03 cycles aux

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  • 8/3/2019 03 Cycles Aux

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    Independent Properties

    Simple Compressible Pure Substance

    (Absence of motion, gravity, surface, magnetic, electrical effects)

    States are defined by 2 independent properties

    Saturated liquid, saturated vapor are in diff state but at same P, T

    In a saturated state, Press & Temp are dependent

    Superheated vapor: any two of P, v, T define the state

    (P, v), (P, T), (v, T)

    Saturated state: the following would define the state but not (P,T)

    (P, v), (v, T)

    (T, x), (P, x)

    Example

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    P-V-T Behavior of Low- and Moderate-Density Gases

    Ideal

    gas eqn

    (See next slide)

    cr

    cr P

    P

    PT

    T

    T==

    , Generalized compressibility chart

    ZRTPvRTPvZ == OR Compressibility factor: Unity for ideal gas

    22dbcbvv

    abv

    RTP++

    =

    Ideal gas can beassumed at low andmoderate densities(See next slide)

    )(

    ,

    )KJ/mol3145.8(

    ,

    MRR

    RTPvmRTPV

    Mmn

    R

    TRvPTRnPV

    =

    ==

    =

    =

    ==

    Reduced properties

    Cubic eqn of state

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    T-v Diagram of water (Opt)

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    N2

    Compressibility Factor for N2 (Opt)

    Ex. 3.8, 9

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    ==

    =

    =

    ==

    2

    1

    2

    121 PdVWW

    PdVWdVAdx

    PAdxFdxW

    ++

    ::

    ::

    WdV

    WdV

    Work done during the process from state 1 to state 2

    Integral can be:1. Graphically or experimentally evaluated OR2. Analytical evaluated when the P-V relation is

    known

    Work done at the Piston

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    Work = Areain a P-V diagram

    =

    2

    121 PdVW

    Graphical Approach to Find Work

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    During any cycle a system (control mass)undergoes, the cyclic integral of heat is equal to

    the cyclic integral of work.

    = WQ

    The First Law of Thermodynamics for a

    Control Mass Undergoing a Cycle

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    Consider different processes between two states

    = WQ 121

    ,

    BCA

    The First Law of Thermodynamics for a

    Change in State of a Control Mass

    depends only on the initial and finalstates not on the path.

    212112 WQEE

    WQdE

    =

    = Energy = +in out

    ( )WQ

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    Energy is present in various forms

    Net change in energy of the system is exactly

    equal to energy transfer as work or heat

    PEKEUE

    E

    ++=

    ++= EnergyPotentialEnergyKineticEnergyInternal

    mgZPEVVmKE == ,2

    1 rr

    212112

    2

    1

    2

    21212 )(

    2)(

    2

    WQZZmgVVmUUEE

    mgdZVVm

    ddUdE

    =++=

    +

    +=

    rr

    WQPEdKEddUdE =++= )()(

    Conservation of Energy

    m

    Uu

    U

    =

    InternalEnergy

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    Consider a constant-pressure process withnegligible kinetic energy and potential energy

    Enthalpy

    12

    111222

    11221221

    1221

    211221

    )()(

    )(

    HHVPUVPU

    VPVPUUQ

    VVPW

    WUUQ

    =

    ++=

    +=

    =

    +=

    Pvuh

    PVUH

    +=

    +=

    Thermodynamic Property Enthalpy

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    Recap 1st law of thermodynamics

    Definition of specific heats at const volume/press

    For solids and liquids, specific heat reduced to:

    Little volume change

    VdPdHQ

    PdVdUWdUQ

    =

    +=+=

    PPP

    P

    vvv

    v

    T

    h

    T

    H

    mT

    Q

    mC

    T

    u

    T

    U

    mT

    Q

    mC

    =

    =

    =

    =

    =

    =

    11

    11

    CdTdudh

    vdPduPvddudh

    ++= )(

    Specific heat atconst volume

    Specific heat atconst pressure

    pv CCC ==

    Const-Vol, Const-Press Specific Heats

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    In general, ),( vTuu =

    However, for a low-density gas(Confirmed by steam table)

    )(~ Tuu

    It is known for an ideal gas)(Tuu =

    dTmCdU

    dTCdudT

    duC

    v

    v

    v

    0

    0

    0

    =

    =

    =

    Similarly, for an ideal gas)(Thh =

    dTmCdH

    dTCdhdT

    dhC

    p

    p

    p

    0

    0

    0

    =

    =

    =

    Specific heats for an ideal gas:

    RTuPvuh +=+=

    U, H, Cv, Cp of Ideal Gases

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    Specific heats for an ideal gas are also functionsof temperature only like internal energy andenthalpy

    )(),( 0000 TCCTCC ppvv ==

    RCC

    RdTdTCdTC

    RdTdudh

    RTuh

    vp

    vp

    +=

    +=

    +=

    +=

    00

    00

    Specific heats for an ideal gas are related as

    follows:

    Specific Heats of Ideal Gases

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    The inequality of Clausius:

    It is a consequence of the 2nd law ofthermodynamics

    The equality holds for reversible cyclesThe inequality holds for irreversible cycles

    0 TQ

    The Inequality of Clausius: Second Law

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    Consider reversible cycles between 1 and 2

    +

    ==

    1

    2

    2

    10

    BA T

    Q

    T

    Q

    T

    Q

    +

    ==

    1

    2

    2

    10 BC T

    Q

    T

    Q

    T

    Q

    For a cycle along A and B:

    For a cycle along C and B:

    =

    2

    1

    2

    1CA T

    Q

    T

    Q

    We can thus define a property entropy like

    revT

    QdS

    revT

    Q

    : path independent

    =

    2

    112

    revT

    QSS

    Entropy A Property of a System

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    Entropy: Thermodynamic propertyTherefore, once entropy is obtained for areversible process, the same entropy can be

    used for an irreversible as long asthermodynamic states are the same

    Zero entropy

    1. All pure substances in the (hypothetical)ideal-gas state at absolute zero temperaturehave zero entropy

    2. However, it is not practically easy to definezero entropy (or reference state) so anarbitrary state is chosen for zero entropy. Water: Saturated liquid at 0.01C Refrigerants: Saturated liquid at -40C

    Entropy A Property of a System

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    Extensive property S Intensive property s(specific entropy)

    At saturation ( )

    fgf

    gf

    xss

    xssxs

    +=

    += 1

    Temperature-entropy diagramEnthalpy-entropy diagram (Mollier diagram)

    Entropy of a Pure Substance

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    0

    0

    41

    434

    334

    23

    212

    112

    =

    =

    =

    =

    =

    =

    SS

    T

    Q

    T

    QSS

    SS

    T

    Q

    T

    QSS

    Lrev

    Hrev

    Consider a heat engine on the Carnot cycle

    Rev. isothermal

    Isentropic

    Rev. isothermal

    Isentropic

    1-a-b-2-1area

    1-4-3-2-1area==

    H

    netth

    Q

    W

    TH th and TL thSimilar argument for a refrigerator

    (see text)

    Entropy Change in Reversible Processes

    *Isentropic = Rev. adiabatic

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    Consider a reversible heat-transfer process

    12: The change of state from saturated liquid tosaturated vapor at constant pressure

    Heat transfer = area 1-2-b-a-1

    T

    h

    T

    q

    T

    qsss

    fg

    rev

    fg ==

    == 21

    2

    112

    23: The change of state fromsaturated vapor to superheatedvapor at constant pressure

    Heat transfer = area 2-3-c-b-2

    ==3

    2

    3

    232 Tdsqq

    Entropy Change in Reversible Processes

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    Lets start with the Gibbs equation.

    PdvduTds +=

    +=

    +=

    ==

    2

    11

    2012

    0

    0

    ln

    gas,idealFor

    vvRdT

    TCss

    dvv

    RdT

    T

    Cds

    v

    R

    T

    PdTCdu

    v

    v

    v

    =

    =

    2

    1 1

    20

    12

    0

    ln

    P

    PRdT

    T

    Css

    dPPRdT

    TCds

    p

    p

    Similarly,

    Method of integration1. Constant specific heat2. Functional form known

    3. Tabulated Standardentropy

    =T

    T

    p

    T dTT

    Cs

    0

    00

    Entropy Change of An Ideal Gas

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    Entropy Change of An Ideal Gas

    const

    or2

    1

    1

    2

    1

    2

    1

    1

    2

    1

    1

    2

    1

    2

    1

    2

    1

    20

    =

    =

    =

    =

    =

    k

    kk

    k

    k

    C

    R

    Pv

    v

    v

    P

    P

    v

    v

    T

    T

    P

    P

    T

    T

    P

    P

    T

    T p

    In an isentropic process, 0ln2

    11

    20

    12 == PP

    RdTT

    Css

    p

    When the specific heat is constant, this equation becomes:

    0lnln1

    2

    1

    20 =

    PPR

    TTCp

    k

    k

    kC

    CC

    C

    R

    p

    vp

    p

    111

    0

    00

    0

    ==

    =

    Ratio of specific heats is defined as

    0

    0

    v

    p

    C

    Ck=

    Ideal gas equation

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    Reversible Polytropic Process for an Ideal Gasnnn

    VPVPPV 2211const ===Polytropic process:

    nn

    n

    n

    VV

    PP

    TT

    V

    V

    P

    P

    =

    =

    =

    1

    1

    2

    1

    1

    2

    1

    2

    1

    2

    1

    2

    In a polytropic process for an ideal gas,

    n

    TTmR

    n

    VPVP

    dV

    V

    PdVWn

    =

    =

    ==

    1

    )(

    1

    1const

    121122

    2

    1

    2

    1

    21

    for n1