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2NE3 formula sheet

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Basic ideas and formulas
-
Property table
𝑇(𝐾) = 𝑇(𝐢) + 273.15
π‘ƒπ‘ π‘Žπ‘‘ > 𝑃 → π‘”π‘Žπ‘ 
𝑉
1
𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 π‘£π‘œπ‘™π‘’π‘šπ‘’ = 𝑣 = π‘š = 𝜌
π‘žπ‘’π‘Žπ‘™π‘–π‘‘π‘¦ = π‘₯ =
β„Žπ‘“π‘” = β„Žπ‘” − β„Žπ‘“
β„Ž =𝑒+𝑃∗𝑣
π‘šπ‘™π‘–π‘žπ‘’π‘–π‘‘ 𝑦 − 𝑦𝑓
=
, 𝑦 π‘π‘Žπ‘› 𝑏𝑒 𝑣, 𝑒, β„Ž, 𝑠
π‘šπ‘‘π‘œπ‘‘π‘Žπ‘™
𝑦𝑓𝑔
Compressed liquid cheat 𝑦 ≈ 𝑦𝑓@𝑇 → 𝑦 π‘π‘Žπ‘› 𝑏𝑒 𝑒, 𝑣, β„Ž
-
Energy
Energy definition → An objects energy is its ability to life a weight
Work definition → A process by which a system can interact with surroundings and exchange energy
Work done to the system is negative
𝐾𝐸 =
π‘šπ‘‰2
2
(π‘˜π½)
Heat transfer to the system is positive
𝑃𝐸 = π‘šπ‘”β„Ž (π‘˜π½)
𝐸 = π‘ˆ + 𝐾𝐸 + 𝑃𝐸 (π‘˜π½)
Conservation of energy → Total amount of energy is constant → Energy does not come in blocks
Adiabatic process → A process during which there’s no heat transfer → well insulated
Ideal gas law
𝑃𝑣 = 𝑅𝑇 → 𝑃𝑉 = π‘šπ‘…π‘‡
Work formula
πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘π‘Ÿπ‘’π‘ π‘ π‘’π‘Ÿπ‘’ → π‘Š = 𝑃(𝑉2 − 𝑉1 )
πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ π‘£π‘œπ‘™π‘’π‘šπ‘’ → π‘Š = 0
𝑉2
𝑃1
πΌπ‘ π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘šπ‘Žπ‘™ πΌπ‘‘π‘’π‘Žπ‘™ π‘”π‘Žπ‘  → π‘Š = π‘šπ‘…π‘‡ ∗ ln ( ) = π‘šπ‘…π‘‡ ∗ ln ( ) π‘“π‘œπ‘Ÿ 𝑛 = 1
𝑉1
𝑃2
π‘π‘œπ‘™π‘¦π‘‘π‘Ÿπ‘œπ‘π‘–π‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘  → π‘Š =
𝑃2 ∗ 𝑉2 − 𝑃1 ∗ 𝑉1
π‘“π‘œπ‘Ÿ 𝑛 ≠ 1
1−𝑛
π‘’π‘™π‘’π‘π‘Ÿπ‘‘π‘–π‘ π‘€π‘œπ‘Ÿπ‘˜ → π‘Š = 𝑉 ∗ 𝐼 ∗ βˆ†π‘‘
𝑄 = π‘šπ‘ ∗ βˆ†π‘‡
First law of thermodynamics
Energy can be neither created nor destroyed, it can only change form
π‘€π‘Žπ‘ π‘  π‘“π‘™π‘œπ‘€ π‘Ÿπ‘Žπ‘‘π‘’ → π‘šΜ‡ =
