Chapter 3. Volumetric Properties of Pure Fluids

Transcription

1 Chapter 3. olumetri roperties of ure Fluids

2 Introdution hermodynami properties (U, H and thus Q, W) are alulated from data data are important for sizing vessels and pipelines Subjets behavior of pure fluids Ideal gas behavior Real gas behavior Generalized orrelation when experimental data are laking

3 3. Behavior of ure Substanes - hase Diagram ritial point Solid hase (subooled) Liquid hase Super-ritial (fluid) phase triple point (superheated) apor hase Gas apor vs. Gas? - Gas : nonondensable (annot beome liquid) - apor : ondensable

4 3. Behavior of ure Substanes - hase Diagram vapor pressure urve aporization urve Component = hase = ( and L) F = Solid hase apor pressure at Liquid hase p vap apor hase

5 3. Behavior of ure Substanes - hase Diagram melting urve fusion urve Solid hase Liquid hase freezing pressure triple point Gas sublimation pressure apor hase sublimation urve sublimation point freezing point

6 3. Behavior of ure Substanes Solid hase triple point Liquid hase Gas apor hase

7 3. Behavior of ure Substanes Solid hase Liquid hase triple point Gas apor hase

8 3. Behavior of ure Substanes hase Diagram Compare with diagram - triple point? - vapor pressure line?

9 3. Behavior of ure Substanes diagram vs. diagram heating heating

10 3. Behavior of ure Substanes

11 Single hase Region Equation relating,, Equation of State f (,, ) 0 d d d olume expansivity Isothermal ompressibility d d d

12 3. irial Equation of State behavior is omplex annot be represented by simple equation Gas phase alone an be orrelated with simple equation term is expanded as a funtion of by a power series a b... a( B' C' D' 3...) At low pressure, trunation after two terms usually provide satisfatory result

13 Ideal-Gas emperature : Universal Gas Constant B, C, D,. depends on omponent, temperature a same for all speies, funtion of temperature only ()*, the limiting value of as 0, is independent of the gas a( B' C' 0, a D' 3...) ( ) * a f ( ) Useful for defining the temperature sale

14 Ideal-Gas emperature : Universal Gas Constant Ideal Gas emperature Sale Make ()* proportional to, (R: proportionality onstant) ( * ) a R Assign the value 73.6 K to triple point of water * ( ) t R73. 6K * ( ) / K * ( ) t 73. 6K / K ( ) 73.6 ( ) Kelvin sale temperature * * t

15 Ideal-Gas emperature : Universal Gas Constant 0 Moleular volume beomes smaller Moleules are separated infinite distanes Intermoleular fores approahes zero Universal Gas Constant () t * =,7.8 m 3 bar/molk R R 8.34 J/mol.K m a/mol.k R * ( ) t 73. 6K R R R R R m bar/mol.k 3 834m ka/mol.k 8.06 m.atm/mol.k.987 al/mol.k Btu/lb mol.r R ft.atm/lb mol.r R ft.psia/lb mol.r

16 wo forms of the irial Equation Compressibility Z Z B' C' R D' 3... B C D Z... 3 B, B : the seond virial oeffiient C, C : the third virial oeffiient irial expression B B R C' C B ( R ) ' D' D 3BC ( R B 3 ) 3

17 Importane of the irial Equation Many form of equation of state have been proposed irial Equation has a firm basis in theory Statistial mehanis B/ : interation between two moleules (two-body interation) C/ : three-body interation Z B C D 3...

18 3.3 he Ideal Gas No interation between moleules (B/, C/, =0) Z or R Ideal gas Internal energy of gases Real gas : funtion of and Ideal gas : funtion of only - ressure dependeny resulting from intermoleular fore - No intermoleular fore in ideal gas no dependeny in - he Equation of State Z or R - Internal Energy U U()

19 Implied roperty Relations for an Ideal Gas Heat apaity at onst. volume is a funtion of temperature only C v du d du d C v () Enthalpy is also a funtion of temperature only H U U R H() Heat apaity at onst. pressure is a funtion of temperature only C dh d dh d C () Heat Capaity Relationship C dh d d( U ) d d( U R ) d C R Caution: C v and C p are not onstant, they vary with while keeping C p = C v + R

20 Equations for roess Calulation for Ideal Gases From the first law of thermodynamis du C d dq dw dq d Combine ideal gas law R / C C R dq C d dq Cd dq C d R d R d R C d R d dw R dw Rd R dw d d,,,

21 Equations for roess Calulation for Ideal Gases - () Isothermal roess du C d dq dw dq d dh C d dq dw s dq dw U 0 H 0 = onst. dq C d R dq Cd R d d dw R d dw Rd R d Q R ln R ln W R ln R ln DO NO memorize the equation! Memorize the sequenes of derivation!

