Isentropic process

download Isentropic process

of 47

  • date post

    05-Apr-2018
  • Category

    Documents

  • view

    218
  • download

    0

Embed Size (px)

Transcript of Isentropic process

  • 7/31/2019 Isentropic process

    1/47

    1

    Lec 18: Isentropic processes,TdS relations, entropy changes

  • 7/31/2019 Isentropic process

    2/47

    2

    For next time: Read: 7-2 to 7-9 Group project subject selection due on

    November 3, 2003

    Outline: Entropy generation and irreversible processes Entropy as a property Entropy changes for different substances

    Important points: Entropy is a property of a system it is not

    conserved and is generated by irreversibleprocesses Know how to identify an isentropic processes Know how to use the tables to find values for

    entropy

  • 7/31/2019 Isentropic process

    3/47

    3

    Recall we had entropy

    2

    1 revintT

    q

    Rlb

    Btuor

    Kkg

    kJ

    m

    s2 - s1 =

    Units

    are

  • 7/31/2019 Isentropic process

    4/47

    4

    Lets look at a simpleirreversible cycle on a p-v

    diagram with two processes

    P

    1

    2

    .

    .A

    B

    Let A be

    irreversible and Bbe reversible

  • 7/31/2019 Isentropic process

    5/47

    5

    Irreversible cycle

    0)T

    QAB

    By Clausius Inequality

    Evaluate cyclic integral

    0T

    Q

    T

    Q

    T

    Q2

    1 B

    2

    1 Acycle

    (non-rev) (rev)

  • 7/31/2019 Isentropic process

    6/47

    6

    Irreversible cycle

    For the reversible process, B, dS=Q/dT,thus:

    0dST

    Q

    T

    Q2

    1

    2

    1 Acycle

    Rearranging and integrating dS:

    2

    1 AT

    QS

  • 7/31/2019 Isentropic process

    7/47

    7

    Second Law of Thermodynamics

    Entropy is a non-conserved property!

    2

    1 A

    12T

    QSSS

    This can be viewed as a mathematicalstatement of the second law (for aclosed system).

  • 7/31/2019 Isentropic process

    8/47

    8

    We can write entropy change as anequality by adding a new term:

    gen

    2

    1 A

    12 S

    T

    QSS

    entropychange

    entropytransfer

    due toheattransfer

    entropyproduction

    orgeneration

  • 7/31/2019 Isentropic process

    9/47

    9

    Entropy generation

    Sgen 0 is an actual irreversible process.

    Sgen = 0 is a reversible process.

    Sgen 0 is an impossible process.

  • 7/31/2019 Isentropic process

    10/47

    10

    TEAMPLAY

    Consider the equation

    You have probably heard, Entropy alwaysincreases.

    Could it ever decrease? What are theconditions under which this could happen(if it can)?

    gen

    2

    1 A

    12 S

    T

    QSS

  • 7/31/2019 Isentropic process

    11/47

    11

    Entropy transfer and production

    What if heat were transferred from thesystem?

    The entropy can actually decrease if

    gen

    2

    1 A

    ST

    Q

    and heat is being transferred awayfrom the system so that Q is negative.

  • 7/31/2019 Isentropic process

    12/47

    12

    Entropy Production

    Sgen quantifies irreversibilities. Thelarger the irreversibilities, the greaterthe value of the entropy production,Sgen .

    A reversible process will have no entropyproduction.

  • 7/31/2019 Isentropic process

    13/47

    13

    Entropy transfer and production

    S2 S1

    > 0, Q could be + or ; if,

    because Sgen is always positive.

    < 0, if Q is negative and

    = 0 if Q = 0 and Sgen = 0.

    = 0 if Q is negative and

    gen

    2

    1 A

    ST

    Q

    gen

    2

    1 A

    ST

    Q

    gen

    2

    1 A

    ST

    Q

  • 7/31/2019 Isentropic process

    14/47

    14

    Isentropic processes

    Note that a reversible (Sgen = 0),adiabatic (Q = 0) process is alwaysisentropic (S1 = S2)

    But, if the process is merely isentropicwith S1 = S2, it may not be a reversibleadiabatic process.

    For example, if Q 0 and gen

    2

    1 A

    ST

    Q

  • 7/31/2019 Isentropic process

    15/47

    15

    Entropy generation

    Consider

    What if we draw our system boundariesso large that we encompass all heat

    transfer interactions? We wouldthereby isolate the system.

    gen

    2

    1 A

    12 S

    T

    QSS

  • 7/31/2019 Isentropic process

    16/47

    16

    Entropy changes of isolated systems

    And then

    gen

    2

    1 A

    12 ST

    QSS

    0

    gen12 SSS

    But Sgen0. So, the entropy of anisolated system always increases. (Thisis the source of the statement, The world

    is running down.)

