPA and DOT announcement that I am talking about.
The particularly interesting thing is that interest in reducing CO2 evaporates in the face of an issue that will determine the bulk of CO2 emissions for many years to come.
(PS not distinguishing between forcings and feedbacks and setting SW feedbacks aside)… they cool the atmosphere but warm the surface by increasing the downward flux at the surface. The atmosphere cools and the surface warms. Depending on direct solar heating of the air, there may at some point emerge a troposphere, which is LW radiatively cooled but convectively heated…
(It’s possible that some overturning could occur absent a greenhouse effect driven by horizontal and/or temporal(?) variations in solar heating, or mechanically forced by tides if that’s sufficient – kinetic energy is generated by cooler air sinking and warmer air rising in an overturning cell – even if locally stable – and that may then help do work by mixing heat downward. Cooler air flowing in from night or higher latitudes might perhaps be heated from below sufficiently for destabilization and some localized overturning. However this works, the inability of the air to radiatively cool means that any heat it picks up from the surface, as well as any direct solar heating, must be (forcibly) mixed downward and/or conducted downward (very slow process) to the cooler parts of the surface; the surface itself in the global time average would emit all the flux that the atmosphere and surface combined absorb from the sun)
…; the surface and troposphere will then warm or cool as a whole (with the distribution of temperature within that grouping determined in part by the physics of convection) based on changes in fluxes at the tropopause. We still have the potential for stratospheric cooling (depending on the spectra of gases and the amount/distribution of direct solar heating …
– PS awhile back there was an extended conversation on that somewhere at RC – things got interesting. for a gray gas with no solar heating, setting aside latitudinal, seasonal, diurnal, and other variations, the temperature profile above the tropopause tends toward an equilibrium profile with cooling with height (weak enough to not be part of the troposphere) toward a skin temperature at the top. Warming below and increasing the amount of a well-mixed gray gas should then, in equilibrium, compress this temperature profile toward TOA, but without changing the skin temperature, so warmer temperatures migrate upward. There could still be some temporary cooling in places due to the equilibration of the stratosphere that occurs while the surface and troposphere are still warming up. When the atmosphere is very optically thin then the whole atmosphere would tend toward the skin temperature except within the troposphere. With solar heating, there is additional absorption that must be balanced by additional emission, requiring higher temperatures (which will be distributed more broadly then the solar heating because of the absorption of the additional emission by other layers or the surface – although if this solar heating would otherwise have occured at the surface or troposphere, having it higher up would of course have a cooling effect at the surface+troposphere, which would be partially cancelled by the additional LW flux from above) – if the density of a greenhouse gas increases, less additional emission per molecule is required so there generally can be some additional stratospheric cooling, which persists into full equilibrium (depending on what the SW feedbacks are), when more greenhouse gas is added. With non-grey gases, it is possible to have stratospheric cooling at full equilibrium without the presence of solar heating (though solar heating could add to this cooling); it is important to consider how the sensitivity of the Planck function to temperature varies with wavelength; at some wavelengths there could be net LW cooling while net LW heating occurs at others, depending on temperature and the spectra of the gases – see http://www.atmosphere.mpg.de/enid/20c.html (net LW cooling – pretty sure this is for clear air but I mention clouds in the following too) – note that this would look differently if the vertical scale were proportional to pressure (upper layers would be very compressed) and even more so if proportional to optical thickness (at least in the troposphere, line broadenning decreases with height – which means the upper atmosphere is somewhat inflated at/near line centers but more compressed away from line centers; meanwhile the line strength also changes with height – anyway at most wavelengths and through most of the atmosphere so far as I know, an optical thickness scale compresses the higher layers even more than a pressure scale (especially for H2O which is concentrated near the surface, but I think it’s the opposite for O3 which has a higher mixing ratio in the stratosphere; cloud layers can dominate the picture where they occur), but pressure is a good place to start from for getting a sense of how this works. We expect a combination of these patterns:
1. Going through the spectrum (or from clear to cloudy conditions), increases in opacity remove net LW cooling from the surface and transfer it at first evenly over the atmosphere or evenly over the distribution of gases and very very thin clouds however they are distributed, and then concentrate this net LW cooling upward toward TOA or toward the tops of thickenning cloud or water vapor layers (where there are cloud layers and the temperature increases with height, there would tend to be warming of the bases of cloud layers; maxima in net LW cooling in between the surface and TOA attributed to H2O vapor are occuring near the effective TOA for H2O optical thickness; CO2 and generally some other gases and clouds at higher levels can reduce the net LW cooling from H2O and transfer some of that net LW cooling to higher levels).
2. the net LW cooling will be larger at higher temperatures and smaller or negative at lower temperatures, so the upper stratosphere and lower tropopshere and surface (places recieving more solar heat directly or via convection, or generally with fewer LW photon emissions and absorptions required to deliver that energy from where solar heat was absorbed) will tend to have greater net LW cooling. Net LW warming or reduced net LW cooling could occur at the surface and surface air in a surface-level inversion (such as nocturnal or polar) or likewise on the cold side of a front.
Generally, net LW warming may occur or be larger in layers that are cool enough relative to the layers above and below – for example, at optically thick points in the spectrum, where the lapse rate, in terms of the Planck function and relative to optical thickness, decreases (tropopause) or where there is a relative minimum in temperature (tropopause-lower stratosphere and upper mesosphere – in more optically thin points in the spectrum, any cooler temperature, as measured in terms of the Planck function, and relative to the whole column from the surface to space (or between cloud/humidity layers where they provide opacity), may have net LW warming – for example, if the atmosphere were optically thin from the surface to space, any layers with a Planck function below half that of the surface (effectively the average of the surface and space) would tend to have net LW cooling. Opposite conditions will tend to support net LW warming. Note that the Planck function is less sensitive to temperature at relatively longer wavelengths – the greatest tendency for net LW warming would be for relatively cold layers at shorter wavelengths; some of those same layers may have net LW cooling at longer wavelengths.
3. Both net LW cooling and heating tend to decrease if opacity is small and is getting smaller because there are fewer molecules to emit and absorb photons – however net LW fluxes passing through can be significant if large enough opacity of the emission/absorption type exists on either side of transparent regions in regions with signficant temperature differences between them. The effect of temperature variations on net LW fluxes is small if the opacity on the scale of thos variations is small.
Both net LW cooling and heating, as well as net LW fluxes, tend to decrease if, on the scale of the temperature variations, opacity is large and getting larger, because large opacity reduces the distances over which photons are exchanged, thus hiding the temperature variations – reducing the net LW fluxes, and reducing the difference between emission and absorption rates by bringing the brightness temperature of the radiation closer to the actual local temperature.
