Guest post by Bart Verheggen, Department of Air Quality and Climate Change , Energy research Institute of the Netherlands (ECN)
In Part I, I discussed how aerosols nucleate and grow. In this post I’ll discuss how changes in nucleation and ionization might impact the net effects.
Cosmic rays
Galactic cosmic rays (GCR) are energetic particles originating from space entering Earth’s atmosphere. They are an important source of ionization in the atmosphere, besides terrestrial radioactivity from e.g. radon (naturally emitted by the Earth’s surface). Over the oceans and above 5 km altitude, GCR are the dominant source. Their intensity varies over the 11 year solar cycle, with a maximum near solar minimum. Carslaw et al. give a nice overview of potential relations between cosmic rays, clouds and climate. Over the first half of the 20th century solar irradiance has slightly increased, and cosmic rays have subsequently decreased. RC has had many previous posts on the purported links between GCR and climate, e.g. here, here and here.
The role of ions
The role played by ions relative to neutral (uncharged) molecules in the nucleation process is still very much under discussion. For instance, based on the same dataset, Yu and Turco found a much higher contribution of ion induced nucleation (to the total amount of particles produced) than Laakso et al did. Evidence for a certain nucleation mechanism is often of an indirect nature, and depends on uncertain parameters. Most literature points to a potential importance of ion induced nucleation in the upper troposphere, but the general feeling is that neutral pathways for nucleation (i.e. not involving ions) are likely to be dominant overall. Most field studies, however, have been performed over land, whereas over the open ocean nucleation rates are generally lower due to lower vapor concentrations. In theory at least, this gives more opportunity for ion induced nucleation to make a difference over the ocean (even though the ion production rate is smaller).
The ion production rate (increasing with altitude from ~10 to ~50 ion pairs per cubic centimeter per second over land) sets a limit to what the particle formation rate due to ion induced nucleation can be. Based on his model for ion induced nucleation, Yu found that at low altitude, the number of particles produced is most sensitive to changes in cosmic ray intensity. At first sight, this may be a surprising result in light of the increasing cosmic ray intensity with increasing altitude. The reason is that high aloft, the limiting factor for particle formation is the availability of sulfuric acid rather than ions. Above a certain GCR intensity, increasing ionization further could even lead to a decrease in ion induced nucleation, because the lifetime of ion clusters is reduced (due to increased recombination of positive and negative ions). In contrast, at low altitude particle formation may be limited by the ionization rate (under certain circumstances), and an increase in ionization leads to an increase in nucleation.
How important is nucleation for climate?
Different modeling exercises have been performed to investigate this question. The strong dependency on input data and assumptions used, e.g. relating to primary particle emissions and nucleation parameterizations, and the different sensitivities tested, hampers an overall assessment. However, it is clear that globally, nucleation is significant for the number of cloud condensation nuclei (CCN) e.g. in the absence of boundary layer nucleation, the number of CCN would be 5% lower (Wang and Penner) or 3-20% lower (Spracklen et al.), and in a recent follow up study, they concluded that the number of cloud droplets would be 13-16% lower (in 2000 and 1850, respectively). Pierce and Adams took a different approach and looked at the variation of predicted number of CCN as a result of using different nucleation schemes. The tropospheric number of CCN varied by 17% (and the boundary layer CCN by 12%) amongst model runs using different nucleation rate parameterizations. Note that the globally averaged nucleation rates differed by a factor of a million (!).
It should be noted that the sensitivity of the number of CCN to nucleation depends greatly on the amount of primary emissions and secondary organic aerosol (SOA) formed. These are very uncertain themselves, which further limit our ability to understand the connection between nucleation and CCN. If there are more primary emissions, there will be more competition amongst aerosols to act as CCN. If more organic compounds partition to the aerosol phase (to form SOA), the growth to CCN sizes will be quicker.
Locally, particle formation has been observed to contribute significantly to the number of CCN; the second figure in Part I gives an example of freshly nucleated aerosols which grew large enough to influence cloud formation. Kerminen et al observed a similar event, followed by activation of part of the nucleated aerosols into cloud droplets, thus providing a direct link between aerosol formation and cloud droplet activation.
How important are cosmic rays for climate?
At the recent AGU meeting (Dec 2008), Jeff Pierce presented results on the potential effects of GCR on the number of CCN (their paper at GRL (sub. required)). Two different parameterizations for ion induced nucleation were used (Modgil et al and an ‘ion-limit’ assumption that all ions go on to form a new particle). They ran their model with both high and low cosmic ray flux, simulating conditions during solar maximum and minimum, respectively. This happens to be comparable to the change in cosmic ray flux over the 20th century (mostly confined to the first half), and amounts to a 20% change in tropospheric ion production. With both mechanisms of ion-induced nucleation, this leads to a 20% change in globally averaged particle nucleation, but only to a 0.05% change in globally averaged CCN. The authors concluded that this was “far too small to make noticeable changes in cloud properties based on either the decadal (solar cycle) or climatic time-scale changes in cosmic rays.” To account for some reported changes in cloud cover, a change in CCN on the order of 10% would be needed. More studies of this kind will undoubtedly come up with different numbers, but it’s perhaps less likely that the qualitative conclusion, as quoted above, will change dramatically. Time will tell, of course.
The bottom line
Freshly nucleated particles have to grow by about a factor of 100,000 in mass before they can effectively scatter solar radiation or be activated into a cloud droplet (and thus affect climate). They have about 1-2 weeks to do this (the average residence time in the atmosphere), but a large fraction will be scavenged by bigger particles beforehand. What fraction of nucleated particles survives to then interact with the radiative budget depends on many factors, notably the amount of condensable vapor (leading to growth of the new particles) and the amount of pre-existing particles (acting as a sink for the vapor as well as for the small particles). Model-based estimates of the effect of boundary layer nucleation on the concentration of cloud condensation nuclei (CCN) range between 3 and 20%. However, our knowledge of nucleation rates is still severely limited, which hampers an accurate assessment of its potential climate effects. Likewise, the potential effects of galactic cosmic rays (GCR) can only be very crudely estimated. A recent study found that a change in GCR intensity, as is typically observed over an 11 year solar cycle, could, at maximum, cause a change of 0.1% in the number of CCN. This is likely to be far too small to make noticeable changes in cloud properties.
sample:
r = interest rate on a lo-an: 7%
f = inflation rate on all prices considered: 2%
h = decay rate of solar panel performance (including loss of panel area): 1.5 %
for a cost of $10/average W for a device at one time, paid with a lo-an:
m0 (cents/kWh) (inflation adjusted constant price) required to pay off lo-an in time t (years):
t(years), m0 (cents/kWh)
15, 11.8
20, 10
25, 9.06
30, 8.46
40, 7.81
50, 7.51
60, 7.35
70, 7.27
80, 7.23
100, 7.20
infinite, 7.18
CORRECTION TO: comment 297 CORRECTION TO PART III:
d/dt[exp(S*t)] = S*exp(S*t)
d/dt[exp(H*t)] = H*exp(H*t)
——
m*P*dt=L*N*dt
m = L*(N/P)
N in order to keep P increasing at rate S*P:
N*dt = P*(S-H)*dt + N*dt*exp(-S*U)*exp(H*U)
N*dt is necessary to increase P by P*S*dt while replacing the loss -P*H*dt and the retirement loss N*exp(-S*U)*dt * exp(H*U)
N in order to keep P0 increasing at rate S*P0:
N*dt = P0*(S-Hs)*dt + N*dt*exp(-S*U)*exp(Hs*U)
——
N = P*(S-H) + N*exp[(H-S)*U]
N * ( 1 – exp[(H-S)*U] ) = P*(S-H)
—————
m0/L0 = N/P = (S-H)/( 1 – exp[(H-S)*U] )
P/N = ( 1 – exp[(H-S)*U] )/(S-H)
P/P0
=
( 1 – exp[(H-S)*U] )/(S-H)
*
(S-Hs)/( 1 – exp[(Hs-S)*U] )
=
(S-Hs)/(S-H) * ( 1 – exp[(H-S)*U] )/( 1 – exp[(Hs-S)*U] )
————–
WHEN U GOES TO INFINITY:
m0/L0 = N/P = (S-H)
P/N = 1/(S-H)
P/P0 = (S-Hs)/(S-H)
————–
WHEN S GOES TO ZERO:
m0/L0 = N/P = -H / ( 1 – exp[H*U] )
P/N = – ( 1 – exp[H*U] ) / H
P/P0
=
Hs/H * ( 1 – exp[H*U] )/( 1 – exp[Hs*U] )
And from a different part of the sensitivity-to-nucleation spectrum: “Simulation of particle size distribution with a global aerosol model: contribution of nucleation to aerosol and CCN number concentrations” by F. Yu and G. Luo (http://www.atmos-chem-phys-discuss.net/9/10597/2009/acpd-9-10597-2009.html)
Upon first glance, they find a much bigger contribution from nucleation to the CCN budget than others have reported. I haven’t read it in detail yet.
