[net-gold] Answer to Fermi Problem Regarding Concentration of Energy by PhotoVoltaic Solar Panels
- From: "David P. Dillard" <jwne@xxxxxxxxxx>
- To: Other Net-Gold Lists -- Educator Gold <Educator-Gold@xxxxxxxxxxxxxxx>, Educator Gold <Educator-Gold@xxxxxxxxxxxxxxxx>, net-gold@xxxxxxxxxxxxx, NetGold <netgold@xxxxxxxxxxxxxxx>, Net-Gold <net-gold@xxxxxxxxxxxxxxxx>, K-12ADMINLIFE <K12ADMIN@xxxxxxxxxxxxxxxxxxx>, K12AdminLIFE <K12AdminLIFE@xxxxxxxxxxxxxxx>, NetGold <netgold@xxxxxxxxxxxxxxxx>, Net-Platinum <net-platinum@xxxxxxxxxxxxxxx>, Net-Gold <NetGold_general@xxxxxxxxxxxxxxxxx>, Temple Gold Discussion Group <TEMPLE-GOLD@xxxxxxxxxxxxxxxxxxx>, Temple University Net-Gold Archive <net-gold@xxxxxxxxxxxxxxxxxxx>, Health Lists -- Health Diet Fitness Recreation Sports Tourism <healthrecsport@xxxxxxxxxxxxxxxx>, Health Diet Fitness Recreation Sports <healthrecsport@xxxxxxxxxxxxxxx>, HEALTH-RECREATION-SPORTS-TOURISM@xxxxxxxxxxxxxxxxxxx
- Date: Fri, 10 Feb 2012 06:41:52 -0500 (EST)
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Date: Thu, 9 Feb 2012 15:07:46 -0800
From: Richard Hake <rrhake@xxxxxxxxxxxxx>
Reply-To: Net-Gold@xxxxxxxxxxxxxxx
To: AERA-L@xxxxxxxxxxxxxxxxx
Cc: Net-Gold@xxxxxxxxxxxxxxx
Subject: [Net-Gold] Answer to Fermi Problem Regarding Concentration of Energy by
PhotoVoltaic Solar Panels
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ABSTRACT: In a previous post "Concentration of
Energy by PhotoVoltaic Solar Panels" [Hake
(2012a) I posed the following "Fermi Problem":
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"What fraction 'f' of the yearly total U.S.
fossil-fueled electrical energy (and thus
fraction of yearly tons of CO2 emission from U.S.
fossil-fueled electrical energy production) could
be eliminated if all rooftops in the U.S. were
covered by photovoltaic solar panels? Guy
Brandenburg of the MathEdCC list estimated f =
0.08 and I estimated f = 1.6 - the details are
given in this post. These solutions are not
within an order of magnitude of each other (as
expected for valid solutions to Fermi Problems),
so Brandenburg's solution must be wrong ;-).
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In Hake (2012a) I warned (paraphrasing) "Before
recommending that all U.S. rooftops be covered
with PhotoVoltaic Solar Panels one should
consider the EMERGY associated with such a plan.
Odum wrote: "EMERGY measures both the work of
nature and that of humans in generating products
and services. . . . The photovoltaic power grid
at Austin, Tex., evaluated by King & Schmandt
(1991), is taking more EMERGY out of society than
it is generating."
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So is the motto of Solar Forward
<http://solarforward.com/>: "SAVING THE PLANET,
ONE SOLAR SYSTEM AT A TIME" a deception? I THINK
NOT because the evaluation of King & Schmandt
(1991), failed to consider the devastating
decrease in EMERGY if alternatives to fossil fuel
are not actuated - see e.g. "Paleoclimate
Implications for Human-Made Climate Change"
[Hansen & Sato (2011)].
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*************************************************
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In a previous post "Concentration of Energy by
PhotoVoltaic Solar Panels" [Hake (2012)] I posed
the following "Fermi Problem":
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******************************************
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What fraction "f" of the yearly total U.S.
fossil-fueled electrical energy (and thus
fraction of yearly tons of CO^2 emission from
U.S. fossil-fueled electrical energy production)
could be eliminated if all rooftops in the U.S.
were covered by photovoltaic solar panels?
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******************************************
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I gave four hints:
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1. According to CARMA (2007) the total electrical energy
production for 2007 in the U.S was 4.19 x 10^9 MWh/y
[MWh = MegaWattHours] of which 0.67 x 4.19 x 10^9 MWh/y
= 2.81 x 10^9 MWh/y = . . . . . . . . . . . .
.2.8 x 10^12 kWh/y was fossil
fueled...........(H1)
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(2) According to the population clock at <http://1.usa.gov/zaQynk>,
the population of the U.S. on 6 Feb 2012 = . . .
. . . . . . . . . . . . 313 x 10^6
...............(H2)
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(3) According to <http://en.wikipedia.org/wiki/Photovoltaic_system>:
"A typical '150 watt' solar panel is about a square
meter in size. Such a panel may be expected to produce
1 kWh/day every day, on average, after taking into account
the weather and the latitude"; thus . . . . . . .
