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Iowa State University *
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Course
433
Subject
Mechanical Engineering
Date
Apr 3, 2024
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4
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1 © 2023 Kenneth Bryden
ME 433 Homework 1 - Power System Fundamentals 1.
Your community is powered by a 100.0 MW biomass-fired power plant with an availability of 95.0%, a capacity factor of 65.0%, and an overall efficiency of 30.0% based on the higher heating value. What is the annual energy production of the power plant (10
3
MWh/y)? 100 MW
1
×
8760 h
y
×
0.65 MWh
!"#$"#
MWh
%&$&%’#(
=
569.4 MWh
y
=
569 × 10
)
MWh/y
2.
A key issue with energy storage devices is the round-trip efficiency of the device. Round trip efficiency is the fraction of energy put into the storage device that is later retrieved. That is ࠵?
*+
=
࠵?
!"#
࠵?
’,
Consider a proposed 25 MW energy storage device based on stacking and unstacking concrete blocks as needed to store and then provide energy (see for example Energy Vault
). The plant requires an input of 1.22 MJ of electricity to produce 1 MJ of energy output. What is the round-trip efficiency of the concrete block energy storage plant (%)? ࠵?
*+
=
1 MJ
!"#
1.22 MJ
’,
= 0.8197 = 82.0%
3.
One form of energy storage is compressed air energy storage (CAES). In a diabatic CAES plant, ambient air is compressed and stored under pressure in an underground cavern or container and the heat generated by compression of the air is lost. When electricity is required, the pressurized air is heated (generally using natural gas) and expanded in an expansion turbine driving a generator for power production. Based on this, there are two energy inputs to a diabatic CAES system—the input electricity and the natural gas used for heating the expanding gases. Thus, the round-trip efficiency of a CAES system is ࠵?
*+
=
࠵?
!"#
࠵?
’,
=
࠵?
!"#
࠵?
-,’,
+ ࠵?
/-&#,’,
Where ࠵?
/-&#,’,
is provided by natural gas. Consider a proposed diabatic compressed air energy storage (CAES) system with a round trip efficiency of 37.0% in which the compressed air is released, heated using a natural gas burner, and then expanded through a gas turbine. To produce 1 MWh of electrical output requires 1.41 MWh of electrical input to the CAES system. a.
How many MWh’s of heat from the natural gas burner are required to produce 1 MWh of electricity out of the system (MWh
heat
)? 0.37 MWh
0-1
MWh
’,
=
1 MWh
0-1
1.41 MWh
-,’,
+ ࠵? MWh
/-&#,’,
2 © 2023 Kenneth Bryden
࠵? =
1 MWh
’,
0.37 MWh
0-1
−
1.41 MWh
-
MWh
0-1
=
1.293 MWh
/-&#,’,
MWh
0-1
= 1.29 MWh
/-&#,’,
/MWh
0-1
b.
If the natural gas burner and heat exchanger have an overall efficiency of 80%, how many MWh of natural gas are needed to produce 1 MWh of electricity out of the system (MWh
heat
)? 1.293 MWh
/-&#,’,
MWh
0-1
×
1 MWh
23,’,
0.8 MWh
/-&#,’,
=
1.616 MWh
/-&#,’,
MWh
0-1
= 1.62 MWh
23,’,
/MWh
0-1
4.
Fuel cells use an electrochemical process to convert the chemical energy in a fuel to electricity. In contrast to heat engines (e.g., gas and diesel engines, coal-fired power systems, and gas turbines) fuel cells generate electricity without combusting the fuel. This process also produces heat. In stationary power application, fuel cells are used for distributed generation (electricity only) and are also configured for combined heat and power (CHP) in which the residual heat generated is used for process heat or building/district heating. CHP systems are common for example ISU’s power plant is a CHP system. In the ISU system based using natural gas in a Rankine cycle power system in which the waste heat from the boilers is utilized to provide heating and cooling as well as electricity to the campus. The efficiency of CHP systems are reported two ways, the electrical efficiency ࠵?
-
=
࠵?
-
࠵?
4"-1
and the overall CHP efficiency ࠵?
567
=
࠵?
-
+ ࠵?
/-&#
࠵?
4"-1
A 220 kW fuel cell combined heat and power (CHP) plant that is fueled with natural gas. The natural gas has a lower heating value of 48.3 MJ/kg. Based on the lower heating value of the fuel, the power plant has a heat rate of 9200 Btu/kWh
e
. The heat rate accounts only the production of electricity from the fuel cell and not the delivery of heat. The capacity factor of the plant is 60%. The fuel cell produces 320,000 Btu/h of heat energy at the rated power of 220 kW. What is the lower heating value combined heat and power efficiency of the plant (%)? ࠵?
