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Raj P. Chhabra

Property Relations and Data

Publication details https://www.routledgehandbooks.com/doi/10.4324/9781315119717-8 Michael J. Moran Published online on: 05 Dec 2017

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Si the specific heats heats specific the found volume, be experimentally. relationships can between mass The and temperature, Pressure, Moran Michael J. 1.3 TS Intr 1.1.14b,From Equation absenceof overall the system in cess effect of the motion reads gravity and formfor reversible aclosed differential system in internally undergoing balance an energy pro- An B tems with volumewith change, Eq

asic milar expressions can be written on aper-mole expressions written be basis. can milar , t uations 1.3.1uations expressed be on aper-unit-mass can basis as Property data are provided in the publications providedthe of in the are data Property oducing enthalpy,

hree additional expressions additional obtained: are hree : systems for which the only significant work in an internally reversible : systems internally foran associated only is whichsignificant in process the work

PROPERTY RELATIONS AND DATA R elations (formerly U.S. the of Bureau Standards), of professional such the groups as Dupont

c for v and and and and () H

δ= P Wp

= () ure c δ= Dow Chemical Dow

U p QT and temperature atrelatively accessible temperature also and low experimen are pressure int

S int

re pV u v re b , the H , the v stances d V d( d UQ elmholtz function, function, elmholtz s , the following, the is obtained: equation . When consideration is limited to to consideration is limited . When , and the the , and . Handbooks and property reference volumes property such and those as . Handbooks d– =δ d– dd dd dd ψ= dd dd dd ψ= UT HT GV uT gv hT = , the , the = = =+ = =+ )( int pV pv American Chemical Society Chemical American d– re d– ps American Society of Heating, Refrigerating, and Refrigerating, Heating, and of Society American Sp sp sv pS SV vi –d −δ –d –d –d sT ST simple compressible systems compressible simple d W d T d d p v ψ T V p

)

nt r

U ev National Institute of Standards and and Standards of Institute National – T S , and the Gibbs function, Gibbs function, the , and simple compressible sys compressible simple , and of corporate enti of corporate , and , which include American (1.3.2d) (1.3.2b) (1.3.2a) G (1.3.2c) (1.3.1d) (1.3.1b) (1.3.1a) (1.3.1c)

H 23 – - - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 respectively, form of the functions ond mixed partial derivatives are equal: ( derivatives equal: are partial ond mixed Maxwell relations Maxwell 24 1.3.2 is an involved, are expressionsSince only four properties of each differential the given by Equations M axwell Solution Derive Maxwell the relation following Equation from 1.3.2a. Example 1.3.1 z General: Function Differentials Exact Relations from TABLE g Gibbs function: ψ Helmholtz function: h Enthalpy: u Internal energy:

= ( ( ( ( R T s s v : The differential: The function of the exact , , , , z elations p v ( T p ) x ) ) ) , y 1.3.1 ) differential exhibiting the general formd general exhibiting the differential given in Table d d d d d g ψ h u z 1.3.1 established. be can = = = = =– u M v T T ( d d d s p dd ( Differential p , x s s d u – + – , v v y – ), ), = )d s p v ∂M d d d s    h u x T d v p + ( ∂

∂ T = s u / s ,

∂y u N    p ( ( s vs x ), ), ) , ,

= y v ψ s

)d ( ) is ∂N + ( y v ,    / T ∂ ∂ ∂x u v ), and )    . Underlying these exact differentials are, . Underlying are, exact these differentials CRC Handbook of Thermal Engineering of Thermal Handbook CRC                   ofiinsMaxwell Coefficients             z d ∂ ∂ ∂ ∂ψ ∂ψ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ v = u T T u y x z h s z g v p h s g v p

g M                               ( s p x v y s T p T v T ( =− ======− x =− = , =− T , v T N M p v p y s ). From such considerations the s p )d x

N (                x , ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ T T T T M v v y p p y )d                p vT s sv y x = =− = =− , whe =          ∂ ∂ ∂ ∂       ∂ v v s ∂ s ∂ ∂ ∂ ∂ N x       p s p s re the sec the re p          y T - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 Specific heatratio Compressibility factor Specific Gibbsfunction Specific Helmholtzfunction Specific enthalpy Specific entropy Specific internalenergy Specific volume Temperature Pressure Property Properties Selected for Definitions Symbols and TABLE among these properties. properties. these among relations and properties of thermodynamic awide uses assortment thermodynamics Engineering S u functions derivations the of as partial defined are analysis and for thermodynamic required often Engineering Thermodynamics Engineering ( pecific T of derivative property relationsderivative derived can be (see, property e.g., Zemansky, 1972). can alsofollowing as regarded be this from single differential expression. Several additional first- simply fourth obtainedcan the be from by manipulation, relations 16 the in property fourThese relations are identified in Since each properties of the In Equation 1.3.2a, By comparison with Equation 1.3.2a, Among the entries in in entries the Among mixed partial derivatives gives Maxwell the relation, , v ) and ) and Table 1.3.2

H 1.3.1, four relations additional obtained can be by property equating such expressions: eats h ( T ,

p and ), respectively, ),

O T ther plays role the of Table Table yblDefinition Symbol

P ψ Z T p g h u k v s 1.3.2 properties. several encountered lists commonly roperties T 1.3.2 heats specific the are ,       p ∂ ∂ ∂ ∂ , u p h s v , and s appears on the right on the , and sappears side eight of the of two coefficients       T vp sT = = pv up = Ts u Ts h cc Table + pv – – M          /    / RT ∂ ∂ ∂ ∂ ∂ ∂ , and – ∂ ∂ v h u s pT g s T v          1.3.1 by brackets. any of Equations As three 1.3.2    vs s , , , =− Velocity ofsound Joule coefficient Joule–Thomson coefficient Isentropic bulk modulus Isothermal bulk modulus Isentropic compressibility Isothermal compressivity Volume expansivity Specific heat,constantpressure Specific heat,constant volume p −= plays role the of          p ∂ψ ∂ ∂ ∂ ∂ ∂ u vv p s             sT v v ∂ ∂ = u v Property = c    v       and and ∂ψ ∂ ∂ ∂ T g       c p p . These intensive properties are intensive are . These properties N , so the equality the , so of second yblDefinition Symbol B μ c c B α η κ β c p v J s Table − − −∂ −∂ − 1 v () () () () vp vp 1 vp v 1 v ∂∂ ∂∂ ∂∂ ∂∂ () 1.3.1 1.3.1 2 () () hT uT Tp () ∂∂ Tv () ∂∂ () ∂∂ ∂∂ vT / / vp vp ∂ ∂ v v u h p v v 25 p s T T s s Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 k 26 Si heats can be similarly expressed. similarly be can heats , the specific heat ratio, heat is specific , the nce nce u and and h can be expressed either on a unit mass basis or aper-mole expressed mass be either on aunit can basis, values specific of the       cT cc cT c c Specific HeatSpecific Relations TABLE Note: k v v p pv p ∂ ∂ == ∂ ∂ =− = = −= = c = = c v p v p c c    T    η 1 µ    v p   

1 T ∂ ∂ ∂ T ∂    J T    = =− T Engineering Thermodynamics See, forexample, Moran,M.J.andShapiro,H.N. u T    h ∂     = ∂ 1.3.3 =− T Tv       T ∂ ∂       v T T T vv ∂    κ ∂ p pp    ∂ ∂    β T    = v p    ∂ ∂ T =    ∂ ps p 2 ∂ T ∂ ∂       ∂ vs T p T    ∂ v T    T T 2    T v ∂ ∂    ∂ ∂ v       p    T    2 ∂ T ∂ 2 vp    − p p ∂ ∂ T v ∂       v ∂ ∂ 2 2 ∂ p T −    s T p v s ∂ v    p       Table ∂    v    T p v       ∂ ∂     s    v p    T 1.3.3 summarizes relations involving 1.3.3 summarizes c c p v k = = , 4thed., Wiley, New York, 2000. =       ∂ ∂ ∂ ∂ c c T T u h v p

      v p

(12) (11) (10) (1) (9) (8) (7) (6) (5) (4) (3) (2) CRC Handbook of Thermal Engineering of Thermal Handbook CRC Fundamentals of c v and and c p . The property property . The (1.3.5) (1.3.3) (1.3.4) Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 equation of state of equation well. as by exhibited gases other which is acharacteristic perature, limit as pressure tends to zero (the ideal gas limit). In this limit (the limit zero to tends limit). gas ideal this pressure as In limit Values for The ally important gases and liquids. These data can be represented in the form the in represented be can data liquids. These and gases important ally have industri for data accumulated been Considerable specific volume,temperature pressure, and P gases exhibit similar behavior. exhibitgases The similar Engineering Thermodynamics Engineering Figure discussions of the Section 1.3 of the apart as inalsoconsidered are reference 1references. Chapter volumes heats the Specific among listed property and provided by handbooks solids for are the and gases, liquids, common data heat Specific measurements. exacting property through macroscopically determined be also can They – v

Introducing Maxwell the relation from The following convenientThe often two are equations relations for properties: among establishing Applying Equation 1.3.6a to each of (∂ ing to to specific volumerespect at fixedtemperature, and again usingMaxwell the relation correspond Example 1.3.3. Equation 11 of With this, Equation 2of

Solution Obtain Equations 2and 11 in Example 1.3.2 ir use is illustrated in Example1.3.2. in is use illustrated ir – T 1.3.1 shows how

R ψ elations . c :

v Identifying Identifying and and . Equations of state can be expressed in tabular, graphical, and analytical forms. analytical and graphical, tabular, expressed be in can of state . Equations c p can be obtained via statistical mechanics using mechanics statistical via obtained be can x , c y p for water vapor varies as a function of temperature and pressure. Other Other pressure. and of for temperature afunction wateras vapor varies , Table    z with with ∂ ∂ T s Table    1.3.3 is obtained Equation from 1, which in turn is obtained in v Table s =− , T    1.3.3 Equation from 1. figure , and       ∂ () ∂ 1.3.3 obtained can be by differentiating Equation 1with ∂∂ ∂ ∂ y z ∂ T ∂ Tv T v Table T    s /∂          xy ∂ ∂ v v sT also gives the variation of gives also variation the vs    ) x y , respectively, Equation 1.3.6b reads    s =− and (∂ and ∂    ∂ 1.3.1 corresponding to ∂ sT ∂ 1 v () x z z s ∂∂             vs incompressible model incompressible ∂ ∂ ∂ ∂    T    x v y v ∂ ∂    ∂ ∂    /∂ T s z x y s       =    ) =− v T ∂ ∂ z , 1 T p =− =−

      v ∂ ∂ 1 T v 1

   sT c    p ∂ ∂ increases with increasing tem increasing with increases v s ψ spectroscopic    ( T , c and the the and v p ), with temperature in the the in temperature with p

