McCabe-Thiele Diagrams for Binary Distillation

Transcription

1 McCabe-Thele Dagrams for Bnary Dstllaton Tore Haug-Warberg Dept. of Chemcal Engneerng August 31st, 2005 F V 1 V 2 L 1 V n L n 1 V n+1 L n V N L N 1 L N L 0 VN+1 Q < 0 D Q > 0 B FIGURE 1: Smplfed pcture of a counter-current dstllaton column. The McCabe Thele dagram for solvng bnary dstllaton problems represents a pllarstone n every chemcal engneerng class. The reason for ths s not ts practcal mportance whch n the era of modern computer technology has become very modest but ts educatonal value based on a very llustratve graphcal pcture of a complcated physcal process. The McCabe Thele dagram n ts most smple form represents a graphcal soluton to the classcal separaton problem of N + 1 deal equlbrum stages connected wth countercurrent vapor and lqud flows as llustrated n Fg.1. The streams are shown as separated flows except at certan dstnct locatons (trays) where complete mxng and thermodynamc equlbrum s assumed. Ths s a useful abstracton but t must be understood that the stuaton n a real plant devates n two respects: Frst of all there s no one-to-one correspondance between a physcal tray (or plate) and a thermodynamc equlbrum stage 1. Secondly, many columns have ther plates replaced by a structured packng whch gves ntmate contact between the vapor and lqud phases at all locatons n the column. The smplfed pcture s stll vald, however, f each tray s consdered to be one theoretcal equlbrum stage regardless of the physcal realzaton of t. 1 It s common practce to talk about tray effcences whch s the fracton of a theoretcal equlbrum stage reached for each physcal plate. Ths effcency may reach values as hgh as but s n many cases much lower. 1

2 2 1. The energy balance To calculate the soluton of the complcated separaton problem n Fg.1 we need 3 sets of mass balances, one for the rectfyng, one for the feed and one for the strppng secton of the column. In addton there s an overall energy conservaton assumpton known as constant molar overflow. The orgn and mportance of ths assumpton s clearfed below. Let n represent any of the nternal stages n the the column except the feed tray. The energy balance for an adabatc stage s Ḣ vap +1 + Ḣlq 1 = Ḣvap + Ḣ lq Snce enthalpy s an Euler homogenous functon of frst order (extensve state varable) the energy balance can also be wrtten h vap +1 V +1 + h lq 1 L 1 = h vap V + h lq L where h, andv and L denote the molar enthalpy and molar flows of the vapor and lqud streams respectvely 2. Combned wth the mass balance V +1 + L 1 = V + L for stage the last equaton becomes ( h lq 1 ) hvap +1 L 1 = ( ) h vap +1 V + ( h lq h+1) vap L Addng and subtractng h vap 1 L 1 and h vap L brngs the equaton on the followng form: ( h lq 1 hvap 1 ) L 1 + ( h vap 1 ) hvap +1 L 1 = ( h vap + ( h lq +1) V ) L + ( h vap +1) L The enthalpy functon has no absolute zero, and t s only the enthalpy dfferences that matter n the energy balance. But, the mportance of the temperature dependency of enthalpy s usually much smaller than the heat of vaporzaton 3. The assumpton that h vap 1 = hvap = h vap +1 s therefore qute legtmate and the energy balance can be smplfed to: ( h lq 1 ) hvap 1 L 1 = ( ) h lq L Fnally, assumng that the heat of vaporzaton s constant for the two-phase mxture makes h lq 1 hvap 1 = hlq and consequently L 1 = L.Thelast argument can easly be extended to yeld L 0 = L 1 = L 2 =...= L n 1 V 1 = V 2 = V 3 =...= V n 2 The dot-notaton has been dropped for the rate symbols. Ths s accordng to common practce n ths feld. 3 Typcal changes per stage are J/mol for sensble heats and J/mol for heats of vaporzaton