Spatial Variability in opacity (such as with clouds or humidity, or maybe ozone) can enhance (via concentration) net LW cooling or heating near the edges of regions with greater opacity (Same as point 1).
Note that net radiant heating is equal to the convergence of net radiation fluxes. Same for convection, etc.
…, also, when optically thin, the net LW heating and cooling rates and gross rates (total emission and absorption rates) will be small (per unit air, not per molecule of greenhouse gas) but the relative difference between emission and absorption could be large if at a much warmer or colder temperature.
Edward G at 349 writes:
“Snapple: How do you develop methane hydrates without triggering a huge “methane gun”? They are apparently going to drill through the hydrate [quasi] stability zone to the free gas zone.”
I’m just telling you what the APGA said.
Patrick 027, I think my 1st reaction to para#1 or post #353 was wrong. It sounds like you are describing the character (orientation, density, etc., etc.) of the radiation emitted by relaxing vibration. While that radiation is not generated by blackbody/Planck function, the radiation itself has the same character of blackbody radiation (except intensity?). Never thought that through before, and it sounds interesting and sensible. Thanks; unless of course I totally missed your point…??
O.K. Got it.
However,
is a tough one. I have to think about it. How/when does it eventually start the warm the atmosphere… so maybe eventually it might get some H2O to help out?
Re 354 Jim Bullis –
1. I stated the necessary caveats (except perhaps externalities in the lifecycles of different vehicle technologies, besides climate-change pollutants if I did mention those or suggested a CO2eq tax or policy to cover that thus putting it into the price signals) when suggesting that it could be overall advantagenous to use P(H)EVs rather than either HEVs or ICE cars, although HEVs could themselves be better than ICE cars. Yes, it depends on the (lifecyle) cost of the vehicle as well as the energy technology vs petroleum, and the concurrent adoption of renewable energy and related technology and greater efficiency (beyond that which is justified by economics absent externalities) that could be compensated by savings from switching some share of transportation over to P(H)EV’s does depend on government policy.
2. Whereever reserve capacity comes from (on short time scales it could come from CSP; generally there’s the possibly of (AA)CAES, also hydroelectric, although use of that as storage may require reduced capacity factor and thus increased cost of hydroelectric power), if the total amount of energy is produced with less CO2eq (and other pollution/land degradation), then that’s good. Why this need to match one-to-one an energy user to an energy source? What about people’s microwaves. We already have them, so I guess we can’t switch from coal to solar now because we turn microwaves on and off?
(Also remember cars could be plugged in not just at night but during the day when parked, and if parked for long hours there could be flexibility in when the energy is delivered (‘smart meters’ or is that the right term?) (PS a potential problem if you don’t know when you’ll need the car again – always a possibility – but in emergencies there could be other options, and weather forecasts could be used so that ‘smart meters’ could tell you when you plug the car in when it will charge the car – if you take that option (pay a higher rate for charge on command when necessary). Then there’s the possibility of ‘gas stations’ that just put charged batteries in and take out discharged batteries for recharging.)
Re 359 Rod B – first part sounds right – intensity is just the flux per unit area per unit solid angle (in steradians, a solid angle in steradians is the area of a unit sphere that it projects onto from the center; all directions fill 4*pi steradians, a hemisphere of directions fills 2*pi steradians.
Second part – I had to cut that off and finish it later – see subsequent comment (if there is no greenhouse effect to begin with then the atmosphere may tend to be as warm as near the surface or warmer if there is any direct solar heating – setting aside horizontal and temporal variations and any mechanical forcings, the atmosphere would be 100 % stratosphere, and the tropopause would effectively be at the surface, and that is where the ‘tropopause level radiative forcing’ would occur. Adding some greenhouse agent (of the emitting-absorbing type, as opposed to scattering or any other kind) would in that situation tend to cool the atmosphere; some of the emission would reach the surface and that would be the forcing at the surface. After ‘stratospheric adjustment’ that forcing would be reduced (because the atmosphere would be cooler) but it would still be positive (because any downward flux is greater than none) so the surface would tend to warm. At some point a troposphere may appear (when the radiative equilibrium profile is unstable to convection) – etc.
360 Patrick 027
The issue is MPGE, which is an attempt to relate electric vehicles to gasoline powered vehicles. The deceit is to ignore the source of electricity altogether and simply pretend that electricity is a fuel that can be compared to actual fuels. This is based on the misunderstood concept of equivalence. Equivalence holds for converting electrical energy to heat but does not hold when converting from heat to mechanical or electrical energy. By setting 33.7 kWhr of electric energy to be equivalent to a gallon of gasoline it can be made to appear that a Nissan Leaf achieves 99 MPGE, and all that is done is to add batteries and an electric motor. The fact is that 33.7 kWhr of heat can be obtained from a gallon of gasoline but this amount of heat has never produced much more than 11 kWhr of electric energy in an automobile heat engine.
One can argue about what the heat engine is, but it is very hard to get more out of renewables than they immediately produce, regardless of whether new loads are connected. So we really are stuck with fossil fuel resources to make electricity. These can vary somewhat, but there is always a massive amount of heat discharged due to the Second Law effect. The EPA does not count this at all.
Hydro-electric can be managed to help alleviate load variations, but in all cases it is fully exploited according to whatever regulatory wisdom dictates. As such, it does not offer anything extra that could be used for electric vehicles.
Certainly CSP is not just left unused waiting for electric vehicles.
Jim Bullis wrote: “… it is very hard to get more out of renewables than they immediately produce, regardless of whether new loads are connected. So we really are stuck with fossil fuel resources to make electricity.”
With all due respect, what in the world are you talking about? The first sentence makes no sense at all, as far as I can tell.
And we are obviously NOT “stuck with fossil fuel resources to make electricity” given that the world added 68 gigawatts of new wind power alone, just in 2010, and given that more solar energy arrives on the Earth’s surface every hour than human civilization uses in a year, etc.
Jim Bullis #355,
> It is not hyperbole if the laws of thermodynamics are ignored as is the > case with the EPA MPGE formula.
This is hyperbole, and it misses the point, derails serious conversation, and distracts from your legitimate (I think) complaint about unfairness to hybrids. What is ignored in the EPA MPGe rating is wells-to-pump-or-plug efficiency; it looks only at the pump-or-plug-to-wheels efficiency of the vehicle itself. As for CO2, it looks only at tailpipe emissions and “does not include emissions from generating electricity”, as is explicitly noted on the sticker. The sticker (which is getting pretty crowded) further points consumers to fueleconomy.gov, where there’s a calculator for total GHG emissions. Maybe you could convince them to add a calculator for primary energy consumption, too.
> It is not an assumption that the reserve capacity of the electric system > determines the response to new loads, not the ‘mix’ that so many like to > assume.