𝐸𝑖𝑛 − πΈπ‘œπ‘’π‘‘ = βˆ†πΈπ‘ π‘¦π‘ π‘‘π‘’π‘š
𝑉∗𝐴
𝑣
(π‘˜π‘”/𝑠)
π‘šπ‘–π‘› − π‘šπ‘œπ‘’π‘‘ = βˆ†π‘šπ‘ π‘¦π‘ π‘‘π‘’π‘š
∑ π‘šπ‘–π‘›
Μ‡ = ∑ π‘šπ‘’π‘₯𝑖𝑑
Μ‡
π‘π‘™π‘œπ‘ π‘’π‘‘ π‘ π‘¦π‘ π‘‘π‘’π‘š → 𝒒 − π’˜ = βˆ†π’– + βˆ†π’Œπ’† + βˆ†π’‘π’† (π‘˜π½/π‘˜π‘”)
Internal energy change of ideal gas → βˆ†π‘’ = 𝐢𝑣 ∗ βˆ†π‘‡
βˆ†β„Ž = 𝐢𝑝 ∗ βˆ†π‘‡
-
Steady flow system or open system
2
π‘˜π½
2
π‘œπ‘π‘’π‘› π‘ π‘¦π‘ π‘‘π‘’π‘š → 𝒒 − π’˜ = π’‰πŸ − π’‰πŸ (π‘˜π‘”)
𝑉 −𝑉
𝑄̇ − π‘ŠΜ‡ = π‘šΜ‡ ∗ (β„Ž2 − β„Ž1 + 2 2 1 + 𝑔(𝑧2 − 𝑧1 ))
Nozzles and Diffusers characteristics →
π‘ŠΜ‡ = 0
𝑄̇ = 0
Turbine and compressor characteristics →
𝑄̇ = 0
βˆ†π‘π‘’ = 0
Pumps → βˆ†β„Ž = 𝑣 ∗ βˆ†π‘ƒ
Throttling valves → β„Ž2 = β„Ž1 , π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘  = 0
Mixing chamber → π‘šΜ‡1 ∗ β„Ž1 + π‘šΜ‡2 ∗ β„Ž2 = (π‘šΜ‡1 + π‘šΜ‡2 )β„Ž3
Μ‡ + π‘šΜ‡1 ∗ β„Ž1 = π‘šΜ‡2 ∗ β„Ž2
Heat exchanger → 𝑄𝑖𝑛
βˆ†π‘π‘’ = 0
π‘šΜ‡1 + π‘šΜ‡2 = π‘šΜ‡3
Μ‡ = π‘šΜ‡ ∗ 𝑐𝑝 ∗ (𝑇2 − 𝑇1 )
𝑄𝑖𝑛
Second law of thermodynamics
Spontaneous process causes a degradation in the quality of energy of that system.
Non-spontaneous process requires additional quality energy and must take it from its surroundings.
Source → A reservoir that supplies energy in the form of heat
Sink → A reservoir that absorbs energy in the form of heat
𝑇𝐻 → β„Žπ‘–π‘”β„Ž π‘‘π‘’π‘šπ‘π‘’π‘Ÿπ‘Žπ‘‘π‘’π‘Ÿπ‘’ π‘šπ‘’π‘‘π‘–π‘’π‘š
𝑇𝐿 → π‘™π‘œπ‘€ π‘‘π‘’π‘šπ‘π‘’π‘Ÿπ‘Žπ‘‘π‘’π‘Ÿπ‘’ π‘šπ‘’π‘‘π‘–π‘’π‘š
𝑄𝐻 → β„Žπ‘’π‘Žπ‘‘ π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ π‘“π‘Ÿπ‘œπ‘š π‘Ž 𝑑𝑒𝑣𝑖𝑐𝑒 π‘‘π‘œ 𝑇𝐻
𝑄𝐿 → β„Žπ‘’π‘Žπ‘‘ π‘‘π‘Ÿπ‘Žπ‘›π‘ π‘“π‘’π‘Ÿ π‘“π‘Ÿπ‘œπ‘š π‘Ž 𝑑𝑒𝑣𝑖𝑐𝑒 π‘‘π‘œ 𝑇𝐿
Heat engines → A device that converts heat to work
π‘Šπ‘›π‘’π‘‘ = |𝑄𝐻 | − |𝑄𝐿 |
πœ‚=
π‘Šπ‘›π‘’π‘‘
𝑄𝐻
=1−
𝑄𝐿
𝑄𝐻
𝑄𝐿 =
𝑇
𝑄
Carnot heat engine → πœ‚π‘π‘Žπ‘Ÿπ‘›π‘œπ‘‘ = 1 − 𝑇𝐻 = 1 − 𝑄 𝐿