= ideal gas model gas ideal

f ( measurements. measurements. v , T ), ca (1.3.6b) (1.3.6a) lled an an lled - 27 - - . Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 28 FIGURE et

al., al., Steam Steam 1.3.1 Table

c s p of water vapor as a function of temperature and pressure. (Adapted from Keenan, J.H. Keenan, from (Adapted pressure. and of temperature afunction of as vapor water —

S.I. Units (English Units) (English Units S.I. cp Btu/lb·°R cp kJ/kg·K 1.5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2 3 4 5 6 7 8 9 0 0 0 40 300 200 100 0 0 0 800 600 400 200 0

0 200

Saturated vapor 1

500 Saturated vapor Ze 2

ro pre

, John Wiley & Sons, New Wiley &Sons, , John York, 1978.) 1000 Ze 5

ssure limit

ro pre 1500

2000 Ibf/in. 2000 10

ssure limit 2

15 3000 °C °F

70 60

20 80

4000 100 90

1000

060 00 25

5000 30

CRC Handbook of Thermal Engineering of Thermal Handbook CRC 6000

8000 201400 1200 MPa 50

10,000 15,000 lbf/in. 15,000 0800 00 100 MPa

90 80 70 60 2 1600 Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 the pressure is the is the pressure the atthe meet vapor line saturated the and liquidline saturated by denoted liquid–vapor two-phase region.state region the The and by denoted state The liquid line. liquid–vapor two-phase region saturated the is the and liquid phase the rating lines tion regions two-phase are regions single-phase the from the separating lines The equilibrium. a single single phase. Between the te p of afunction graph The P FIGURE Engineering Thermodynamics Engineering the corresponding temperature is called the the is called temperature corresponding the pressure, For aspecified temperature. of constant aline is also pressure ofregion, constant aline present.Accordingly, liquid–vapor are two-phase phases long as the both in as constant remains (Sectionscale 1.1,Entropy). Law The BasicSecond Thermodynamics, Definitions; and of Concepts the Kelvintemperature of point defining triple waterin is used The diagram. phase onpoint the labeled line along the equilibrium coexist in can phases – – mperature plane, called the the plane, called mperature v v Figure When a phase change occurs during constant pressure heating or cooling, the temperature temperature or cooling, the heating pressure constant during change aphase occurs When – – T T

relationship for water. 1.3.2 S urface . Any state represented by a point on a saturation line is a line by on apoint represented asaturation . Anystate 1.3.2 has three regions labeled solid, liquid, and vapor where the substance exists only in exists only 1.3.2 in regionsvapor substance where labeled and solid,the liquid, three has f

(Figure 1.3.2c) is a saturated liquid state. The saturated vapor line separates the vapor the separates vapor 1.3.2c) line (Figure saturated The liquidstate. is asaturated

(b Pr Pressure ) essure-specific volume–temperature surface and projections for projections and water scale).to (not surface volume–temperature essure-specific critical pressure p pressure critical Solid S S (a) p V L

= Te

phase diagramphase f Tr ( mp Liquid

Pressure Figure v iple point , erature

L Speciﬁc volume Speciﬁc T - ) is as Solid V phase regionsphase lie Critical 1.3.2b shows pressure– onto the projection surface of the the point Va

c Solid–vapor is the temperature the , and po Liquid urface in three-dimensional space. three-dimensional in urface r

. The projection onto. The the

Triple line Triple

Liquid – Liquid saturation temperature (c) vapor f Pressure

g Vapor two-phase point Critical

Solid Temperature triple line triple Tr Sp – va f Liquid iple line ec Critical po point iﬁc volume T regions, where two coexist in phases r S c critical point critical olid–va g p – . The triple line projects onto projects a line triple . The saturation state g v critical temperature T temperature critical is a saturated vapor state. The vapor The state. is asaturated . For a specified temperature, temperature, . For aspecified plane is shown plane in po r Va po r Figure . At the critical point, point, critical . At the T T T c > < T T 1.3.2 shows the . The line sepa line . The c c

Figure c . Three . Three satura

1.3.2c. 29 - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 from solid vapor to from (sublimation):

respectively. Dividing by the total mass of the mixture respectively. mixture of mass the by total Dividing the vapor in the mixture, mixture, vapor the in entropy: V volumeis vapor. total of vapor The anyis such a saturated phase the mixture liquid and rated for pressure its temperature. saturation the as is known line liquid for left region saturated of the to itsthe The pressure. temperature saturation the than greater the as is known vapor line the is called pressure corresponding the 30 FIGURE of case For water, the where v of the mixture is mixture of the

= When a mixture of liquid and vapor coexists in equilibrium, the liquid phase is a satu liquid phase the equilibrium, of vapor in liquid and coexists a mixture When

V f

+ v

V 1.3.3 fg g

= ; or, alternatively,

v g

−

v Ph f . Expressions similar in form can be written for internal energy, for enthalpy, internal written be formcan in and . Expressions similar ase diagram for (not water scale). to diagram ase compressed liquid region liquid compressed Figure m g / m , be symbolized by symbolized , be 1.3.3 solid change liquid(melting): phase to from the illustrates

Pressure region vapor superheated mv Solid Sublimation

a = a a ΄

m ′ – f b v ′ bc f –

+ c saturation pressure ux hx

sx ′ m Melting vx ; and from liquid to vapor liquidto (vaporization): from ; and b =+ =− =+ =− =+ =− ΄ =+ =− g Liqui v () ux () () hx sx () vx g 1 1 1 1 Te ff ff , where ff ff Tr because the liquid is at a pressure higher than the the than higher liquidis atapressure the because mp d x iple point c , called the the , called ΄ v h s u eratur ux vx hx sx g g g g fg fg fg fg + + a + + m ˝ because the vapor exists at a temperature vapor exists the atatemperature because e and and v h s u b CRC Handbook of Thermal Engineering of Thermal Handbook CRC

Va . The region to the right of the saturated saturated of right region the the to . The Va and letting the the letting and quality po v po denote mass and specific specific volume, and mass denote riza po Critical c r ˝ int tion , the apparent specific specific volume apparent , the mass fraction mass a ″ – b ″ – c ″ . During During . (1 (1 a (1 (1 of the – .3.7b) .3.7d) .3.7a) .3.7c) b – c - ; Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 reads equation Clapeyron Engineering Thermodynamics Engineering whe evaluated from be to temperature The properties. not independent are thus and constant remain pressure and any such temperature change phase the et Thermodynamics FIGURE in diagram conventionallyare employed. include These the several selections are that there may used, gravity. motionwith be any and such pair While excluding associated coordinates, any the twoproperties intensive as independent properties intensiveThe of simple with apure, states compressible graphically represented system be can G for few these is negative, for substances water in illustrated as curve solid–liquid on melting. specific The volumeslope the the saturated decreases substances, of (d and increases, also enthalpy, specific in the specific increase volume placean with when changecases, aphase takes In change.the most signsphase specific ofon and the volume enthalpy accompanying the changes 1.3.8 for melting. and sublimation written be can Equation formto in change. phase Expressions the similar held during constant by temperature the

raphical al., al., T (°C) depends diagram on aphase shows equation Clapeyron slope line the The of asaturation that re (dre 100 200 300 400 500 600 700 800 NBS/NRC Steam 0 23 1.3.4 23 p /d

T Figure 800 R )

x epresentations

= 10% sat Te , Prentice-Hall, Englewood Cliffs, NJ, 1996 based on data and formulations from Haar, L. L. Haar, from formulations and NJ, on data 1996 Englewood based Cliffs, , Prentice-Hall, is the slope of the saturation pressure–temperature curve at the point determined determined point atthe curve pressure–temperature slope is the saturation of the Clapeyron equation

20% mperature–entropy diagram for R.E., water. Jones, J.B. (From Dugan, and diagram mperature–entropy

1.3.4, the 1000

Table 1200 p

30% /d T )

s p = 3000 MPa sat

. Hemisphere, Washington, DC, 1984.) DC, Washington, . Hemisphere, 40% 1400 is positive. However, meltingof case the of a few the ice and in other 45 45 h – s

50% 1000 in (Mollier) diagram p

– 1600

allows the change in enthalpy during a phase change aphase atfixed allows enthalpy during change in the

v 60%

300 –

h h    = 2600 kJ/kg 2600 = T

data pertaining to the phase change. For vaporization, the change. phase the For the to vaporization, pertaining data

d d 1800

T p 70%   

100 sa t

= 80%

s (kJ/kg·K) liquid Saturated 2000 Tv p 67 67

()

–

x gf = 90% = T gf − and and

30 −

0.01 Figure

v h = 2200 kJ/kg 2200 = h

10 p

– v Figure 1.3.5, the and diagrams in in diagrams

2400 υ = 0.10 m3 3 1.3.3.

p = 1 MPa /kg 2800 h = 3200kJ/kg 89

3000 2600 3400 Figure p 3600

0.3 – h

3800 1.0 0.1 diagram diagram 4000 0.003 1.3.2, the

Engineering Engineering p = 0.001 MPa 0.01

0.0003 0.03 shown shown (1.3.8)

10 10 10 0 100 200 300 400 500 600 700 800 T 31 – s

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 of the coordinates. of the temperature, respectively.temperature, Values of 32 In t In reduced shown in chart ibility The C in FIGURE

and from Haar, L. et al., al., et L. Haar, from Engineering Thermodynamics ompressi Figure

hese expressions,hese h (kJ/kg·K) 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000 p – v 1.3.5

pressure,

– 45 45 600°C T 1.3.6. considered next compressibility factor one the use as compressibility charts The b relation for a wide range of common gases is illustrated by the generalized compress relationby generalized the for awideis ofillustrated gases range common

ility p

= 1000 MPa υ = 0.001 m 700°C 3 /kg

C p 800°C NBS harts thalpy–entropy (Mollier) diagram for R.E., water. Jones, J.B. (From Dugan, and (Mollier) diagram thalpy–entropy R , reduced

Figure / 900°C

is the universal gas constant, and and constant, gas universal is the NRC Steam Steam NRC 80% , Prentice-Hall, Englewood Cliffs, NJ, 1996 based on data and formulations formulations and NJ, ondata 1996 Englewood based Cliffs, , Prentice-Hall, 1.3.7. compressibility factor, the chart, this In temperature, temperature, p

R 80% 300 == Table p 67 67 p p c c

and and ,, s x , Hemisphere, Washington, DC, 1984.) DC, Washington, , Hemisphere,

