CE 107
CLIMATE CHANGE MITIGATION
Spring 2019
Assignment 4 Solutions
4.1. Exploring the climate-change consequences of natural gas from shale
PART (a)
Find: the carbon intensity for this power plant, in units of gC/kWh, accounting only for the carbon
emitted as CO2 from combustion of methane.
Use unit conversion:
12 gC
1000 gCH4 1 kgCH4 1 M Jth 3.6 M Je
·
·
·
·
= 98 gC/kW h
16 gCH4 1 kgCH4
50 M Jth 0.55 M Je 1 kW h
PART (b)
Find: Compute the global warming potential of methane for a time horizon of T = 30 years.
We can use the GWP equation which depends on ratio of radiative efficiency and persistence
functions of methane over CO2 . Note that when used in the GWP equation, radiative efficiency
must be normalized on a mass basis and not a concentration basis (as given in the problem).
Therefore, we must divide the numerator by the molecular weight of methane, and divide the
denominator by the molecular weight of CO2 to get the proper units. Rearranging our terms, we get:
RT
M WCO2 · ∆fCH4 0 RCH4 (t)dt
GW PCH4 ,30 =
RT
M WCH4 · ∆fCO2 0 RCO2 (t)dt


M WCO2 = 44 g/mol



M W
CH4 = 16 g/mol
where :

∆fCH4 = 3.63e−4 W/m2 /ppb



∆f
2
2
CO2 = 0.0137 W/m /ppm = 1.37e−5 W/m /ppb
Next, we can integrate our persistence function of methane for the T=30 years time horizon:
Z
30
Z
RCH4 (t)dt =
0
30
30
e−t/12.4 = −12.4e−t/12.4
0
= 12.4(1 − e−30/12.4 ) = 11.30 y
0
The solution to integrating the CO2 persistence function over our time horizon was given in the
problem statement:
Z 30
RCO2 (t)dt = 20.0 y
0
Plugging all our parameters into the GWP equation and solving:
GW PCH4 ,30 =
(44 g/mol)(3.63e−4 W/m2 /ppb)(11.30 y)
= 41
(16 g/mol)(1.37e−5 W/m2 /ppb)(20.0 y)
For two emissions of equal mass, the radiative impact of methane after 30 years is 41 times more
than that of CO2 . Note that the GWP of methane depends a lot on the time scale
(GW PCH4 ,20 = 84, GW PCH4 ,100 = 28) because it has a much shorter persistance time than CO2 .
PART (c)
Find: the effective carbon intensity of generating electricity with natural gas assuming that 5% of
the methane from fracking escapes by leakage and venting, and 95% is used to produce electricity.
First, determine the mass of leaked CH4 per kWh:
1000 gCH4 0.05 gCH4 leaked
1 M Jth 3.6 M Je
6.89 gCH4 leaked
·
·
·
=
50 M
0.95 gCH
burned 0.55
M Je 122:13:47
kW h GMT -06:00 kW h
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CE 107
CLIMATE CHANGE MITIGATION
Spring 2019
Convert to gC(e):
12 gC(e)
6.89 gCH4 leaked 41 gCO2 (e)
77.0 gC(e)
·
·
=
kW h
1 gCH4
44 gCO2 (e)
kW h
Calculate total carbon intensity:
98.2 gC(e) 77.0 gC(e)
175 gC(e)
+
=
kW h
kW h
kW h
A 5% methane leak increases the carbon intensity of electricity produced from natural gas by
roughly three quarters (assuming a 30-year time horizon). It is very important to limit methane
leaks to the atmosphere!
4.2. A sustainable fossil carbon budget for California
PART (a)
Find: per capita daily allocation of fossil-C emissions (kgC per person per day)
4.1 GtC 1012 kgC
1y
1
·
·
·
= 1.0 kgC/pers/d
y
1 GtC
365 d 10.9e9 ppl
PART (b)
Find: the corresponding per capita daily emissions of fossil carbon for gasoline (transportation),
electricity (residential), and natural gas (residential) individually as well as their total.