Yes, this is likely the best way to look at it under present circumstances. My previous comment failed to acknowledge that, sorry.
What I meant to dispute was not the marginal generator approach, but the assumption that the generator dispatched will be a coal plant. In some places it may be. But the power dispatched to a plug near you will likely come from natural gas.
“[T]he near-term marginal electricity mix for vehicles and fuels in California will come from natural gas-fired power plants”, according to McCarthy and Yang (2010), who use “an hourly electricity dispatch model to simulate and investigate operation of the current California grid and its response to added vehicle and fuel-related electricity demands in the near term.” Much of this is from inefficient plants, so the “the marginal electricity emissions rate will be higher than the average rate from all generation”, yet McCarthy and Yang find that “alternative vehicle and fuel platforms still reduce emissions compared to conventional gasoline vehicles and hybrids, through improved vehicle efficiency.”
McCarthy, Ryan, and Christopher Yang (2010). “Determining marginal electricity for near-term plug-in and fuel cell vehicle demands in California: Impacts on vehicle greenhouse gas emissions.” Journal of Power Sources 195, no. 7: 2099-2109. 16/j.jpowsour.2009.10.024
re your #356: And yet, discuss it we did, at great length, while it was still being planned, and I linked to it.
364 SecularAnimist,
I am pointing out that renewables do not provide reserve capacity. This is because they are fully used immediately to their full capacity. Once the renewable equipment is installed and integrated, the fuel is obviously not an issue, and thus, no sane power planner would reject taking all that can be produced. After that, there is no possibility of additional production from such renewables. This includes hydro, wind, and solar. The same is true for nuclear equipment. All these things run flat out as soon as possible, and that is the end of it. So when an aggregate of electric vehicle loads starts to emerge, these renewable facilities continue to operated unchanged due to such new loads.
This is not the case for coal and natural gas facilities, which offer substantial capability to provide increase in output. They will respond to such loads simply by use of more fuel.
As each new resource is added, it is integrated into the system, and that has a nearly immediate effect of cutting back on coal facility operation. That is great for reducing CO2. But plugging in the electric vehicle reverses this effect, thus causing an increase of CO2. That is why I say that the coal facilities will provide the marginal response to new loads.
It seems hard for folks to understand that renewables do not correspond with electric vehicles. These are two separate decision events. It is entirely desirable, if affordable, to alleviate CO2 by expanding renewable facilities. But until these get to the point that fossil fuel power plants are scrapped, there will be no correspondence between EVs and renewable sources.
I recall the IEA said that coal will be substantially in the mix beyond 2030 and this date extends out depending on how many EVs there are. My observations of the cost of renewables suggests that it will be longer than that. What you say about what the ‘world added’ might be true, but it should not be assumed that the ‘world’ will continue to descend into bankruptcy at a rate needed to soon scrap all related fossil fuel systems.
When there is reserve capacity at the ready from renewables, that will be the time to promote the electric vehicle. At that point of course, there will be no meaning at all in the formula for MPGE.
This is of course the ultimate condemnation of the government policies trying to promote electric vehicles with this fake ‘MPGE’. If the basis of running such electric vehicles is indeed renewables, there really is no CO2 implication of such an MPGE system. All that is left is the sham comparison with gasoline powered vehicles, but if that is the purpose, it would have to be done on a common heat input comparison basis.
Here’s one for you Ed: Connecticut has the highest electricity rates in the country. They also get over 50% of their power from…..drum roll….nuclear.
http://www.eia.gov/cneaf/nuclear/state_profiles/connecticut/CT.html
http://www.eia.gov/cneaf/electricity/epm/table5_6_a.html
And btw, the average residential elec rate in illinois is 10.14 c/kwh. In Minnesota – where over 9% of our electricy is generated by wind – the average if only 10.35.
Of course, none of this has anything to do with the exhorbitant cost of NEW nuclear power plants.
#366–
Jim, very interesting comments. But I have to say I don’t think I fully understand.
What I think you are saying is that EVs are strongly linked to reserve generation capacity, and that this capacity is predominantly fossil–you said coal, which is, as we are all aware, terrible, and CM said natgas–better, but not great.
Fine, and I think we’d all stipulate that, should EVs take off, the result would have to be significantly increased electrical demand.
But why reserve capacity? If EV demand ‘normalizes,’ then why wouldn’t the power used reflect the overall mix? Is there something particularly unpredictable about EV loads? Is their time structure particularly awkward somehow? Please elaborate, if you would.
This leads us to a second, and related, point. The awkward thing about renewables is not just *not* having power when you need it, but also having power when you *don’t.* And here EVs can help, since if they take off, they can serve as a semi-flexible load which can absorb high off-peak renewable output–especially with smart grid load management. In fact, if I understand correctly, that’s precisely the idea behind the EV initiative currently getting under way in Denmark–use the load capacity (if that’s a real term) of EVs to improve grid stability, better use the wind output, and hence reduce emissions.
Is there something wrong with their logic–or with my understanding of it?
Re 363 Jim Bullis – CSP stores solar energy as heat at sufficient temperature to be able to run a heat engine (Solar ponds can also store heat). It isn’t expected to store up heat over multiple days but presumably it could react to load variations on shorter timescales.
Re 366 Jim Bullis – If we already had electric cars, and we were adding microwaves, TVs, computers, air conditioners, etc, then would you say that it will be coal (or nat gas) that powers the later as they can respond to that variable demand? In aggregate just how variable will the electric car load be?
Your argument just boils down to –
1. the need for some reserve capacity to hand fluctuations in load or otherwise balance load and supply. This would be true with or without P(H)EV’s, though the amount may be different.
2. the economics of renewables/nuclear vs storage vs transmission vs coal/gas vs petroleum, as this affects the extent to which EVs, PHEVs, and HEVs displace basic ICEs and which ones come to dominate, and the fuel economies of ICEs and (P)HEVs, and the balance between using storage and transmission to match supply and demand vs hanging on to coal and gas, and to what extent hydroelectric might reduce their capacity factor, etc.
You’re concerned that adding EVs will prolong coal(or gas) but not adding EVs would prolong petroleum usage (until we run out, and then what?). Increases in efficiency can help both situations.
Using more EVs and using more renewable energy, etc, are different decisions, but we could make both decisions. We don’t have to just pick one.
Re my 354 – cross sections – I can make this a little clearer:
A substance has some absorption cross section (acs) per unit amount. This can be converted to absorption cross section per molecule (or atom, etc.), or acsm for short. Let N be number of molecules per unit volume.
acsm * N = acs per unit volume = acsv.
acsv * x = acs per unit area normal to x = acsa(x) = optical thickness over distance x (see why below). Notice that multiplying acsa by an area A gives a total acs in the volume equal to x * A.