  • 7/31/2019 Isentropic process

    17/47

    17

    Entropy

    )ss(xss fgf

    )T(s)p,T(s f

    It is tabulated just like u, v, and h.

    Also,

    And, for compressed or subcooled liquids,

  • 7/31/2019 Isentropic process

    18/47

    18

    The entropy of a pure substance is determined from the tables, just as forany other property

  • 7/31/2019 Isentropic process

    19/47

    19

    Ts Diagram for Water

  • 7/31/2019 Isentropic process

    20/47

    20

    TEAMPLAY

    Use the tables in your book

    Find the entropy of water at 50 kPa and500 C. Specify the units.

    Find the entropy of water at 100 C anda quality of 50%. Specify the units.

    Find the entropy of water at 1 MPa and

    120 C. Specify the units.

  • 7/31/2019 Isentropic process

    21/47

    21

    Ts diagrams

    pdVw

    Work was the area under the curve.

    Recall that the P-v diagram was veryimportant in first law analysis, and that

  • 7/31/2019 Isentropic process

    22/47

    22

    For a Ts diagram

    revintT

    QdS

    TdSQ revint

    2

    1

    revint TdSQ

    Rearrange:

    Integrate:

    If the internally reversible process also isisothermal at some temperature To:

    STdSTQ o

    2

    1

    orevint

  • 7/31/2019 Isentropic process

    23/47

    23

    On a T-S diagram, the area under the process curve represents theheat transfer for internally reversible processes

    d

  • 7/31/2019 Isentropic process

    24/47

    24

    Entropy change of a thermalreservoir

    For a thermal reservoir, heat transfer occursat constant temperaturethe reservoirdoesnt change temperature as heat isremoved or added:

    TQ

    S

    Since T=constant:

    T

    QS

    Applies ONLY tothermalreservoirs!!!!

  • 7/31/2019 Isentropic process

    25/47

    25

    The Tds Equations

  • 7/31/2019 Isentropic process

    26/47

    26

    Derivation ofTds equations:

    dQ dW = dU

    For a simple closedsystem:

    dW = PdV

    The work is given by:

    dQ = dU + PdV

    Substituting gives:

  • 7/31/2019 Isentropic process

    27/47

    27

    More derivation.

    For a reversible process:

    TdS = dQ

    Make the substitution for Q in the energyequation:

    PdV+dU=TdS

    Or on a per unit mass basis:

    Pdv+du=Tds

  • 7/31/2019 Isentropic process

    28/47

    28

    Entropy is a property. The Tds expression

    that we just derived expresses entropy interms of other properties. The propertiesare independent of path.We can use theTds equation we just derived to calculatethe entropy change between any twostates:

    Tds = du +Pdv

    Tds = dh - vdP

    Starting with enthalpy, it is possible todevelop a second Tds equation:

    Tds Equations

    L t l k t th t h

  • 7/31/2019 Isentropic process

    29/47

    29

    Lets look at the entropy changefor an incompressible

    substance:

    dT

    T

    )T(cds

    We start with the first Tds equation:

    Tds = cv(T)dT + Pdv

    For incompressible substances, v const, sodv = 0.

    We also know that cv(T) = c(T), so we canwrite:

  • 7/31/2019 Isentropic process

    30/47

    30

    Entropy change of anincompressible substance

    dTT

    )T(css

    2

    1

    T

    T

    12

    1

    212

    T

    Tlncss

    Integrating

    If the specific heat does not vary with

    temperature:

  • 7/31/2019 Isentropic process

    31/47

    31

    TEAMPLAY

    Work Problem 7-48

  • 7/31/2019 Isentropic process

    32/47

    32

    Entropy change for an ideal gas

    dTcdh p And

    dpp

    RTdTcTds p

    Tds = dh - vdp

    Start with 2nd Tds equation

    Remember dh and v for an ideal gas?

    v=RT/p

    Substituting:

  • 7/31/2019 Isentropic process

    33/47

    33

    Change in entropy for an ideal gas

    p

    dpR

    T

    dTcds p

    Dividing through by T,

    Dont forget, cp=cp(T)..a function oftemperature! Integrating yields

    1

    2

    T

    T

    p12pplnR

    TdT)T(css

    2

    1

    Entropy change of an ideal gas

  • 7/31/2019 Isentropic process

    34/47

    34

    Entropy change of an ideal gasfor constant specific heats:

    approximation

    Now, if the temperature range is solimited that c

    p constant (and c

    v

    constant),

    1

    2pp

    T

    Tlnc

    T

    dTc

    1

    2

    1

    2p12

    p

    plnR

    T

    Tlncss

    Entropy change of an ideal gas

  • 7/31/2019 Isentropic process

    35/47

    35

    Entropy change of an ideal gasfor constant specific heats:

    approximation

    Similarly it can be shown from

    Tds = du + pdv

    that

    1

    2

    1

    2v12

    vvlnR

    TTlncss

  • 7/31/2019 Isentropic process