Another interesting aerosol-climate paper is the following review about possible feedbacks of natural aerosol sources to a changing climate: Atmospheric aerosols in the earth system: a review of interactions and feedbacks (http://www.atmos-chem-phys-discuss.net/9/11087/2009/acpd-9-11087-2009.html)
Both these papers are currently under (open) review.
Here, I think, is the abstract for the Pierce paper that was presented at the AGU (mentioned in the original post):
Pierce, J. R.; Adams, P. J.
Can cosmic rays affect cloud condensation nuclei by altering new particle formation rates?
Geophys. Res. Lett., Vol. 36, No. 9, L09820
http://dx.doi.org/10.1029/2009GL037946
13 May 2009
CALCULATIONS FOR SOME SOLAR ELECTRICITY COSTS AND SOLAR COLLECTOR PERFORMANCE:
EXPLANATIONS:
——————————————————–
m0 is the price of electricity necessary to either:
1.
pay off debt by time t (with payments made over time as electricity sales revenue allows), where collectors are bought entirely with debt – with
annual interest r = 7 %
annual inflation f = 2 %
and the same inflation rate applies to both electricity price m and collector price L
2.
pay for a rate of purchasing new collectors without any debt.
————-
L0,the price per unit of collector, is $10 / average W
Some combinations that result in $10 / average W
(assuming high fill factors or concentration of sunlight into time intervals – more generally, that the average efficiency of conversion is not much lower than the rated efficiency of conversion ):
$1.50 / peak W with average insolation of about 150 W/m2
$1.70 / peak W with average insolation of about 170 W/m2
$2.00 / peak W with average insolation of about 200 W/m2
$2.50 / peak W with average insolation of about 250 W/m2
$3.00 / peak W with average insolation of about 300 W/m2
where the average insolation is the average solar power per unit area of collector; for geometric concentration, this only includes direct solar radiation (as opposed to diffuse radiation).
————-
L0 and m0 are given in constant inflation-adjusted (real) values.
m0 is linearly proportional to L0 – except in the case that L0 is changing while a debt is being paid for purchases based on previous L0 values. In the following, L0 is assumed constant.
——————————————————–
P0 is a quantity of solar collectors measured by average power produced at installation (when new). When not otherwise specified, P0 refers to the quantity of collectors currently in operation at a time t, and can be used as a relative measure of the area of collectors and (with adjustment for collector orientations and spacing) a relative measure of the area occupied by collectors.
P is the average power produced by collectors in operation at a time t.
Thus,
P/P0 is the average efficiency of operational collectors relative to efficiency of new collectors,
and
P0/P is the area required per unit average power supply relative to the area per unit average power supply for new collectors.
Some maintenance and other costs could be proportional to the area of the collectors and the area they occupy, so it is of interest to know P0/P and P/P0.
—————-
-h, -he, and -hs are annual fractional losses.
(h, he, and hs are defined as gains, they have negative values.)
-h is the annual fractional loss in average power P produced by a set of collectors since and includes the effects of decreases in performance of collectors in operation and the loss of collectors from operation (such as from severe storm damage or fire), but not including retirement of collectors at a set age.
-hs is the annual fractional loss in operational P0 (the loss of operational collectors from collectors in operation) that is not due to retirement at a set age.
-he is the annual fractional loss in P from operational P0 that remains operational.
For small h, he, and hs values, h is approximately equal to the sum of he and hs.
The same set of h, he, and hs values are used for each table where applicable; those used in the calculations for a table are restated for that table for convenience.
—————
Where applicable, the retirement age U of operational collectors was chosen as the time when performance (P per unit operational P0) will have dropped by a factor of e from the initial value (P/P0 = 1 for newly installed collectors).
——————————————————–
CASES CONSIDERED:
I. ONE TIME INSTALLMENT at time t = 0
A set of collectors (of quantity “installed P0”) is bought and installed at time t = 0.
Cummulative Energy since installation / installed P0, [(W*yr) / W] = years
– this is the time in years it would take for a set of collectors with constant P (h = 0) to produce the same energy that a set of actual collectors (with nonzero h) would produce from installation at time 0 to time t.
Average ( P/ installed P0 ) from installation to time t since installation
– this is the Cummulative Energy at time t per unit P0 divided by the time t.
—————————
II. Continual purchasing and installation of new collectors at a rate N = new P0 per unit time, measured in (W/year).
A. CONSTANT N
N = 1 installed W per year.
Starting at time t = 0, where P and P0 at time t=0 are both 0.
