General Thermodynamcs for Process Smulaton Dr. Jungho Cho, Professor Department of Chemcal Engneerng Dong Yang Unversty
Four Crtera for Equlbra μ = μ v Stuaton α T = T β α β P = P l μ = μ l1 l 2 Thermal Equlbrum Mechancal Equlbrum Condton, Phase Equlbra (VLE, LLE) G ξ T, P = 0 Chemcal Equlbrum Fugacty (or chemcal potental) s defned as an escapng tendency of a component n a certan phase nto another phase. Thermodynamc Models bult n Smulator
Basc Phase Equlbra Relatons Vapor-lqud equlbrum calculatons The basc relatonshp for every component n vapor-lqud equlbrum s: v ( T P y ) f l ( T P x ) where : the fugacty of component n the vapor phase v fˆ l fˆ f ˆ,, = ˆ,, : the fugacty of component n the lqud phase (1) Thermodynamc Models bult n Smulator
Basc Phase Equlbra Relatons There are two methods for representng lqud fugactes. - Equaton of state method - Lqud actvty coeffcent method Thermodynamc Models bult n Smulator
Equaton of State Method The equaton of state method defnes fugactes as: f ˆ v v = φˆ y P (2) f ˆ l l = φˆ x P (3) where: φ v φ l y x P s the vapor phase fugacty coeffcent s the lqud phase fugacty coeffcent s the mole fracton of n the vapor s the mole fracton of n the lqud s the system pressure Thermodynamc Models bult n Smulator
Equaton of State Method We can then rewrte equaton 1 as: ˆ φ y = ˆ φ x v l (4) Ths s the standard equaton used to represent vaporlqud equlbrum usng the equaton-of-state method. φ v and φ l are both calculated by the equaton-of-state. Note that K-values are defned as: y K = (5) x Thermodynamc Models bult n Smulator
Lqud Actvty Coeffcent Method (VLE) The actvty coeffcent method defnes lqud fugactes as: fˆ l The vapor fugacty s the same as the EOS approach: v f ˆ = φˆ = γ x f (6) v y 0 P (7) where: γ s the lqud actvty coeffcent of component 0 f v φˆ s the standard lqud fugacty of component s calculated from an equaton-of-state model We can then rewrte equaton 1 as: ˆv φ y P = x γ f 0 (8) Thermodynamc Models bult n Smulator
Lqud Actvty Coeffcent Method (LLE) For Lqud-Lqud Equlbrum (LLE) the relatonshp s: f ˆ = l1 fˆ l 2 (9) where the desgnators 1 and 2 represent the two separate lqud phases. Usng the actvty coeffcent defnton of fugacty, ths can be rewrtten and smplfed as: x l1 l1 x l 2 γ l 2 γ = (10) Thermodynamc Models bult n Smulator
K-values The k-values can be calculated from: K = y x = γ l φ sat P sat l V exp RT ˆv φ P ( sat P P ) (11) Or K = y x = ˆ φ ˆ φ l v (12) Thermodynamc Models bult n Smulator
Example 2: Ideal Raoult s Law The precedng equaton reduces to the followng deal Raoult s law: y P = x P vap Example : Pxy plot at constant T(75 o C). (P n kpa, T n o C) 2945.47 lnp vap 1 = 14.2724, lnp 2972.64 vap 2 = 14.2043 t + 224 t + 209 Soluton At 75 o C, P vap 1 = 83. 21kPa and P vap 1 = 41. 98kPa The total pressure Vapor phase composton, vap vap P x1 P1 + x2p2 P= P 2 + P P x = vap ( vap vap, ) 1 2 1 y x P vap 1 = = vap P2 + 1 2 1 x P 1 1 1 1 vap vap ( P P ) x P vap Thermodynamc Models bult n Smulator
Pxy Dagram at Constant Temperature 100 90 a vap P 1 vap P 2 Pressure (kpa) 80 70 60 50 40 30 Lqud P x 1 c b c b P y 1 Vapor d 20 0 5 1.0 x 1, y 1 Thermodynamc Models bult n Smulator