𝐿
𝐻
𝑄𝐿
𝑄𝐻
π‘Š
πœ‚
−π‘Š
𝑇
= 𝑇𝐿
𝐻
Refrigerators and pump → A device transfers heat from low temperature source to high temperature one
Refrigerators
𝐢𝑂𝑃𝑅 =
𝑄𝐿
π‘Šπ‘–π‘›
Carnot refrigerators 𝐢𝑂𝑃𝑅 =
Heat pump
𝐢𝑂𝑃𝐻𝑃 =
=
𝑄𝐿
𝑄𝐻 −𝑄𝐿
𝑄𝐻
π‘Šπ‘–π‘›
Carnot heat pump 𝐢𝑂𝑃𝐻𝑃 =
𝑄𝐿
𝑄𝐻 −𝑄𝐿
=
1
= 𝑄𝐻
𝑄𝐿
1
= 𝑄𝐻
𝑄𝐿
𝑄𝐻
𝑄𝐻 −𝑄𝐿
𝑄𝐻
𝑄𝐻 −𝑄𝐿
=
1
−1
=
π‘Šπ‘›π‘’π‘‘,𝑖𝑛 = |𝑄𝐻 | − |𝑄𝐿 |
−1
= 𝑇𝐻
𝑇𝐿
−1
1
π‘Šπ‘›π‘’π‘‘,𝑖𝑛 = |𝑄𝐻 | − |𝑄𝐿 |
𝑄
1− 𝐿
𝑄𝐻
1
𝑄
1− 𝐿
𝑄𝐻
=
1
𝑇
1− 𝐿
𝑇𝐻
𝐢𝑂𝑃𝐻𝑃 = 𝐢𝑂𝑃𝑅 + 1
Reversible process →Process that can be reversed without leaving trace on the system and surrounding
Reasons → Friction, expansion of gas, heat transfer, mixing substances
The Carnot Cycle process →
𝑄𝐻
𝑄𝐿
=
𝑇𝐻
𝑇𝐿
The efficiency of a real heat engine is always less than the efficiency of a reversible one
The efficiency of all reversible Carnot heat engines operating between the same two reservoirs are same
Entropy
The energy of the universe is constant
The entropy of the universe is increasing
2 𝛿𝑄
Increase of entropy principle → βˆ†π‘† = 𝑆2 − 𝑆1 = ∫1
𝑇
+ 𝑆𝑔𝑒𝑛
𝑆𝑔𝑒𝑛 → Always a positive quantity → Its value depends on the process → Not a property of a system
The total entropy of an isolated system during a process is always increases, or in the limiting case of a
reversible process, remains constant
Entropy change formulas
2 𝛿𝑄
π‘…π‘’π‘£π‘’π‘Ÿπ‘ π‘–π‘π‘™π‘’ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘  → Δ𝑆 = ∫1
π΄π‘‘π‘–π‘Žπ‘π‘Žπ‘‘π‘–π‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘  → S2 − 𝑆1 = 𝑆𝑔𝑒𝑛
𝑇
𝑄
πΌπ‘ π‘’π‘›π‘‘π‘Ÿπ‘œπ‘π‘–π‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘  → 𝑆2 − 𝑆1 = 0
πΌπ‘ π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘šπ‘Žπ‘™ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘  → βˆ†π‘† = 𝑇
0
Gibbs equations
∫ 𝑑𝑠 = ∫
𝑃∗𝑑𝑣
𝑇
+∫
𝑑𝑒
𝑇