Operational collectors installed at time t = ti are retired at time ti + U, where U is the retirement age.
————-
B. EXPONENTIAL GROWTH with Retirement at U
P, operational P0, and N all increase exponentially,
where s is the annual fractional increase.
There is a retirement age of U as in the case “CONSTANT N”.
It is assumed that exponential growth has been ongoing for some time so that retirements are already occuring at time t = 0.
————-
C. EXPONENTIAL GROWTH, no retirement at a set age
As above, but without a retirement age – the only loss of operational P0 is from nonzero hs.
————-
For both EXPONENTIAL cases:
For any starting situation, the ratio of P to N and P0 to N will tend to approach constant values over time if N is increasing exponentially, so that P and P0 will eventually increase at the same rate s.
IF P and P0 = 0 at time t and N starts at some nonzero number and increases exponentially, P/N and P0/N will start at 0 and increase over time.
To create a situation where the calcuations for the cases above would apply immediately, one could initially buy and install some P0 at time t = 0, and then increase P and N exponentially from then on; in that case, the calculation of m0 would not include the initial purchase, and there would be a discontinuity at time t = U if there are retirements.
——————————————————–
Formulas used are found at:
295
https://www.realclimate.org/index.php/archives/2009/04/aerosol-effects-and-climate-part-ii-the-role-of-nucleation-and-cosmic-rays/langswitch_lang/it#comment-123843
Note corrections to PART III in:
302
https://www.realclimate.org/index.php/archives/2009/04/aerosol-effects-and-climate-part-ii-the-role-of-nucleation-and-cosmic-rays/langswitch_lang/it#comment-125193
________________________________________________________________________________________
PRICE OF ELECTRICITY FOR PAYING OFF DEBT BY TIME t, WHERE COLLECTORS ARE BOUGHT or FIRST BOUGHT AT t = 0:
___________________________________________
ONE TIME INSTALLMENT
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ m0 , (cents/kWh)
_ _ _ _ _|___ to pay off debt at time t
_____ 15 |_ 11.02 ,_ 11.16 ,_ 11.38 ,_ 11.53 ,_ 11.75 ,_ 12.14
_____ 20 |__ 9.24 ,__ 9.40 ,__ 9.63 ,__ 9.79 ,_ 10.03 ,_ 10.44
_____ 25 |__ 8.22 ,__ 8.39 ,__ 8.64 ,__ 8.80 ,__ 9.06 ,__ 9.50
_____ 30 |__ 7.58 ,__ 7.76 ,__ 8.02 ,__ 8.19 ,__ 8.46 ,__ 8.92
_____ 40 |__ 6.86 ,__ 7.04 ,__ 7.33 ,__ 7.52 ,__ 7.81 ,__ 8.31
_____ 50 |__ 6.49 ,__ 6.69 ,__ 6.99 ,__ 7.20 ,__ 7.51 ,__ 8.03
_____ 60 |__ 6.29 ,__ 6.50 ,__ 6.82 ,__ 7.03 ,__ 7.35 ,__ 7.90
_____ 70 |__ 6.18 ,__ 6.40 ,__ 6.72 ,__ 6.94 ,__ 7.27 ,__ 7.83
_____ 80 |__ 6.12 ,__ 6.34 ,__ 6.67 ,__ 6.89 ,__ 7.23 ,__ 7.80
_____ 90 |__ 6.08 ,__ 6.31 ,__ 6.64 ,__ 6.87 ,__ 7.21 ,__ 7.78
____ 100 |__ 6.06 ,__ 6.29 ,__ 6.63 ,__ 6.85 ,__ 7.20 ,__ 7.77
____ 120 |__ 6.04 ,__ 6.27 ,__ 6.61 ,__ 6.84 ,__ 7.19 ,__ 7.77
____ 150 |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.18 ,__ 7.76
____ 200 |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.18 ,__ 7.76
____ 250 |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.18 ,__ 7.76
____ INF |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.18 ,__ 7.76
___________________________________________
CONSTANT N
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ m0 , (cents/kWh)
_ _ _ _ _|___ to pay off debt at time t
_____ 15 |_ 17.67 ,_ 17.83 ,_ 18.08 ,_ 18.25 ,_ 18.50 ,_ 18.93
_____ 20 |_ 13.95 ,_ 14.12 ,_ 14.38 ,_ 14.55 ,_ 14.81 ,_ 15.25
_____ 25 |_ 11.76 ,_ 11.93 ,_ 12.20 ,_ 12.37 ,_ 12.64 ,_ 13.09
_____ 30 |_ 10.33 ,_ 10.51 ,_ 10.78 ,_ 10.96 ,_ 11.23 ,_ 11.70
_____ 40 |__ 8.62 ,__ 8.80 ,__ 9.08 ,__ 9.27 ,__ 9.56 ,_ 10.05
_____ 50 |__ 7.66 ,__ 7.85 ,__ 8.15 ,__ 8.34 ,__ 8.64 ,__ 9.15
_____ 60 |__ 7.09 ,__ 7.29 ,__ 7.59 ,__ 7.79 ,__ 8.10 ,__ 8.63
_____ 70 |__ 6.72 ,__ 6.93 ,__ 7.24 ,__ 7.45 ,__ 7.77 ,__ 8.31
_____ 80 |__ 6.49 ,__ 6.70 ,__ 7.02 ,__ 7.23 ,__ 7.56 ,__ 8.11
_____ 90 |__ 6.33 ,__ 6.55 ,__ 6.87 ,__ 7.09 ,__ 7.42 ,__ 7.98
____ 100 |__ 6.23 ,__ 6.45 ,__ 6.78 ,__ 7.00 ,__ 7.34 ,__ 7.90
____ 120 |__ 6.12 ,__ 6.34 ,__ 6.68 ,__ 6.90 ,__ 7.25 ,__ 7.82
____ 150 |__ 6.05 ,__ 6.28 ,__ 6.62 ,__ 6.85 ,__ 7.20 ,__ 7.78
____ 200 |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.19 ,__ 7.77
____ 250 |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.19 ,__ 7.77
____ INF |__ 6.03 ,__ 6.26 ,__ 6.61 ,__ 6.84 ,__ 7.18 ,__ 7.76
________________________________________________________________________________________
PRICE OF ELECTRICITY FOR PAYING FOR NEW COLLECTORS, NO DEBT:
___________________________________________
CONSTANT N
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ m0 , (cents/kWh)
_ _ _ _ _|___ to pay for new collectors, no debt
