Thermodynamics II. Lec 1: Vapor Liquid Equilibrium-part 1. Dr.-Eng. Zayed Al-Hamamre
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1 Thermodynamcs II Lec : Vapor Lqud Equlbrum-part Dr.-Eng. Zayed Al-Hamamre Content Introducton The nature of equlbrum Phase Rule n VLE Raoult s law Bubble and dew ponts calculatons 2
2 VLE Calculatons Purpose of ths lecture: To demonstrate how Raoult s law can be used n the predcton of the VLE behavor of deal mxtures Hghlghts Phase rules gves the number of varables we need n order to determne the ntensve state of a system at equlbrum Raoult s law can be used for constructng Pxy, Txy dagrams and performng dew pont and bubble pont calculatons 3 Introducton Composton changes are the desred outcome, not only of chemcal reactons, but of a number of ndustrally mportant mass-transfer operatons Processes such as dstllaton, absorpton, and extracton brng phases of dfferent composton nto contact, and when the phases are not n equlbrum, mass transfer between the phases alters ther compostons. Both the extent of change and the rate of transfer depend on the departure of the system from equlbrum. Thus, for quanttatve treatment of mass transfer the equlbrum T, P, and phase compostons must be known The most commonly encountered coexstng phases n ndustral practce are vapor and lqud, although lqud/lqud, vapor/lqud, and lqud/sold systems are also found 4
3 Introducton Flash dstllaton Flash dstllaton s a sngle stage separaton technque. A lqud mxture feed s pumped through a heater to rase the temperature and enthalpy of the mxture. It then flows through a valve and the pressure s reduced, causng the lqud to partally vaporze. Once the mxture enters a bg enough volume the lqud and vapor separate. Because the vapor and lqud are n such close contact up untl the "flash" occurs, the product lqud and vapor phases approach equlbrum. 5 Introducton Flash (or partal) evaporaton Is the partal vaporzaton that occurs when a urated lqud stream undergoes a reducton n pressure by passng through a throttlng valve or other throttlng devce. Ths process s one of the smplest unt operatons. If the throttlng valve or devce s located at the entry nto a pressure vessel so that the flash evaporaton occurs wthn the vessel, then the vessel s often referred to as a flash drum. 6
4 Introducton 7 The nature of equlbrum Equlbrum s a statc condton n whch no changes occur n the macroscopc propertes of a system wth tme. Ths mples a balance of all potentals that may cause change. Example An solated system consstng of lqud and vapor phases n ntmate contact eventually reaches a fnal state wheren no tendency exsts for change to occur wthn the system. The temperature, pressure, and phase compostons reach fnal values whch thereafter reman fxed 8
5 Vapor-lqud equlbrum (VLE) Consder a bnary (.e., 2-component) system wth 2-phases: What do we know? T vap, P vap y A, y B y A + y B = x A + x B = y A x A T lq, P lq x A, x B At equlbrum: T vap = T lq P vap = P lq Gbbs Phase Rule: degrees of freedom = # components (C) # phases (P) + 2 For a bnary, 2 phase system: = 2 We can specfy only 2 ntensve varables (all others are fxed, by VLE) 9 Phase Rule for Intensve Varables For a system of phases and N speces, the degree of freedom s: F = N # varables that must be specfed to fx the ntensve state of the system at equlbrum Phase Rule Varables: The system s characterzed by T, P and (N-) mole fractons for each phase Requres knowledge of 2 + (N-) varables Phase Rule Equatons: At equlbrum = = These relatons provde (-)N equatons for all N speces The dfference s F = 2 + (N-) -(-)N = 2- +N Phase rules gves the number of varables we need n order to determne the ntensve state of a system at equlbrum 0
6 Phase Rule n VLE: Sngle Component Systems For a two phase (p=2) system of a sngle component (N=): F = 2- + N F = = Therefore, for the sngle component system, specfyng ether T or P fxes all ntensve varables. VLE for Pure Components 800 Pressure: kpa Temperature: K 420 Acetontrle Ntromethane Specfy P and T 2 graphs n one: T vs. x A T vs. y A 2 phase regon superheated vapor urated vapor lne urated lqud lne T A z A subcooled lqud x A Temperature composton dagram for ethanol water y A A subcooled lqud feed of composton z A, heated to temperature T A, wll separate spontaneously nto 2 phases, of composton x A and y A 2