𝑇𝑑𝑠 = π‘‘β„Ž − 𝑣𝑑𝑃
𝑇
𝑠2 − 𝑠1 = πΆπ‘Žπ‘£ ∗ ln (𝑇2 )
liquids and solids
1
Ideal gas
𝑇
𝑣
𝑇
𝑃
𝑣
𝑃
βˆ†π‘  = 𝐢𝑣,π‘Žπ‘£ ∗ ln (𝑇2 ) + 𝑅 ∗ 𝑙𝑛 (𝑣2 ) = 𝐢𝑃,π‘Žπ‘£ ∗ ln (𝑇2 ) − 𝑅 ∗ 𝑙𝑛 (𝑃2 ) = 𝐢𝑃,π‘Žπ‘£ ∗ ln (𝑣2 ) + 𝐢𝑣,π‘Žπ‘£ ∗ 𝑙𝑛 (𝑃2 )
1
1
1
1
1
1
𝑇2
𝑇1
Constant specific heats, ideal gas, isentropic process
-
𝑃 π‘˜−1
π‘˜
𝑃1
= ( 2)
𝑣
𝑣2
= ( 1 )π‘˜−1
Isentropic efficiency for steady flow devices
Turbine → 𝑀𝑠 = β„Ž1 − β„Ž2𝑠 = 𝐢𝑝 ∗ ( 𝑇1 − 𝑇2𝑠 )
πœ‚=
π‘€π‘Ž
𝑀𝑠
=
π‘€π‘Ž = β„Ž1 − β„Ž2π‘Ž = 𝐢𝑝 ∗ ( 𝑇1 − 𝑇2π‘Ž )
β„Ž1 −β„Ž2π‘Ž
β„Ž1 −β„Ž2𝑠
π‘€π‘Ž < 𝑀𝑠
Compressor, pump, nozzles
𝑀
πœ‚π‘π‘œπ‘šπ‘π‘Ÿπ‘’π‘ π‘ π‘œπ‘Ÿ = 𝑀 𝑠 =
π‘Ž
β„Ž2𝑠 − β„Ž1
β„Ž2π‘Ž −β„Ž1
πœ‚π‘π‘’π‘šπ‘ =
𝑣1 ∗(𝑃2 −𝑃1 )
β„Ž2π‘Ž −β„Ž1
𝑉2
πœ‚π‘›π‘œπ‘§π‘§π‘™π‘’ = 𝑉2π‘Ž
2
2𝑠
Compressor work
π‘€π‘–π‘ π‘’π‘›π‘‘π‘Ÿπ‘œπ‘π‘–π‘ (𝑃 ∗ 𝑣 π‘˜ = π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘) =
π‘˜∗𝑅∗(𝑇2 −𝑇1 )
1−π‘˜
π‘€π‘π‘œπ‘™π‘¦π‘‘π‘Ÿπ‘œπ‘π‘–π‘ (𝑃 ∗ 𝑣 𝑛 = π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘) =
𝑃
π‘€π‘–π‘ π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘šπ‘Žπ‘™ (𝑃 ∗ 𝑣 = π‘π‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘) = −𝑅 ∗ 𝑇 ∗ ln (𝑃1 )
2
Power cycle design and analysis
Carnot cycle
1-2 Isothermal reversible heating in a boiler
2-3 Isentropic expansion in turbine
3-4 Isothermal reversible condensation in a boiler
4-1 Isentropic compression by a compressor
𝑛∗𝑅∗(𝑇2 −𝑇1 )
1−𝑛
Problems
1. Limited maximum temperature → limits the thermal efficiency
2. High moisture content in turbine →quality of steam decreases (2-3), droplets cause erosion when x<0.9
3. Compression of liquid-vapor mixture → not practical for pump handles two phases
Rankine cycle
1-2 Isentropic compression in a pump
2-3 Constant pressure heat addition in a boiler
3-4 Isentropic expansion in turbine
