Obtaining U and G based on A above arrow line: )

Transcription

1 Suary or ch,,3,4,5,6,7 (Here soe olar propertes wthout underlne) () he three laws o herodynacs - st law: otal energy o syste (SYS) plus surroundng (SUR) s conserved. - nd law: otal change o entropy o the unverse (SYS+SUR) s postve. -3 nd law: he entropy o a perect crystal s zero. () Math prelnares () Eact derentals M M y y M: any olar property () egendre transors du ds d + μdn () Euler's and G-D heore M M (,, ) ; dm (,, ), 0 nm M any total property n syste; n: total nuber o oles (3) Fundaental property relatons or a closed syste derved ro the st, nd and 3 rd laws () () r r r r r r r r μ dn U dn S dn + dn γ : any arbtrary phase μ U H A G G γ G ; only G n n, S n S, n, n,, () Relatonshp o olar U to olar G, H, A by usng egendre transoratons. Eployng the graphc plot: - S U A H G - Obtanng U and G based on H below arrow lne: ( U H ) ( G H S) Obtanng U and G based on A above arrow lne: ( U A+ S) ( G A+ )

2 (4) Mawell relatons and ore therodynac propertes (4-basc relatons 8-derentatons 4-Mawell equatons) (5) Energy and entropy balances, heat engnes and heat pups (6) ractcal therodynac property relatons (ost useul) (), dependence o μ (checal potental o the coponent) () U, S, H n ters o (,, ) and n ters o (C p, C v ). () (C p, C v ) n ters o,, dependence. (v) General strategy or reducng dcult dervatves (.e. contanng S) to,, (or C p and C v ) that are easurable state varables by usng Jacob procedure..e. Usng the or o and and M M, M C M C, M M: olar property

3 3 Jacob procedure: Step : I dcult dervatves contan G, H, U, A (say olar property, M), always brng M n the nuerator. (A) I M n the denonator, usng y y M M (B) I M s the constran, usng y M y M y E: p H C H H,, μ Step : Usng undaental property relatons, whch contan only,,, and S as state varables (canoncal varables), or those M n the nuerator. e.g.:, v du ds d C d d +, p dh ds d C d d + + [ ] ( ) G H d S d d d d R R R + + [ ] A U d Sd d d d R R R +

4 Eaple: H, μ H H, C Step 3: Elnate dervatve o S ( S, or ds) by Mawell relaton and by relaton o S wth C p, C v (a) I S dervatve wth respect to (wrt) pressure and volue S, X, X or S, X, X (b) I S dervatve wrt teperature S C, X p or S C, X Step 4: I any S constrans ( S ) are let ro applcaton o Mawell relaton, urther elnate S constrans by usng the trple product rule and repeat step 3. 4

5 (7) herodynac property relatons or deal gases () Ideal gas voluetrc propertes (no olecular volue, etcetera) () U(), H(), C p (), C v (), s a uncton o teperature only () S (a uncton o, ), or S (a uncton o, ), or S (a uncton o, ) (8) Aulary unctons () α, Epansblty (epanson coecent) o any substance () K, Isotheral copressblty o any substance (9) Resdual propertes (.e. departure uncton o deal gas ro real gas ture) () R IG R Δ M M M e.g.: H R Δ ; Δ S R Asde: Δ R( Z) (resdual volue s equal to B() at zero pressure). () he property derence (ΔH, ΔS) calculatons usng deal gas relatonshp and resdual propertes. (0) artal olar property o the coponent ( M ) () M ( nm ) n M: olar property n syste,, n j () M nm n M (,, n) or M Σ M (,, ) 5

6 () Gbbs-Duhe Equaton () or M M 0 d + d dm (,, ) M: olar property,, nm nm 0 d + d n dm (,, n) nm: total property n, n, () Relatons usng Gbbs energy (G) () Gbb's - Helholtz equaton G R, H R G R H R () G n ters o H and (or, ) d d d + R (3) artal olar analoges o those undaental property relatons e.g. du ds d ; dg S d + d (4) Fugacty and ugacty coecent denton and calculaton () Dentons dg, dg dg Rd l n,, or a whole syste, Rd l n or a substance, Rd l n or a real ture 6

7 () Relatons or (whole syste),, ( substance), (real ture) and. Fugacty s a new dened uncton n ters o,, : ' dl n( ) d d d, R R R R NOE: EOS Calculaton o s easy n ters o d. However, Eperental easureent s easy n ters o d. By lookng up the ollowng equatons n the nd Ed. rausntz book (3.4.-4) R Rln ), d RlnZ n,, n j ( or real ture R ln( ), d lnz R n or whole syste dln( ), d lnz or substance R n It eans that the values n ters o,, could be calculated usng EOS ore easly than those o (,, ). volue. Reeber only here the volue () s total 7

8 (5) hase equlbru wth ugacty α β (),, or a substance () or the coponent n a real ture α β (6) Behavor o luds () () () Clausus-Clapeyron Equaton to calculate the heat o phase transton Correspondng state theore (plots) or calculaton o Z, H, S, o luds Crtcal luds- A scalng law (e.g. Z C 3/8 or all luds obeyng vdw EOS (7) Behavor o real gaseous tures () () () CS theore or tures (one-lud theory) Cubc n volue (or n copressblty actor) EOS odels (vdw, R, SRK) Adoptng Mng rules or the ture calculaton 8