22 Equations for roess Calulation for Ideal Gases - () Isobari roess du C d dq dw dq d U C d dh Cd dq dw s H Cd H Q = onst. d dw R dw Rd R d Q W H C d R( ) dw d

23 Equations for roess Calulation for Ideal Gases - (3) Isohori roess du C d dq dw dq d U C dh Cd dq dw s H C U Q d d = onst. d dw R dw Rd R d Q U W 0 C d dw d

24 Equations for roess Calulation for Ideal Gases - (4) Adiabati roess du C d dq dw dq d U C dh Cd dq dw s H C Q 0 d d dq=0 dq dq dq C C C R d R d R d C R d d 0 0 d 0 d d d R C R C C C d d d integration integration integration R / R / C C C / C

25 Equations for roess Calulation for Ideal Gases - (4) Adiabati roess C R / C R / C C / C C C )/ ( C C C )/ (. onst. )/ ( onst onst. C C / Q 0 C d C W hese equations are restrited to onst. heat apaities and reversible, adiabati proesses

26 Equations for roess Calulation for Ideal Gases - (4) Adiabati roess Other expression for WORK alulation C d C W R C R C W ) ( )/ ( )/ ( R R C W Use instead of or

27 Equations for roess Calulation for Ideal Gases - (4) Adiabati roess alues of C p /C v Monatomi gases :.67 (He, Ne, Kr, ) Diatomi gases :.4 (H, N, O, ) Simple polyatomi gases :.3 (CO, SO, NH 3, CH 4, )

28 Equations for roess Calulation for Ideal Gases - (5) olytropi roess olytropi : urning many ways onst. 0 onst R onst onst onst isobari isothermal adiabati isohori Can be used to represent a in-between proesses

29 Equations for roess Calulation for Ideal Gases - (5) olytropi roess Using ideal gas equation, onst. R / onst. ( )/ onst. ( )/ R W d ( )R Q ( )( ) ( )/ onstant heat apaity

30 Irreversible roesses Equations in ()-(5) Only valid for mehanially reversible, losed system for ideal gases roperties hanges (U, H) are also same regardless of the proess State roperties Heat and Work amount depends on the nature of proess (reversible/irreversible, losed/open) ath funtion For irreversible proesses, the following proedures are ommonly employed W is determined For reversible proess Effiieny (h) is multiplied

31 Example 3. Air is ompressed from an initial state of bar and 5 o C to a final state of 5 bar and 5 o C by three different mehanially reversible proesses in a losed system. (a) Heating at onstant volume followed by ooling at onstant pressure. (b) Isothermal ompression. () Adiabati ompression followed by ooling at onstant volume. Assume air to be an ideal gas with the onstant heat apaities, C v = 5/R and C p = 7/R. Calulate W, Q, U, and H.

32 Example 3. solution = bar = 98.5 K = m 3 = 5 bar = 98.5 K = m 3 Choose the system as mol of air (basis) C v = C p = J/molK Sine is same, U = H = 0 Also, U = Q + W = 0 Q = -W (although the absolute value of Q & W an be different path funtion)

33 Example 3.3 An ideal gas undergoes the following sequene of mehanially reversible proesses in a losed system. (a) From an initial state of 70 o C and bar, it is ompressed adiabatially to 50 o C. (b) It is then ooled from 50 to 70 o C at onstant. () Finally, it is expanded isothermally to its original state. Calulate W, Q, U, and H for eah of the three proesses and for the entire yle. Assume air to be an ideal gas with the onstant heat apaities, C v = 3/R and C p = 5/R.

34 Example 3.3 solution Choose the system as mol of gas (basis), R = 8.34 J/molK C v =.47 C p = J/molK = C p /C v = 5/3 (a) Adiabati ompression, Q = 0 U = W = C v =.47(50-70) = 998 J H = C p = 0.785(50-70) = 663 J from /(-) = onst =.689 bar (b) Const proess = bar Q = H = C p = 0.785(70-50) = -663 J U = C v =.47(70-50) = -998 J W = U Q = -998 (-663) = 665 J or W = - () Isothermal proess U = H = 0 onst Q = -W = Rln( 3 / ) = Rln( / ) = (8.34)(343.5)ln(.689/) = 495 J

35 3.4 Appliation of the irial Equations irial Equation Infinite Series 3 Z B' C' D'... Only useful for engineering purpose when onvergene is rapid - wo or three terms Derivatives of ompressibility Z Z B' C' 3D' ; 0 B'... runation equation Z R B R Compressibility-fator graph for methane

36 irial Equation for Engineering urpose runated two terms Z B' similar auray but onvenient Z B runated three terms Z B' C' Z R B C more aurate Extended irial equation (Benedit-Webb-Rubin Equation) R B0 R A0 C0 / a exp 6 3 br a 3 eight parameters A0, B0, C0, a, b,,,