_____ 15 |__ 7.89 ,__ 8.01 ,__ 8.19 ,__ 8.31 ,__ 8.50 ,__ 8.82
_____ 20 |__ 5.99 ,__ 6.11 ,__ 6.30 ,__ 6.42 ,__ 6.61 ,__ 6.93
_____ 25 |__ 4.85 ,__ 4.98 ,__ 5.16 ,__ 5.29 ,__ 5.48 ,__ 5.81
_____ 30 |__ 4.10 ,__ 4.22 ,__ 4.40 ,__ 4.53 ,__ 4.73 ,__ 5.07
_____ 40 |__ 3.15 ,__ 3.27 ,__ 3.46 ,__ 3.60 ,__ 3.80 ,__ 4.16
_____ 50 |__ 2.58 ,__ 2.71 ,__ 2.90 ,__ 3.04 ,__ 3.25 ,__ 3.62
_____ 60 |__ 2.20 ,__ 2.33 ,__ 2.53 ,__ 2.67 ,__ 2.89 ,__ 3.28
_____ 70 |__ 1.93 ,__ 2.06 ,__ 2.27 ,__ 2.41 ,__ 2.64 ,__ 3.04
_____ 80 |__ 1.73 ,__ 1.86 ,__ 2.08 ,__ 2.22 ,__ 2.46 ,__ 2.88
_____ 90 |__ 1.57 ,__ 1.71 ,__ 1.93 ,__ 2.08 ,__ 2.32 ,__ 2.75
____ 100 |__ 1.45 ,__ 1.59 ,__ 1.81 ,__ 1.96 ,__ 2.21 ,__ 2.67
____ 120 |__ 1.27 ,__ 1.41 ,__ 1.64 ,__ 1.80 ,__ 2.06 ,__ 2.67
____ 150 |__ 1.08 ,__ 1.23 ,__ 1.47 ,__ 1.65 ,__ 2.00 ,__ 2.67
____ 200 |__ 0.90 ,__ 1.06 ,__ 1.33 ,__ 1.59 ,__ 2.00 ,__ 2.67
____ 250 |__ 0.85 ,__ 1.06 ,__ 1.33 ,__ 1.59 ,__ 2.00 ,__ 2.67
____ INF |__ 0.85 ,__ 1.06 ,__ 1.33 ,__ 1.59 ,__ 2.00 ,__ 2.67
___________________________________________
EXPONENTIAL GROWTH with Retirement at U
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | m0 , (cents/kWh) _____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ to pay for new collectors, no debt______ _ _ _ __| ____ |_ by factor of 10
____ 0.0 |__ 0.85 ,__ 1.06 ,__ 1.33 ,__ 1.59 ,__ 2.00 ,__ 2.67 |_ 0.0 |
____ 0.1 |__ 0.93 ,__ 1.15 ,__ 1.42 ,__ 1.69 ,__ 2.09 ,__ 2.76 |_ 0.1 | 2303.7
____ 0.2 |__ 1.01 ,__ 1.23 ,__ 1.51 ,__ 1.78 ,__ 2.18 ,__ 2.85 |_ 0.2 | 1152.4
____ 0.5 |__ 1.28 ,__ 1.51 ,__ 1.81 ,__ 2.07 ,__ 2.47 ,__ 3.14 |_ 0.5 |_ 461.7
____ 1.0 |__ 1.77 ,__ 2.00 ,__ 2.33 ,__ 2.58 ,__ 2.97 ,__ 3.63 |_ 1.0 |_ 231.4
____ 1.5 |__ 2.30 ,__ 2.53 ,__ 2.87 ,__ 3.11 ,__ 3.49 ,__ 4.13 |_ 1.5 |_ 154.7
____ 2.0 |__ 2.84 ,__ 3.08 ,__ 3.41 ,__ 3.66 ,__ 4.02 ,__ 4.65 |_ 2.0 |_ 116.3
____ 3.0 |__ 3.95 ,__ 4.18 ,__ 4.52 ,__ 4.75 ,__ 5.11 ,__ 5.72 |_ 3.0 |__ 77.9
____ 4.0 |__ 5.05 ,__ 5.28 ,__ 5.62 ,__ 5.85 ,__ 6.20 ,__ 6.80 |_ 4.0 |__ 58.7
____ 5.0 |__ 6.14 ,__ 6.37 ,__ 6.71 ,__ 6.94 ,__ 7.29 ,__ 7.88 |_ 5.0 |__ 47.2
____ 6.0 |__ 7.22 ,__ 7.45 ,__ 7.79 ,__ 8.02 ,__ 8.37 ,__ 8.96 |_ 6.0 |__ 39.5
____ 7.0 |__ 8.29 ,__ 8.52 ,__ 8.86 ,__ 9.10 ,__ 9.44 ,_ 10.02 |_ 7.0 |__ 34.0
____ 8.0 |__ 9.35 ,__ 9.58 ,__ 9.93 ,_ 10.16 ,_ 10.50 ,_ 11.08 |_ 8.0 |__ 29.9
____ 9.0 |_ 10.40 ,_ 10.63 ,_ 10.98 ,_ 11.21 ,_ 11.56 ,_ 12.14 |_ 9.0 |__ 26.7
___ 10.0 |_ 11.44 ,_ 11.67 ,_ 12.02 ,_ 12.25 ,_ 12.60 ,_ 13.18 | 10.0 |__ 24.2
___ 15.0 |_ 16.52 ,_ 16.74 ,_ 17.09 ,_ 17.32 ,_ 17.67 ,_ 18.25 | 15.0 |__ 16.5
___ 20.0 |_ 21.37 ,_ 21.60 ,_ 21.95 ,_ 22.18 ,_ 22.52 ,_ 23.10 | 20.0 |__ 12.6
___ 30.0 |_ 30.50 ,_ 30.73 ,_ 31.08 ,_ 31.31 ,_ 31.65 ,_ 32.23 | 30.0 |___ 8.8
___________________________________________
EXPONENTIAL GROWTH, no retirement at a set age
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | m0 , (cents/kWh) _____ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ to pay for new collectors, no debt______ _ _ _ __| ____ |_ by factor of 10
____ 0.0 |__ 0.57 ,__ 0.80 ,__ 1.15 ,__ 1.38 ,__ 1.72 ,__ 2.30 |_ 0.0 |
____ 0.1 |__ 0.69 ,__ 0.92 ,__ 1.26 ,__ 1.49 ,__ 1.84 ,__ 2.42 |_ 0.1 | 2303.7
____ 0.2 |__ 0.80 ,__ 1.03 ,__ 1.37 ,__ 1.61 ,__ 1.95 ,__ 2.53 |_ 0.2 | 1152.4
____ 0.5 |__ 1.14 ,__ 1.37 ,__ 1.72 ,__ 1.95 ,__ 2.29 ,__ 2.87 |_ 0.5 |_ 461.7
____ 1.0 |__ 1.71 ,__ 1.94 ,__ 2.28 ,__ 2.51 ,__ 2.86 ,__ 3.44 |_ 1.0 |_ 231.4
____ 1.5 |__ 2.27 ,__ 2.50 ,__ 2.84 ,__ 3.08 ,__ 3.42 ,__ 4.00 |_ 1.5 |_ 154.7
____ 2.0 |__ 2.83 ,__ 3.06 ,__ 3.41 ,__ 3.64 ,__ 3.98 ,__ 4.56 |_ 2.0 |_ 116.3
____ 3.0 |__ 3.94 ,__ 4.17 ,__ 4.52 ,__ 4.75 ,__ 5.10 ,__ 5.68 |_ 3.0 |__ 77.9
____ 4.0 |__ 5.05 ,__ 5.28 ,__ 5.62 ,__ 5.85 ,__ 6.20 ,__ 6.78 |_ 4.0 |__ 58.7
____ 5.0 |__ 6.14 ,__ 6.37 ,__ 6.71 ,__ 6.94 ,__ 7.29 ,__ 7.87 |_ 5.0 |__ 47.2
____ 6.0 |__ 7.22 ,__ 7.45 ,__ 7.79 ,__ 8.02 ,__ 8.37 ,__ 8.95 |_ 6.0 |__ 39.5
____ 7.0 |__ 8.29 ,__ 8.52 ,__ 8.86 ,__ 9.10 ,__ 9.44 ,_ 10.02 |_ 7.0 |__ 34.0
____ 8.0 |__ 9.35 ,__ 9.58 ,__ 9.93 ,_ 10.16 ,_ 10.50 ,_ 11.08 |_ 8.0 |__ 29.9
____ 9.0 |_ 10.40 ,_ 10.63 ,_ 10.98 ,_ 11.21 ,_ 11.56 ,_ 12.14 |_ 9.0 |__ 26.7
___ 10.0 |_ 11.44 ,_ 11.67 ,_ 12.02 ,_ 12.25 ,_ 12.60 ,_ 13.18 | 10.0 |__ 24.2
___ 15.0 |_ 16.52 ,_ 16.74 ,_ 17.09 ,_ 17.32 ,_ 17.67 ,_ 18.25 | 15.0 |__ 16.5
___ 20.0 |_ 21.37 ,_ 21.60 ,_ 21.95 ,_ 22.18 ,_ 22.52 ,_ 23.10 | 20.0 |__ 12.6
___ 30.0 |_ 30.50 ,_ 30.73 ,_ 31.08 ,_ 31.31 ,_ 31.65 ,_ 32.23 | 30.0 |___ 8.8
_________________________________________________________________________________________
OTHER CALCULATIONS:
___________________________________________
ONE TIME INSTALLMENT
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ P / installed P0
_ _ _ _ _|___
_____ 15 |__ 0.93 ,__ 0.90 ,__ 0.86 ,__ 0.83 ,__ 0.80 ,__ 0.74
_____ 20 |__ 0.90 ,__ 0.87 ,__ 0.82 ,__ 0.79 ,__ 0.74 ,__ 0.67
_____ 25 |__ 0.88 ,__ 0.84 ,__ 0.78 ,__ 0.74 ,__ 0.69 ,__ 0.60
_____ 30 |__ 0.86 ,__ 0.81 ,__ 0.74 ,__ 0.70 ,__ 0.64 ,__ 0.55
_____ 40 |__ 0.82 ,__ 0.76 ,__ 0.67 ,__ 0.62 ,__ 0.55 ,__ 0.45
_____ 50 |__ 0.78 ,__ 0.70 ,__ 0.61 ,__ 0.55 ,__ 0.47 ,__ 0.36
_____ 60 |__ 0.74 ,__ 0.66 ,__ 0.55 ,__ 0.48 ,__ 0.40 ,__ 0.30
_____ 70 |__ 0.70 ,__ 0.61 ,__ 0.49 ,__ 0.43 ,__ 0.35 ,__ 0.24
_____ 80 |__ 0.67 ,__ 0.57 ,__ 0.45 ,__ 0.38 ,__ 0.30 ,__ 0.20
_____ 90 |__ 0.64 ,__ 0.53 ,__ 0.40 ,__ 0.34 ,__ 0.26 ,__ 0.16
____ 100 |__ 0.61 ,__ 0.50 ,__ 0.37 ,__ 0.30 ,__ 0.22 ,__ 0.13
____ 120 |__ 0.55 ,__ 0.43 ,__ 0.30 ,__ 0.23 ,__ 0.16 ,__ 0.09
____ 150 |__ 0.47 ,__ 0.35 ,__ 0.22 ,__ 0.16 ,__ 0.10 ,__ 0.05
____ 200 |__ 0.37 ,__ 0.25 ,__ 0.13 ,__ 0.09 ,__ 0.05 ,__ 0.02
____ 250 |__ 0.29 ,__ 0.17 ,__ 0.08 ,__ 0.05 ,__ 0.02 ,__ 0.01
___ 1000 |__ 0.01 ,__ 0.00 ,__ 0.00 ,__ 0.00 ,__ 0.00 ,__ 0.00
_ -he (%):__ 0.45 ,__ 0.50 ,__ 0.50 ,__ 0.60 ,__ 0.76 ,__ 1.01