GASOLINE:
350.6e6 bbl 159 L 0.750 kg gas 0.87 kgC
1y
1
·
·
·
·
·
= 2.5 kgC/pers/d
y
1 bbl
L
1 kg gas 365 d 39.54e6 ppl
ELECTRICITY:
90.12e9 kW h 1 y 0.070 kgC
1
·
·
·
= 0.44 kgC/pers/d
y
365 d
kW h
39.54e6 ppl
NATURAL GAS:
431e9 f t3 1.19 mol CH4
12 gC
1 kg
1y
1
·
·
·
·
·
= 0.43 kgC/pers/d
3
y
ft
1 mol CH4 1000 g 365 d 39.54e6 ppl
TOTAL:
2.52 + 0.44 + 0.43 = 3.4 kgC/pers/d
PART (c)
Find: overall percentage reduction in fossil carbon emissions must be achieved in California during
the 21st century to align emissions with the daily per capita allowance of 1 kgC/pers/d.
First calculate total per capita emissions taking into account industrial, commercial, institutional,
and other transportation purposes related emissions:
total = 2 × 3.38 = 6.77kgC/pers/d
Calculate the total percent reduction required:
6.77 − 1.03
∗ 100 = 85% reduction
6.77
We must cut our emissions by roughly a factor of seven during the 21st century in order to achieve
the RCP4.5 scenario goal.
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CE 107
CLIMATE CHANGE MITIGATION
Spring 2019
4.3. Growth trajectories: Wind, solar PV, and total generated electricity in the
U.S.
PART (a)
Find: average growth-rate coefficients, r, for total generation, for wind generation, and for solar PV
generation of electricity for the period 2008-2017 using the growth data in the given table in the
problem set.
Let t∗ = t − tref (i.e., the time elapsed since 2008).
∗
G(t∗ ) = G0 er·t
Take the natural log of both sides of the equation:
∗
ln(G(t∗ )) = ln(G0 er·t )
∗
ln(G(t∗ )) = ln(G0 ) + ln(er·t )
ln(G(t∗ )) = ln(G0 ) + r · t∗
Note that the equation is now in the form y = b + mx. Apply linear regression to solve for the slope
and the intercept. The slope will give you r and the intercept will give you ln(G0 ). You should
obtain the following values:
• Total generation: G0,T = 4071e9 kW h/y, rT = −1.42e−4 y −1 (essentially zero growth)
• Wind generation: G0,W = 66.2e9 kW h/y, rW = 0.161 y −1
• Solar PV generation: G0,S = 1.70e9 kW h/y, rS = 0.440 y −1
Full solutions in Excel and MATLAB are posted to bCourses.
PART (b)
Find: year where wind power will generate 20% of total U.S. electricity with constant growth
constant found in PART (a).
Rearrange terms to find fraction of wind generation from total generation and set it equal to 0.20:
G0,W · exp(rW · t∗ )
= 0.20
G0,T · exp(rT · t∗ )
Solve for t∗ :
t∗ = ln(0.2
G0,T
1
)·
G0,W (rW − rT )
Substituting in the appropriate values, we find t∗ = 15.6 years, or around the year 2024 .
NOTE: An acceptable alternative approach is to apply the growth rate using the year 2017 as the base year. This
approach also results in an answer of 2024.
Refer to the Excel or MATLAB solution for the full calculations.
PART (c)
Find: year where solar PV will generate 20% of total U.S. electricity with constant growth constant
found in PART (a).
Using the same approach as in Part (b), we find t∗ = 14.0 years, or around the year 2022 .
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CE 107
CLIMATE CHANGE MITIGATION
Spring 2019
NOTE: An acceptable alternative approach is to apply the growth rate using the year 2017 as the base year. This
approach also results in an answer of 2022.
Refer to the Excel or MATLAB solution for the full calculations.
Both wind and solar have demonstrated remarkable growth over a short time scale. Over the decade
analyzed, wind and solar have grown by 4.6X and 39X, respectively. In order to reach 20% of total
electricity, wind must increase by another factor of 3.2, and solar by a factor of 10. Assuming
constant growth rates, we can realistically expect both wind and solar to achieve 20% contributions
within the next decade!