—
transmitted intensity over distance x
= I = I0*exp(-optical thickness) = I0 * exp(-acsv * x),
where I0 is the intensity at x = 0, and we are only considering the original photons at x=0 in that direction, not any from emission, etc.
The absorptivity of the path from 0 to x is then 1 – exp(-optical thickness).
Taking the derivative, the transmission changes at a rate:
dI/dx = -acsv * I0 * exp(-acsv * x) = -acsv * I
Over a differential distance dx, the change in I is:
dI = -acsv * dx * I
The fraction absorbed, or absorptivity, of dx is thus -dI/I, or acsv*dx, which is the acs per unit area normal to x – this is an area per area, and is the fraction of area covered by the cross sections of the material within dx.
At LTE, emission cross section = absorption cross section and thus, absent other things like scattering, emmissivity of a path x = absorptivity of a path x = 1 – exp(-ecsv*x) = 1 – exp(-acsv*x) = 1 – exp(-optical thickness).
———-
The above described how ecs and acs affect intensity. Intensity = flux per unit area per unit solid angle, and is for a particular direction.
Let solid angle be ω. Let the angle from normal (to a flat area being considered) be θ and the angle around normal be λ. From geometry, dω = sin(θ) * dθ * dλ. (See spherical coordinates).
Let flux per unit area be dF.
The dF from an intensity I(θ,λ) from the direction θ = 0 (thus, I(0,_))within the range of directions dω is equal to I(0,_) * dω. For an area Aθλ facing into some nonzero θ and some value of λ, dFθλ = I(θ,λ)*dω, but the same amount of radiation going through an area A that faces θ = 0 is dF = cos(θ) * dFθλ.
For an area facing θ = 0, the F from radiation coming from one side is the integral of dF over the hemisphere of directions (hemisphere of solid angle) from that side, where:
dF
= dFθλ * cos(θ)
= I(θ,λ) * cos(θ) * dω
= I(θ,λ) * cos(θ) * sin(θ) * dθ * dλ
= 1/2 * I(θ,λ) * sin(2θ) * dθ * dλ
Integrate from λ = 0 to 2*π and from θ = 0 to π/2 (note that to integrate over the whole sphere of solid angle, go from θ = 0 to π – that is, pole to pole. Note that the dF for positive I will be negative for θ > π/2, and the result wil be the net flux per unit area. Alternatively you can of course flip the spherical coordinates around to find the flux per unit area from the other side and then subtract the fluxes to find a net flux, or one can integrate net intensity over one hemisphere, where net I(θ,λ) = I(θ,λ) – I(π-θ,λ+π).
If I(θ,λ) is isotropic over the hemisphere, then:
dF = 1/2 * I * sin(2θ) * dθ * dλ
Integrating:
F = Integral(θ = 0 to π/2 , λ = 0 to 2*π) of [1/2 * I * sin(2θ) * dθ * dλ]
= Integral(θ = 0 to π/2) of [π * I * sin(2θ) * dθ]
= -π/2 * I * [cos(2*π/2)-cos(2*0)]
= -π/2 * I * [-1 – 1]
= π * I
Notice if we took out the factor of cos(θ), we’d just be integrating intensity over solid angle, and the result would be:
dFθλ
= I(θ,λ) * dω
= I(θ,λ) * sin(θ) * dθ * dλ
Integrating an isotropic I over a sphere:
Total Fθλ (whatever that is!) = Integral(θ = 0 to π , λ = 0 to 2*π) of [I * sin(θ) * dθ * dλ]
= Integral(θ = 0 to π) of [2*π * I * sin(θ) * dθ]
= -2*π * I * [cos(π)-cos(0)]
= -2*π * I * [-1 – 1]
= 4*π * I
Taking the factor I out, this is the area of a unit sphere, and the total solid angle of all directions; the integration to any limits would give the solid angle within those limits.
Multiplying F by area yields a flux. If an area A(θ,λ) (facing the direction (θ,λ)) emits an intensity I(θ,λ) in a direction θ,λ (note I is now defined as going toward rather than coming from that direction), then A(θ,λ)*I(θ,λ) is an ‘intensity-area’. IF θ of the area = zero, then the ‘intensity-area’ can be weighted by cos(θ) (as dFθλ was before to get dF) and integrated over solid angle to find the flux; the integrand would be:
A * dF
= A * dFθλ * cos(θ)
= A * I(θ,λ) * cos(θ) * dω
= A * I(θ,λ) * cos(θ) * sin(θ) * dθ * dλ
However, if areas facing different directions each emit in that direction, then a total flux being emitted can just be found by integrating the ‘intensity-area’ over all directions, where A and I vary however they do as a function of direction; the integrand would be:
A(θ,λ) * dFθλ
= A(θ,λ) * I(θ,λ) * dω
= A(θ,λ) * I(θ,λ) * sin(θ) * dθ * dλ
(Applying this to the former case where a single constant area faces θ = 0, A(θ,λ) would equal A*cos(θ) ).
If A and I are both isotropic, the result for integrating over the sphere is
A * Total Fθλ
= Integral(θ = 0 to π , λ = 0 to 2*π) of [A * I * sin(θ) * dθ * dλ]
= 4*π * I*A
Note that a flat surface of area A emitting isotropic I emits a flux A*F = π *I*A. Thus the flux emitted when the same amount of emitting area emits the intensity I in each direction is equal to 4 times the flux from a flat isotropically-emitting surface. This is exactly the same relationship that would be found if we took A to be a circle and compared the isotropic intensity emission from a circle to that of a sphere with the same radius. Which goes back to the earlier discussion of thinking of molecules as emitting (and absorbing) as blackbody spheres of the same radii as their cross sections (when the cross sections are isotropic) – of course the cross section is the effective area, as a larger area may be covered with reduced opacity for the same effect, but the math above shows that the emission from the 3-dimensional ‘fuzzball’ will be the same as that from the equivalent opaque sphere, as we didn’t have to specify the shape or distribution in that integration, and know it has the equivalent effect of a more compact opaque area.
… of course all of that applies (at LTE for emission and absorption cross sections being equal) to any particular frequency, and thus to ‘spectral flux’, spectral flux per unit area’, ‘spectral intensity’, and ‘spectral intensity-area’, and any other related variables that might be convenient to some problem. Then integrate over the spectrum, etc.
368 Kevin McKinney 369 Patric027 365 CM
Thanks for clear points made.
I don’t know what you mean by ‘normalizes’. However, maybe I should be more specific about the timing factor which might be something I see differently than some others see it.
