Further Analysis of Reversible Machines P M V Subbarao Professor Mechanical Engineering Department Innovation of A New Property of A System!!!!

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

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

All Reversible Machines Working Between Same Reservoirs should have Same performance

Further Algebra of A Reversible Engine Model Use a generalized indices…

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.

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,

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

S (gaseous state) > S (liquid state) > S (solid state) S o (J/Kmol) H 2 O (liq) H 2 O (gas) S o (J/Kmol) H 2 O (liq) H 2 O (gas) Entropy, S : A Measure of State of Matter For a given substance

Entropy and Order of Molecules of Matter S˚ (Br2 liq) < S˚ (Br2 gas) S˚ (H2O solid) < S˚ (H2O liquid)

Increase in molecular complexity generally leads to increase in S. Entropy, S : Molecular Complexity

Standard Molar Entropies

Entropy and Temperature S increases slightly with T S increases a large amount with phase changes

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

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 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 , the shaded area

Nature of Reversible Machines For all reversible heat engines: For all reversible heat pumps and refrigerators: Therefore all reversible machines in this universe :

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. ss hh h s

Enthalpy Vs Entropy Diagram

Temperature- Entropy Diagram

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.

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.

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