_ _ _ _ _|____________________________________________________
t, years |_ P / operational P0
_ _ _ _ _|___
_____ 15 |__ 0.93 ,__ 0.93 ,__ 0.93 ,__ 0.91 ,__ 0.89 ,__ 0.86
_____ 20 |__ 0.91 ,__ 0.90 ,__ 0.90 ,__ 0.89 ,__ 0.86 ,__ 0.82
_____ 25 |__ 0.89 ,__ 0.88 ,__ 0.88 ,__ 0.86 ,__ 0.83 ,__ 0.78
_____ 30 |__ 0.87 ,__ 0.86 ,__ 0.86 ,__ 0.83 ,__ 0.80 ,__ 0.74
_____ 40 |__ 0.83 ,__ 0.82 ,__ 0.82 ,__ 0.78 ,__ 0.74 ,__ 0.67
_____ 50 |__ 0.80 ,__ 0.78 ,__ 0.78 ,__ 0.74 ,__ 0.68 ,__ 0.60
_____ 60 |__ 0.76 ,__ 0.74 ,__ 0.74 ,__ 0.70 ,__ 0.63 ,__ 0.54
_____ 70 |__ 0.73 ,__ 0.70 ,__ 0.70 ,__ 0.65 ,__ 0.59 ,__ 0.49
_____ 80 |__ 0.70 ,__ 0.67 ,__ 0.67 ,__ 0.62 ,__ 0.55 ,__ 0.44
_____ 90 |__ 0.67 ,__ 0.64 ,__ 0.64 ,__ 0.58 ,__ 0.51 ,__ 0.40
____ 100 |__ 0.64 ,__ 0.61 ,__ 0.60 ,__ 0.55 ,__ 0.47 ,__ 0.36
____ 120 |__ 0.58 ,__ 0.55 ,__ 0.55 ,__ 0.48 ,__ 0.40 ,__ 0.30
____ 150 |__ 0.51 ,__ 0.47 ,__ 0.47 ,__ 0.40 ,__ 0.32 ,__ 0.22
____ 200 |__ 0.41 ,__ 0.37 ,__ 0.37 ,__ 0.30 ,__ 0.22 ,__ 0.13
____ 250 |__ 0.32 ,__ 0.28 ,__ 0.28 ,__ 0.22 ,__ 0.15 ,__ 0.08
___ 1000 |__ 0.01 ,__ 0.01 ,__ 0.01 ,__ 0.00 ,__ 0.00 ,__ 0.00
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
_ _ _ _ _|____________________________________________________
t, years |_ operational P0 / installed P0
_ _ _ _ _|___
_____ 15 |__ 0.99 ,__ 0.97 ,__ 0.93 ,__ 0.91 ,__ 0.89 ,__ 0.86
_____ 20 |__ 0.99 ,__ 0.96 ,__ 0.90 ,__ 0.89 ,__ 0.86 ,__ 0.82
_____ 25 |__ 0.99 ,__ 0.95 ,__ 0.88 ,__ 0.86 ,__ 0.83 ,__ 0.78
_____ 30 |__ 0.99 ,__ 0.94 ,__ 0.86 ,__ 0.83 ,__ 0.80 ,__ 0.74
_____ 40 |__ 0.98 ,__ 0.92 ,__ 0.82 ,__ 0.79 ,__ 0.74 ,__ 0.67
_____ 50 |__ 0.98 ,__ 0.90 ,__ 0.78 ,__ 0.74 ,__ 0.69 ,__ 0.61
_____ 60 |__ 0.97 ,__ 0.89 ,__ 0.74 ,__ 0.70 ,__ 0.64 ,__ 0.55
_____ 70 |__ 0.97 ,__ 0.87 ,__ 0.70 ,__ 0.66 ,__ 0.59 ,__ 0.49
_____ 80 |__ 0.96 ,__ 0.85 ,__ 0.67 ,__ 0.62 ,__ 0.55 ,__ 0.45
_____ 90 |__ 0.96 ,__ 0.84 ,__ 0.64 ,__ 0.58 ,__ 0.51 ,__ 0.40
____ 100 |__ 0.95 ,__ 0.82 ,__ 0.61 ,__ 0.55 ,__ 0.47 ,__ 0.37
____ 120 |__ 0.94 ,__ 0.79 ,__ 0.55 ,__ 0.49 ,__ 0.41 ,__ 0.30
____ 150 |__ 0.93 ,__ 0.74 ,__ 0.47 ,__ 0.41 ,__ 0.32 ,__ 0.22
____ 200 |__ 0.90 ,__ 0.67 ,__ 0.37 ,__ 0.30 ,__ 0.22 ,__ 0.13
____ 250 |__ 0.88 ,__ 0.61 ,__ 0.29 ,__ 0.22 ,__ 0.15 ,__ 0.08
___ 1000 |__ 0.61 ,__ 0.14 ,__ 0.01 ,__ 0.00 ,__ 0.00 ,__ 0.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ Cummulative Energy since installation / installed P0, [(W*yr) / W] = years
_ _ _ _ _|___
_____ 15 |_ 14.45 ,_ 14.24 ,_ 13.92 ,_ 13.72 ,_ 13.42 ,_ 12.94
_____ 20 |_ 19.03 ,_ 18.66 ,_ 18.12 ,_ 17.77 ,_ 17.26 ,_ 16.45
_____ 25 |_ 23.50 ,_ 22.93 ,_ 22.11 ,_ 21.58 ,_ 20.82 ,_ 19.63
_____ 30 |_ 27.85 ,_ 27.05 ,_ 25.90 ,_ 25.17 ,_ 24.12 ,_ 22.50
_____ 40 |_ 36.25 ,_ 34.87 ,_ 32.94 ,_ 31.73 ,_ 30.02 ,_ 27.44
_____ 50 |_ 44.23 ,_ 42.16 ,_ 39.30 ,_ 37.54 ,_ 35.09 ,_ 31.47
_____ 60 |_ 51.82 ,_ 48.96 ,_ 45.06 ,_ 42.69 ,_ 39.45 ,_ 34.77
_____ 70 |_ 59.04 ,_ 55.30 ,_ 50.26 ,_ 47.25 ,_ 43.20 ,_ 37.46
_____ 80 |_ 65.91 ,_ 61.20 ,_ 54.97 ,_ 51.30 ,_ 46.42 ,_ 39.67
_____ 90 |_ 72.44 ,_ 66.71 ,_ 59.23 ,_ 54.89 ,_ 49.19 ,_ 41.46
____ 100 |_ 78.65 ,_ 71.84 ,_ 63.08 ,_ 58.06 ,_ 51.57 ,_ 42.93
____ 120 |_ 90.18 ,_ 81.08 ,_ 69.71 ,_ 63.38 ,_ 55.38 ,_ 45.12
____ 150 | 105.44 ,_ 92.72 ,_ 77.46 ,_ 69.29 ,_ 59.31 ,_ 47.11
____ 200 | 126.29 , 107.42 ,_ 86.17 ,_ 75.43 ,_ 62.95 ,_ 48.63
____ 250 | 142.52 , 117.77 ,_ 91.43 ,_ 78.78 ,_ 64.65 ,_ 49.18
____ INF | 199.50 , 142.36 ,_ 99.50 ,_ 82.83 ,_ 66.17 ,_ 49.50
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ Average ( P/ installed P0 ) from installation to time t since installation
_ _ _ _ _|___
_____ 15 |__ 0.96 ,__ 0.95 ,__ 0.93 ,__ 0.91 ,__ 0.89 ,__ 0.86
_____ 20 |__ 0.95 ,__ 0.93 ,__ 0.91 ,__ 0.89 ,__ 0.86 ,__ 0.82
_____ 25 |__ 0.94 ,__ 0.92 ,__ 0.88 ,__ 0.86 ,__ 0.83 ,__ 0.79
_____ 30 |__ 0.93 ,__ 0.90 ,__ 0.86 ,__ 0.84 ,__ 0.80 ,__ 0.75
_____ 40 |__ 0.91 ,__ 0.87 ,__ 0.82 ,__ 0.79 ,__ 0.75 ,__ 0.69
_____ 50 |__ 0.88 ,__ 0.84 ,__ 0.79 ,__ 0.75 ,__ 0.70 ,__ 0.63
_____ 60 |__ 0.86 ,__ 0.82 ,__ 0.75 ,__ 0.71 ,__ 0.66 ,__ 0.58
_____ 70 |__ 0.84 ,__ 0.79 ,__ 0.72 ,__ 0.68 ,__ 0.62 ,__ 0.54
_____ 80 |__ 0.82 ,__ 0.77 ,__ 0.69 ,__ 0.64 ,__ 0.58 ,__ 0.50
_____ 90 |__ 0.80 ,__ 0.74 ,__ 0.66 ,__ 0.61 ,__ 0.55 ,__ 0.46
____ 100 |__ 0.79 ,__ 0.72 ,__ 0.63 ,__ 0.58 ,__ 0.52 ,__ 0.43
____ 120 |__ 0.75 ,__ 0.68 ,__ 0.58 ,__ 0.53 ,__ 0.46 ,__ 0.38
____ 150 |__ 0.70 ,__ 0.62 ,__ 0.52 ,__ 0.46 ,__ 0.40 ,__ 0.31
____ 200 |__ 0.63 ,__ 0.54 ,__ 0.43 ,__ 0.38 ,__ 0.31 ,__ 0.24
____ 250 |__ 0.57 ,__ 0.47 ,__ 0.37 ,__ 0.32 ,__ 0.26 ,__ 0.20
___ 1000 |__ 0.00 ,__ 0.00 ,__ 0.00 ,__ 0.00 ,__ 0.00 ,__ 0.00
___________________________________________
CONSTANT N
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ P/N , [W / (W/yr)] = years
_ _ _ _ _|___