7 Bolng pont, dew pont, bubble pont Pure lquds have a bolng pont; mxtures have a bolng range, delmted by ther bubble pont and dew pont.. Consder a sub cooled bnary lqud that s 40 mol% ethanol. What s ts bubble pont? What s the composton of the frst bubble? dew pont bubble pont x E,ntal y E,ntal bolng range 2. Consder a superheated bnary vapor that s 40 mol% ethanol. What s ts dew pont? What s the composton of the frst drop? 3. What s the bolng range of ths mxture? Fgure 2 3 Temperature composton dagram for ethanolwater 3 Useful defntons Bolng/bubble pont T bp : temperature at whch the average lqud molecule has just enough knetc energy to escape from the surface of the lqud nto the gas phase Recall that knetc energy follows a Boltzmann dstrbuton, so molecules wth hgher than average knetc energy can stll escape from the surface at T < T bp, by evaporaton Saturated lqud: a lqud at ts bolng/bubble pont Dew pont T dp : temperature at whch the average vapor molecule has just enough knetc energy to condense Saturated vapor: a vapor at ts dew pont Vapor pressure: pressure at whch the lqud and vapor phase are n equlbrum at a gven temperature Azeotrope: a constant-bolng mxture,.e., a mxture that behaves lke a sngle component 4
8 Useful defntons Saturated lqud and Subcooled lqud If a substance exsts as a lqud at the uraton temperature, t s called a urated lqud. If the temperature of the lqud s lower than the uraton temperature, t s called a subcooled lqud. Saturated vapor and Superheated vapor If a substance exsts entrely as vapor at the uraton temperature, t s called a urated vapor. When the vapor s at a temperature greater than the uraton temperature, t s sad to exst as superheated vapor. 5 Correlatons for Vapor Pressures Antone Equaton: The Antone equaton may have slghtly dfferent forms: ln P A B T C where A, B, and C are expermentally derved parameters. temperature and pressure are n the followng unts: T [=] o C and P * [=] mm Hg. The Antone equaton s generally reasonably good over a moderate range of temperatures. It doesn't do very well near the crtcal pon 6
9 Correlatons for Vapor Pressures P, or the vapour pressure of component, s commonly represented by Antone Equaton B ln P A T C For acetontrle (Component ): ln P / kpa T / C 224 For ntromethane (Component 2): ln P 2 / kpa T / C 209 These functons are the only component propertes needed to characterze deal VLE behavour CHEE 3 Lecture Correlatons for Vapor Pressures DIPPR Correlaton: More accurate yet s the DIPPR correlaton of vapor pressures: lnp* = A + B/T + ClnT + DT E where the constants A, B, C, D and E are contaned n the DIPPR database for each compound. Ordnarly, E = 6. The DIPPR database also has a calculator package to do the nterpolaton at any desred temperature so that the coeffcents don't need to be coped down. 8
10 Phase Rule n VLE: Ideal Bnary Mxtures (General Case) For a two phase (=2), bnary system (N=2): F = = 2 Therefore, for the bnary case, two ntensve varables must be specfed to fx the state of the system. 9 Phase Rule n VLE: Bnary Systems (Pxy dagrams) Example: Acetontrle () / Ntromethane (2) system 90 Acetontrle() - 75C 80 Pressure, kpa x,y y x 20
11 Phase Rule n VLE: Bnary Systems (Txy dagrams) Alternatvely, we can specfy a system pressure and examne the VLE behavor as a functon of temperature and composton. Acetontrle() 70kPa Temp, deg C x,y y x CHEE 3 Lecture Tetrahydrofuran (THF) / Carbon tetrachlorde Negatve devaton from Raoult s law (dotted lne) Intermolecular (attractve) forces between unlke molecules are stronger than lke molecules Unlke molecules lke each other! 22
12 Chloroform/ Tetrahydrofuran Strong negatve devaton from Raoult s law due to hydrogen bondng Azeotrope (constant bolng mxtures): A bolng lqud at ths composton produces a vapor of exactly the same composton No separaton s possble by dstllaton at azeotropc composton dagram Mnmum-pressure or negatve azeotrope 23 Furan / Carbon tetrachlorde A slghtly postve devaton 24
13 Ethanol / toluene Strong postve devaton wth a maxmum pressure azeotrope Intermolecular forces between lke molecules are stronger than unlke molecules Unlke molecules don t lke each other! 25 26