4-1 Constant pressure heat rejection in a
condenser
π‘€π‘π‘’π‘šπ‘,𝑖𝑛 = β„Ž2 − β„Ž1 = 𝑣 ∗ (𝑃2 − 𝑃1 )
Boiler π‘žπ‘–π‘› = β„Ž3 − β„Ž2
π‘€π‘‘π‘’π‘Ÿπ‘,π‘œπ‘’π‘‘ = β„Ž3 − β„Ž4
Condenser π‘žπ‘œπ‘’π‘‘ = β„Ž4 − β„Ž1
πœ‚π‘‘β„Ž = 1 −
π‘žπ‘œπ‘’π‘‘
π‘žπ‘–π‘›
=
𝑀
𝑀𝑛𝑒𝑑
π‘žπ‘–π‘›
π‘Ÿπ‘π‘€ = 𝑀 π‘π‘’π‘šπ‘
π‘‘π‘’π‘Ÿπ‘π‘–π‘›π‘’
Brayton cycle
1-2 Isentropic compression
2-3 Constant pressure heat addition
3-4 Isentropic expansion
4-1 Constant pressure heat rejection
π‘žπ‘–π‘› = β„Ž3 − β„Ž2 = 𝐢𝑝 (𝑇3 − 𝑇2 )
π‘žπ‘œπ‘’π‘‘ = β„Ž1 − β„Ž4 = 𝐢𝑝 (𝑇1 − 𝑇4 )
𝑀𝑑 = β„Ž4 − β„Ž3
𝑀𝑐 = β„Ž2 − β„Ž1
πœ‚π‘π‘Ÿπ‘Žπ‘¦π‘‘π‘œπ‘› =
𝑀𝑛𝑒𝑑
π‘žπ‘–π‘›
=1−
π‘žπ‘œπ‘’π‘‘
π‘žπ‘–π‘›
𝑇
= 1 − 𝑇1
2
𝑀𝑑 + 𝑀𝑐 = π‘žπ‘–π‘› + π‘žπ‘œπ‘’π‘‘
𝑇2
𝑇1
𝑃
π‘˜−1
π‘˜
= (𝑃2 )
1
𝑃
π‘˜−1
π‘˜
= (𝑃3 )
4
𝑇
= 𝑇3
4
𝑃
Pressure ratio → π‘Ÿπ‘ = 𝑃2
πœ‚π‘π‘Ÿπ‘Žπ‘¦π‘‘π‘œπ‘› = 1 −
1
1
π‘˜−1
π‘Ÿπ‘ π‘˜
The thermal efficiency increases with both pressure ratio and the specific ratio (k)
𝑀
Back work ratio → 𝑀𝑐
𝑑
Brayton cycle with regeneration
Point 5 must have a lower temperature than point 4, so that heat can regenerate. Heat flows high to low
Point 6 must have a higher temperature than point 2, so that heat can regenerate.
π‘žπ‘Ÿπ‘’π‘”π‘’π‘›,π‘šπ‘Žπ‘₯ = β„Ž5′ − β„Ž2 = β„Ž4 − β„Ž2
π‘žπ‘Ÿπ‘’π‘”π‘’π‘›,π‘Ÿπ‘’π‘Žπ‘™ = β„Ž5π‘Ž − β„Ž2
𝑇 −𝑇
πΆπ‘œπ‘›π‘ π‘‘π‘Žπ‘›π‘‘ 𝐢𝑝 π‘Žπ‘ π‘ π‘’π‘šπ‘π‘‘π‘–π‘œπ‘› → πœ€ = 𝑇5 −𝑇2 → 𝑒𝑓𝑓𝑒𝑐𝑑𝑖𝑣𝑒𝑛𝑒𝑠𝑠
4
2
πœ€=
β„Ž5π‘Ž −β„Ž2
β„Ž4 −β„Ž2
< 0.85 π‘‘π‘¦π‘π‘–π‘π‘Žπ‘™π‘™π‘¦
𝑇
π‘˜−1
πœ‚π‘Ÿπ‘’π‘”π‘’π‘› = 1 − (𝑇1 ) ∗ π‘Ÿπ‘ π‘˜
3
Ideal vapor compression refrigeration cycle
1 – 2 Isentropic compression in a compressor
3 – 4 Throttling in an expansion device
State 3 and state 4 have the same enthalpy
2 – 3 Constant pressure heat reject
4 – 1 Constant pressure heat absorption in an evaporator
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