_____ 15 |_ 14.45 ,_ 14.24 ,_ 13.92 ,_ 13.72 ,_ 13.42 ,_ 12.94
_____ 20 |_ 19.03 ,_ 18.66 ,_ 18.12 ,_ 17.77 ,_ 17.26 ,_ 16.45
_____ 25 |_ 23.50 ,_ 22.93 ,_ 22.11 ,_ 21.58 ,_ 20.82 ,_ 19.63
_____ 30 |_ 27.85 ,_ 27.05 ,_ 25.90 ,_ 25.17 ,_ 24.12 ,_ 22.50
_____ 40 |_ 36.25 ,_ 34.87 ,_ 32.94 ,_ 31.73 ,_ 30.02 ,_ 27.44
_____ 50 |_ 44.23 ,_ 42.16 ,_ 39.30 ,_ 37.54 ,_ 35.09 ,_ 31.47
_____ 60 |_ 51.82 ,_ 48.96 ,_ 45.06 ,_ 42.69 ,_ 39.45 ,_ 34.77
_____ 70 |_ 59.04 ,_ 55.30 ,_ 50.26 ,_ 47.25 ,_ 43.20 ,_ 37.46
_____ 80 |_ 65.91 ,_ 61.20 ,_ 54.97 ,_ 51.30 ,_ 46.42 ,_ 39.67
_____ 90 |_ 72.44 ,_ 66.71 ,_ 59.23 ,_ 54.89 ,_ 49.19 ,_ 41.46
____ 100 |_ 78.65 ,_ 71.84 ,_ 63.08 ,_ 58.06 ,_ 51.57 ,_ 42.73
____ 120 |_ 90.18 ,_ 81.08 ,_ 69.71 ,_ 63.38 ,_ 55.38 ,_ 42.73
____ 150 | 105.44 ,_ 92.72 ,_ 77.46 ,_ 69.29 ,_ 57.14 ,_ 42.73
____ 200 | 126.29 , 107.20 ,_ 85.97 ,_ 71.55 ,_ 57.14 ,_ 42.73
____ 250 | 133.81 , 107.20 ,_ 85.97 ,_ 71.55 ,_ 57.14 ,_ 42.73
___ 1000 | 133.81 , 107.20 ,_ 85.97 ,_ 71.55 ,_ 57.14 ,_ 42.73
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
_ _ _ _ _|____________________________________________________
t, years |_ operational P0 / N , [W / (W/yr)] = years
_ _ _ _ _|___
_____ 15 |_ 14.94 ,_ 14.78 ,_ 14.45 ,_ 14.34 ,_ 14.18 ,_ 13.92
_____ 20 |_ 19.90 ,_ 19.60 ,_ 19.03 ,_ 18.84 ,_ 18.57 ,_ 18.12
_____ 25 |_ 24.84 ,_ 24.38 ,_ 23.50 ,_ 23.21 ,_ 22.79 ,_ 22.11
_____ 30 |_ 29.78 ,_ 29.12 ,_ 27.85 ,_ 27.45 ,_ 26.85 ,_ 25.90
_____ 40 |_ 39.60 ,_ 38.44 ,_ 36.25 ,_ 35.55 ,_ 34.54 ,_ 32.94
_____ 50 |_ 49.38 ,_ 47.58 ,_ 44.23 ,_ 43.18 ,_ 41.67 ,_ 39.30
_____ 60 |_ 59.11 ,_ 56.54 ,_ 51.82 ,_ 50.36 ,_ 48.28 ,_ 45.06
_____ 70 |_ 68.79 ,_ 65.32 ,_ 59.04 ,_ 57.13 ,_ 54.41 ,_ 50.26
_____ 80 |_ 78.42 ,_ 73.92 ,_ 65.91 ,_ 63.49 ,_ 60.10 ,_ 54.97
_____ 90 |_ 88.00 ,_ 82.36 ,_ 72.44 ,_ 69.49 ,_ 65.37 ,_ 59.23
____ 100 |_ 97.54 ,_ 90.63 ,_ 78.65 ,_ 75.14 ,_ 70.26 ,_ 62.53
____ 120 | 116.47 , 106.67 ,_ 90.18 ,_ 85.46 ,_ 79.01 ,_ 62.53
____ 150 | 144.51 , 129.57 , 105.44 ,_ 98.79 ,_ 83.60 ,_ 62.53
____ 200 | 190.32 , 164.21 , 125.74 , 104.67 ,_ 83.60 ,_ 62.53
____ 250 | 209.77 , 164.21 , 125.74 , 104.67 ,_ 83.60 ,_ 62.53
___ 1000 | 209.77 , 164.21 , 125.74 , 104.67 ,_ 83.60 ,_ 62.53
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ P / operational P0
_ _ _ _ _|___
_____ 15 |__ 0.97 ,__ 0.96 ,__ 0.96 ,__ 0.96 ,__ 0.95 ,__ 0.93
_____ 20 |__ 0.96 ,__ 0.95 ,__ 0.95 ,__ 0.94 ,__ 0.93 ,__ 0.91
_____ 25 |__ 0.95 ,__ 0.94 ,__ 0.94 ,__ 0.93 ,__ 0.91 ,__ 0.89
_____ 30 |__ 0.94 ,__ 0.93 ,__ 0.93 ,__ 0.92 ,__ 0.90 ,__ 0.87
_____ 40 |__ 0.92 ,__ 0.91 ,__ 0.91 ,__ 0.89 ,__ 0.87 ,__ 0.83
_____ 50 |__ 0.90 ,__ 0.89 ,__ 0.89 ,__ 0.87 ,__ 0.84 ,__ 0.80
_____ 60 |__ 0.88 ,__ 0.87 ,__ 0.87 ,__ 0.85 ,__ 0.82 ,__ 0.77
_____ 70 |__ 0.86 ,__ 0.85 ,__ 0.85 ,__ 0.83 ,__ 0.79 ,__ 0.75
_____ 80 |__ 0.84 ,__ 0.83 ,__ 0.83 ,__ 0.81 ,__ 0.77 ,__ 0.72
_____ 90 |__ 0.82 ,__ 0.81 ,__ 0.82 ,__ 0.79 ,__ 0.75 ,__ 0.70
____ 100 |__ 0.81 ,__ 0.79 ,__ 0.80 ,__ 0.77 ,__ 0.73 ,__ 0.68
____ 120 |__ 0.77 ,__ 0.76 ,__ 0.77 ,__ 0.74 ,__ 0.70 ,__ 0.68
____ 150 |__ 0.73 ,__ 0.72 ,__ 0.73 ,__ 0.70 ,__ 0.68 ,__ 0.68
____ 200 |__ 0.66 ,__ 0.65 ,__ 0.68 ,__ 0.68 ,__ 0.68 ,__ 0.68
____ 250 |__ 0.64 ,__ 0.65 ,__ 0.68 ,__ 0.68 ,__ 0.68 ,__ 0.68
___ 1000 |__ 0.64 ,__ 0.65 ,__ 0.68 ,__ 0.68 ,__ 0.68 ,__ 0.68
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ operational P0 / P
_ _ _ _ _|___
_____ 15 |__ 1.03 ,__ 1.04 ,__ 1.04 ,__ 1.05 ,__ 1.06 ,__ 1.08
_____ 20 |__ 1.05 ,__ 1.05 ,__ 1.05 ,__ 1.06 ,__ 1.08 ,__ 1.10
_____ 25 |__ 1.06 ,__ 1.06 ,__ 1.06 ,__ 1.08 ,__ 1.09 ,__ 1.13
_____ 30 |__ 1.07 ,__ 1.08 ,__ 1.08 ,__ 1.09 ,__ 1.11 ,__ 1.15
_____ 40 |__ 1.09 ,__ 1.10 ,__ 1.10 ,__ 1.12 ,__ 1.15 ,__ 1.20
_____ 50 |__ 1.12 ,__ 1.13 ,__ 1.13 ,__ 1.15 ,__ 1.19 ,__ 1.25
_____ 60 |__ 1.14 ,__ 1.15 ,__ 1.15 ,__ 1.18 ,__ 1.22 ,__ 1.30
_____ 70 |__ 1.17 ,__ 1.18 ,__ 1.17 ,__ 1.21 ,__ 1.26 ,__ 1.34
_____ 80 |__ 1.19 ,__ 1.21 ,__ 1.20 ,__ 1.24 ,__ 1.29 ,__ 1.39
_____ 90 |__ 1.21 ,__ 1.23 ,__ 1.22 ,__ 1.27 ,__ 1.33 ,__ 1.43
____ 100 |__ 1.24 ,__ 1.26 ,__ 1.25 ,__ 1.29 ,__ 1.36 ,__ 1.46
____ 120 |__ 1.29 ,__ 1.32 ,__ 1.29 ,__ 1.35 ,__ 1.43 ,__ 1.46
____ 150 |__ 1.37 ,__ 1.40 ,__ 1.36 ,__ 1.43 ,__ 1.46 ,__ 1.46
____ 200 |__ 1.51 ,__ 1.53 ,__ 1.46 ,__ 1.46 ,__ 1.46 ,__ 1.46
____ 250 |__ 1.57 ,__ 1.53 ,__ 1.46 ,__ 1.46 ,__ 1.46 ,__ 1.46
___ 1000 |__ 1.57 ,__ 1.53 ,__ 1.46 ,__ 1.46 ,__ 1.46 ,__ 1.46
___________________________________________
EXPONENTIAL GROWTH with Retirement at U
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | P/N , [W / (W/yr)] = years _____ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ ______ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __| ____ |_ by factor of 10
