WEBVTT
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Now, let's turn back to more traditional description of 2nd law.
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The 2nd law states that 100% conversion from heat to work in heat engines is impossible
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and there's always a limitation. Here's a heat engine.
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We have a high temperature heat source at temperature TH or T2
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and a low temperature heat sink at TL or T1.
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Heat is extracted from the high temperature source by QH
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and released to the low temperature sink by QL, doing work W.
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That is the basic operating principle of the heat engine.
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Then, we want to calculate the maximum work that can be obtained from this machine.
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Take the machine as our system. The machine is reversible.
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The reversibility is assumed to calculate the maximum work and the highest efficiency of this machine.
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And the machine is steady state, since the equipment itself does not change over time.
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And it is a closed system, since there is no material exchange.
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To find out the maximum work and the highest efficiency,
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let's start from the generalized entropy equation.
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The first two things in this equation is related to the material transfer,
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so they are zero since the system is closed.
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The lost work is also zero, since the process is reversible.
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And the entropy change of the system is zero,
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since the machine is steady-state, meaning that the initial and the final state are the same.
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So the equation becomes sigma delta Q over T is zero.
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There are two heats, QH and QL.
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So the equation is QH over T2 + QL over T1 is zero. So QL is -T1 over T2 times QH.
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Let's go to the 1st law then.
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The first law states that sum of heat change and work change is the internal energy change.
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Since the machine itself is steady states, internal energy does not change, thus delta U is zero.
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Insert the previously derived a equation, QL equal to -T1 over T2 times QH.
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Then, the work can be expressed with QH and temperatures.
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By convention the work done on the system is W.
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So the work done by the system is -W and it is QH times (1-T1/T2).
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This work is reversible work, since the machine operates reversibly,
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and it is the maximum work done by the system.
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The efficiency is generally defined as output divided by input.
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The output is what we want to get, so here for the heat engines,
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the output is the work done by the machine.
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The input is heat, so the efficiency of heat engine is -Wrev over QH and it is 1-(T1 over T2).
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So the efficiency only depends on the temperature T1 and T2.
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And the efficiency of 1, the 100 % conversion from heat to work is impossible
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since the temperature is a positive value.
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We never can get absolute zero temperature. Here is the actual constitution of heat engines.
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Let's start from the high temperature heat source. It's a boiler.
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Starting from the point 1, the liquid boils at temperature TH.
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The boiler provides the heat for the vaporization.
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At the exit of boiler, it's now vapor at temperature TH. It goes into turbin at temperature TH.
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In turbin, work is done by the vapor by its expansion.
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Since the turbin operates under adiabatic, steady state and reversible condition,
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the expansion of vapor is isoentropic.
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At the exit of turbin, the temperature of vapor is decreased to TL due to the expansion.
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Then, the vapor goes into the condenser at temperature TL.
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There, the vapor becomes liquid,
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since condensation reaction happens in condenser at temperature TL.
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The latent heat is removed by the cooling water.
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At the exit of condenser, it is now liquid at temperature TL.
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Then, it goes into compressor at temperature TL. Here, the liquid are compressed.
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The compressor pumps the liquid isoentropically, thus at the exit of compressor,
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the liquid temperature is increased to TH.
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Then, the liquid at temperature TH go back to the boiler completing a cylcle.
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The energy conversion process can be usefully displayed
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on a set of axes such as temperature-entropy diagram.
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Start from point 1. It is the liquid at temperature TH.
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In the boiler, the liquid vaporize at constant temperature TH.
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The vapor has higher entropy than the liquid, so the entropy increase from S1 to S2.
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Then, at turbin, the vapor expands isoentropically,
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in other words, at constant entropy S2, the temperature decreases TL.
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Next at condenser, vapor condensed into liquid at constant temperature TL.
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Finally, at compressor, the liquid compressed isoentropically, increasing temperature back to TH.
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The heat, if operated reversibly, is temperature times entropy change by definition,
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so the heat QH supplied at boiler, is TH times (S2-S1). It is the heat source.
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Likewise, the heat QL removed at condenser, is TL times (S1-S2).
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The work done by this heat engine is the difference between QH and QL. So it is (TH-TL) times (S2-S1)
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The efficiency of heat engine is work divided by heat supply.
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So it is (TH-TL) times (S2-S1) divided by TH times (S2-S1).
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Cancel out the same thing. Then the efficiency is (TH-TL) over TH, as before.