Energy, Entropy, and Availability Balances Phase Equilibria. Nonideal Thermodynamic Property Models. Selecting an Appropriate Model

Transcription

1 Lecture 4. Thermodynamcs [Ch. 2] Energy, Entropy, and Avalablty Balances Phase Equlbra - Fugactes and actvty coeffcents -K-values Nondeal Thermodynamc Property Models - P-v-T equaton-of-state models - Actvty coeffcent models Selectng an Approprate Model

2 Thermodynamc Propertes Importance of thermodynamc propertes p and equatons n separaton operatons Energy requrements (heat and work) Phase equlbra : Separaton lmt Equpment szng Property estmaton Specfc volume, enthalpy, entropy, avalablty, fugacty, actvty, etc. Used for desgn calculatons Separator sze and layout Auxlary components : Ppng, pumps, valves, etc.

3 Energy, Entropy and Avalablty Streams n n, z, T, P, h, s, b, v : : : (Surroundngs) Heat transfer n and out Q, Q, T n s Separaton process (system) S rr, LW Balances out T s : : : Streams out n, z, T, P, h, s, b, v One or more feed streams flowng nto the system are separated nto two or more product streams that t flow out of the system. n z T P h s Molar flow rate Mole fracton Temperature Pressure Molar enthalpy Molar entropy T b Molar avalablty 0 (W s ) n (W s ) out v Specfc volume Shaft work n and out

4 Energy Balance Contnuous and steady-state state flow system Knetc, potental, and surface energy changes are neglected Frst law of thermodynamcs (conservaton of energy) (stream enthalpy flows + heat transfer + shaft work) leavng system - (stream enthalpy flows + heat transfer + shaft work) enterng system = 0 ( nh Q W ) ( nh Q W ) 0 s out of n to system system s

5 Entropy Balance The frst law provdes no nformaton on energy effcency Second law of thermodynamcs (stream entropy flows + entropy flows by heat transfer) leavng system - (stream entropy flows + entropy flows by heat transfer ) enterng system = producton of entropy by the process out of system Q Q ns ns S T T s n to s system rr - Producton of entropy - Irreversble ncrease n the entropy of the unverse - Quanttatve measure of the thermodynamc neffcency of a process

6 Avalablty (Exergy Exergy) Balance The entropy balance contans no terms related to shaft work The entropy s dffcult to relate wth power consumpton Avalablty (exergy) : Avalable energy for complete converson to shaft work Stream avalablty functon : b h T0s a measure of the maxmum amount of stream energy that can be converted nto shaft work f the stream s taken to the reference state n to system (Entropy balance) T 0 -(Energy balance) T 1 T nb Q W nb Q 1 W LW 0 0 s s Ts out of Ts system (stream avalablty flows + avalablty of heat + shaft work) enterng system enterng system - (stream avalablty flows + avalablty of heat + shaft work) leavng system = loss of avalablty (lost work)

7 Lost Work, Mnmum Work, and Second Law Effcency Lost work, LW T0S rr - The greater ts value, the greater s the energy neffcency - Its magntude depends on the extent of process rreversbltes - Reversble process : LW = 0 Mnmum work of separaton, W mn Mnmum shaft work requred to conduct the separaton Equvalent to the dfference n the heat transfer and shaft work W nb nb The second-law effcency Wmn LW W mn mn out of system (mnmum work of separaton) (equvalent actual work of separaton) n to system

8 Phase Equlbra The phase equlbra of the gven system provde possble equlbrum compostons (separaton lmt) Equlbrum : Gbbs free energy for all phases s a mnmum G G ( T, P, N1, N2,..., N C ) dg SdT VdP dn components dg dn (at constant T & P) p p ( ) ( ) system phases components P, T N (1) dn dn (1) (2) p2 ( p)... (conservaton of moles of each speces, no reacton) ( N ) The chemcal potental of a partcular speces n a multcomponent system s dentcal n all phases at physcal equlbrum.

9 Fugactes and Actvty Coeffcents Chemcal potental Unts of energy Not easy to understand physcal meanng More convenent quanttes Fugacty : pseudo-pressure f C exp( / RT ) Equalty of chemcal potentals equalty of fugactes Fugacty coeffcent Rato of fugacty and pressure Reference : deal gas Actvty a 0 / f Rato of fugactes Reference : deal soluton f P f / yp f / At equlbrum, f f f (1) (2) ( N ) a a a (1) (2) ( N ) 0 Actvty coeffcents a / x f / x f (1) (2) ( N ) Rato of actvty and composton Departure from deal soluton behavor T T T P P P (1) (2) ( N )

10 K-Values Phase equlbrum rato : rato of mole fractons of a speces present n two phases at equlbrum K-value (vapor-lqud equlbrum rato; K-factor) : for the vapor-lqud case K y / x Dstrbuton coeffcent (lqud-lqud equlbrum rato) : for the lqud-lqud case K x / x (1) (2) D Relatve volatlty : for the vapor-lqud case j K / K j Relatve selectvty :for the lqud-lqud case K K / j D Dj

11 Phase Equlbrum Calculatons (VLE) f f For vapor-lqud equlbrum V L Ideal gas + Ideal soluton y P x P sat K y x P P satt Ph-Ph approach : equaton-of-state form of K-value V y P L x P Gamma-Ph approach : actvty coeffcent form of K-value o o V yp Lx f L L L L L K V P V K L V f

12 Nondeal Thermodynamc Property Models No unversal equatons are avalable for computng, for nondeal mxtures, values of thermodynamc propertes such as densty, enthalpy, entropy, fugactes, and actvty coeffcents as functons of T, P, and phase composton. (1) P-v-T equaton-of-state models (2) Actvty coeffcent or free-energy models P-v-T equaton-of-state models Nondealty s due to (1) the volume occuped by the molecules and (2) ntermolecular forces among the molecules e.g. the van der Waals equaton RT a P v b v 2

13 Useful Equatons of State Mxng rules a C C C 0.5 yyj( aa j) b 1 j1 1 yb

14 Models for Actvty Coeffcents ( T, x, x2,..., x 1 C )

15 Notes on Usng Phase Equlbrum Low pressure VLE Models Gamma-Ph approach recommended Poyntng correcton (modfed Raoult s law) requred for medum pressure Cannot be appled when T or P condton exceeds crtcal T, P Hgh pressure VLE Ph-Ph approach recommended Specal care should be taken for polar components (alcohols, water, acds, amnes, etc.) Check bnary nteracton parameters matrx If parameters exst, use them If parameters do not exst, Try to obtan by regresson of expermental data Use group contrbuton t method (e.g. UNIFAC) Specal applcatons Specalzed models requred Polymer soluton Polymer soluton Electrolyte soluton Bomolecular applcatons

16 Selectng an Approprate Model (LG): lght gases (PC): polar organc compounds (E): electrolytes (HC): hydrocarbons (A): aqueous solutons If the mxture s (A) wth no (PC) - If (E) are present modfed NRTL equaton - If (E) are not present a specal model If the mxture contans (HC), coverng a wde bolng rage The correspondng-states method of Lee-Kesler-Plöcker If the bolng range of a mxture of (HC) s not wde - For all T and P the P-R equaton - For all P and noncryogenc T the S-R-K equaton - For all T, but not P n the crtcal regon the Benedct-Webb- Rubn-Starlng method If the mxture contans (PC) - If (LG) are present the PSRK method - If (LG) are not present a sutable lqud-phase method