____ 0.0 | 133.81 , 107.20 ,_ 85.97 ,_ 71.55 ,_ 57.14 ,_ 42.73 |_ 0.0 |
____ 0.1 | 122.44 ,_ 99.40 ,_ 80.40 ,_ 67.67 ,_ 54.64 ,_ 41.32 |_ 0.1 | 2303.7
____ 0.2 | 112.48 ,_ 92.45 ,_ 75.41 ,_ 64.11 ,_ 52.32 ,_ 39.99 |_ 0.2 | 1152.4
____ 0.5 |_ 89.10 ,_ 75.63 ,_ 63.14 ,_ 55.11 ,_ 46.23 ,_ 36.38 |_ 0.5 |_ 461.7
____ 1.0 |_ 64.41 ,_ 56.90 ,_ 49.05 ,_ 44.21 ,_ 38.43 ,_ 31.46 |_ 1.0 |_ 231.4
____ 1.5 |_ 49.64 ,_ 45.05 ,_ 39.81 ,_ 36.66 ,_ 32.69 ,_ 27.60 |_ 1.5 |_ 154.7
____ 2.0 |_ 40.13 ,_ 37.10 ,_ 33.41 ,_ 31.21 ,_ 28.35 ,_ 24.51 |_ 2.0 |_ 116.3
____ 3.0 |_ 28.91 ,_ 27.32 ,_ 25.24 ,_ 24.00 ,_ 22.32 ,_ 19.95 |_ 3.0 |__ 77.9
____ 4.0 |_ 22.61 ,_ 21.62 ,_ 20.29 ,_ 19.49 ,_ 18.39 ,_ 16.78 |_ 4.0 |__ 58.7
____ 5.0 |_ 18.59 ,_ 17.92 ,_ 16.99 ,_ 16.43 ,_ 15.65 ,_ 14.48 |_ 5.0 |__ 47.2
____ 6.0 |_ 15.80 ,_ 15.32 ,_ 14.64 ,_ 14.22 ,_ 13.63 ,_ 12.74 |_ 6.0 |__ 39.5
____ 7.0 |_ 13.76 ,_ 13.39 ,_ 12.87 ,_ 12.54 ,_ 12.08 ,_ 11.38 |_ 7.0 |__ 34.0
____ 8.0 |_ 12.20 ,_ 11.91 ,_ 11.49 ,_ 11.23 ,_ 10.86 ,_ 10.29 |_ 8.0 |__ 29.9
____ 9.0 |_ 10.97 ,_ 10.73 ,_ 10.39 ,_ 10.18 ,__ 9.87 ,__ 9.40 |_ 9.0 |__ 26.7
___ 10.0 |__ 9.97 ,__ 9.77 ,__ 9.49 ,__ 9.31 ,__ 9.06 ,__ 8.66 | 10.0 |__ 24.2
___ 15.0 |__ 6.91 ,__ 6.81 ,__ 6.68 ,__ 6.59 ,__ 6.46 ,__ 6.25 | 15.0 |__ 16.5
___ 20.0 |__ 5.34 ,__ 5.28 ,__ 5.20 ,__ 5.14 ,__ 5.06 ,__ 4.94 | 20.0 |__ 12.6
___ 30.0 |__ 3.74 ,__ 3.71 ,__ 3.67 ,__ 3.64 ,__ 3.60 ,__ 3.54 | 30.0 |___ 8.8
U, years : 221.61 , 199.10 , 198.50 , 165.17 , 131.83 ,_ 98.50
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | operational P0 / P _____ _ _ _ _ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ ______ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __| ____ |_ by factor of 10
____ 0.0 |__ 1.57 ,__ 1.53 ,__ 1.46 ,__ 1.46 ,__ 1.46 ,__ 1.46 |_ 0.0 |
____ 0.1 |__ 1.54 ,__ 1.51 ,__ 1.44 ,__ 1.45 ,__ 1.45 ,__ 1.45 |_ 0.1 | 2303.7
____ 0.2 |__ 1.51 ,__ 1.48 ,__ 1.42 ,__ 1.43 ,__ 1.43 ,__ 1.44 |_ 0.2 | 1152.4
____ 0.5 |__ 1.44 ,__ 1.42 ,__ 1.37 ,__ 1.38 ,__ 1.40 ,__ 1.41 |_ 0.5 |_ 461.7
____ 1.0 |__ 1.34 ,__ 1.33 ,__ 1.29 ,__ 1.32 ,__ 1.34 ,__ 1.37 |_ 1.0 |_ 231.4
____ 1.5 |__ 1.27 ,__ 1.27 ,__ 1.24 ,__ 1.26 ,__ 1.29 ,__ 1.33 |_ 1.5 |_ 154.7
____ 2.0 |__ 1.21 ,__ 1.22 ,__ 1.20 ,__ 1.22 ,__ 1.26 ,__ 1.29 |_ 2.0 |_ 116.3
____ 3.0 |__ 1.15 ,__ 1.16 ,__ 1.14 ,__ 1.17 ,__ 1.20 ,__ 1.24 |_ 3.0 |__ 77.9
____ 4.0 |__ 1.11 ,__ 1.12 ,__ 1.11 ,__ 1.13 ,__ 1.16 ,__ 1.20 |_ 4.0 |__ 58.7
____ 5.0 |__ 1.09 ,__ 1.10 ,__ 1.09 ,__ 1.11 ,__ 1.13 ,__ 1.17 |_ 5.0 |__ 47.2
____ 6.0 |__ 1.08 ,__ 1.08 ,__ 1.08 ,__ 1.09 ,__ 1.12 ,__ 1.15 |_ 6.0 |__ 39.5
____ 7.0 |__ 1.07 ,__ 1.07 ,__ 1.07 ,__ 1.08 ,__ 1.10 ,__ 1.13 |_ 7.0 |__ 34.0
____ 8.0 |__ 1.06 ,__ 1.06 ,__ 1.06 ,__ 1.07 ,__ 1.09 ,__ 1.12 |_ 8.0 |__ 29.9
____ 9.0 |__ 1.05 ,__ 1.06 ,__ 1.06 ,__ 1.07 ,__ 1.08 ,__ 1.11 |_ 9.0 |__ 26.7
___ 10.0 |__ 1.05 ,__ 1.05 ,__ 1.05 ,__ 1.06 ,__ 1.07 ,__ 1.10 | 10.0 |__ 24.2
___ 15.0 |__ 1.03 ,__ 1.04 ,__ 1.03 ,__ 1.04 ,__ 1.05 ,__ 1.07 | 15.0 |__ 16.5
___ 20.0 |__ 1.02 ,__ 1.03 ,__ 1.03 ,__ 1.03 ,__ 1.04 ,__ 1.05 | 20.0 |__ 12.6
___ 30.0 |__ 1.02 ,__ 1.02 ,__ 1.02 ,__ 1.02 ,__ 1.03 ,__ 1.04 | 30.0 |___ 8.8
___________________________________________
EXPONENTIAL GROWTH, no retirement at a set age
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | P/N , [W / (W/yr)] = years _____ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ ______ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __| ____ |_ by factor of 10
____ 0.0 | 199.50 , 142.36 ,_ 99.50 ,_ 82.83 ,_ 66.17 ,_ 49.50 |_ 0.0 |
____ 0.1 | 166.33 , 124.62 ,_ 90.50 ,_ 76.50 ,_ 62.06 ,_ 47.16 |_ 0.1 | 2303.7
____ 0.2 | 142.64 , 110.83 ,_ 83.00 ,_ 71.07 ,_ 58.44 ,_ 45.04 |_ 0.2 | 1152.4
____ 0.5 | 100.00 ,_ 83.25 ,_ 66.50 ,_ 58.62 ,_ 49.75 ,_ 39.70 |_ 0.5 |_ 461.7
____ 1.0 |_ 66.83 ,_ 58.91 ,_ 50.00 ,_ 45.41 ,_ 39.90 ,_ 33.16 |_ 1.0 |_ 231.4
____ 1.5 |_ 50.25 ,_ 45.63 ,_ 40.10 ,_ 37.09 ,_ 33.33 ,_ 28.50 |_ 1.5 |_ 154.7
____ 2.0 |_ 40.30 ,_ 37.28 ,_ 33.50 ,_ 31.37 ,_ 28.64 ,_ 25.00 |_ 2.0 |_ 116.3
____ 3.0 |_ 28.93 ,_ 27.33 ,_ 25.25 ,_ 24.02 ,_ 22.39 ,_ 20.10 |_ 3.0 |__ 77.9
____ 4.0 |_ 22.61 ,_ 21.62 ,_ 20.30 ,_ 19.50 ,_ 18.40 ,_ 16.83 |_ 4.0 |__ 58.7
____ 5.0 |_ 18.59 ,_ 17.92 ,_ 17.00 ,_ 16.43 ,_ 15.65 ,_ 14.49 |_ 5.0 |__ 47.2
____ 6.0 |_ 15.80 ,_ 15.32 ,_ 14.64 ,_ 14.22 ,_ 13.63 ,_ 12.74 |_ 6.0 |__ 39.5
____ 7.0 |_ 13.76 ,_ 13.39 ,_ 12.87 ,_ 12.54 ,_ 12.08 ,_ 11.38 |_ 7.0 |__ 34.0
____ 8.0 |_ 12.20 ,_ 11.91 ,_ 11.49 ,_ 11.23 ,_ 10.86 ,_ 10.29 |_ 8.0 |__ 29.9
____ 9.0 |_ 10.97 ,_ 10.73 ,_ 10.39 ,_ 10.18 ,__ 9.87 ,__ 9.40 |_ 9.0 |__ 26.7
___ 10.0 |__ 9.97 ,__ 9.77 ,__ 9.49 ,__ 9.31 ,__ 9.06 ,__ 8.66 | 10.0 |__ 24.2
___ 15.0 |__ 6.91 ,__ 6.81 ,__ 6.68 ,__ 6.59 ,__ 6.46 ,__ 6.25 | 15.0 |__ 16.5
___ 20.0 |__ 5.34 ,__ 5.28 ,__ 5.20 ,__ 5.14 ,__ 5.06 ,__ 4.94 | 20.0 |__ 12.6
___ 30.0 |__ 3.74 ,__ 3.71 ,__ 3.67 ,__ 3.64 ,__ 3.60 ,__ 3.54 | 30.0 |___ 8.8
_ -hs (%):__ 0.05 ,__ 0.20 ,__ 0.50 ,__ 0.60 ,__ 0.75 ,__ 1.00
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
__ s (%) | operational P0 / P _____ _ _ _ _ _ _ _ _ _ _ _ _ _ _| s (%)| time (yrs) to grow
_ _ _ _ _|___ ______ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __| ____ |_ by factor of 10