= 90% = 0.01 100 T R T T R Z c , and , and are given are for several in substances s = T (kJ/kg·K) T c RT pv pseudoreduced v

R ′ 10 = CRC Handbook of Thermal Engineering of Thermal Handbook CRC p ()

c 0.10 RT and and cc v 3 200°C 89 89 T p c

denote the critical pressure and and pressure critical the denote p specific specific volume, = 1 MPa

0.003 300°C

500°C 1.0 T 400°C 600°C = 100°C

Z 0.3 , is plotted versus, is plotted the 700°C

0.1 Table p v = 0.001 MPaυ = 100m 10 m 3 R ′ /kg , where 3 A.9. The /kg 0.01 0.03 (1.3.10) 10 10 (1 2600 2800 3000 3200 2000 2200 2400 3400 3600 3800 4000 .3.9) - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 (From FIGURE FIGURE Engineering Thermodynamics Engineering et Thermodynamics

al., al., p (MPa)

Compressibility factor, Z = pvlRT 1000 0.01 0.02 0.04 0.06 0.08 1.20 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.00 1.10 100 200 400 600 800 0.1 0.2 0.4 0.6 0.8 10 20 40 60 80 .1 NBS/NRC Steam

2 4 6 8 Obert, E.F. Obert, 0 0 0 1.3.7 1.3.6

3.00 s = 0.5 kJ/kg·K T = 0°C 2.00 0.001

1.60 1.0 1.40 1.0

1.20 L. Haar, from formulations and NJ, on data 1996 Englewood based Cliffs, , Prentice-Hall, Pr 100°C Concepts of Thermodynamics Ge 1.00 1.5

0.90 for R.E., water. Jones, J.B. (From Dugan, and diagram essure–enthalpy neralized compressibility chart chart compressibility neralized T 0.80 R =1.00 0.70 2.0 Table T 0.60 x = 10% 1000 1000 200°C R

2.0 =0.50 1.05 2.5 0.45 20% s , Hemisphere, Washington, DC, 1984.) DC, Washington, , Hemisphere, 1.10 0.40 3.0 0.35 30% 300°C

1.15 3.0 0.30 T 40%

=100°C 3.5 1.20

0.25 50% 200°C

R 300°C

= 0.20 2000 2000 1.30 60% 4.0 4.0

. McGraw-Hill, New York,. McGraw-Hill, 1960.) 0.002 Reduc

1.40 70% 400°C h 1.50 80% (kJ/kg) ed ()

5. 4.5 pressure ( TT

06 1.60 x = 90% RC

== 5.0

1.80 υ = 0.00125 m

2.00 5.5 Joule- omson inversion 3000 Tp

3000

P 6.0 500°C T R

R ) 07. .0 , = 3.50

2.50 6.5 600°C 7.0 υ = 10 m 7.5

3 RC

/kg 700°C

T pp

R /kg s 800°C =15.0 = 8.0 kJ/kg·K 10.00 7.00 5.00 08 , vv ′ RC

= 900°C 4000 4000

υ = 0.01 m

09. .0 9.0 8.5 1000°C

9.5

3 T

/kg

C 10.5 10.0 1100°C

for for = 110 kJ/kg·K 01 Engineering Engineering

T 0.10 1200°C =

1.0 p 5000 5000 R . ≤ 0. 1000 0.04 0.06 0.08 0.1 0.2 0.4 0.6 0.8 .1 2 4 6 8 10 20 40 60 80 100 200 400 600 800 0.01 0.02

10 33 0

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 ity often may be approximated by expressing the equation in terms of the compressibility of factor the by terms expressing in may equation approximated ity often be the be expected to represent the representthe to expected be of or density, pressure, interval by can an giving equation indicated where is the often applicability of realm The states. phase, at least qualitatively. particular to Every is ofrestricted equation state cal in character. Most are developed character.Most in are cal for the but gases, some describe empiri mainly are and physical way significance the exhibit in equations of little fundamental these general, In series. behavior yet and of avoid substances virial complexities afull the in inherent result is a this first few be the accurately can coefficients way, ofOnly found interest. realms however, the and principle,in be simple by found, also molecules. can coefficientsfitting The first fewonly only the andcoefficients far calculated gases consistingfor have of been relatively consideration derived from force ofthe molecules.Thus mechanics the fieldsaround statistical from called are

greater accuracy, variables other than accuracy, than other greater variables including the critical compressibility factor as an independent variable: independent compressibility factor an as critical including the restricting the correlation to substances having essentially the same same havingthe essentially substances to correlation the restricting removed be can by of inaccuracy source This point. critical of the vicinity the in is inaccurate chart 0.23 from 0.33, to varies compressibility factor actually for the substances critical different the As isof atmostorder 5% on the forchart and most is much ranges less. deviation developing the of of in values those observed 30used For gases the from the chart, the shown in isotherms reduced 34 * expressions as known These are in is aseries other the and expansionseries pressure, in enjoy basis. givesthat One infinite compressibility factor an the as a theoretical sions written be can of intervals at least for equation, expressed an certain factor be as ibility might shown in isotherms the Considering E a asubstitute as for used not be should compressibility data consideration, essential generalized able is an accuracy. accuracy When or equation) tabular, allows (graphical, form(see,1966) equation in and e.g., Reynolds, 1979). any form in data of use generalized The factor acentric

quations using using To determine To determine table Over 100 equations of state have been developed in an attempt to portray accurately the Over the accurately 100 have of equations state developed portray to been attempt an in Figure Generalized compressibility data are also available in tabular form(see, available also tabular e.g., in are compressibility data Reid Sherwood, and Generalized T , or an equation of equation state. , or an R = 1.3.7 gives value acommon of about point. 0.27 for compressibility critical factor the atthe truncated virial coefficients virial T

/( of T Z c for hydrogen, helium, and neon above a neon and forhelium, hydrogen, + 8) and +8) and

(see, e.g., Reid 1966). Sherwood, and S tate equation valid only at certain states. only atcertain valid equation P R = p 1 /( . In principle, the virial coefficients can be calculated using expressionsbe calculated coefficients can virial principle, the . In p p Figure v Z – c + 8), where temperatures are in in are +8), temperatures where , p v =+ – – Z T v virial expansions virial 1 – =+ behavior faithfully. When it is not stated, the realm of applicabil realm behavior the it faithfully. is not When stated, 1.3.7 severalof data the gases. to fitted curves best represent the T 1 BT data for agiven data provided as by substance computer software, Figure ˆˆ Z )( () () () c p have been proposed as a third parameter—for example, the parameter—for have athird as proposed been BT , v )( () () () v pC , and , and 1.3.7, it is plausible compress of the variation the that ++ T ++ R of 5, the reduced temperature and pressure should be be should pressure and temperature reduced of 5, the T CT Tp for evaluated simply be to gases reason with and v , and the coefficients the , and 23 23 K DT DT ˆ and pressures are in atm. in are pressures and CRC Handbook of Thermal Engineering of Thermal Handbook CRC v + p + Z * c values, which is equivalent to p – Z BC v ˆ

= , –

f p T ˆ ( – behavior liquid of the , T D v R ˆ p – , , and and T … p data in particular particular in data R and and , Z T . Two expres c ). To achieve B , C cal , p culated culated D, – v … – Z T - - - - -

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 Webb–Rubin explicit is also equation pressure, in stants, has been successful in predicting the the predicting in successful been has stants, For example,constants. Benedict–Webb–Rubin the which involves equation, eight adjus of number adjus agreater requires (or normally pressure accuracy density) Increased increases. mole fraction Engineering Thermodynamics Engineering Va thus specific is andvolume and It of gives temperature afunction as pressure of equations state. Redlich–Kwong two-constant of best is consideredthe by the equation be to many literature. the from compressibility with data comparing or compressibility chart result on ageneralized the superimposing and properties, reduced the and The The evaluation for ofSystems, property multicomponent presented. aspects some general systems are Section 1.3, in is discussed Model. Section 1.3, Gas In Ideal of mixtures gas Multicomponent ideal case The plausible for to obtain gas mixtures. estimates relations adapted for components are pure for methods evaluating section, the this In data. values empirical with property predicted ing by established compar be can technique of of any particular validity physical realm principles. The not fundamental derived from are and character in empirical are properties mixture for predicting however. of mixtures, available properties the are for Mostsuch Means air. techniques predicting as cases special only in reported are of mixtures relative properties the the present, amounts varying agiven from formed be components set of by pure can of mixtures variety unlimited Since an G the representing have having equations 50 or more developed constants and been literature, for engineering the in Kessler and by (1975). is Lee discussed 12 constants found be can equations Many multiconstant 1967).e.g., Goldfrank, the Benedict–Webb–Rubin and Amodification of Cooper involvingequation having severalstate adjus accuracy. better achievingA.9.aim of the equationhave with the of proposed been Modified forms Valuesarguments. for Redlich–Kwong the provided for in several are constants substances of molecular noterms with rigorous justificationin nature, in empirical is primarily equation This weights,molecular such that weight to unity. molecular apparent is equal The as lues of the Benedict–Webb–Rubin constants for various gases are provided in the literature (see,lues literature Benedict–Webb–Rubin of providedthe the in for are gases various constants Equations of state can be classified be the adjusof number can ofby state Equations The total number of moles of mixture, of number moles total of mixture, The of some equations than Redlich–Kwong better two-constant performs the equation Although

r M elative ixtures p amounts of the components present can be described in terms of terms in described be components of present can the amounts =+ y i RT of component v p – v –    T BRT behavior of different substances. different of behavior table −− A i constants, two-constant equations tend to be limited in accuracy as as accuracy in limited be to tend equations two-constant constants, is is Tv y C i n 22

=    p =+

n = i 1 nn / n 12 n vb . The sum of the mole fractions of all components present mole of of sum the all fractions . The + RT , My − is () bR p the sum of the number of number of moles sum the the components, of the = – ++ − v Ta v ∑ i – = 36 vv j − 1 T M () behavior of is the mole average is the fraction componentof the ii nn M + + ji a bT = a

v α ∑ i = j 12 1 ++ vT

32 light hydrocarbons c table    1e constants they involve. they constants The vv γ 22    explicit xp mole fractions mole    − . The Benedict– . The in pressure: in γ   

table (1.3.12) (1.3.13) (1.3.14) (1 p Table . The . The .3.11) con – table v – 35 n T i - - :