Under the present policy of government encouragement, I see a fairly fast emergence of electric vehicles, such that the existing installed equipment will be used to respond to new loads. Nuclear will not expand in time to matter, the wind will not blow harder, it will not rain more, and the sun will not shine brighter because of such fast emergence of electric vehicles. EV loads should be, as an aggregate, quite accurately predictable, but it does not matter, the renewables will be tapped out regardless. When parts of the ‘mix’ are tapped out, any variation in load does not involve carrying forward the same ‘mix’ ratios.
The predictability of EV loads will make coal the especially likely choice by power planners, since the coal facilities can be efficiently operated. In any case, it will be fossil fuel facilities since these have the reserve capacity, but coal is still far cheaper than natural gas on a day to day operation basis. Both kinds of fossil fuel equipment stand ready so cost of the fuel is the primary basis of the choice. This is fundamental reasoning; it transcends any study about hourly dispatch or whatever. There are factors that affect cost of fuel however, and some places the transportation of coal can make natural gas come out cheaper.
Obviously, government that is heedless of the burden of the cost of electric power can impose restrictions on coal use which would also shift the choice to natural gas. I consider these kinds of government action to be transitional due to the effect of political force when the populace figures out how they are being duped.
Many seem to be thinking that renewables will expand to keep pace. If that were remotely possible, I would have to reconsider my general argument.
But above and beyond all this verbage, the outstanding fact is the hyperbolically distorting rating system now to be used by the EPA that gives the Nissan Leaf a 99 MPGE rating to make it look like it is a solution that is so superior that no other technology need be worried about. I say the Nissan Leaf will mostly cause a response from coal fired facilities and these will release more CO2 than would be released from a gasoline fueled hybrid that rates about 33 MPG.
If and when the renewables take over this will of course be a different story, but certainly the MPGE then will not be correctly rated either, though it will be hyperbolically wrong in the other direction.
CSP could be involved in a mix and rapid load changes could be dealt with, but I don’t see this as relevant in the scheme of things where a broadly expanding load would develop due to plug-ins.
It seems fundamentally to the point that any comparison of technologies be based on sound physics. The objectionable EPA formula skips the key point of thermodynamics about the fact that much heat has to be thrown away in a heat engine. That is highly objectionable; how such an objection can be called a ‘hyperbolic argument’ is beyond my understanding.
And here it really matters in a practical way as well. What I see here is a standard setting agency that is perpetuating wrong thinking about the function of electricity as a carrier of energy from a primary point of conversion in a heat engine to a point of use. This leads to wildly wrong advertising that would make snake oil salesmen proud. Maybe folks here do not pay much attention to the automobile market developments.
Apparently the functions of engines and motors have not been emphasized in physics education as they once were. The relationship of this science to the technology on which the industrial revolution was based was given more emphasis in former times. Perhaps for good reason, that seems to have been de-emphasized in the educational system now. Now that we are back to a big energy crisis, that may need to change.
Thanks for helping get some hopefully better explanations out there.
FUN WITH THE PLANCK FUNCTION PART II – the I hope I got the algebra right edition!:
In terms of photon energy E:
Bν(T,ν) = |d[Bint(T,ν)]/dν|
= ( 2*h*ν^3 / c^2 ) / ( exp[h*ν/(kB*T)] – 1 )
E = h*ν:
BE(T,E) * |dE| = Bν(T,ν) * |dν|
dE = h * dν
BE(T,E) = Bν(T,ν) * |dν|/|dE|
= 1/h * ( 2*h*(E/h)^3 / c^2 ) / ( exp[E/(kB*T)] – 1 )
= ( 2*E^3 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
Graphically, BE(T,E) behaves similarly to Bν(T,ν):
let V#*h = E# = E/T
BE(T,E)
= T^3 * ( 2*E#^3 / (h^3 * c^2) ) / ( exp[E#/kB] – 1 )
Etc. for putting in terms of angular frequency, which is 2*π*ν
———————————————-
In terms of ln(ν), ln(E), ln(λ):
d[ln(ν)]/dν = 1/ν
d[ln(ν)] = dν / ν , ν = dν/d[ln(ν)]
d[ln(λ)] = dλ / λ , λ = dλ/d[ln(λ)]
d[ln(E)] = dE / E , E = dE/d[ln(E)]
———————
Blnλ[T,ln(λ)] = Bλ(T,λ) * dλ/d[ln(λ)]
= λ * ( 2*h*c^2 / λ^5 ) / ( exp[h*c/(λ*kB*T)] – 1 )
= ( 2*h*c^2 / λ^4 ) / ( exp[h*c/(λ*kB*T)] – 1 )
Blnν[T,ln(ν)] = Bν(T,ν) * dν/d[ln(ν)]
= ν * ( 2*h*ν^3 / c^2 ) / ( exp[h*ν/(kB*T)] – 1 )
= ( 2*h*ν^4 / c^2 ) / ( exp[h*ν/(kB*T)] – 1 )
BlnE[T,ln(E)] = BE(T,E) * dE/d[ln(E)]
= E * ( 2*E^3 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
= ( 2*E^4 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
In the above, replace λ, ν, and E with exp[ln(λ)], exp[ln(ν)], and exp[ln(E)] to put the expression in terms of the desired variable. This isn’t shown because it is unnecessarily cumbersome.
—————————–
E/(T*h) = ν/T = c/(λ*T) = V#,
E/T = h*ν/T = h*c/(λ*T) = V#*h = E#
——————————
Blnλ[T,ln(λ)]
= ( 2*h*c^2 / (c/(V#*T))^4 ) / ( exp[h*V#/kB] – 1 )
= T^4 * ( 2*h*V#^4 / c^2 ) / ( exp[h*V#/kB] – 1 )
Blnν[T,ln(ν)]
= T^4 * ( 2*h*V#^4 / c^2 ) / ( exp[h*V#/kB] – 1 )
BlnE[T,ln(E)]
= T^4 * ( 2*E#^4 / (h^3 * c^2) ) / ( exp[E#/kB] – 1 )
= T^4 * ( 2*h*V#^4 / c^2 ) / ( exp[h*V#/kB] – 1 )
Graphing notes: Graph is vertically stretched in proportion to T^4, and is translated in the positive direction along the ln(ν) or ln(E) axis, or in the negative direction along the ln(λ) axis, by the change in ln(T). At the peak, the Planck fucntion is proportional to T^4, as is the area under the graph.