37 Example 3.8 Reported values for the virial oeffiients of isopropanol vapor at 00 o C are: B = -388 m 3 /mol C = -6,000 m 6 /mol Calulate and Z for isopropanol vapor at 00 o C and 0 bar by: (a) the ideal-gas equation (b) Seond virial oeffiient, B () Seond virial oeffiient, B, and third virial oeffiient, C

38 Example solution = K and R = 83.4 m 3 bar/mol K (a) For ideal gas, Z = (b) Using B, R R ( )( ) 3, 934m 0 B 3, , 546m 3 3 / mol / mol () with B and C, use the iteration or solver to solve for Z R B C

39 3.5 Cubi Equations of State behavior of liquid & vapor over wide range Equation must not be so omplex olynomial equation at least ubi equation Cubi EOS are simplest equation real behavior ubi EOS representation

40 he van der Waals Equation of State J. D. an der Waals First proposed in 873 Won Novel rize in 90 R b a R R b a Ideal gas : no volume, no interation an der Walls gas volume + interation

41 he van der Waals Equation of State Saturated liquid and vapor = ; Critial Region > ; monotonially dereasing funtion < ; Liquid region < ; apor region < ; Unrealisti Behavior < ; Real behavior : two phase region

42 he van der Waals Equation of State R b a Molar volume annot be smaller than b b often alled hard sphere volume At = and >, only single root exists At <, three roots exist Smaller : liquid-like volume Middle : no signifiane Larger : vapor-like volume b

43 Limitation of an der Waals EOS Inaurate ritial point predition Z R For all fluids Z = 0.4 to 0.9 for real fluid (mainly hydroarbons) Inaurate vapor pressure predition a parameters are not optimized to fit vapor pressure Only historial interest

44 Cubi Equation of State - History Improvement over an der Waals EOS Redlih and Kwong (949) - Slightly different volume dependene in attrative term - Improved ritial ompressibility (0.3333) and Seond irial Coeffiient - apor pressure and liquid density are still inaurate R b a R b / a ( b ) Wilson (964) - First introdued dependeny of a parameter R ( b ) r a( ) ( b ) [ (.57.6)( )] r a() = a () Aentri fator ln ( at 0. 7 )

45 Cubi Equation of State - History Soave (97) More refined temperature dependeny of a parameter Improved vapor-pressure alulation Also alled Soave-Redlih-Kwong (SRK) EOS R b / a ( b ) eng-ronbinson (976) m( / R ( )a b ( b ) m ) R ( b ) ( a( ) b ) b( b )

46 Cubi Equation of State - History Equation of State with more parameters Smidt and Wenzel (980) R ( v b ) v atel and eja (98) a( ) ( 3 )bv 3b Hamens and Knapp (980) R a( ) ( v b ) v bv ( ) b R ( v b ) v a( ) (b )v b Results an be improved But meaning of the third parameter is not lear It is not easy to write a meaningful mixing rule for the third parameter Note: a() = a ()

47 Cubi Equation of State - History a() law improvement Stryjek-era (986) Soave (984) Carrier (988) 0.5 ( ) r C ( r ) C (/ r ) C ( r ) C ( r Mathias and Copeman (983) ( )(0.7 0 ) r r ) 3 3 r Numerous funtional forms have been proposed

48 Cubi Equation of State - History Ref) J. O. alderrama, Ind. Eng. Chem. Res., 4,603 (003) - Referene in textbook, page 9

49 A Generi Cubi Equation of State Several hundred ubi equation of state have been proposed Most EOS an be redued to a generi form; R ( b) ( ( b)( h) ) R a( ) ( b) ( b)( b) and : pure number (same for all substane) a() and b : substane dependent (a() = f( r,, ) and b = f(, )) e.g., for van der Waals EOS, a() = a, = = 0

50 A Generi Cubi Equation of State

51 Determination of Equation-of-State arameters How to determine a and b parameters? An EOS should represent behavior of pure fluid wo onditions at ritial point ; r 0 ; r 0 At ritial point ; =, =, = page derivation (an der Waals EOS) a 7R 64 b R 8 Z R 0.375

52 Determination of Equation-of-State arameters For other EOS (RK, SRK, R), similar equation an be obtained R a 64 7 R b 8 R a R b

53 Determination of Equation-of-State arameters For better representation of vapor pressure, a parameters are assumed to be temperature dependent R a ) ( ) ( ) ( r r a R a

54 heorem of Corresponding State: Aentri Fator a() law an be fitted using vapor pressure data ( a( ) r ) R C ( r ) C a ( (/ r Is there any method not using vapor pressure? Corresponding state theorem aentri fator ( ) ) r )