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 compo rule the to According literature. engineering found the Several in rules. are mixture m

as tively pressure whe whe whe The The 36 whe whe Using Using pressure n pressure the among from quantity unknown constants developed values rules of combining for Forthe equation. example, the empirical with mixture are of equation state the in appearing constants The overall mixture. of volumes components: occupiedby individual the component that assuming mined as tively volume requiring only the determination of a mole fraction averaged critical temperature of amole averaged temperature fraction critical determination only the requiring rules. by one of calculated several mixture properties literature. found the also in are of equations state other in values for constants the mixture of the gas mixture can then be obtained by solving obtained be then can mixture gas of the i / m Additional means for predicting the the forAdditional predicting means The The The The Another approach is to regard the mixture as if it if were as component asingle having pure critical mixture the approach is regard to Another , the pressure of a gas mixture is expressible by individual exerted of the a sum of pressures as mixture agas pressure , the re the volumes the re re pressures the re re re relative , where T nents a Z T additive volume rule volume additive c p V i c is the compressibility factor of the mixture and the compressibility factors the and compressibility factor is mixture the of the and and and and ,i p – p and temperature temperature and and and a c v and temperature temperature and : and and – : amounts of the components present also can be described in terms of terms in described be can components of present also the amounts T m p b relation for a gas mixture can be estimated by applying an equation of state to the by the to of applying equation state an estimated be can relation for mixture agas c i , the mixture compressibility factor mixture , the is the mass of mass component is the i p are the values of the constants for values component constants the of the are b c ,i for use in the Redlich–Kwong equation are obtained using relations form of Redlich–Kwong for the the obtained in use are equation are the critical temperature and critical pressure of component pressure critical and temperature critical the are V p 1 1 , , V p 2 2 , etc. , , etc. , T postulates that the volume the that postulates T of the mixture. The additive pressure rule can be expressed be alterna can additive rule The pressure mixture. of the of the mixture. The additive expressed The be alterna volume can rule mixture. of the , , are evaluated by considering the respective components to be at the respective evaluated by atthe components the be to considering are are evaluated by considering the respective components to be at the respective evaluated by atthe components the be to considering are i a occupies the entire volume of the mixture at the temperature temperature atthe volume entire occupies mixture the of the T cc = ==    V pp ∑∑ ∑∑ i i == == =+ j j =+ 1 1 Z p Z i and and VV ya – yT = = 12 i ii 12 v p – ∑ i 12 , volume ∑ i T i , = = m j j pp , 1 relation of a mixture are provided by empirical provided by are empirical relation of amixture 1    is the total mass of mass mixture. total is the 2 yZ yZ + + Z ii , ii py is obtained as for as component. asingle The pure is obtained V cc Z 3, by 3,         … TV … V pT Kay’s rule = V = , , , temperature , temperature i ] of a gas mixture is expressible sum the of as mixture agas ]

CRC Handbook of Thermal Engineering of Thermal Handbook CRC pV 1 pT TV i j 1

ii p nR ii b , i . Combination rules for obtaining for. Combination obtaining rules

T is perhaps the simplest the ofis perhaps these, . T , and total number of number moles total , and mixture values mixture additive pressure pressure additive mass fractions: fractions: mass i T , respectively. c and critical critical and Z determined determined i are deter are (1.3.17b) (1.3.18b) (1.3.18a) (1.3.17a) (1.3.15) (1.3.16) T . - - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8

volume is known. explicit of equation state when approach is expedited an specific in analytical or analytically. The of temperature at some specific atsome specific known.volume is of temperature is known, pressure 1-a: Since temperature is constant at is constant 1-a: Since temperature be found explicit ofbe equation state when an in Engineering Thermodynamics Engineering on perature and pressure, and temperature and specific specific volume. and temperature and pressure, and perature choices variables: independent of tem the of particular sions terms changes in for such property of regions. single-phase states between determined be entropy can and and heat Using specific appropriate E compressibility factorswhere the for example, requires the variation of variation for the example, requires expressions, these Adding result is the at is constant b-2: Since temperature a-b: at is constant Since pressure knowledge of variation of requires the This gram. dia property accompanying shown steps the in three using the determined be 2can 1and states

performing the integrals is expedited when an equation of state explicit of equation when state is known. is expedited an pressure in integrals the performing

Equation 1.3.19.Equation available: one information found on the The by depends integration p variation of [ variation tion of [ and temperature temperature and valuating ly one of Equations 3 Equations Similar considerations apply 4of 2to Equations to Similar Example 1.3.3 Solution Obtain Equation 1of Taking Equation 1ofTaking Equation As changes in specific enthalpy and internal energy are related through through related are energy internal and enthalpy specific changesAs in T ( ∂p . With Equation 1.3.3 and Maxwell the relation corresponding to

∆ / v h ∂T – ′ 2 h and 4 and

− , ) T

v h ∆ – ( 1 ∂v T u u and and of the mixture. of the 2 p , ′ /

of of − ∂T ] with specific volume at temperatures specific ] with volumetemperatures at and

u Table ) Table Table 1 u would found explicit of be equation state when an in p ] with pressure at temperatures attemperatures pressure ] with 2

∆ −

1.3.3 and Equations 3and 4of u s 1.3.4 respectively, become, 1.3.4between enthalpy specific change a representative in as the case, 1 need be found by integration. The other can be evaluated from evaluated from be can found other be by The need integration. h p Z hh hh 21 ′ i 2b , the second integral of Equation 1 vanishes, and of1vanishes, Equation integral second , the a1 are determined assuming that component that assuming determined are –– T −= −= p T c 1 h hu , the first integral of Equation 1 in Equation 1 of integral first , the – h v 2 ba with temperature at a fixed atafixed specific volume temperature with , the first integral of Equation 1 vanishes, and Equation 1 of vanishes, integral first , the v 2 −= =+ – – T hc () h p ∫ data, the changes in specific enthalpy, energy, specific changes in the internal data, p ∫ p p 1 1 2 ′ ′ 21 . The required integrals may be performed numerically numerically may performed be integrals required . The     vT vT T T ∫ v −∂ 1 2 up −∂ and and c p p () () () with temperature at a fixed pressure pressure atafixed temperature with Tp Table () c vT vT ,d p as a function of temperature atsome fixed of temperature afunction as 22 ∂ ∂ vp ′ T 1.3.4. To evaluate 1 – Table and and p T p     T 11 d d 1 v and and p p T 1.3.4. 2

: T 2 Table . An analytical approach to analytical . An Table ψ ( h T u

1.3.4 provides expres , = 1.3.4 and vanishes, i 2 v p exists at the pressure pressure exists atthe

− ) from ) from and and

1 pv with Equation with v ′ by c , and the varia the , and v Table as a function afunction as h 2

− p

h ′ 1.3.1, 1.3.1, , and the the , and 1 (1.3.19) would would 37

3, 3, - - - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 Property diagram: Property expressions: Preferred data: Independent Properties: ∆ TABLE h, h, ∆ u, u, 1.3.4 ∆ s Expressions h v s ( ( ( T,p T T , dd dd ∆= ∆= , p p h s p ): hc s p ΄ ), ): = = Temperature andPressure c       cv c ∫∫ T p ( p ∂ p ∂ ∂ ∂ T c T T T s h , p p    p    dd dd ) p Tv p T 1 a     +− − T T − − ∫∫ + + c T           p       ∂ (    ∂ ∂ ∂ ∂ T ∂ ∂ ∂ T v T ∂ v ∂ p s h , p T T p    v          ΄ p )       T p T p ∂ d ∂ d     T p p v p    p 2 b     T p (2) (2 (1) (1 ′ ′ ) ) u s p ( ( ( T,v T T , , dd dd ∆= ∆= Temperature andSpecific Volume T v v ): u s uc s ): ), = = c       v c cT T ∫∫ ( v v ∂ T ∂ ∂ ∂ c T T T s , 1 u v ν v    dd    dd ) vT TT vT T    + 2 T + T       ∫∫ + + ∂ ∂ ∂ ∂          T    T p p ∂ ∂ ∂ ∂ ∂ ∂          T v s p u v v v ∂ ∂          − T p v d    p d v v v    v υ − ΄ a b pv c υ    ( υ T , υ ΄ ) (4 (3 (4) (3)

′ ′ ) )

CRC Handbook of Thermal Engineering Thermal of Handbook CRC 38 Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 thermodynamic functions. thermodynamic the specific Gibbs function is found from the definition the Gibbsfunction is from specific found the Table p properties the combination. function by Taking and the function differentiation calculated from representations of the Helmholtz function. Helmholtz of representations the from calculated (1969) et al. by of Keenan sible. water tabulated properties (1984) et al. The Haar and have been pos are properties consistency thermodynamic and the among in extreme accuracy is established, suchknown as by for usingfor values is tested it accuracy are function evaluate to which accepted the properties Engineering Thermodynamics Engineering with with functions The state. dynamic provides is one that acomplete thermo function of the description thermodynamic A fundamental F more times, formore example, times, one or differentiated be formto functional assumed the involvesgenerally requiring data property This sense. in a valuesconditionsleast-squares satisfied observed be and/or of property selected set a that requiring by determined are function fundamental considerations. coefficientsthe The of andpractical theoretical the basis both on of formis specified 50 functional or more.ber The aset of and adjus of properties independent pair appropriate of the mined by further differentiation differentiation by further mined and specific entropy specific and undamental In principle, all properties of interest can be determined from a fundamental thermodynamic thermodynamic afundamental from determined be of can interest properties principle, all In

Equation 3of earlierond of the expressions establishes Equation 3of to corresponds Equationthese of first The 1 of Since

The development of a fundamental function requires the selection of a functional form in terms terms formin selection the of afunctional development requires The function of afundamental Introducing expressions these for u 1.3.1. By definition, , p T , and , and and and

v v v T known, the specific enthalpy can be found from the definition the be from found can enthalpy specific the known, velocity of sound of velocity and and are independent, the coefficients of d coefficients are the independent, Table hermodynamic T 1.3.3 and Equations 1and 2of , being the independent variables, are specified to fix the state. The pressure pressure The state. the fix to specified are variables, independent the , being s at this state can be determined by of differentiation determined be can state atthis p ψ –    

T = v u –

   u T ( – ∂ ∂ s and specific heat data. When all coefficients all When havebeen evaluated, data. heat specific and T

, s c F and and v Ts v    unctions

), ), = v du , so specific internal energy is obtained as is energy obtained internal , so specific

(∂ h − dd and and ( s Joule–Thomson cT u s dd       v , = /∂ uc     p ∂ ∂ ∂ ∂    dd T T u ), ), v =+ s ds ∂ ∂ )       ψ T v uT s T v . in Equation 1.3.2a, and collecting terms, = v (    =ψ = = T v     T , Table    T c T v T ∂ ∂ v    ), and

u v + ∂ ∂    +    T Table p ∂ ∂ T 1.3.3 and Equation 4of    T u v s    and d and ∂ ∂ data. Once asui Once data. +− v    T p g g T pT −