____ 0.0 |_ 10.02 ,__ 3.51 ,__ 2.01 ,__ 2.01 ,__ 2.01 ,__ 2.01 |_ 0.0 |
____ 0.1 |__ 4.01 ,__ 2.67 ,__ 1.84 ,__ 1.86 ,__ 1.89 ,__ 1.92 |_ 0.1 | 2303.7
____ 0.2 |__ 2.81 ,__ 2.26 ,__ 1.72 ,__ 1.76 ,__ 1.80 ,__ 1.84 |_ 0.2 | 1152.4
____ 0.5 |__ 1.82 ,__ 1.72 ,__ 1.50 ,__ 1.55 ,__ 1.61 ,__ 1.68 |_ 0.5 |_ 461.7
____ 1.0 |__ 1.43 ,__ 1.42 ,__ 1.34 ,__ 1.38 ,__ 1.43 ,__ 1.51 |_ 1.0 |_ 231.4
____ 1.5 |__ 1.29 ,__ 1.30 ,__ 1.25 ,__ 1.29 ,__ 1.34 ,__ 1.41 |_ 1.5 |_ 154.7
____ 2.0 |__ 1.22 ,__ 1.23 ,__ 1.20 ,__ 1.23 ,__ 1.28 ,__ 1.34 |_ 2.0 |_ 116.3
____ 3.0 |__ 1.15 ,__ 1.16 ,__ 1.15 ,__ 1.17 ,__ 1.20 ,__ 1.26 |_ 3.0 |__ 77.9
____ 4.0 |__ 1.11 ,__ 1.12 ,__ 1.11 ,__ 1.13 ,__ 1.16 ,__ 1.21 |_ 4.0 |__ 58.7
____ 5.0 |__ 1.09 ,__ 1.10 ,__ 1.09 ,__ 1.11 ,__ 1.13 ,__ 1.17 |_ 5.0 |__ 47.2
____ 6.0 |__ 1.08 ,__ 1.08 ,__ 1.08 ,__ 1.09 ,__ 1.12 ,__ 1.15 |_ 6.0 |__ 39.5
____ 7.0 |__ 1.07 ,__ 1.07 ,__ 1.07 ,__ 1.08 ,__ 1.10 ,__ 1.13 |_ 7.0 |__ 34.0
____ 8.0 |__ 1.06 ,__ 1.06 ,__ 1.06 ,__ 1.07 ,__ 1.09 ,__ 1.12 |_ 8.0 |__ 29.9
____ 9.0 |__ 1.05 ,__ 1.06 ,__ 1.06 ,__ 1.07 ,__ 1.08 ,__ 1.11 |_ 9.0 |__ 26.7
___ 10.0 |__ 1.05 ,__ 1.05 ,__ 1.05 ,__ 1.06 ,__ 1.07 ,__ 1.10 | 10.0 |__ 24.2
___ 15.0 |__ 1.03 ,__ 1.04 ,__ 1.03 ,__ 1.04 ,__ 1.05 ,__ 1.07 | 15.0 |__ 16.5
___ 20.0 |__ 1.02 ,__ 1.03 ,__ 1.03 ,__ 1.03 ,__ 1.04 ,__ 1.05 | 20.0 |__ 12.6
___ 30.0 |__ 1.02 ,__ 1.02 ,__ 1.02 ,__ 1.02 ,__ 1.03 ,__ 1.04 | 30.0 |___ 8.8
“Some combinations that result in $10 / average W
(assuming high fill factors or concentration of sunlight into time intervals – more generally, that the average efficiency of conversion is not much lower than the rated efficiency of conversion ):”…
To clarify: Those combinations assume the average efficiency is close to the rated efficiency at 1000 W/m2 insolation (a standard full sun, presumably mostly direct solar radiation) AT THE TIME of purchase or installation – when the collector is new.
The last row of one of the tables was erroneously zero: Correction:
ONE TIME INSTALLMENT
__ -h (%):__ 0.50 ,__ 0.70 ,__ 1.00 ,__ 1.20 ,__ 1.50 ,__ 2.00
_ _ _ _ _|____________________________________________________
t, years |_ Average ( P/ installed P0 ) from installation to time t since installation
_ _ _ _ _|___
___ 1000 |__ 0.20 ,__ 0.14 ,__ 0.10 ,__ 0.08 ,__ 0.07 ,__ 0.05
__________________________________________________
I also included hs in some tables when the table values did not directly depend on it, and only depended on it indirectly through U (U was determined from he, which was determined from the combination of h and hs values).
This applies to these tables:
m0 for:
1. CONSTANT N (with and without debt) (note that U only affects the calculation for t larger than U)
2. EXPONENTIAL GROWTH with Retirement at U
and
P/N for:
1. CONSTANT N (note that U only affects the calculation for t larger than U)
2. EXPONENTIAL GROWTH with Retirement at U
In general, m0 only depends on hs through U, and when it depends on U. The same is true of P/N. P0/N depends on hs, so P0/P depends on h and hs.
re #1242 John Reisman
You suggest that James, who is concerned for species other than his own and who believes that we have a severe problem of human overpopulation, should contribute to its solution by topping himself.
I fear that such a single selfless act would do little good unless it served as encouragement for a further 2 to 4 billion others to volunteer in like manner. The rest of us might then thrive. However, we’ll probably get the same or a worse outcome with BAU without the painful necessity of having to make the choice of becoming volunteers, not that this will eliminate pain in any way.
Before setting the high moral tone for the rest of us, I would invite you to contemplate the following thought experiments and then apply your exemplary moral standards to them.
1) You are on a lifeboat with 9 others. You have no means of feeding yourself. You know that it is 99.9% probable that you won’t reach landfall unless you eat one or more of your fellow passengers. What do you do?
a) All agree to starve.
b) Volunteer yourself as the first meal.
c) Gang up with a few others to kill the remainder as soon as possible before they come up with the same idea.
d) Democratically (or otherwise) work out a pecking order to establish the rank in which you become the consumed rather than the consumer, eating, may it be said, in a fruglal but sustainable manner on humane grounds and thus minimising the death toll.
2) Same lifeboat and personnel. This time you know that the only landfall is sterile and won’t sustain you. However, you have with you some seed corn and a few chickens which could potentially save you when you reach land. From your comments about James, I gather that you think he’d eat people and leave the non human food resource to ensure the survival of a few. You seem to deem this immoral and so I gather you’d head straight for the wheat and chickens so dooming everyone by failing to take difficult choices because of moral squeamishness. On reflection, I probably misjudge you. You’d take the easy option and eat the chickens only and argue for a vegetarian future, despite problems of future pernicious anaemia.
I hope things don’t come to this but it seems likely that they will unless we get our act together very quickly indeed.
Douglas, 308, no, I’m concerned that James is making up anything he can to keep the conclusion that he wants to see maintained: Nuclear Power is teh bomb! Uh, sorry, The Best Thing Since Sliced Bread.
Oracle seems to agree: troller woosnam