9 (8) Addtonal or Crtcal luds, EOS, and hase equlbru Fgure : An deal lud sochore (equal densty curves) C.. 3 Ideal lud behaves lke gure Fgure : A real lud sochore (equal densty curves) ρ 5 ρ ρ4 3 Real lud behaves lke gure ρ ρ At the sochore nlecton locus: 0 ρ, C v 0, Fgure 3: here are nu ponts occurred at curves o C v vs ρ. C v constant shown one o the sochore nlecton locus ponts Fgure 3 ρ 9

10 (9) Representaton o the -- curve by a cubc volue EOS odel C.. sat a b apor phase c < c Sy S At the crtcal pont (C..) o any lud: n n c 0 (n,, 3, 4) : oles o the phase ; Sj : olar volue o the phase j at saturated pressure ( sat ). y: a ed phase. q: qualty o vapor oles t : + t S t + sy sy S ( q) + ( q) where t q sy S a b 0

11 At phase equlbru: < Mechancal Stablty 0 Checal Stablty ust be tted by Mawell equal area rule For any ture: I μ n μ I μ μ n, p G I G For a coponent: μ S μ S G G Integraton o S da d It gves A A S d then A S + σ S A + σ A A S S σ S ( ) S S p σ ( ) d 0 It s the Mawell equal area rule What s the physcal eanng o the Mawell equal area rule?

12 Any cubc EOS adopted or the phase equlbru calculaton ust ollow the Mawell equal area rule..e.: he area o ( sat ) ( S - ),.e. two does, gven by the EOS curve ust be equal to the area o ( sat )( S - ),.e. cross rectangle, represented by the eperental curve (bold lne). hen, ndng a sutable vapor pressure ( sat ) s possble to get vapor phase volue ( S ) and lqud phase volue ( ). C.. cubc EOS curve sat < c Sy S S A - curve at constant A σ cubc EOS curve

13 For eaple: Supposed that DW cubc EOS s good to represent the data R a b Integrate the above equaton ro S to at a certan σ, then: Mawell equal area relaton or Checal Stablty n phase equlbru: S R d b R n S b b a σ S d ( ) 0 + a σ S l ( ) 0 EQ() S σ R b a ( ) EQ() σ R S b a ( S ) EQ(3) EQ(), EQ(), EQ(3) or relatons o c σ, S, ndeed ested at σ <. In other words, t s possble to nd a sutable pressure ( ) n ulllent o the Mawell equal area relaton, then, two roots o vapor phase volue ( S ) and lqud phase volue ( ) could be solved by usng EQ(), EQ(), EQ(3), sultaneously. 3

14 Dscusson: Even the eperental ---y data could be tted by the EOS, such as the van der Waals (vdw) EOS as ollows, but the accuracy can not be always satsed: R a b Z R b a R At crtcal pont, t s easy to nd the a and b values ro the values o c and c. hen, substtute the nto the Z equaton o the vdw EOS. c c 3 Z c R 8 c Accordng to the correspondng state theory, however, or ost real luds the eperental Z c data ranged ro 0.3 to 0.7. Agan, the ollowng good EOS odels dd not gve a correct answer or the Z C value o the polar water copound: For nstance, eperental data o Z c : Z But: eng-robnson EOS: / Z c water c water SRK EOS: Z / c water 4

15 (0) Introducton o a dgtalzed coputer project Suppose there s a 3-phase equlbru proble contanng I,, and I Method : rausntz approach ( - ethod) (apor phase: ; qud phase: ) At phase equlbru o bnary syste, our phase equatons (n(π-)4) could be orulated as ollows: y I or I or I I I or ( I ) and ( ) or ( I ) whch ay be not accurate as vapor phase. Method : γ approach (γ - ethod) A ew odels, NR, Wlson, Unac, Unquac, etc..e. vapor phase usng ; lqud phase (I, ) usng γ : y,, y γ (,, ; ), I, I I I γ (,, γ (,, ; ; ) ),, (, ) (, ) γ γ I 5

16 (A) Calculaton procedure o the coputer language prograng (lease reer to Sandler s book, MAHCAD, MAHAB, BASIC progras) FORRAN an progra (bubble pont-pressure calculaton by gvng, ) Coon Coon, etc. a, b, a, b, A, B a A, B R b R Input ( c, c, ω); crtcal property and accentrc actor Input (, k j, δ j ); bubble pont teperature and bnary nteractonal paraeters Input ( ); bubble pont lqud-phase coposton o the coponent (0.0 to.0) Σ Check pont: y S k k ( ) + k ( ) Σ I S hen k k ( ) Σ y k hen, ncreasng the tred bubble pont pressure () by usng a try and error ethod. 6

17 (B) Flow Chart an progra Guess ntal, change, and nput data Bubble pont pressure calculaton,, y K,K subroutnes Fugacty subroutne Coon a, b, a, b, δ, A, B pck Z cale,, KA subroutne K K prnt nteractve values, y, Fugacty subroutne Coon a, b, a, b, δ, A, B pck Z cale Z,Z, Z3,, y Z,Z, Z3 Cubc (at,,, y known) Coon Coon a, b, a, b, δ, A, B Calc, a & b, calc, A and B solve or Z, Z, Z 3 (each te) large or vapor sall or lqud, separately 7