Notice:
Blnλ[T,ln(λ)] = Blnν[T,ln(ν)] = BlnE[T,ln(E)]
which makes sense because:
ln(ν)
= ln(E/h) = ln(c/λ)
= ln(E) – ln(h)
= ln(c) – ln(λ)
d[ln(ν)]/d[ln(E)] = 1
d[ln(ν)]/d[ln(λ)] = -1
|d[ln(ν)]| = |d[ln(E)]| = |d[ln(λ)]|
—————————–
summary of earlier forms:
Bλ(T,λ)
= ( 2*h*c^2 / λ^5 ) / ( exp[h*c/(λ*kB*T)] – 1 )
Bν(T,ν)
= ( 2*h*ν^3 / c^2 ) / ( exp[h*ν/(kB*T)] – 1 )
BE(T,E)
= ( 2*E^3 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
Photon intensity β:
E = h*ν = h*c/λ
βλ(T,λ) = Bλ(T,λ) / E
= ( 2*c / λ^4 ) / ( exp[h*c/(λ*kB*T)] – 1 )
= T^4 * ( 2*V#^4 / c^3 ) / ( exp[h*V#/kB] – 1 )
βν(T,ν) = Bν(T,ν) / E
= ( 2*ν^2 / c^2 ) / ( exp[h*ν/(kB*T)] – 1 )
= T^2 * ( 2*V#^2 / c^2 ) / ( exp[h*V#/kB] – 1 )
βE(T,E) = BE(T,E) / E
= ( 2*E^2 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
= T^2 * ( 2*E#^2 / (h^3 * c^2) ) / ( exp[E#/kB] – 1 )
βlnE[T,ln(E)] = BlnE[T,ln(E)] / E
= T^3 * ( 2*E#^3 / (h^3 * c^2) ) / ( exp[E#/kB] – 1 )
= ( 2*E^3 / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
= BE(T,E)
Shapes of graphs: same as for Bλ, Bν, BE, BlnE, except the stretching along the vertical axis is reduced by a factor of the ratio of final to initial T. At the peaks (which are not located at the same points in the spectrum as for Bs where s is λ, ν, E, or lnE), the photon intensity is proportional to T^4, T^2, T^2, and T^3, respectively, and the area under the graph is proportional to T^3 in every case.
In Part III I will give solutions for the points in the spectrum where the peaks are.
There was a person on public TV who said that there are 13 federal regulatory agencies for nuclear power and 27 congressional committees to regulate nuclear power. Did I remember that correctly?
Kevin #368,
As far as I understand, your logic is fine — hence my caveat “under present circumstances.” As EV use grows, utilities learn to predict and plan for the added electricity demand, and consumers learn when to charge and even to sell back to the grid, more EV demand may be met with baseload power and the marginal-mix assumption may break down.
But meanwhile, the marginal-generator approach appears to be taken in U.S. studies of the near-term impact by folks at Argonne and Oak Ridge (Elgowainy et al., 2010; Hadley and Tsvetkova, 2008), who presumably know their stuff.
IEA chief economist Fatih Birol: global CO2 emissions 2010 estimated to be 30.6 billion tonnes, up 5.5% over 2009. Lord Stern: worrying development, if continued would mean world is back on ‘business-as-usual’ track leading to 50% chance of a 4C average temperature rise by 2100.
Jim Bullis:
This isn’t how it’s playing out in the Pacific Northwest, and I imagine that the people who run BPA would object to being labelled “insane”.
Can’t wait for the next ‘Unforced Variations’. Just a heads up for a new AJAX climate science data visualisation project over at Skeptical Science:
http://www.skepticalscience.com/Interactive-History-of-Climate-Science.html
I think there will be more versions in the future, although getting this one off to a start, took a while.
Re 374 Edward Greisch – I don’t know the specifics for nuclear, but I’ve also heard of some perhaps problematic overlaps in our food/ag system (USDA vs FDA, and ?), and also perhaps some crazy redundancy(? been awhile, not sure if it’s the right word) in our (U.S.) intelligence (Daily Show or Colber Report some time ago). Certainly if restructuring would improve efficiency that would be good – I hope though that this isn’t an excuse to just throw out or weaken sensible regulations. Might nuclear power and the others be more effectively regulated with a more efficient structure (more enforcement, less red tape perhaps)?
Re 372 Jim Bullis – so basically you are concerned about the relative rates of implementation? I can understand that. I’ll just say that a short term increase in emissions could be okay if sufficiently short with the end result being a sufficient decrease. For example, if we started using more fossil fuel energy to produce solar and wind (and nuclear) power infrastructure now, in a few years (depending on specifics) we could be back to the same fossil fuel use while still building that infrastructure, and then we’d be reducing fossil fuels.
#372, 375, 377–
I note that, according to the EIA, coal-fired capacity has declined since 1998, while most other sources have increased–renewables most rapidly by far. So why would that trend reverse because of growing demand for electricity?
Moreover, coal’s proportion of generation is larger than its proportion of capacity–again, uniquely so–which suggests that it has less spare capacity than other energy sources.
http://www.eia.gov/cneaf/electricity/epa/figes2.html
The EIA also says that “The construction of new coal plants has been discouraged by increasing costs for capital-intensive projects, concerns over possible future CO2 and other environmental restrictions, and the prospect that natural gas prices will remain low over the long-term.”
http://www.eia.gov/cneaf/electricity/epa/epa_sum.html
So, though this data is not as directly on-point as I’d like, there seems to be some reason to think that coal will not be much boosted by EVs, should they become really popular. Better data welcomed–!
Interesting–the EIA ‘reference case’ expects that it will take the US until 2035 to reach 2005 CO2 emission levels once again, *without* any regulations restricting CO2 emissions:
http://www.eia.gov/forecasts/aeo/chapter_executive_summary.cfm
It also notes that emissions levels are ‘sensitive’ to such regulation.
Well, I should hope so. . .
In other (and much less cheerful) news the confusingly-similarly-named IEA announced that 2010 saw record emissions at the global level, and that we really have pretty much run out of stopping space:
http://www.cbc.ca/news/technology/story/2011/05/30/iea-carbon-emissions.html
377 dhogza
I should have been more diligent and explained what I mean by ‘integrated’.
The BPA system is not integrated, meaning there is not sufficient infrastructure to utilize the wind and hydro resources they have under the range of conditions that have been demonstrated by Mother Nature.
In the context of energy, including both CO2 issues and oil issues, this is a massive waste. Who the pathological entities are is not clear, but I would expect it would be the bungled system of government that oversees this whole thing. Note, I am not advocating revolution, but I am advocating more awareness of the energy facts and the impact these have on the country as a whole.
A first step in setting things right might be a national recognition in law that water resources are owned by the country as a whole. That would start to make the solution to CO2 a little more accessible. CO2 is clearly a problem that we all own. Maybe we could even stop vilifying sectors of our system, but instead, unlock regional fiefdoms over resources.
382 Kevin McKinney,
So are you willing to reconsider the notion of continental water distribution?
380 Kevin McKinney
The reality is that both coal and natural gas systems have a lot of spare capacity in the USA. Some of the present spare here is that the recession has reduced the use of electricity. Natural gas has been somewhat more utilized due to cheaper natural gas and government pressure to use more of that.