3 2. OPERATING LINES 3 whch s known as the constant molar overflow assumpton 4. The same assumpton s vald for the bottom secton as well, but not across the feed tray, where a dscontnuty occurs because of the feed stream enterng from outsde the control volume. 2. Operatng lnes Ths secton (and the rest of the paper) s restrcted to the dstllaton of bnary mxtures. In the top secton of the column an enrchment of the lghter of the two components takes place and n the bottom secton the heaver component s enrched 5. The next task s to express the mass balances of the column and have them ntegrated wth the energy balance from the prevous secton nto somethng called an operatng lne. The composton of the lghter component s desgnated by y and x for the vapor and lqud phases respectvely. By conventon these symbols stand for the lghter component The rectfyng secton. We shall wrte the mass balances as one total balance and one component balance. For the top secton these equatons are V = D + L 1 y V = x 0 D + x 1 L 1 whch means the heaver component s never consdered explctly 6. The equatons are easly solved for the vapor composton: D L 1 y = x 0 + x 1 V V D L 1 = x 0 + x 1 D + L 1 D + L 1 The constant molar overflow assumpton yelds L 0 = L 1 =... = L n 1 L. Furthermore, t s customary to defne the external reflux rato of the column as R L/D [0, : 1 y = x R + x R 1, [1, n] 1 + R Ths s the so-called operatng lne representng the combned mass and energy balances n the rectfyng secton of the column. The equaton s vald for all of the nternal stages = 1, 2,...,n and n partcular for = 1 whch yelds y 1 = x 0 = x D. A bt surprsng maybe, 4 Vald for any number of the chemcal components n the mxture. 5 Ths means that the temperature profle of the column decreases from the bottom towards the top. 6 For bnary mxtures t can easly be elmnated from the gven equatons.

4 4 y 1 y 2 y 3 y 4 slope = R 1+R x 0 1+R x 3 x 2 x 1 x D FIGURE 2: Rectfyng secton of a benzene toluene column (R = 2). but from Fg.1 we see that the condenser has no separatng effect t only condenses the vapor nto a lqud phase 7,8. For = 2 the operatng lne connects the vapor composton from stage 2 wth the lqud composton from stage 1, but here the calculaton stops. We cannot move on before the lqud composton from stage 1 s known! The answer to ths problem les n the assumpton of thermodynamc equlbrum from whch we can derve x 1 = x eq (y 1 ), x 2 = x eq (y 2 ) and so on. Wth ths relatonshp n mnd we got all three ngredents of the McCabe Thele graphcal method: (1) Constant molar overflow assumpton (2) Operatng lne (3) Equlbrum curve The dagram s conventonally drawn wth the heaver component to the left and the lghter component to the rght. That means the equlbrum curve s located above the man 45 dagonal n the dagram. Fg.2 shows a typcal example for an almost deal mxture of benzene (lght) and toluene (heavy). 7 Ths s referred to as a total condenser. In some cases the column s equpped wth a partal condenser whch makes x D x 0. The condenser must then be treated lke an addtonal equlbrum stage, see the dscusson of the strppng secton below. 8 Prove that the heat duty of the condenser s n the range R Q/(D vap h) R + 1for partal versus total operaton.

5 2. OPERATING LINES 5 The starcase (red) shows the nteracton between the equlbrum curve (vertcal step) and the operatng lne (horsontal step). Note that the operatng lne (black) crosses the equlbrum lne (blue) at one pont n the dagram. The column cannot operate to the left of ths cross-over pont because that would be n the one-phase regon of the phase dagram. Hence, the number of theoretcal stages approaches nfnty as the column strves towards a sngular condton The strppng secton. The operatng lne for the bottom secton follows the same recepe as for the top secton. The two mass balances are prncpally the same, V + B = L 1 y V + x B B = x 1 L 1 and agan we solve for the vapor composton on the left hand sde: B L 1 y = x B + x 1 V V B V + B = x B + x 1 V V The constant molar overflow assumpton yelds V n+1 = V n+2 =...= V N+1 V. Furthermore, we may characterze the heat duty 10 of the reboler as S V/(V + B) 0, 1] and wrte 1 S 1 y = x B + x 1, [n + 1, N + 1] S S where S s the fracton of the downcomng lqud whch s beng evaporated. Alternatvely we can defne S as the heat duty of the reboler dvded by the heat whch s needed to evaporate all the downcomng lqud. The operatng lne s vald for all the nternal stages = n + 1, n + 2,...,N and also for the reboler = N + 1. The strppng secton has therefore one external equlbrum stage n addton to the nternal stages of the column (the stuaton s smlar to that of a partal condenser n the rectfyng secton). The cross-over between the operatng lne and the man dagonal s gven by y N+2 = x N+1 = x B. The ndex = N + 2 s strctly speakng outsde the control volume but t does stll serve as an easy-toremember constructon pont of the dagram, see Fg.3. At ths pont t s possble to count the number of theoretcal stages from the specfcatons of the top and bottom products together wth the condenser and reboler dutes. It s more common, however, to specfy 9 Ths s barely vsble n the current dagram but an nfntly large magnfyng glass would reveal an nfnte number of successvely smaller steps. 10 Prove that the heat duty of the reboler s Q = S (1 S ) 1 B vap h.