(    Table = 1.3.4 may obtained. be d T v p

v h , d v – p v must vanish, giving, respectively, ) listed in in ) listed    1.3.4. Withsimilar considerations, Ts ∂ ∂ T p . The specific heat heat specific . The    table ψ v     ( T v , Table table coefficients that may that coefficients num v ) as arepresentative) as case, Table fundamental function function fundamental 1.3.1 fundamental are ψ h (

= T 1.3.4. sec The

, u

c v + ), shown as in v

can be deter be can pv . S imilarly, - 39 - - - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 enthalpy. behavior for This is pressure definitionthe explicitof because is somewhatin greater temperature atfixed pressure with ofenthalpy energy,volume, specific variation is slight. The entropy pressure with and internal Section 1.4. is considered reaction in chemical to should exercised. approach followed be The care special process, when composition changes due the composition in chemical are changes during there When cancels. datum the property, specific involving aparticular in performed only differences are calculations If state. atthis set zero to liquiddata: saturated from atliquidstates data followingthe property set of for equations estimating water at0.01 liquid to saturated entropyand specific seencorrespond ofto is energy water internal specific the substance. to substance from vary suchand datums such of use the particular, In familiar. be to assumed are howused and are they model such pressure, that and temperature both with enthalpy varies constant, common specific heat is often shown is heat often specific simplycommon as Th b As Bornakke and Sonntag (1996). Sonntag and Bornakke available,wide is ofalso range substances for as, example, Steam ASME the ofreferences chapterprovide a of examples. properties this forthe Computer software retrieving energy, such enthalpy,analyses, internal as entropy. and various The liquids. The and gases important are available Tabular practically for of specific presentations volume,temperature pressure, and T 40 Di heat specific 1.3.6to provide practice. engineering mechanical in encountered commonly substances negligible, giving side hand of1.3.20c right Equation on the term is often second The pressure. saturation responding hermodynamic fferentiation of Equation 1.3.22 with respect to temperature at fixed pressure gives fferentiationpressure of1.3.22 atfixed Equation temperature to respect with is model is also applicable to solids. Since internal energy varies only with temperature, the the temperature, only with varies energy applicableis model is also solids. to Since internal Liquid water data (seeLiquid water data Specific internal energy, enthalpy, and entropy data are determined relative to arbitrary datums, datums, relative arbitrary energy,to internal determined enthalpy,are Specific anddata entropy In the absence of saturated liquid data, or as an alternative to such data, the the such to alternative data, an or as liquiddata, absenceof saturated the In The sample The efore, subscript the table can be employed: be can s is assumed known. s is assumed ° c C (32.02 C v is also a function of only temperature: of only afunction temperature: is also steam table steam steam table steam

D h ( ata T, p T, ° F), the triple point temperature. The value of each of these properties is value of The properties of each these temperature. point triple F), the

) R f

≈ denotes the saturated liquid state at the temperature temperature atthe liquidstate saturated the denotes etrieval

Table h v , data presented in in presented data f data for a greater range of states. The form of the formof the of range The for states. agreater data ( u I T ncompre , ), Example1.2.3 in which is used evaluate to s 1.3.5d) suggests that at fixed temperature the variation of specific 1.3.5d)of specific the variation temperature suggests atfixed that , and , and h table () Tp ,– h s normally include other properties useful for thermodynamic for useful thermodynamic include properties other s normally is exhibited generally by liquid data and provides basis for and the by isgenerally exhibited liquiddata ssi h ≈+ () ble mode uT vT sT Tp hT () () () , ff )() () , , , ps pv pu c Table =+ . Specific heat data for several liquids and solids are are data and forseveral solidsliquids heat . Specific ≈ ≈ uT ≈ l: vp () Referring to to Referring      f 1.3.5 representative available of are data for f f   () () () T T T uu v c pv v =

CRC Handbook of Thermal Engineering of Thermal Handbook CRC ( pT T cons

sa ) t ()

= T

du tant Table /   dT

. Al

1.3.5a, the datum state for 1.3.5a, state datum the steam steam Table though specific specific though volume is linear interpolation linear h A.5 and and A.5 table 1 . T , and , and table Table s included in the included the in incompressible incompressible p s and s and Figure s (1993) and sat c

is p

= (1.3.20d) (1.3.20b) (1.3.20a) (1.3.20c) the cor the

figure (1.3.22) (1.3.21) c s 1.3.4 v . The . The with with s, s, - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 rsue(a)Temp. ( Pressure (bar) Temp. ( Sample Steam Sample TABLE 0.20 0.10 0.08 0.06 0.04 8 6 5 4 0.01 ° 1.3.5 )Pressure (bar) C) Table 0.01072 0.00935 0.00872 0.00813 0.00611 60.06 45.81 41.51 36.16 28.96 Data ° C) Liquid ( Liquid ( Saturated Saturated 1.0172 1.0102 1.0084 1.0064 1.0040 1.0002 1.0001 1.0001 1.0001 1.0002 Specific Volume (m Specific Volume (m Specific Volume v v f f

× ×

10 10 3 3 ) ) (a) Properties ofSaturated Water (Liquid–Vapor): Temperature Table (b) Properties ofSaturated Water (Liquid–Vapor): Pressure Table Vapor ( Vapor ( Saturated Saturated 120.917 137.734 147.120 157.232 206.136 7.649 14.674 18.103 23.739 34.800 3 3 /kg) /kg) v v g g ) ) Liquid ( Liquid ( Saturated Saturated 251.38 191.82 173.87 151.53 121.45 33.59 25.19 20.97 16.77 0.00 Internal Energy (kJ/kg) Internal Energy (kJ/kg) u u f f ) ) Vapor ( Vapor ( Saturated Saturated 2456.7 2437.9 2432.2 2425.0 2415.2 2386.4 2383.6 2382.3 2380.9 2375.3 u u g g ) ) Liquid ( Liquid ( Saturated Saturated 5.025. 697082 7.9085 8.1502 8.2287 0.8320 8.3304 0.6493 8.4746 0.5926 2609.7 0.5210 2584.7 2358.3 0.4226 2577.0 2392.8 2567.4 251.40 2403.1 2554.4 191.83 2415.9 173.88 2432.9 151.53 121.46 36 42521. .228.9501 9.0003 9.0257 0.1212 9.0514 0.0912 0.0761 2516.1 0.0610 2512.4 2482.5 2510.6 2487.2 2508.7 2489.6 33.60 2491.9 25.20 20.98 16.78 .120. 514000 9.1562 0.0000 2501.4 2501.3 0.01 h h Enthalpy (kJ/kg) Enthalpy (kJ/kg) f f ) ) Evap. Evap. ( ( h h fg fg ) ) Vapor ( Vapor ( Saturated Saturated h h g g ) ) Liquid ( Liquid ( Saturated Saturated Entropy (kJ/kg∙K) Entropy (kJ/kg∙K) s s f f ) ) (Continued) Vapor ( Vapor ( Saturated Saturated

s s

g g ) )

Engineering Thermodynamics Engineering 41 Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 T Source 20 Sat. Steam Sample TABLE 80 80 140 120 200 160 Sat. 200 T ( ( ° ° C) C) :

1.3.5 Wiley, New Moran, M.J.andShapiro,H.N.,

( Table C

1.0006 1.0280 1.0784 1.1555 1.1973 v v 23.739 27.132 30.219 33.302 36.383 York, 1969. ontinued

× (m

10 3 Data /kg) 3 (m 3 /kg) p )

p =

0.06bar = 25bar Fundamentals ofEngineering u u 334.29 587.82 83.80 849.9 959.1 2425.0 2487.3 2544.7 2602.7 2661.4

(kJ/kg) (kJ/kg) = =

2.5 0.006

P ( MPa

P ( MPa T sat T

h h 336.86 590.52 = 86.30 852.8 962.1 sat 2567.4 2650.1 2726.0 2802.5 2879.7

223.99 (kJ/kg) (kJ/kg)

= 36.16 (d) Properties ofCompressed Liquid Water (c) Properties ofSuperheated Water Vapor ° Thermodynamics C) ° C) s s (kJ/kg∙K)

(kJ/kg∙K) 8.3304 8.5804 8.7840 8.9693 9.1398 .910.9995 0.2961 .77106 333.72 1.0268 1.0737 .39106 586.76 1.0768 1.7369 .241.1530 2.3294 .56125 1147.8 1.2859 2.5546 , 4thed. ν

× , Wiley, New York, 2000asextracted fromKeenan, J.H.etal., v

10 (m 4.526 4.625 5.163 5.696 6.228 3 (m 3 /kg) 3 /kg) u u 83.65 848.1 2473.0 2483.7 2542.4 2601.2 2660.4

(kJ/kg) (kJ/kg) p p

= 0.35bar 50bar = =

5.0

0.035 h h 338.85 592.15 1154.2 2631.4 2645.6 2723.1 2800.6 2878.4

88.65 853.9

(kJ/kg) (kJ/kg) P ( MPa

P ( MPa T sat

T = sat 263.99

= 72.69 Steam TableSteam ° C) s ° s

C) (kJ/kg∙K) (kJ/kg∙K) 0.2956 1.0720 1.7343 2.3255 2.9202 7.7158 7.7564 7.9644 8.1519 8.3237

CRC Handbook of Thermal Engineering Thermal of Handbook CRC 42 Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8

useful: frequently tions are The data for a heat following heat of number relasubstances. gas-specific gas-specific ideal ideal measurements. spectroscopic from together data models with of matter ular usingcalculated molec be can also heats result.gas-specific heats Ideal gas-specific ideal pressure, introduced. should exercisedis be model gas when lesterror ideal invoking significant the model. of predictions the Accordingly, the from substantially caution behavior may depart actual of the low states, At pressure. other limit reduced the model in approaches gas the real alone. The given of by 1.3.25 temperature functions Equation enthalpy are and energy (2) and internal the on temperature. atlowof of energy Joule, primarily air density depends who showed internal the that work the with observations beginning conclusion by This experimental is supported only on temperature. whose is of equation state Engineering Thermodynamics Engineering constant. as taken provided Appendix in whe whe compressibility chart, generalized of the Inspection I 1.3.2a Equation as Also, d to reduces Re T words, for pressures that are lowwords, are relative for that pressures to for when states many

with reasonable accuracy that that reasonablewith accuracy deal c , the compressibility factor approaches a value of 1. Within the indicated limits, it may be assumed it may assumed compressibility be factor, the limits, approaches avalue of indicated 1. the Within ferring to Equation 3 Equation to ferring Specific heat data for gases can be obtained by direct measurement. When extrapolated to zero zero to When extrapolated measurement. direct by be obtained data can for heat gases Specific considerations allowThese for an When the incompressible the 1.3.19 model Equation When is applied, form the takes re re re re