. . . . . . . . . . . 365 kWh/(yrm^2)
.............(H3)
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(4) For Fermi Problem strategies see, e.g. :
a. <http://en.wikipedia.org/wiki/Fermi_problem>,
b. "Consider a Spherical Cow" [Harte (1988)],
c. "Consider a Cylindrical Cow" [Harte ( 2001)],
d. "Street-Fighting Mathematics [Mahajan (2010)].
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As far as I know (please correct me if I'm wrong) the
only discussion-list subscriber to post a solution to the above
Fermi problem was Guy Bandenburg (2012) of the MathEdCC list,
who estimated the total U.S. solar electrical energy generation
if all U.S. rooftops were covered by solar panels as
2.2 x 10^11 kwhr/y, thus f = 0.22 x 10^12 /
2.8x10^12, or . . f = 0.08 <---------
.........(B)
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Here is the correct ;-) solution:
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A. Divide the total U.S.population of 313 x 10^6
from "H1" into two categories,
each with two subcategories:
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1. 75% = 0.75 x 313 x 10^6 = 235 x 10^6 living
in families averaging 4 persons each for a total
of 235/4 x 10^6 = 59 x 10^6 families
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a. 0.75 x 59 x 10^6 = 44 x 10^6 of whom live in
single-family houses of 1 or 2 stories, each with
roof area Am^2, hence a total roof area . . . . .
. . . . . . . . .44A x 10^6 m^2.............(1),
where A = *average* roof area of U.S. single-family
houses in m^2.
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b. 0.25 x 59 x 10^6 = 15 x 10^6 of whom live in
multi-family buildings of more than 2 stories, each with
roof areas 3A, hence a total roof area . . . . .
. . . . . . . . . . . . .45A x 10^6
m^2.........(2)
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2. 25% = 0.25 x 313 x 10^6 = 18 x 10^6 living singly
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a. 0.25 x 18 x 10^6 = 4.5 x 10^6 of whom live in
houses of 1 or 2 stories, each with roof area A,
hence a total roof area . . . . . . . . . . . .
. . . . . . . . . . . . . . . .4.5A x 10^6
m^2............ (3)
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b. 0.75 x 18 x 10^6 = 14 x 10^6 of whom live in
in buildings of more than 2 stories, each with roof area 3A,
hence a total roof area . . . . . . . . . . . . .
. . . . . . . .. . . . . . . . .42A x 10^6
m^2.............(4)
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B. Adding Eq. (1, 2, 3, 4), yields a total U.S. roof area
A(total) = 44A + 45A + 4.5A + 42A = . . . . . .
. . . . . . . . . . .136A x 10^6 m^2.........(5)
where A = average of roof area of U.S. single-family
houses in m^2.
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C. My own middle-class-residence roof has an area A = 170 m^2.
Assuming half that value for the *average* U.S. single-family
house A yields, from Eq. (5)
A(total) = 136 x 10^6 x 85 m^2 = . . . . . . . .
. . . . . . . . . . . .12 x 10^9
m^2................(6)
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Note that according to Wikipedia <http://bit.ly/AA8dXS>,
A (total 50 states + DC land area) = 9.8 x 10^6 km^2 = 9.8 x 10^12 m^2,
so the U.S. has plenty more land for rooftops ;-).
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D. Eq. (6) and H3 yield for the total U.S. solar
electrical energy generation if all rooftops
were covered by solar panels =
(12 x 10^9 m^2) x 365 kWh/(yr m^2)
= 4.4 x 10^3 x 10^9.. . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 4.4 x 10^12
kWh/y.........(7)
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D. Thus the fraction "f" of the yearly total U.S. fossil-fueled
electrical energy (and thus fraction of yearly tons of
CO^2 emission from U.S. fossil-fueled electrical energy
production) that could be eliminated if ALL rooftops in the U.S.
were covered by photovoltaic solar panels is, from Eq. (7) and H3
f = 4.4 x10^12 (kWh/y)/ 2.8 x 10^12 (kWh/y) . . .
. . . . . . . . . . . . . . . . . f = 1.6
.......(8) <-------
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Note that Brandenburg's solution f = 0.08 of Eq.
(B) and my solution f = 1.6 of Eq. (8) are not
within an order of magnitude of one another as
expected for valid Fermi Problem solutions, so
Brandenburg's solution must be wrong ;-).
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Note that in "Concentration of Energy by
PhotoVoltaic Solar Panels" [Hake (2012)] I warned:
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"Before recommending, on the basis of one's
answer to the above Fermi problem, that all U.S.
rooftops be covered with PhotoVoltaic Solar
Panels one should consider the EMERGY associated
with such a plan - see "Environmental Accounting:
Emergy and Environmental Decision Making" [Odum
(1995)]. On page 1, Odum writes: "EMERGY, spelled
with an 'm,' measures both the work of nature and
that of humans in generating products and
services. By selecting choices that maximize
EMERGY production and use, policies and
judgements can favor those environmental that
maximize real wealth, the whole economy, and the
public benefit." See esp. pp. 156-157 "Solar
Voltaic Power." On page 156-157 "Solar Voltaic
Power." Odum writes" "EMERGY evaluations of solar
voltaic arrays have yet to show any net EMERGY
yield. The power grid at Austin, Tex., evaluated
by King & Schmandt (1991), is taking more EMERGY
out of society than it is generating. . ."