6 6 y 5 y 6 slope = 1 S y 7 y 8 x B x 7 x 6 x 5 x 4 FIGURE 3: Strppng secton of a benzene toluene column (S = 2/3). Tray numbers are rounded to the nearest nteger. the feed stream rather than the reboler duty. To manage ths we need the operatng lne for the feed tray whch s dscussed n the next secton The feed tray. The operatng lne for the feed tray (q-lne) requres four mass balances and two energy balances (ncludng the constant molar overflow assumpton). The extra mass and energy balances are needed to characterze the physcal state of the feed stream. To begn wth we have the component balances y n V n = x D D + x n 1 L n 1 x n L n = y n+1 V n+1 + x B B for the rectfyng and strppng sectons respectvely. At the feed tray there wll n general be sudden jumps n all the stream varables, but for an optmal placement of the feed tray the ntensve state varables (temperature and composton) are contnuous. Hence, we shall requre that y n = y n+1 y and that x n = x n 1 x n the equatons above. Summaton plus reorganzaton yelds: y (V n+1 V n ) = x (L n L n 1 ) (x D D + x B B) } {{ } x F F

7 2. OPERATING LINES 7 Note that the total mass balance x F F = x D D+ x B B s substtuted on the rght sde. The q-lne s frst wrtten on the prelmnary form: y = x L n L n 1 F x F V n+1 V n V n+1 V n The feed stream s subsequently replaced by the mass balance for the feed tray F = (L n L n 1 ) (V n+1 V n ). A lttle manpulaton yelds y = x L ( ) n L n 1 Ln L n 1 V n+1 V n x F x L V x F ( L V 1 1 V n+1 V n ) where V and L measure the dfferences n the vapor and lqud flows across the feed tray (top-down). Ths could serve as our fnal result, but t s customary to characterze the feed qualty by L q L V where q = 1 V = 0 (saturated lqud feed) and q = 0 L = 0 (saturated vapor feed). Hence, the feed qualty can be nterpreted as the fracton of saturated lqud n the feed. The defnton of the feed qualty s easly nverted to gve L/ V = q/(q 1) whch puts the q-lne on the fnal form: q y = x q 1 x 1 F q 1 Ths s the last of the three operatng lnes used n a McCabe-Thele dagram, but t should be mentoned that the optmal feed tray locaton causes a slght redundancy n the calculaton of the dagram: E.g. n most cases the composton specfcatons x F and x D are gven n addton to the reflux rato R and the feed qualty q. Ths specfes the upper part of the column and the condtons of the strpper wll be fxed once x B or S s known (the other parameter s fxed by the common cross-over pont of the operatng lnes). Anyway, the calculaton s complete n the sense that t reveales all the sgnfcant fgures of a bnary dstllaton column (see also Fg.4): (1) Compostons of the top and bottom products (2) Composton and qualty of the feed stream (3) Reflux rato (and condenser duty) (4) Reboler duty 2.4. Extreme specfcatons. The condtonof mnmum reflux s reached then ) the cross-over between the operatng lnes s located at the phase boundary as llustrated n Fg.2 or ) the upper operatng lne s tangent to the equlbrum curve whle the cross-over pont s nsde the phase dagram

8 slope = q q 1 N top = N bot = FIGURE 4: Complete McCabe-Thele dagram for a benzene toluene column wth parameters x D = 0.9, x F = 0.55, x B = 0.1, R = 2and q = 0.2 (consstent wth S = 2/3). (apples to mxtures wth concave equlbrum curves). In both cases the number of theoretcal stages approaches nfnty. The other extreme s then the reflux s nfnte and the operatng lnes for the column overlap wth the man dagonal. Ths yelds the smallest number of theoretcal stages for a gven separaton. There are also cases when the strppng secton or the rectfyng secton s mssng or neffectve. These cases nclude batch dstllaton (no strpper) and degassng unts (no rectfyer). Mathematcally, there s also the possblty that a secton becomes neffectve because of an extreme feed condton. Ths s readly verfed from the q-lne whch has slope ±1 for q = ±. The frst case (q = ) corresponds to a lqud feed so cold that t condenses all the vapor comng up from the reboler (neffectve rectfyng secton). The second case (q = ) corresponds to a superheated vapor so hot that t evaporates all the lqud comng down from the condenser (neffectve strppng secton).