G k RR

= as =

p M / c v M odel . is the is the table T specific ′ of of R exactly is large, the value is large,compressibility the of factor the Table s B and C. As the variation of variation the C. As s Band Z gas constant. Other forms of this expression of are use forms this common in Other constant. gas given by Equation 1.3.25, and thus the specific internal energy depends givendepends energy by 1.3.25, Equation internal specific the thus and

h = 1.3.4, concluded be it (∂ can that 21

1 −= , that is, that hc u VnR pV c ideal gas model gas ideal

= pv pv =−

cT T = == cT T pv ∆= T ∫ 1 2 d == av )() () p s k s e2 c kR , a RT = , and for many states with temperatures high relative high to temperatures for with , and states many () − () TT Tp 1 T c T ∫ =+ nd d 1 2 , av , or d cT cT e () T ln Tv c u pv 12 Vm +−

Figure = T T = vp of each real gas: of (1) real each isof equation state the +−

2 1 d c () T ( k T RT c R

() R with temperature is slight, temperature with − 21 )d RT

pp 1.3.7, shows when that 1

, the c , the p

u 1 /∂

hange in specific entropy is specific hange in v ) T vanishes identically for vanishes agas Z is close 1. to other In Table p R c A.9 provides is small, and and is small, is frequently frequently is (1.3.26b) (1.3.25b) (1.3.26a) (1.3.25a) (1. (1. 3.24) 3.23) 43 - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 uT h defined by Expressions similar in form to Equations 1.3.26 Equations formto in 1.3.38 basis. on amolar Expressions through written similar be can sT

sT of 44 Variable SpecificHeats for Expressions Gas Ideal TABLE developed:be also entropy can specific same or with as A re gives 1.3.28 s vT vT pT pT 2 () () r () () r

r r = Th ()

() () () 22 22

21 21 Table s For processes of an ideal gas between states having the same specific entropy, specific same having the states between gas ideal ofFor an processes With the ideal gas model, Equations 1 through 4of 1through model, gas Equations ideal With the ,, ,, 1 2 1 1 2 vs ps lation between the specific volume and temperatures for two states of an ideal gas having the having gas two an for ideal of states specific temperatures andvolume lation the between −= −= = uT = −= 1.3.6 −= () () v v p Tc p p 1.3.6, respectively. 2of Equation 1 2 r 1 2 () p[ () = Tv Tp

11 11

ex ∫ ∫ T T T T 1 1 2 2 s cT v ° ∫ p ∫ () ( () T T T TT 1 T T 2 1 2 )/ cT d d cT v R T T p () T () ] dl sT dl TR () TR + ∆ − 22 h ,, , n ps n ∆ v v 1 2 p p u 1 2 −= , and , and p p v v () 1 2 p p sT 1 2 Tp 1 2 11 == == () ∆ = pT pT s vT vT Table r2 ex r1 ex r2 r1 () = () () () (6) (5) (4) (3) (2) (1) p p ∫ T 0 TsT sT     1.3.6 expressed be alternatively can using sT sT cT () p () () T () () 21 TuT uT ss h s sT sT () T T T T 1 2 2 21 ss () () () () 1 2 1 2

= −− Th 21 22 22

= = 21 21 s d ,, ,, R R 1 vs CRC Handbook of Thermal Engineering of Thermal Handbook CRC T ps       −= −= Table     v () v p p

2 1 −= 1 2 −= () ()

   Tc    Constant SpecificHeats k () () − kk () Tv 1 − 1.3.4 4 give 1through Equations Tp

1 11 11 R cT v p () ln () TT 21 21 c p − p c − v p 1 2 T nln ln nln ln

T T T T 2 1 1 2 + − R R v v 1 2 p p s 1 2 2

s 1 , Equation , Equation (1.3.29b) (1.3.29a) (1.3.27) (1. s 3.28) ° (6 (5 (4 (3 (2 (1 ( T ′ ′ ′ ′ ′ ′ ) ) ) ) ) ) ) Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 the ideal gas model is assumed. The expressions The model gas is assumed. for ideal the when vides several they take expressions forms special applicable the and polytropicto processes model. gas ideal of use togetherthe the with practice in it most appears generally p An internally reversible by expression the internally described An process P Equations 1.3.29Equations in listed are Thermodynamics Engineering gas. Tabulations of achievable it by as agas is compressed. cooling determine to ate evaluations 1.1.10 Equations with 1.2.12, and respectively. some applications, In it may appropri be Equations 1 Equations Table provides e.g., see, also such software data; from retrieval (2000). Shapiro of use and data The Moran rocess olytropic Using Equation 1. Interpolating with with Interpolating Example 1.3.4 Table At and 960 520 Solution exitwith the velocity for an isentropic expansion to 15 lbf/in. Using data from

Table generally,agreement cannot expected be however. for example, See, Brayton the cycle data of velocities calculated using oretical value. In this particular application, is agreement in there good between each case exit velocity is 2% about less than velocity the for an isentropic expansion, maximum the the pic expansion is found follows: as Example 1.3.5 illustrates both the polytropic process and the reduction in the compressor work the reduction in the and polytropic Example process 1.3.5 the both illustrates When the ideal gas-specific heats are assumed constant, Equationsto 6 1 of constant, assumed are heats gas-specific ideal the When A.8 for nozzle Example1.3.4. of the in Example1.2.2 is illustrated and and 1 3.7. 1. A.8 provides a tabular display A.8 provides of atabular : The exit: The velocity is given by Equation 1.2 ′

n is v to 6 to P e rocesses = = the the ′ 2312.5 n , respectively. heat specific The ° by fitting pressure-specific volume data. data. volume bypressure-specific fitting    Table R, R, h 10 3.29 polytropic exponent p , s Table u r ft data, data, , and , and ft/s A.8,evaluate exit the velocity for nozzle the of Example 1.2.2 and compare a and    2 +− A.8 gives, A.8 respectively, 2 h s Table () Table p e 3.6124.27 231.06 pT

= for several other common gases are provided in for are gases several common other r data from rr

119.54 Btu/lb. Withthis, exit the velocity is 2363.1 () ei A.8 data and, A.8 in as Example 1.2.2, assuming 1.3.6 respectively. 6, 5and Equations as == pT vv . Although this expression can be applied with real gas data, data, gas expression real with applied be this . Although can ei () =+ Table h , p p u e i 2 , () A.8, the specific the A.8, enthalpy the exitat for an isentro Btu/lb s c ° 10.61 2 h , p is taken as constant in Example1.2.2. in constant as is taken i p ()

.9 = hh r

, and , and 231.06 and Btu/lb f ie       ∫ − 778.17 150 15 pv d v 1B    r versus temperature for air as an ideal ideal an as for air versus temperature and and 2 = tu . tlb ft pv 1.061 ⋅ n

= ∫ f

constant is called a is called constant    vp    d 32.174 h e have application work to

124.27 Btu/lb. Then 1l bf bft/s lb c Table Table p ⋅ constant. Such constant.

ft /s. actual The Table 2 A.2. Property Property A.2.    1.3.6 become 1.3. polytropic 7 pro - - 45 - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 to the mixture as a whole as component to each and mixture to the of equation gas Applying state ideal the mixture. of the volume atthe separately temperature and Dalton model Dalton the as known 46 When applied to an ideal gas mixture, the additive pressure rule (Section additive rule the 1.3, pressure p mixture, gas ideal an to applied When I pressure T volume temperature Vatthe deal

: Solution of airkJ/kg flowing. Repeat for (2) an isothermal compression and (3) an isentropic compression. airthe with apolytropic n process undergoes atoperates A compressor steady state with air entering at 1 bar, 20 Example 1.3.5

3. 2. G 1.

as of component i W Area 1–2Area Using Equation 5 of air flowing. 1–2 Using Equation 3 mass of air flowing. balanceAn energy to reduces give the workthe required, unit per mass of air flowing.) Also, Equation 1 area behind(The 1–2 process given in Figure data, data, For an isentropic compression, An energy rate balance energy An at steady stateand enthalpy data Table from cv

M s – mh ixtures h a 2 – =− s

in b in = ′ –

464.8 kJ/kg ( kJ/kg 464.8 () a – Figure 21 in Figure b in s − , is the pressure that component iwould that pressure n exert, is if the h of Table ′ of of Table ′ of W m 1.3.8 magnitude the work represents of the required, unit per of mass cv , Q . According to this model, each gas in the mixture acts as if it if exists as acts mixture the model, in gas each this to . According where2 m cv . Forming a ratio, the partial pressure of component pressure iis partial aratio, the . Forming = =− = T 1.3.8 magnitude the work represents of the required, unit per of 2    nR n =− =+ s

1. 0. = 163. − 1.3. 1.3. W

T 463 K). Then 3 3 m 1 163.9 1 cv    W     denotes the exit the s denotes state. WithEquation 1.3.29a and p 1    9k m 7 together with7 together Equation 1.2.12b, 7 together with7 together Equation 1.2.12b, cv − 8.314 28.97 J/k Q hh    =− =− =− +− 21 p cv p p g − i () 2 1 426.3 == 135.3k = RT       kJ/k () 8.314 28.97

0 nn = n n − ln and an energy rate and an balance energy to give reduces W

i 1.3, determine work the and heat transfer, each in 1/ gK 1.3.8, area magnitude 1–2–a–b, the of represents py cv p p ⋅ 2 i     1 J/k 293.2 ,       m () pV g () 293 293 i =− p

= CRC Handbook of Thermal Engineering of Thermal Handbook CRC K () =− ln 464.8 nR 5    30. Q T 15 − , cv 8k −= () pV ′ of mW ii 293.2 J/k 0. ° Table 31 == C and exiting at 5 bar. (1) If g = .3 nR    cv 1.3. A.8 gives A.8 i moles occupied the full moles full occupiedthe T m − , where 171. 7 gives gives − – 6k 135. v – T J/kg. T Relations) is 2 p 3kJ/k