55 heorem of Corresponding State: Aentri Fator Redued roperties : How far from ritial point? / r / heorem of Corresponding State (Simplest Form) All fluids, when ompared at the same redued temperature and pressure, have approximately the same ompressibility fator and all deviate from ideal gas behavior to about the same degree. Only valid for simple fluid (argon, krypton and xenon) Systemati deviations are observed r

56 heorem of Corresponding State: Aentri Fator Aentri Fator Introdued by K. S. itzer and oworkers lot of log ( r sat ) vs. (/ r ) d d log (/ r ) r sat S Slope varies depending on the hemial speies

57 heorem of Corresponding State: Aentri Fator Aentri Fator Basis : = 0 for Ar, Kr, Xe sat.0 log( r ) 0.7 his parameter represent the degree how far from simple gases Example: CH 4 : 0.0, C H 5 : 0.00, C 3 H 8 : 0.5, C 4 H 0 : 0.00,., C 0 H : 0.49 See able B. for,,,, and Z for many hemial speies r log(/ r sat ) - (/0.7) (/ r ) Slope varies depending on the hemial speies S=-.3 (for Ar, Kr, Xe)

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59 heorem of Corresponding State: Aentri Fator hree-arameter heorem of Corresponding State All fluids having the same value of, when ompared at the same r and r, have about the same value of Z, and all deviate from ideal gas behavior to about the same degree. Use of aentri fator in Cubi EOS

60 How to Solve Cubi EOS? Solution ehnique Root formula for Cubi equation Iterative alulation - Newton-Raphson iteration or Seant iteration - Suessive Substitution Use the solver funtion - Engineering alulator - Computer program: Mathematia, Matlab, et Initial guess - For gas phase : Ideal Gas Root ( = R/) - For liquid phase : Hard Sphere olume ( = b)

61 Solution Method (3) Suessive Substitution apor or apor-like Roots Start with Ideal Gas Root :=R/ R b a( ) ( b b)( b) Liquid and Liquid-Like Roots Start with : =b b ( b)( R b) b a( )

62 Example 3.9 Given that the vapor pressure of n-butane at 350 K is bar, find the molar volume of (a) saturated-vapor and (b) saturated-liquid n- butane using the Redlih/Kwong equation

63 Example solution and from able B., = 45. K and = bar From able 3., r r a / r R / m bar / mol R b m / mol R b a( ) ( b b )( b ) ( ) (a) For vapor-like root, start with = R/ = / = m 3 /mol Using solver with initial value of = m 3 /mol, then = 555 m 3 /mol (b) For liquid-like root, start with b = m 3 /mol Using solver with initial value of = m 3 /mol, then = 33.3 m 3 /mol

64 3.6 Generalized Correlations for Gases itzer orrelations for the Compressibility fator Z Z 0 Z Z 0 and Z are given as a generalized funtion of r and r - hey are tabulated - See Appendix E (able E. E.4) Lee-Kesler Method (975) - Given by hart (Z 0 and Z ) : able E. E.4 - Basis : Benedit/Webb/Rubin Equation For quantum gases (hydrogen, helium, neon) - Small moleules - Do not onform orresponding behavior - Use effetive ritial parameters ( ) Errors - -3 perent for nonpolar or slightly polar small moleules (mainly hydroarbons) - Larger errors for polar/assoiating omponents, large moleules

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66 3.6 Generalized Correlations for Gases itzer orrelations for the Seond irial Coeffiient Z R B R r Bˆ r Z Z 0 Z Bˆ B 0 B B 0. 7 B r r Not reommended Not aurate for highly polar or assoiating moleules

67 3.7 Generalized Correlation for Liquids Cubi EOS Not aurate for liquid phase molar volume (5-0 % error) Lee-Kesler orrelation Aurate liquid volume but not aurate for polar speies Saturated molar volume Rakett equation (978) sat Z ( r / 7 ) Z sat r r Z [( r ) / 7 ]

68 Example 3.0 Determine the molar volume of n-butane at 50 K and 5 bar by eah of the following: (a) he ideal-gas equation (b) he generalized ompressibility-fator orrelation

69 Example solution (a) For ideal gas, R ( )( 50 ) 5 696,. m 3 / mol (b) and from able B., = 45. K and = bar r r from able E. and E. for Z 0 and Z, Z 0 = 0.865, Z = 0.038, and = 0.00 (able B.) Z = Z 0 + Z = (0.00)(0.038) = ZR ( )( )( 50 ) 5 480,. 7m 3 / mol

70 Homework Homework 3.8, 3.0, 3.4, 3.0, 3.3, 3.44, 3.68 Due: Other Reommend roblems 3., 3.9, 3., 3.5, 3.6, 3.7, 3.8, 3., 3., 3.3, 3.4, 3.7, 3.36