14 27 Smple Models For VLE: Raoult's Law Appled to vapor/lqud equlbrum, to fnd by calculaton the temperatures, pressures, and compostons of phases n equlbrum The two major assumptons are: o o The vapor phase s an deal gas. The lqud phase s an deal soluton The frst assumpton means that Raoult's law can apply only for low to moderate pressures. The second mples that t can have approxmate valdty only when the speces that comprse the system are chemcally smlar The deal soluton represents a standard to whch real-soluton behavor may be compared. Ideal-soluton behavor s often approxmated by lqud phases wheren the molecular speces are not too dfferent n sze and are of the same chemcal nature. 28
15 Smple Models For VLE: Raoult's Law Raoult s Law for deal phase behavor relates the composton of lqud and vapor phases at equlbrum through the component vapor pressure, P. or y x P P Where x s a lqud phase mole fracton, y s a vapor phase mole fracton, and P s the vapor pressure of pure speces at the temperature of the system. The product y P s known as the partal pressure of speces. If expermental data are not avalable, estmaton of VLE can stll be done. The smplest method assumes deal vapor and deal lqud phases. It s not very accurate for some mxtures, but s a good frst approxmaton, and the only method that we wll deal wth n ths ntroductory course. 29 Smple Models For VLE: Raoult's Law The Raoult s law, smple model for VLE, provdes a realstc descrpton of actual behavor for a relatvely small class of systems. Nevertheless, t s useful for dsplayng VLE calculatons n ther smplest form, and t also serves as a standard of comparson for more complex systems. A lmtaton of Raoult s law s that t can be appled only to speces of known vapor pressure, and ths requres the speces to be subcrtcal,.e., to be at a temperature below ts crtcal temperature. Raoult s Law feature for speces wth mole fracton approachng unty o An mportant and useful feature of Raoult s law s that t s vald for any speces present at a mole fracton approachng unty,.e. one provded only that the vapor phase s an deal gas. o Chemcal smlarty of the consttuent speces s not here a requrement. 30
16 Smple Models For VLE: Raoult's Law Raoult's Law can be appled to each component that dstrbutes between the two phases n the system. For example, f we have a bnary mxture of A and B, then we can wrte Raoult's Law for both components. If we do that and then add the two equatons together, we obtan a convenent relatonshp between the lqud mole fractons and the total pressure: 3 Smple Models For VLE: Raoult's Law Raoult's law can be used to solve VLE problems n place of the Txy and Pxy dagrams or to generate Txy and Pxy. 32
17 Dew pont and Bubble pont Calculatons 33 Dew pont and Bubble pont Calculatons 34
18 Dew pont and Bubble pont Calculatons 35 Dew pont and Bubble pont Calculatons 36
19 Dew pont and Bubble pont Calculatons 37 Dew pont and Bubble pont Calculatons For bubble pont calculatons P For a bnary system: y P P x P x P-x curve s lnear to composton Characterstc of deal soluton For dew pont calculatons x P P y P y P 38
20 VLE Calculatons For now, we are gong to employ these calculatons only for dentfyng the state and composton of bnary and deal mxtures As we are gong to see later n the course, the aforementoned VLE calculatons are also applcable to non-deal or/and mult-component mxtures The calculatons revolve around the use of 2 key equatons: ) Raoult s law for deal phase behavour: P y * P x * P () 2) Antone s Equaton ln( P ) A B T C (2) 39 BUBL P Calculaton (T, x known) - Calculate P and from Antone s Equaton P 2 - For the vapour-phase composton (bubble) we can wrte: y +y 2 = (3) - Substtute y and y 2 n Eqn (3) by usng Raoult s law: x * P P x * P P x* P P ( x )* P P (4) - Re-arrange and solve Eqn. (4) for P - Now you can obtan y from Eqn () - Fnally, y 2 = -y 40
21 DEW P Calculaton (T, y known) P P 2 - Calculate and from Antone s Equaton - For the lqud-phase composton (dew) we can wrte: x +x 2 = (5) - Substtute x and x 2 n Eqn (5) by usng Raoult s law: y * P y2 * P P P y * P ( y P P )* P 2 2 (6) - Re-arrange and solve Eqn. (6) for P - Now you can obtan x from Eqn () - Fnally, x 2 = -x 4 BUBL T Calculaton (P, x known) Snce T s an unknown, the uraton pressures for the mxture components cannot be calculated drectly. Therefore, calculaton of T, y requres an teratve approach, as follows: - Re-arrange Antone s equaton so that the uraton temperatures of the components at pressure P can be calculated: T B C A ln( P) - Select a temperature T so that T T' T2 -Calculate P ( T' ) and P2 ( T' ) - Solve Eqn. (4) for pressure P - If P P', then P =P; If not, try another T -value -Calculate y from Raoult s law (7) 42