= i , the , the

2 K. 425 Area Area g . (1.3.30) partial r

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 Engineering Thermodynamics Engineering where ated at the condition at which the gas exists in the mixture. On a On mixture. the condition exists gas in atwhich atthe the ated is gas evalu each contributionfrom component of the provided gases, the that respective properties pressure. The internal energy, enthalpy, and entropy of the mixture can be determined as the sum of the of sum the the as energy, determined be can enthalpy, internal The entropy mixture of and the ∫ − Note: n n n General Polytropic Processes: TABLE a n −= ∫

1 p

p ∫ ≠ = = = ∫ 2 1 1 2 2 1 p 1

pv 1 1 ±∞ 0: constantpressure 2 2 y pv = vp d vp i dl

is the mole of is the component fraction dl d    : constantspecificv − 5 a one-inlet,one-exit controlvolume atsteadystateundergoes apolytropicprocess,Equations3,5,and3 and 2 For polytropicprocessesofclosedsystemswherev = = v v = 1.3. ′ 2 1 ∫∫ areapplicablewithEquations1.2.12aandbtoevaluate thepower. Also notethatgenerally = =− n pv pv    pv 1 22 2 11 1 − n pv np 11 n vp ′ 1 − pv 1 n , 4 − d 11 11 − 7 n −     n 1 1 ′ n are applicable with Equation 1.1.10 to evaluate the work. When each unit of mass passing through () pv v = − v     pv 11 n 1 2 1 22    np − p p p p    2 1 1 2 − n 1 2 p p    = pv 1 2 ()

   olume ±∞ d s s − = constant = 1 () v nn

. −

T T 1 pv

    = constant =     n onstant = n

C = –1 n n = k (5) (4) (3) (2) (1) = 0 n = 1 υ i − ∫ T n n n n n n . The sum of the partial pressures equals the mixture mixture the equals pressures partial of sum the . The −= ∫ 1 p

p ≠ = = = = = ∫ 2 ∫ 1 1 2 2 1

pv 1 1 k 1: constanttemperature ±∞ 0: constantpressure 1 2 pv 2 = : constantspecificentrop d vp vp dl d    : constantspecificv dl = = v v = olume changeistheonlywork mode,Equations2,4, 2 1 = =− n RT RT    RT 1 − () nn nRT n nR 1 − 12 RT 1 = − 21 − n n        1 − 12 1 n v () v T T T     − n TT 1 2 1 1 2 21    p p −    Ideal Gas − 1 2 n p p    1 () = k n p p    − 1 1 () molar nn    olume υ − 1 = constant () nn

p − y when = constant 1 a         basis, n = n k isconstant ±∞ = –1 n = 0 n = 1 s (3 (2 (5 (4 (1 ′ ′ ′ ′ ′ ) ) ) ) ) ′ , 47 - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 The s The 48 and and mass Whe FIGURE entropy depends onentropy depends c expressed similarly: components are the v in place of in The internal energy and enthalpy of an ideal gas depend only on temperature, and thus the the thus and only on depend gas temperature, ideal enthalpy of and an energy internal The h n working n onworking a and and i pecific heats heats pecific terms appearing in Equations 1.3.31 are evaluated at the temperature of the mixture. Since 1.3.31 Equations in mixture. of the appearing temperature evaluated at the are terms 1.3.8 mass fractions mass

u , In h ternally reversibleternally compression processes. , c s v a p b mass two , and and c p 5 1 , and , and independent properties, the the properties, independent ba in place of in ba basis, expressions similar in form to Equations 1.3.31 form to Equations in basis, expressions using similar written be can c p r r for an ideal gas mixture in terms of the corresponding specific heats heats of specific corresponding of the terms in mixture gas for ideal an

Isothermal c v , respectively. , 2 ΄ U Hn moles 22 S ======∑∑ ∑∑ ∑∑ i i i ======and and j j j 11 s 11 11 cy cy pi nu vi ns ii ii ii = = hh mole fractions mole ∑ ∑ or n i i or or = = j j =1.3 1 1 sy uy s i Isentropic c c terms are evaluated either at the temperature temperature evaluated either atthe are terms vi pi

i i j j j CRC Handbook of Thermal Engineering of Thermal Handbook CRC yh ii ii ii s u , respectively, using and

1 v u , h , s (1.3.31d) (1.3.31b) (1 (1.3.31e) (1.3.31c) , c .3.31a) p , and , and u i

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 Engineering Thermodynamics Engineering of function pressure of the water vapor equals water of vapor the pressure equals water vapor, and air respectively. humidity ratiohumidity (mixture) dry-bulb temperature the to and the partial pressure pressure partial the and is the apply usually principles moist to particular, In air. mixture gas component. Ideal it if were as apure is treated air water dry vaporwhich and the in air of dry refers amixture to applications for is interest practical many of particular mixture gas ideal An M frompressure T temperature fixed where molar-specificwhereGibbsfunction the componentof G expressions the for Inserting case, former of a gas phase of dry air and saturated water vapor in equilibrium with aliquidwater with phase. water vaporequilibrium in saturated and air of of phase agas dry sample would consist original the below dew the temperature, point atatemperature state librium equi cooled afinal to condensation of the present.When some in results water of vapor the initially the is called saturated becomes mixture. exposed by the to such indicated athermometer temperature is the mixture liquidwater, with covered the and awick with saturated the mixture pressure such 1atm. as pressure mixture the relative the ratio, and humidity, humidity for the wet-bulbvalue usually aspecified of temperatures, applicable, and so the mixture pressure pressure applicable, mixture so the and

= oist Saturated air Saturated When a sample of moist air is cooled at constant pressure, the temperature at which the sample atwhich the temperature the pressure, asample is ofcooled atconstant moistWhen air Psychrometric charts Psychrometric

H –

p A TS ′ is some specified pressure. Equation 1.3.34 is obtained by integrating Equation Equation pressure. integrating 1.3.2d is someby 1.3.34 specified is obtained ir , i can be expressed be alternatively can as ( specific humidity specific is a mixture of dry air and saturated water vapor. For saturated air, the partial partial water the vapor. air, saturated For and saturated air of dry is amixture p are plotted with various moist air parameters, including the dry-bulb and and dry-bulb including the parameters, moist various with air plotted are i of component H gT and and ii Gn dew point temperature point dew ) and the the ) and () = =− p p ,, ′ sat pg ∑ ∑∑ to to S i i = == j j ( given by 1.3.31b Equations Gibbs function, the cinto and T 1 11 T . The makeup of moist air can be described in terms of the of the terms in described be can of moist makeup . The air S p = = ng ), which is the saturation pressure of water corresponding of water pressure corresponding ), saturation which is the i i = = . ii ii , or at the temperature and volume of the mixture. In the the In volume and mixture. of the temperature , or atthe relative humidity gT hT p ii ii ∑ ∑ is the sum of the partial pressures pressures partial of sum the is the i i () () () () = = j j Tp Tp 1 1 ,l , ns ns pR ii ii ′ ′ i Tn () () + + Tx Tp i RT j , , Tx . Cooling below the dew point temperature below. Cooling dew the temperature point wet-bulb i i i p ln ii is is n sT

. The bulb of a . The () () () pp g i , pp ( T p , temperature of an air–water vapor air–water of an temperature i ′ p ′

i )

h i ( T wet-bulb thermometer wet-bulb ) – Ts moist air moist p i a ( and and T , p Dalton model model Dalton i ). Gibbs The p v . Moist air . Moist air of the dry dry of the (1 (1 (1 .3.34) .3.32) .3.33) 49 is is at - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 changes in specific enthalpy and specific entropy between states 1 and 2 are and 2 1 states between entropy and specific enthalpy specific changes in factoribility (Reynolds, 1979). Using superscript the (see, e.g., for equation compress Reid the 1966) Sherwood, calculable and using and ageneralized charts, correction entropy and atleast approximately, enthalpy correction obtained, generalized of by the inspection be can corrections model. gas ideal The using the change determined respective property the ing by principle in correct determined be can entropy two between enthalpy and states changes in The G 50 Fundamentals of Classical Thermodynamics FIGURE eneralized 1.3.9

–– h* – h C Ge – harts –1.0 Figure neralized enthalpy correction chart. (Adapted from Van (Adaptedfrom Wylen, R.E., Sonntag, G.J. and chart. correction enthalpy neralized RTc 2.0 3.0 4.0 5.0 6.0 7.0 1.0 0 . . . 0.40.5 0.3 0.2 0.1

for 0.75 h s 1.3.9 1.3.10, and respectively. form available also tabular in are Such data 21

sss E −= 21 nthalpy

0.80 −=

hh Saturated vapor liquid Saturated 0.90 0.85 2 * 21 , ∗∗ −− E −− 0.90 hR 0.95 ntropy sR 1c ∗ , 3rd ed., English/SI., Wiley, English/SI., , 3rd ed., New York, 1986.) 0.92 0.94 0.96

Re 1.00 T 0.98 ,     1.0 duce 1.20    and 1.20    

1.30    ss 1.50 hh 2.00 ∗∗ d pressure R RT

− ∗∗ T F − R ugacity * 2.0 c to identify ideal gas property values, the property gas ideal identify to    21 CRC Handbook of Thermal Engineering of Thermal Handbook CRC    − 2 . .501 20 10 4.05.0 3.0 ( −    P R    ss ) hh R − 0.50 1.70 1.30 1.20 0.98 0.80 RT 2.80 2.40 0.70 1.06 1.02 0.90 0.94 1.50 4.00 1.90 1.10 0.60 − c 0.75 1.25 1.15 0.85 0.55    3.00 2.60 0.65 2.20 2.00 1.40 0.92 1.60 0.96 1.08 1.80 1.00 1.04        1

   

30 (1.3.35b) (1.3.35a) - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 ) ( function with pressure at fixed temperature (from temperature atfixed pressure with function plays pressure as gas gas. for To ideal the Gibbs develop specific of the consider variation the this, fugacity The temperature. correction charts. correction entropy and enthalpy correction generalized enter to the used be can also rule mixture some other using obtained be and pressure pressure and temperature reduced using the value. quantities The actual the gas value ideal obtain to to the applied be must that correction is the term gas behavior. ideal underlined second change assuming The right the side on each of term expressionrespective the property represents underlined first The Fundamentals of Classical Thermodynamics FIGURE Engineering Thermodynamics Engineering ss ∗ Figure − 1.3.10 R 1.3.11 gives the at state 1 would be read from the respective correction chart or chart respective correction the 1would from atstate read be p

1 at state 1, atstate respectively. Similarly, Ge

T – – neralized entropy correction chart. (Adapted from Van from (Adapted Wylen, R.E., Sonntag, G.J. and chart. correction entropy neralized * –

R2 s s – and and

R 10.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 0 ...... 2. 1.0 0.5 0.4 0.3 0.2 0.1 0.75 f fugacity p plays a similar role in determining the specific Gibbsfunction specific real for a the role determining plays in a similar