New EV loads will readily be met with expanded use of either spare capacity, but the cheapest of the two is still the coal type. For a while the existing systems will be adequate, and the cost of electricity will stay low. If there is a lot of resistance to expanding coal systems, then the price of electricity will be pushed up. The natural gas people are eagerly hoping for this supply sitution because it will give them the opportunity to play their trading games, and then we should expect the cost of electricity to go up even more. This is not a new story for California, where similar events transpired ten years or so ago.
The largest utility in California. PG&E went into bankruptcy and so did a very forward thinking provider of advanced efficient natural gas systems, Calpine. A fortunate supply of hydro and access to coal based generation kept rates to the public from getting out of hand.
Re my 373: In Part III I will give solutions for the points in the spectrum where the peaks are.
No I won’t. Can’t be done (analytically)(so far as I can tell). I thought I could at least give the peaks for the Planck function per unit frequency, per unit ln(frequency), etc, as relative to the peak per unit wavelength, but I forgot about the variable in the exponential term, and so when I did that before – several months ago or more (?) – I was in error. If you see that comment just ignore that part (although wherever those peaks are (per unit frequency interval, per unit ln(frequency) interval, per uni wavelength interval), the 3 %, 4 %, and 5 % increases in Planck function for a 1 % temperature increase is still true (for a linear approximation based on the T^3, T^4, and T^5 proportionalities)).
Anyway, here’s what I did:
All of these Planck functions have a common form:
Bs, Blns, βs, or βlns =
s^n * A / (exp(B*s^m) – 1)
Taking the derivative d/ds, assuming n and m are nonzero:
n * s^(n-1) * A / (exp(B*s^m) – 1)
– s^n * A / (exp(B*s^m) – 1)^2 * (m * B * s^(m-1) * exp(B*s^m) )
=
n * s^(n-1) * A / (exp(B*s^m) – 1)
– s^n * A * (m * B * s^(m-1) * exp(B*s^m) ) / (exp(B*s^m) – 1)^2
=
s^(n-1) * A*n / (exp(B*s^m) – 1)
–
s^(n-1+m) * A*B*m * exp(B*s^m) / (exp(B*s^m) – 1)^2
Set derivative to zero and solve for s = sp (p stands for peak):
sp^(n-1) * A*n / (exp(B*sp^m) – 1)
=
sp^(n-1+m) * A*B*m * exp(B*sp^m) / (exp(B*sp^m) – 1)^2
n
=
sp^m * B*m * exp(B*sp^m) / (exp(B*sp^m) – 1)
sp^m
=
n/(m*B) * (exp(B*sp^m) – 1)/exp(B*sp^m)
=
n/(m*B) * [1 – exp(-B*sp^m)]
sp = (n/(m*B) * [1 – exp(-B*sp^m)])^(1/m)
For m=1: sp = n/B * [1 – exp(-B*sp)]
For m=-1: 1/sp = -n/B * [1 – exp(-B/sp)]
for m = 1:
sp = n/B * [1 – exp(-B*sp)]
let sp = 0:
n/B * [1-1] = 0. sp = 0 is a solution for m=1. Note that letting sp go to infinity is not a solution.
for m = -1:
1/sp = -n/B * [1 – exp(-B/sp)]
Take the limit as sp goes to infinity
-n/B * [1 – 1] = 0. As sp approaches infinity, 1/sp approaches 0, so the solution works (but note that the sides of the equation approaches 0 from opposite sides).
These solutions correspond to the limit of infinite wavelength, zero frequency and energy, and is a minimum of the Planck function. Solving for the maximum may require something other than analytical solving.
Anyway, in case anything can be done with the above:
Let B = C/T
Bs, Blns, βs, or βlns = s^n * A / (exp(C*s^m / T) – 1)
sp = (T * n/(m*C) * [1 – exp(-C*sp^m / T)])^(1/m)
For: s = , n = , m = , A = , C = ,
Bλ , λ , -5 , -1 , 2*h*c^2 , h*c/kB
Bν , ν , 3 , 1 , 2*h/c^2 , h/kB
BE , E , 3 , 1 , 2/(h^3 * c^2) , 1/kB
BlnE, E , 4 , 1 , 2/(h^3 * c^2) , 1/kB
βλ , λ , -4 , -1 , 2*c , h*c/kB
βν , ν , 2 , 1 , 2/c^2 , h/kB
βE , E , 2 , 1 , 2/(h^3 * c^2) , 1/kB
βlnE same as BE
Alternative: put in terms of L#, V#, or E#
ν = c/λ , E = h*ν
L# = λ*T
V# = ν/T = c/(λ*T) = c/L#
E# = E/T = h*ν/T = h*V#
For: s= , n = , m = , A = , B =
Bλ , L# , -5 , -1 , T^5 * 2*h*c^2 , h*c/kB
Bλ , V# , 5 , 1 , T^5 * 2*h/c^3 , h/kB
Bλ , E# , 5 , 1 , T^5 * 2/(h^4*c^3) , 1/kB
Bν , L# , -3 , -1 , T^3 * 2*h*c , h*c/kB
Bν , V# , 3 , 1 , T^3 * 2*h/c^2 , h/kB
Bν , E# , 3 , 1 , T^3 * 2/(h^2*c^2) , 1/kB
BE , V# , 3 , 1 , T^3 * 2/c^2 , h/kB
BE , E# , 3 , 1 , T^3 * 2/(h^3*c^2) , 1/kB
BlnE, E# , 4 , 1 , T^4 * 2/(h^3*c^2) , 1/kB
βλ , L# , -4 , -1 , T^4 * 2*c , h*c/kB
βλ , V# , 4 , 1 , T^4 * 2/c^3 , h/kB
βλ , E# , 4 , 1 , T^4 * 2/(h^4*c^3) , 1/kB
βν , V# , 2 , 1 , T^2 * 2/c^2 , h/kB
βν , E# , 2 , 1 , T^2 * 2/(h^2*c^2) , 1/kB
βE , V# , 2 , 1 , T^2 * 2/(h*c^2) , h/kB
βE , E# , 2 , 1 , T^2 * 2/(h^3*c^2) , 1/kB
βlnE same as BE
βE/E, E# , 1 , 1 , T * 2/(h^3*c^2) , 1/kB
This last one doesn’t look too meaningful, but if we hold s constant and vary T, the change in the Planck function / s would be proportionally the same as that of the Planck function itself. With s = E, and putting βE/E in terms of E#, we see a linear proportion to T, which means that at the peak βE/E, the Planck function is linearly proportional to T. (One could also find where it is proportional to any T^n by multiplying/dividing Planck functions by the sufficient power of E, ν, or λ and solving for the maximum point).