22 DEW T Calculaton (P, y known) Same as before, calculaton of T, x requres an teratve approach: - Re-arrange Antone s equaton so that the uraton temperatures of the components at pressure P can be calculated from Eqn. (7): - Select a temperature T so that T T' T2 - Calculate P ( T' ) and P ( T' ) from Antone s Eqn. 2 - Solve Eqn. (6) for pressure P - If P P', then P =P; If not, try another T -value -Calculate x from Raoult s law 43 P, T Flash Calculaton P P 2 - Calculate and from Antone s Equaton - Use Raoult s law n the followng form: y x * P P ( x )* P P 2 (8) - Re-arrange and solve Eqn. (8) for x - Now you can obtan y from Eqn (),.e., y x * P P 44
23 Example Bubble-Pont Calculaton: What s the composton of the ntal vapor formed when a mxture of 40 mol% n-hexane and 60 mol% n-octane s heated to the bubble pont at 2 o C? At what pressure does ths occur? Soluton: Ths s a bubble-pont calculaton because we know the pressure and the lqud composton as shown on the sketch below. The pure-component vapor pressures for hexane and octane at 2 o C P * C6 = 3059 mm Hg P * C8 = 666 mm Hg The total pressure s calculated by addng together the two expressons for Raoult's Law P = x P * + x 2 P * 2 = x (P * -P * 2) + P * 2 = (0.4)( ) = mm Hg Then we can use ths pressure to calculate the vapor composton: y = x P * (T)/P = (0.4)(3059)/623.2) = Example Dew-Pont Calculaton:What s the dew pont pressure of a mxture contanng 40 mol% n-pentane and 60 mol% n-hexane at 2 o C and what s the composton of the frst droplets of lqud that begn to condense? Soluton: The vapor pressures are the same as n the above bubble-pont problem because the temperature s stll 2 o C. However, now we want to elmnate x from the two Raoult's Law equatons because we know y values. Therefore we can use the expresson derved above: y A / P A *(T) + y B / P B *(T) = /P = 0.4/ /666 = / mm Hg Thus, P = mm Hg/ = mm Hg We then fnd the composton from Raoult's Law x A = y A P / P A * = (0.4)(969.3)/3059 =
24 Example 47 Constructon of Pxy dagrams 48
25 Example * (a) Generaton of Pxy Data P kpa P kpa x x2 P (kpa) y y Example (a) Constructon of a Pxy Plot lqud P (kpa) VLE x y vapor
26 Constructon of Txy dagrams The constructon of Txy, dagram requres multple P, T, Flash calculatons, each one of whch provdes a set of equlbrum y, x values for a gven value of temperature (at fxed P) The results can be tabulated as shown below: T ( o C) T 2 x P P P 2 P2 y x* P P 0 0 T.0.0 Ths type of calculatons can also be performed by keepng P constant and varyng x or y from 0.0 to.0 CHEE 3 Lecture Example (b) Generaton of Txy Data P kpa A 3.96 A T degc B B T degc C C T (degc) P P2 x x2 y
27 Example (b) Constructon of a Txy Plot Txy dagram for -chlorobutane () and chlorobenzene (2) at P = 90 kpa (assumng valdty of Raoult's law) VLE vapor T (degc) lqud x y x,y 53 VLE Calculatons - Summary Why? To completely dentfy the thermodynamc state of a mxture at equlbrum (sngle phase, 2 phases..?) How? Through the calculaton of ts P, T, and composton - The type of calculaton that we need to perform s subject to the varables we are lookng to evaluate - These calculatons are classfed as follows: Specfed/Known Varables Unknown Varables Calculaton T, x P, y BUBL P T, y P, x DEW P P, x T, y BUBL T P, y T, x DEW T P, T x, y P, T Flash 54
28 Example Ex 0. acetontrle () / ntromethane (2) a) P-x-y dagram BUBL P For gven T, x P, y At 75 C, x = 0.6 P xp x2p2 P = = 66.7 kpa yp xp y = /66.7 = No teratons needed! 55 Example cont. At 75 C, y = 0.6 For gven T, y P, x P y / P y2 / yp xp P 2 P = (0.6/ /4.98) - = 59.7 kpa x = /83.2 = No teratons needed! Bubblepont dewpont Ether BUBL P or DEW P gves the same P-x-y dagram 56
29 Example cont. T-x-y dagram BUBL T For gven P, x T, y dewpont Snce T s unknown, vapor pressures are also unknown. Iteratons are requred. Bubblepont 57 Example cont. Re-arrange Antone s equaton so that the uraton temperatures of the components at pressure P can be calculated: T B C A ln( P) (7) - Select a temperature T so that -Calculate P ( T' ) and P2 ( T' ) - Solve Eqn. (4) for pressure P x * P P x * P P T x* P P T' T2 ( x )* P P (4) - If P P', then P =P; If not, try another T -value -Calculate y from Raoult s law 58
30 Example cont. BUBL T results for P= 70 kpa, x = DEW T results for P= 70 kpa, y =
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