Example 3: Slghtly Non-deal System For systems whch the lqud phase behaves nondeally: y P = γ x P vap Relaton between actvty coeffcent and excess Gbbs energy s as: lnγ = ex ( ng / RT ) n T, P, n j As an example, excess Gbbs energy expresson s as: G ex = RT Ax 1 x 2 Therefore, 1 γ and γ 2 becomes. 2 2 γ = ( ) γ = ( ) 1 exp Ax 2 So, = A( 1 x ) 2 exp Ax 1 [ ] 2 vap ( 2 ) vap x P Ax x P P exp 1 1 1 + exp 1 2 2 y 1 exp = 2 [ A( 1 x ) ] P 1 x 1 P vap 1 Thermodynamc Models bult n Smulator
Predcton wth Margules Equatons 160 Unstable 140 120 Pressure (kpa) 100 80 60 A = 0.0 A = 0.5 A = 1.0 A = 2.0 A = 3.0 40 20 0.0 0.2 0.4 0.6 0.8 1.0 x 1, y 1 Thermodynamc Models bult n Smulator
Devatons from Raoult s Law (1 of 2) In general, you can expect non-dealty of unlke molecules. Ether the sze and shape or the ntermolecular nteractons between components may be dssmlar. For short, these are called sze and energy asymmetry. Energy asymmetry occurs between polar and non-polar molecules and also between dfferent polar molecules. In the majorty of mxtures, actvty coeffcents s greater than unty. The result s a hgher fugacty than deal. The fugacty can be nterpreted as the tendency to vaporze. If compounds vaporze mere than n an deal soluton, then they ncrease ther average dstance. So actvty coeffcents s greater than unty ndcate repulson between unlke molecules. If the repulson s strong, lqud-lqud separaton occurs. Ths s another mechansm that decreases close contact between unlke molecules. If the actvty coeffcent s larger than unty, the system s sad to show postve devatons from Raoult s law. Negatve devatons from Raoult s law occur when the actvty coeffcent s smaller than unty. Thermodynamc Models bult n Smulator
Devatons from Raoult s Law (2 of 2) 90 80 Sub-cooled Lqud Pressure (kpa) 70 60 50 postve negatve deal 40 Super-heated Vapor 30 0.0 0.2 0.4 0.6 0.8 1.0 x 1, y 1 Thermodynamc Models bult n Smulator
Isothermal Flash Calculatons Lqud feed T,P,F z T & P Flash Drum Heater Valve L, x Thermodynamc Models bult n Smulator
Equlbrum Flash Vaporzaton The equlbrum flash separator s the smplest equlbrum-stage process wth whch the desgner must deal. Despte the fact that only one stage s nvolved, the calculaton of the compostons and the relatve amount of the vapor and lqud phases at any gven pressure and temperature usually nvolves a tedous traland-error soluton. Buford D. Smth, 1963 Thermodynamc Models bult n Smulator
Flash Calculaton (1 of 4) MESH Equaton Materal Balance Equlbrum Relatons Summaton of Compostons Enthalpy(H) Balance Thermodynamc Models bult n Smulator
Flash Calculaton (2 of 4) Overall Materal Balance F = V + L (1) Component Materal Balance Fz = Vy + Lx (2) Equlbrum Relatons y = K x (3) Thermodynamc Models bult n Smulator
Flash Calculaton (3 of 4) Summaton of Compostons x = 1 Defnng (4a) y = 1 (4b) φ =V F (5) Combnng (1) through (5), we obtan: ( 1 K ) ( K 1) ( ) z F φ = + = 0 1 φ (6) Thermodynamc Models bult n Smulator
Flash Calculaton (4 of 4) From deal Raoult s law y P = x vap K-value can be rewrtten as: From Antone equaton y K = = x log P vap ( kpa) P P = vap P A t B ( o C) + C (7) (8) (9) Thermodynamc Models bult n Smulator
Antone Coeffcents log P vap ( kpa) = A t B ( o C) + C Benzene Toluene A 6.01788 6.08436 B 1203.677 1347.620 C 219.904 219.787 Thermodynamc Models bult n Smulator
Rachford-Rce Functon 0.10 0.05 φ = 0.707 F(φ) 0.00-0.05-0.10 ( 1 K ) ( K 1) ( ) z F φ = + = 0 1 φ -0.15 0.0 0.2 0.4 0.6 0.8 1.0 φ Thermodynamc Models bult n Smulator
Flash Calculaton Results (1 of 3) Vapor Flowrate (K-mole/hr) V = Fφ = ( 100 ) (0.707) = 70. 7 (1) Lqud Flowrate (K-mole/hr) L = F V = 100 70.7 = 29.3 (2) Thermodynamc Models bult n Smulator
