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Further Analysis of Reversible Machines P M V Subbarao Professor Mechanical Engineering Department Innovation of A New Property of A System!!!!
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Performance of Reversible Machines : SSSF W out,C Q HC Q LC HTR (Source) LTR (Sink) Carnot Engine W out,S Q HS Q LS HTR (Source) LTR (Sink) Stirling Engine W out,R Q HR Q LR HTR (Source) LTR (Sink) Regenerative Engine
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Compound Machine using two Reversible Machines W out,C Q HC Q LC HTR (Source) Carnot Engine W out,S Q HS Q LS LTR (Sink) Stirling Engine
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All Reversible Machines Working Between Same Reservoirs should have Same performance
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Further Algebra of A Reversible Engine Model Use a generalized indices…
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During a time duration For any reversible cycle with n number heat transfer processes Where, i shows i th process and n is the number of process in a cycle. But Q i is a path function and depends on process.
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k is an initial state and k+1 is a final state. m is the number state Points in a cycle. Therefore for all reversible cycles. Therefore,
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Cyclic integral of this new quantity is zero! Any quantity whose cyclic integral is zero is a property ! For a reversible process an infinitesimal change in this new property is: Boltzman named S as Entropy of a substance. A New cyclic Integral During a reversible process
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S (gaseous state) > S (liquid state) > S (solid state) S o (J/Kmol) H 2 O (liq) 69.95 H 2 O (gas) 188.8 S o (J/Kmol) H 2 O (liq) 69.95 H 2 O (gas) 188.8 Entropy, S : A Measure of State of Matter For a given substance
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Entropy and Order of Molecules of Matter S˚ (Br2 liq) < S˚ (Br2 gas) S˚ (H2O solid) < S˚ (H2O liquid)
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Increase in molecular complexity generally leads to increase in S. Entropy, S : Molecular Complexity
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Standard Molar Entropies
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Entropy and Temperature S increases slightly with T S increases a large amount with phase changes
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Entropy Change during a Reversible Process From the definition of the entropy, it is known that Q=TdS during a reversible process. The total heat transfer during this process is given by Q reversible = TdS Therefore, it is useful to consider the T-S diagram for a reversible process involving heat transfer On a T-S diagram, the area under the process curve represents the heat transfer for a reversible process T S A reversible adiabatic process
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Carnot Cycle Show the Carnot cycle on a T-S diagram and identify the heat transfer at both the high and low temperatures, and the work output from the cycle. S T THTH TLTL S 1 =S 4 S 2 =S 3 1 2 3 4 AB 1-2, reversible isothermal heat transfer Q H = TdS = T H (S 2 -S 1 ) area 1-2-B-A 2-3, reversible, adiabatic expansion isentropic process, S=constant (S 2 =S 3 ) 3-4, reversible isothermal heat transfer Q L = TdS = T L (S 4 -S 3 ), area 3-4-A-B 4-1, reversible, adiabatic compression isentropic process, S 1 =S 4 Net work W net = Q H - Q L, the area enclosed by 1-2-3-4, the shaded area
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Nature of Reversible Machines For all reversible heat engines: For all reversible heat pumps and refrigerators: Therefore all reversible machines in this universe :
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Process : h-s Diagram : Mollier Diagram Enthalpy-entropy diagram, h-s diagram: it is valuable in analyzing steady-flow devices such as turbines, compressors, etc. h: change of enthalpy from energy balance (from the first law of thermodynamics) s: change of entropy from the second law. A measure of the irreversibilities during an adiabatic process. ss hh h s
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Enthalpy Vs Entropy Diagram
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Temperature- Entropy Diagram
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TdS -- Equations For a control mass containing a pure compressible substance undergoing a reversible process (no change in KE & PE) dU= Q rev - W rev = TdS - pdV TdS = dU + pdV, or Tds = du + pdv ( per unit mass) This is the famous Gibbsian equation Eliminate du by using the definition of enthalpy h=u+pv dh = du + pdv + vdp, thus du + pdv = dh - vdp Tds = du + pdv, also Tds = dh - vdp Important: these equations relate the entropy change of a system to the changes in other properties: dh, du, dp, dv. Therefore, they are independent of the processes.
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Entropy change of an incompressible substance For most liquids and all solids, the density is not changed as pressure changes, that is, dv=0. Gibbsian equation states that Tds=du+pdv=du, du=CdT. For an incompressible substance C p =C v =C is a function of temperature only. Integrating from state 1 to state 2 Where, C avg is the averaged specific heat over the given temperature range.
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Entropy change during change of Phase Consider steam is undergoing a phase transition from liquid to vapor at a constant temperature. Determine the entropy change s fg =s g -s f using the Gibbsian equations and compare the value to that read directly from the thermodynamic table. For a change from saturated liquid to saturated vapor
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