4.4. Wind power analysis
PART (a)
Find: the total available annual-average power density in wind at Russel, Kansas in [W/m2 ]
First, adjust the wind speeds to account for a hub height of 95 meters by using the following
equation:
95
U95 = U10 ( )0.14
10
Next, calculate the wind power density at each wind speed using the following equation:
1
W = ρair U 3
2
Multiply each of the resulting wind power densities by its corresponding frequency and sum:
X
Wtot =
Wi fi
i
You should find an annual average wind power density of approximately 480 W/m2 .
Refer to the Excel spreadsheet or MATLAB code posted to bCourses for the full solution.
PART (b)
Find: the total annual electrical energy [kWh/y] that would be produced by this wind turbine,
assuming that it is available to operate during all hours.
Using the data in the Vestas V90-2.0 Table in the problem set, apply linear interpolation to
determine the power generation for each wind speed. Multiply by the corresponding frequencies and
sum:
X
Ptot =
Pi fi
i
This results in an annual average power generation of 805 kW. Convert from power [kW] to energy
[kWh] by multiplying by the number of hours in one year (8760 hrs = 1 y). You should find an
annual electricity production of 7.1 million kW h/y .
Refer to the Excel spreadsheet or MATLAB code posted to bCourses for the full solution.
PART (c)
Find: the annual average efficiency of the turbine for converting wind power to electricity
First, determine the area swept by the turbine blades (note that using the given rotor swept area of
6362 m2 given in the problem is also acceptable):
π(44 m)2 = 6082 m2
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CE 107
CLIMATE CHANGE MITIGATION
Spring 2019
Multiply the result by the power density in order to determine the power:
(479 W/m2 )(6082 m2 ) = 2.915e6 W = 2915 kW
Divide the annual average power generation by the annual available wind power:
812 kW/y
= 28%
2915 kW/y
NOTE: if the given wind swept area in the problem is used (6362 m2 ), then the average efficiency
will be 26% which is an acceptable answer as well.
PART (d)
Find: the “wind capture” of this turbine [kWh/y per kW]
Divide the annual electricity production by the turbine’s rated power capacity:
7.05e6 kW h/y
kW h/y
= 3530
2000 kW
kW
4.5. Assigned Reading 4: J. H. Williams et al. (2012) The Technology Path to
Deep Greenhouse Gas Emissions Cuts by 2050: The Pivotal Role of Electricity
Science 335 53-59
PART (a)
From Table 1 in the paper, the authors calculate that to meet the California GHG reduction goals,
the annual percapita annual emissions would correspond to about 1.5 tonnes CO2 (e)/person per
year. Converting this to kgC(e) per person per day:
Ecapita =
1.5 tonnes CO2 e 1000 kg
12 kgC(e)
1y
·
·
·
= 1.1 kgC(e)/person per day
person · y
1 tonne 44 kgCO2 (e) 365 d
PART (b)
The authors recommend the following sequence in terms of strategy deployment:
1. Energy Efficiency
2. Decarbonization of Electricity
3. Electrification
The authors argue that decarbonization must precede electrification because switching from fuels to
electricity before the grid is substantially decarbonized negates the emissions benefits of
electrification. They further argue that agressive energy efficiency should precede decarbonization
since large deployment of electrified technologies without high energy efficiency measures would
reduce utility load factors and increase electricity costs, making the bulk requirements for
decarbonizing electricity much more difficult.
PART (c)
Convert the carbon intensity of electricity given in the paper to gC(e)/kWh:
CIelec =
0.025 kgCO2 (e) 1000 g
12 gC(e)
·
·
= 6.8gC(e)/kW h
kW h
1 kg 44 gCO2 (e)
The authors propose that this target can be achieved through a mixture of (i) high renewable
deployment, (ii) higher proportion of electricity derived from nuclear power plants within the power
mix, (iii) and high utilization of carbon capture and storage (CCS) in fossil fuel power plants.
PART (d)
According to the authors, the most important finding of their research is that California’s set
reduction goals cannot be achieved without widespread electrification. Without electrification, the
maximum emission reductions achievable are about 50% below the 1990 level in 2050 baseline (the
target is 90% below the 2050 baseline).
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