Flash Calculaton Results (2 of 3) Mole Fracton at the lqud phase x = z 1+ φ 1 ( K ) x = 0.4479 = 0. 5521 B Mole Fracton at the vapor phase y = K x = x T Kz 1+ φ 1 ( K ) y = 0.6631 = 0. 3369 B y T (3) (4) (5) (6) Thermodynamc Models bult n Smulator
Flash Calculaton Results (3 of 3) 200 180 x B = 0.4329 Pressure (kpa) 160 140 120 100 y B = 0.6429 80 60 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mole Fracton of Benzene Thermodynamc Models bult n Smulator
PRO/II Keyword Input for Flash Calculaton TITLE PROBLEM=PRBLEM-1A,PROJECT=CLASS,USER=JHCHO DIMENSION METRIC,PRES=ATM PRINT INPUT=ALL,PERC=M,FRAC=M COMPONENT DATA LIBID 1,BENZENE/2,TOLUENE THERMODYNAMIC DATA METHOD SYSTEM=IDEAL STREAM DATA PROP STREAM=1,TEMP=25,PRES=1,RATE=100,COMP=1,60/2,40 UNIT OPERATION DATA FLASH UID=F01 FEED 1 PROD V=1V,L=1L ISO TEMP=100,PRES=1.2 END Thermodynamc Models bult n Smulator
PRO/II Output Summary for Flash Calculaton STREAM ID 1 1L 1V NAME PHASE LIQUID LIQUID VAPOR FLUID MOLAR FRACTIONS 1 BENZENE 0.6000 0.4476 0.6629 2 TOLUENE 0.4000 0.5524 0.3371 TOTAL RATE, KG-MOL/HR 100.0000 75.6710 24.3290 TEMPERATURE, C 25.0000 100.0000 100.0000 PRESSURE, ATM 1.0000 1.2000 1.2000 ENTHALPY, M*KCAL/HR 0.0865 0.2800 0.2681 MOLECULAR WEIGHT 85.1285 85.8632 82.8433 MOLE FRAC VAPOR 0.0000 0.0000 1.0000 MOLE FRAC LIQUID 1.0000 1.0000 0.0000 Thermodynamc Models bult n Smulator
PRO/II BVLE Analyss Thermodynamc Models bult n Smulator
Dew & Bubble Pont Calculaton Dew Pont s the very state at whch condensaton s about to occur. Dew Pont Temperature Calculaton at a Gven Pressure Dew Pont Pressure Calculaton at a Gven Temperature Vapor Fracton s 1 at Dew Pont Bubble Pont s the very state at whch vaporzaton s about to occur. Bubble Pont Temperature Calculaton at a Gven Pressure Bubble Pont Pressure Calculaton at a Gven Temperature Vapor Fracton s 0 at Bubble Pont Thermodynamc Models bult n Smulator
Ex-1: Bubble Pont Falure Case Calculate the bubble pont pressure at 85 o C of the followng stream. Dd you get a converged soluton? If not, why? Use SRK for your smulaton. Component Mole % C1 65 C2 15 C3 15 IC4 5 Save as Flename: EX-1.np Thermodynamc Models bult n Smulator
Dfference between Gas and Vapor For gas, T > T c For vapor, T < T c T: System temperature, T c : Crtcal temperature Methane Gas but not Methane Vapor Water Vapor but not Water Gas Thermodynamc Models bult n Smulator
Ex-2: C7 Plus Heavy Cut Characterzaton Calculate the bubble pressure at 45 o C and dew temperature at 1.5bar of the followng stream. Regard C6+ as NC6(1), NC7(2) and NC8(3) and compare the results. Use SRK for your smulaton. Component Mole % C1 5 C2 10 C3 15 IC4 10 NC4 20 IC5 15 NC5 20 C6+ 5 Save as Flename: EX-2A.np for NC6, EX-2B.np for NC7, EX-2C.np for NC8 Thermodynamc Models bult n Smulator
Results for EX-2 Characterzaton of heavycut s very mportant n the calculaton of dew pont temperature. EX-2A.np, EX-2B.np, EX-2C.np Bubble P Dew T C6 Plus at 45 o C at 1.5bar 18.505 30.519 NC6 18.561 42.783 NC7 18.669 59.585 NC8 Thermodynamc Models bult n Smulator
Results for EX-2A (C6+ NC6) Thermodynamc Models bult n Smulator
Results for EX-2B (C6+ NC7) Thermodynamc Models bult n Smulator
Results for EX-2C (C6+ NC8) Thermodynamc Models bult n Smulator
The End of General Thermodynamcs The End. Thermodynamc Models bult n Smulator