567
=
࠵?
̇
-
+ ࠵?
̇
/-&#
࠵?
̇
4"-1
࠵?
567
=
࠵?
̇
-
࠵?
̇
4"-1
+
࠵?
̇
/-&#
࠵?
̇
4"-1
࠵?
567
= ࠵?
-
+
࠵?
̇
/-&#
࠵?
̇
4"-1
3 © 2023 Kenneth Bryden
࠵?
567
=
3412
࠵?࠵?
+
࠵?
̇
/-&#
࠵?
̇
4"-1
࠵?
567
=
kWh
-
9200 Btu
4"-1
∙
3412 Btu
4"-1
kWh
4"-1
+
320,000 Btu
/-&#
220 kWh
-
∙
kWh
-
9200 Btu
4"-1
= 52.90%
5.
A rural community in California is considering a using wood chips from pine trees killed by pine bark beetles and drought as fuel for a community-based biomass-fired 40.0 MW electric power generation plant. Assume that the power plant will have a capacity factor 85.0% and an overall efficiency of 29.0% based on the dry basis higher heating value of the wood chips. The stover is dried before being used to fuel the power plant. The dry basis higher heating value of the wood chips is 20.2 MJ/kg. a.
How much electricity will be delivered annually (MWh/y)? 40 MW
8&#-0
1
×
8760 h
y
×
0.85 MWh
0-1
MWh
8&#-0
=
297,800 MWh
0-1
y
= 298,000 MWh
0-1
/y
b.
How many kg’s of wood chips (dry basis) will be required to deliver 1 MWh of electricity (kg/MWh)? MWh
#/
0.29 MWh
-
×
3600 MJ
MWh
×
kg
08(
20.2 MJ
=
614.5 kg
08(
MWh
-
= 615 kg
08(
/MWh
-
c.
How Mg’s of wood chips (dry basis) will be required each year to fuel the power plant (Mg/y)? 297,800 MWh
-
y
×
614.5 kg
08(
MWh
-
×
Mg
08(
10
)
kg
08(
=
183,000 Mg
08(
y
= 183,000 Mg
08(
/y
6.
Consider a plan to build a wind farm that is composed of 22 wind turbines each with a rated capacity of 2.50 MW. Each wind turbine has wind energy conversion efficiency of 39.0%, and a combined gearbox/generator efficiency of 98.0%. The array efficiency is 74.0%. It is estimated that the wind farm will generate 190 GWh of electricity annually. Each wind turbine requires 300 hours of maintenance annually. a.
What is the availability of an individual wind turbine (%)? (8760 − 300) h
8760 h
= 96.6%
b.
What is the capacity factor of the proposed wind farm (%)? 190 GWh
y
×
10
)
MWh
GWh
×
1
22 turbines
×
1
2.5 MW
8&#-0
×
y
8760 h
= 0.3944 = 39.4%
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7.
A wood chip fueled Rankine cycle 54.0 MW electric power generation plant uses chipped pine bark beetle deadfall as fuel. The overall efficiency of the power plant is 31% and the capacity factor of 84.0%. The as-received wood fuel is 45% carbon, has a higher heating value of 16.0 MJ/kg, and the wood is not harvested sustainably. Carbon emissions are taxed at the rate of $25/Mg
CO2e
. The life of the system is 25 years and the interest rate is 5.5%. Assume an operations and maintenance cost of $0.025/kWh. The initial cost of the system is $4000/kW. The cost of wood chips is $7.20 per MBtu. a.
What is the heat rate of the power plant (Btu/kWh)? ࠵?࠵? =
3412 Btu
’,
kWh
’,
×
kWh
’,
0.31 kWh
0-1
=
11,000 Btu
’,
kWh
0-1
= 11,000 Btu
’,
/kWh
0-1
b.
What is the cost of fuel per kWh for the electricity generated ($/kWh)? ࠵?
9
=
$7.20
MBtu
×
11,000 Btu
kWh
×
MBtu
10
:
Btu
=
$0.0792
kWh
= $0.0792/kWh
c.
How much electrical energy is generated annually by the power plant (MWh/y)? 54 MW
8&#-0
1
×
0.84 MW
0-1’;-8-0
MW
8&#-0
×
8760 h
y
=
397,400 MWh
y
= 397,000 MWh/y
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