E# = (n/(m*B) * [1 – exp(-B*E#^m)])^(1/m)
= kB * [1 – exp(-E#/kB)]
One solution is E# = 0. If βE/E is not zero at E = 0 then this might be a maximum – even if not, at least it would (help) show that the Planck functions (which all vary with T in the same proportion at the same point in the spectrum) approach a linearly proportionality with T in the limit of infinite wavelength, or at zero frequency or energy. (I’ve only been able to show this graphically up to this point):
βE/E
= ( 2*E / (h^3 * c^2) ) / ( exp[E/(kB*T)] – 1 )
= T * ( 2*E# / (h^3 * c^2) ) / ( exp[E#/kB] – 1 )
lim(E# goes to 0) of E#/(exp[E#/kB] – 1) = …
ratio of derivatives: 1/(1/kB*exp[E#/kB] – 1); the limit as E# goes to 0 is 1/(1/kB – 1) … but kB isn’t unitless!
Well there is an approximation to the Planck function for larger L# (small V# or E#) which could help. There is also an obvious approximation at the other end of the spectrum – take the 1 out of the denominator:
Bs, Blns, βs, or βlns ~= s^n * A / exp(B*s^m) for large B*s^m
… if you’re going to use any of that, double check the algebra
Patrick, I dunno if anyone is reading your notes or not but I find them unreadable. Dunno if it’s worth your effort but I personally would be far more inclined to read your posts if you used latex a la (I hope this still works):
Go here (below) enter your equations and they’ll translate to html readable TeX for you.
http://www.codecogs.com/latex/eqneditor.php
[Response: We are using QuickLatex so you should use latexpage and tags for inline equations to get this:
. – gavin]
Re 388 – thanks, will try it (probably won’t redo what I already did though unless I just summarize results (ie just give the resulting equations, no work shown).
Re my 373 – Shapes of graphs: same as for Bλ, Bν, BE, BlnE, except … – bad wording; the shapes of the graphs would generally be different; I meant how the graph changes with changes in T.
Jim Bullis, 384-5:
Thanks for the chuckle on remodeling N. American hydrology, but no–I always agreed with you about the urgency of the proble; I just didn’t (and, I’m afraid, still don’t) think that your proposal had good odds of either implementation or success–success outright, or in terms of emissions ROI.
As to the spare coal capacity, I follow your logic better now–but can you point me to more information on this capacity question? Not so much that I’m doubting you, as that I’d really like to do a bit more digging in order to understand this more deeply–and I didn’t have great luck in my initial search for more.
Oops–“proble” = “problem.”
390 Kevin McKinney
Thanks for feedback, and the odds of implementation are indeed daunting. But so is the notion that we can turn off the industrial revolution, it is just that so many seem not to see how important that is. When the reality of cutting off coal sinks in, we will be looking at a need for more creative thinking.
I learned serious details from reading the Annual Reports of NRG Energy (stock ticker NRG).
As I recall, the 2008 – 2009 reports were the most explanatory. The last one seems to have been worked over more by the PR staff.
Arch Coal (ACI), Peabody (BTU) also offer some insight, though I have been following all types of energy for some years now.
I don’t think I have time to use QuickLatex tonight, but only a few equations here:
Here’s an interesting version of the Planck function from a book about solar cells (Jenny Nelson, “The physics of solar cells” – browsed with google books ):
(this is for n=1 (the real component of the index of refraction, not to be confused with the ‘n’ I used in 386); the full expression is proportional to n^2)
A = 2 / (h^3 * c^2)
βE = A*E^2 / ( exp[(E-Δμ)/(kB*T)] – 1 )
But from prior work above:
βE = A*E^2 / ( exp[E/(kB*T)] – 1 )
The first version is for a system that is not at LTE, though components of it are at a partial LTE (‘quasi thermal equilibrium’). Note that if
E-Δμ is replaced by E, the formula reverts to blackbody emission at temperature T.
Δμ is the chemical potential between two populations of electrons (or electrons and holes – however you look at at). Specifically, at complete LTE, the electrons are distributed among all available energy levels – in all bands – according to the Fermi distribution, with a single common Fermi level μ0:
f1(E1,μ0,T1) = 1/( exp[(E1-μ0)/(kB*T1)] + 1 )
where f1 is the fraction of available states at energy E1 that are occupied (at temperature T1).
… I’m skipping over a few things because of time. But the point is that
1. one would expect some kind of proportionality of photon emission or absorption to f1*(1-f2) for a transition from E1 to E2 (which is emission or absorption depending on which is larger).
2. Δμ is the difference between the Fermi levels – actually, I think they’re called ‘quasi-Fermi levels’, between two energy bands that are not in equilibrium with each other (a situation that can arise when electrons are excited from one band to another, then allowed to reach a (quasi?-)Fermi distribution within each band at the temperature of the material).
I think it can be shown (making use of the fact that for a given f1 at E1, the distance to the ‘quasi-Fermi level’ is proportional to T1) that, where E = E1-E2, where E1 and E2 are energy levels in bands 1 and 2, and the electron distribution within each band fits the Fermi distribution for the same temperature T, that the same f1 and f2 of E1 and E2 can be fit to a different Fermi distribution with a common Fermi level and a temperature Teff (effective temperature), such that replacing exp[(E-Δμ)/(kB*T)] in the Planck function with exp[E/(kB*Teff)] gives the same result – in other words, (E-Δμ)/T = E/Teff. Which suggests that the rate of photon emission does depend somehow on the occupancy of states, as expected. It’s an interesting example of how a system not in complete LTE can still be described, in this case as an equivalent system at LTE (but note that different Teff values will be found for different E values, so the system is simply one set of pairs of energy levels with the same energy difference).
PS I used ^ for exponents as in an Excel spreadsheet. A number of letters or numbers that follow a letter would be writted as subscripts. I discovered the use of ‘beta’ for photon intensity in the book mentioned above; I used the subscript E for beta and B based on the usage of subscripts for frequency and wavelength. Not everything is exactly in standard notation but I tried to keep it close. My use of the # symbol was my own idea.
… I proofread the above but in a bit of hurry, so I hope it’s okay…
PS didn’t have time to post more recent CO2eq/kWh for solar power – earlier I think some values were posted that may have been out of date; they’re not that big.
#392–Thanks!
The current Climate Policy magazine (former climateprogress) http://thinkprogress.org/romm/issue/ let’s some Yahoo-comments through, I hear, but I don’t very much like (can’t dislike) the otherwise forced coupling to twitter. Maybe the articles will get shorter in there so one may also comment shortly, as there have been some articles which do need an answer that is over 4 rows in lenght.