Saturated vapor R2 . Mixture values for . Mixture 0.80 T

R1 Saturated liquid Saturated coefficient, and reduced pressure pressure reduced and

0.90 0.85 0.90

, 3rd. ed., English/SI., Wiley, English/SI., ed., , 3rd. New York, 1986.) 0.95 0.92 Reduc 1.00 0.94

1.04 0.96 1.10

1.20 0.98 1.60 1.40 ed f/p pressure ( T Table , as a function of reduced pressure and reduced reduced and pressure of reduced afunction , as R T ) ( c hh and and ∗ 03 0.90 1.3.1) 0.80 0.70 0.94 − 0.60 0.50 2.00 1.20 0.98 1.60 3.00 1.06 1.02 1.40 1.10 P R p p 04. .0 ) 0.96 0.75 0.85 0.55 1.00 1.80 0.65 2.50 c R1 1.30 0.92 1.08 1.15 1.04 determined by applying Kay’s determined or rule 1.80 corresponding to the temperature temperature the to corresponding RT 05 01 20 10 .0 c

and

() ss 0.94 1.06 1.02 0.98 * − table 30 R ) ( hh at state 2would atstate ∗ or calculated, or calculated, − RT c

and 51 T 1

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 As p As 52 consisting of nonreacting mixtures. For asingle mixtures. consisting of nonreacting of multicomponent systems properties of the aspects some general presented are section this In M given value. gas by ideal the then is temperature fixed gas the value at fromideal gas value real the of departure the coefficient, fugacity of the terms In where gives temperature atfixed gas, integration ideal For an FIGURE Fundamentals of Classical Thermodynamics ulticomponent ressure is reduced at fixed temperature, temperature, atfixed is reduced ressure C 1.3.11 ( T ) is a function of To integration. ) is afunction evaluate

neralized fugacity coefficient chart. (From (From R.E., Van and Wylen,Sonntag, G.J. chart. coefficient fugacity neralized Fugacity coefficient (f/P) ystems 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

0.60

0.65

0.70

Saturation line Saturation

0.75

0.90 0.80

1.00

0.94

1.10 1.20 1.50 0.85 2.00 0.50.40.30.20.1 gR gR

gg , 3rd ed., English/SI., Wiley, English/SI., , 3rd ed., New York, 1986.)

0.92 =+ 0.90 =+

Reduc −=

   0.96 0.94 Tf ∂ ∂ . . . 4.05.0 3.0 2.0 1.0 Tp f

/ g 1.00 p p ln ln ed tends to unity, and the specific Gibbsfunction is unity, to specific tends the and    T RT pressure ( - phase phase = g T for a real gas, fugacity replacespressure, gas, fugacity for areal ln R CT v CT () () p f CRC Handbook of Thermal Engineering of Thermal Handbook CRC

multicomponent P R ) 0.60 0.96 2.00 0.75 0.92 0.85 1.04 1.15 1.45 1.00

1.80 10.00 1.08 1.25 1.35 1.60

6.00 1.10 1.30 1.40 0.94 0.98 1.20 1.50 1.06

1.70 15.00 1.02 0.90 0.80 0.70 1.90 020 10 system consisting of 30 (1.3.36) j

Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 mathematically mathematically molal property molal property in general on temperature, pressure, and mixture composition: mixture and pressure, on temperature, general in whe whe of number molesthe of mixture: component each the present in extensivecomponents, an property whe an energy, volume,changes in given enthalpy, internal are by entropy on mixing and respectively, function, givesGibbs component. pure component pure Engineering Thermodynamics Engineering

d the subscript subscript d the When pure components, each initially at the same temperature and pressure, are mixed, the the mixed, are pressure, and temperature same atthe components, initially each pure When Selecting the extensiveSelecting the property re re the re re re V v i i , , U u partial molar property molar partial i i , , h H i , and , and i , homogeneous of degree one degree of homogeneous X i S n . i i is an intensive property of the mixture and not simply a property of the of the not simply and aproperty mixture of intensive the is an property l , and , and denotes that all all that denotes s i denote the molar-specific energy, the denote volume,internal enthalpy,and entropy of G i denote the respective partial molal properties. respective partial the denote

X n Hn i Gn X Vn is by definition by is ∆= ’s except ∆= X may be regarded as a function of temperature, pressure, and and pressure, of temperature, afunction as may regarded be ∆= ∆= Hn Un = == == to be volume, internal energy, volume, be to enthalpy, internal entropy, the and Vn Sn mixin mixin mixin mixin ∑ ∑∑ ∑∑ i i i = == == X j j j 1 11 11 Xn i g g g g in the the in = ii ii ii = n VU G HS    i ∑ are held fixed during differentiation. As differentiation. held fixed during are ∑ ∑ ∑ ∑ , i i ∂ i ∂ i i = = = , = = j j j j j 1 1 n 1 1 1 X i n    ’s, is expressible function the as i i i i ii Tp () X () () () ,, Uu Hh Vv Ss ii ii

i n ii ii i l j − −

j − − nU nS ii ii

X XT

= ij

() X ( T, ,, pn p, n 12 ,, 1 nn , n … 2 ,… , , n j , the partial partial , the ). Since X i (1.3.40d) depends depends (1.3.40b) (1.3.40a) (1 (1.3.37) (1.3.38) (1.3.39) .3.40c) X i 53 is is th th Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 from which can be derived be which can from solution of mixture: or solid) aggregation liquid, state (gas, the and as pressure, temperature, Li potential of molal Gibbs the function partial The C 54 components present in components present in intensive the consistingfix of to state ofa specified equilibrium system at trarily is condition for having phase alower the to Anecessary potential. stance chemical for sub that potential system: moveto chemical tends higher having a substance phase the the from of the pressure mole its per Gibbs of function gas equals 1.3.42 suggests 1.3.33 of Equations comparison and mixture, gas ideal For Gibbs molar function. an the equals Fo Ideal Solution Ideal Whe

The phase rule does not address the relative amounts that may be present in the various phases. various the relative may present in the be that not does address amounts rule phase The For water asingle as component, for example, that the chemical potential of each component has the same value every in same of phase. component each the has potential chemical the that hemical ke temperature and pressure, the chemical potential, potential, chemical the pressure, and ke temperature r a The The of the is a measure potential chemical The When written in terms of chemical potentials, Equation 1.3.37 Equation potentials, of chemical terms for reads Gibbs in function the written When With With • • •

n the mixture composition is constant, Equation 1.3.44 composition Equation 1.3.1a. is constant, Equation to reduces mixture n the there is a unique temperature: 0.01 temperature: is aunique there is fixed. For phase a each system consisting liquidof ice, water,and equilibrium, water at vapor Fo example, temperature maintained—for two are phases Fo say temperature Fo single component system single component Gibbs phase rule phase Gibbs is the product of its mole fraction and the fugacity of the pure component, pure of fugacity productof the its is the the mole and fraction r three phases, phases, r three r two phases, r asingle phase, G ,

= P i

, otential H

− : The The : i = µ

th gas of the mixture. of gas the th TS ii a gT nd nd Lewis–Randall rule Lewis–Randall P () and P P

H =

, = phases: gives number the

2 P = p

3 pressure, while maintaining asingle phase. maintaining while pressure, i

and and = U and and ; that is, the chemical potential of component potential chemical is, the ; that

and and pV , Equation 1.3.42, Equation to reduces F F

= F , Eq

f = ii

1: O dd =

F 0: T = UT

2 = uation 1.3.42 expressed be as can yf

U +

° 2: T =− µ nly one intensive property can be varied independently if independently if varied be can nly one intensive property here are no degrees of freedom; each intensive property of of intensive each of freedom; no property degrees are here C (32.02 C

i N ii th component ofth amulticomponent system is the =− states that the fugacity fugacity the that states i ==

− TS wo intensive properties can be varied independently, varied wo be intensive can properties Gn i F

G P () Sp evaluated at the mixture temperature and the partial partial the and temperature mixture evaluated atthe Lewis–Randall r of independent intensive properties that may arbi of be intensive that independent properties . =µ F N ∑ Vn pV ° escaping tendency i

i = F) and a unique pressure: 0.6113 aunique and F)    = j dd s called the the s called 1

1 ∂ ∂ Vn +µ G n and and ii ∑ +µ i i =    j ∑ 1 Tp i μ

= j ,, 1 F i is an is an ii CRC Handbook of Thermal Engineering of Thermal Handbook CRC

= n l i

G or

ule − degrees of freedom of degrees

= i intensive P pressure.

. n

f μ i of component each of a substance in a multiphase of in a substance ; th at is, the chemical potential potential chemical at is, the property. i in an ideal gas mixture mixture gas ideal an in

kP phase equilibrium equilibrium phase (or the a (0.006 atm). N f i , at the same same , atthe nonreacting nonreacting i in an an in variance chemical chemical (1.3.44) (1.3.45) (1.3.42) (1. (1.3.41) ideal ideal 3.43) ). ). - - Downloaded By: 10.3.98.104 At: 04:24 25 Sep 2021; For: 9781315119717, sec1_3, 10.4324/9781315119717-8 solution ideal enthalpy of and an energy are for solutions. ideal internal The 1.3.18a Equations Comparing mixture. of the 1.3.46, and the entropy, in nents would increase an however, result in such is irreversible. aprocess because solution. ideal components forman to pure The mixing 1.3.40a,Equations energy, b,is no cshow volume, change in and there that or enthalpyon internal whe The following characteristics are exhibited by an ideal solution: byideal exhibited an following are The characteristics whe the Lewis–Randall rule. Lewis–Randall the case. Some modeled liquidsolutions with be special can also 1.3, important Model, Gas is an Ideal adequatelymodeled by Lewis the are atlow pressures moderate to mixtures Many gaseous mixture. of the pressure and temperature the Engineering Thermodynamics Engineering The volumeThe solution ideal of an is re re re u V i i and and is the volume component is the pure that h i denote, respectively, component enthalpy of and pure energy internal molar the Un == ∑∑ V i == j 11 == ∑∑ ii i uH == j 11 ,i - nv Randall rule. The ideal gas mixtures considered in Section considered in mixtures gas ideal The rule. Randall ii i would occupy when at the temperature and pressure pressure and would occupy temperature when atthe i j i j V i nh ii () ideal solutio adiabatic () deal s additive volume rule volume additive v V ii olutio mixing of different pure compo of pure different mixing n =

, u U n ii =

, Hh is seen to be exact be to is seen ii = . With these, . With these, (1.3.46) (1.3.47) i at 55 -