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On my blog entry "Concentration of Energy by
PhotoVoltaic Solar Panels" at
<http://bit.ly/Aimt1K> I advertise the motto of
"Solar Forward" <http://solarforward.com/>:
"SAVING THE PLANET, ONE SOLAR SYSTEM AT A TIME."
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Considering the EMERGY evaluation by King &
Schmandt (1991), is the above motto a deception?
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I THINK NOT because the evaluation of King &
Schmandt (1991), as set forth in Odum (1995,
failed to consider the devastating decrease in
EMERGY if alternatives to fossil fuel are not
actuated - see e.g. "Paleoclimate Implications
for Human-Made Climate Change" [Hansen & Sato
(2011)], who wrote:
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"The paleoclimate record makes it clear that a
target to keep human made global warming less
than 2°C, as proposed in some international
discussions, is not sufficient - IT IS A
PRESCRIPTION FOR DISASTER. . . .[[my CAPS]]. . .
Assessment of the dangerous level of CO2, and the
dangerous level of warming, is made difficult by
the inertia of the climate system. The inertia,
especially of the ocean and ice sheets, allows us
to introduce powerful climate forcings such as
atmospheric CO2 with only moderate initial
response. But that inertia is not our friend - it
means that we are building in changes for future
generations that will be difficult, if not
impossible, to avoid."
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Richard Hake, Emeritus Professor of Physics, Indiana University
Honorary Member, Curmudgeon Lodge of Deventer, The Netherlands
President, PEdants for Definitive Academic References
which Recognize the Invention of the Internet (PEDARRII)
<rrhake@xxxxxxxxxxxxx>
Links to Articles: <http://bit.ly/a6M5y0>
Links to SDI Labs: <http://bit.ly/9nGd3M>
Blog: <http://bit.ly/9yGsXh>
Academia: <http://iub.academia.edu/RichardHake>
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REFERENCES [URL's shortened by <http://bit.ly/> and accessed on 09 Feb 2012.]
Brandenburg, G. 2012. "Re: Fermi Problems in Math
Education?" MathEdCC post of Feb 7, 2012 7:16 AM;
online on the OPEN! MathEdCC archives at
<http://bit.ly/wOxmrU>.
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CARMA. 2007 CARbon Moniorinf for Action, "Center
for Global Development"; online at
<http://carma.org/region/detail/202>.
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Hansen, J.E. & M. Sato. 2011. "Paleoclimate
Implications for Human-Made Climate Change," 20
July, online at
<http://arxiv.org/abs/1105.0968v2>; to appear in
Berger, Mesinger and Sijaci, eds., "Climate
Change at the Eve of the Second Decade of the
Century: Inferences from Paleoclimate and
Regional Aspects: Proceedings of Milutin
Milankovitch 130th Anniversary Symposium"
(Springer, in press), a popularization of this
paper is online as a 213 kB pdf at
<http://1.usa.gov/AuzXMw>.
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Hake, R.R. 2012. "Concentration of Energy by
PhotoVoltaic Solar Panels" online on the OPEN!
AERA-L archives at <http://bit.ly/wdkAoL>.
[Transmitted to Math discussion lists as "Fermi
Problems in Math Education?"] Post of 6 Feb 2012
09:59:03-0800 to AERA-L and Net-Gold. The
abstract and link to the complete post were
transmitted to several discussion lists and are
also on my blog "Hake'sEdStuff" at
<http://bit.ly/Aimt1K> with a provision for
comments.
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Harte, J. 1988. "Consider a Spherical Cow: A
Course in Environmental Problem Solving."
University Science Books. Amazon.com information
at <http://amzn.to/zRek6q>. Note the "Look
Inside" feature.
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Harte, J. 2001. "Consider a Cylindrical Cow: More
Adventures in Environmental Problem Solving."
University Science Books. Amazon.com information
at <http://amzn.to/xecQhv>. Note the "Look
Inside" feature.
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King, R.J. & J. Schmandt. 1991. "Ecological
Economics of Alternative Transportation Fuels,"
Report to Texas State Dept. of Energy,
Unpublished report, L.B.J. School of Public
Affairs, Univ. of Texas, Austin, 26 pp.
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Mahajan, S. 2010. "Street-Fighting Mathematics:
The Art of Educated Guessing and Opportunistic
Problem Solving." MIT Press, publisher's
information at <http://bit.ly/ghF5XY> where the
book may be downloaded for FREE as a 770 kB pdf
by clicking on "Download This Book" in the
left-hand column. Author's information at
<http://streetfightingmath.com/>. Amazon.com
information at <http://amzn.to/xqOt3Q>.
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Odum, H.G.1995. "Environmental Accounting: Emergy
and Environmental Decision Making." Wiley,
publisher's information at
<http://bit.ly/wSeH9b>. Amazon.com information at
<http://amzn.to/ycGgJ2>, note the searchable
"Look Inside" feature." An expurgated Google
Preview is online at <http://bit.ly/w6aRIZ>.
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