bequette module ideal binary distillation

IDEAL BINARY DISTILLATION

After studying this module, the student should be able to

Develop the dynamic modeling equations for ideal hinary distillation

Solve for the steady-slale

Linearize and find the slale space model

Understand the dynmnic behavior of distillation columns

Use MATLAB for steady-state and dynamic simulation

'fhe major sections of this module are:

MODULE

10

iVI 10.1

iVI I 0.2

iVI 10.3

iVI 10.4

iVI I0.5

iVI 10,6

iVI 10,7

Background

Conceptual Description (If l)isti Ilation

Dynamic Material Balances

Solving the Stc<ldy-Statc l~quations

Solving the Nonlinear l)ynamic Equations

State-Space l,inear Distillation Models

Multipl iei tyBehavior

M10.1 BACKGROUND

Distillation is a common separation technique for liquid slre-anlS cOlltaining two or morecomponents and is one of the more important unit operations in chl~mical manufacturingprocesses. Design and control of distillation is important in order to produce productstreams of required purity, either for sale or for lise in other chemical processes.

597

598

vapor

y

x

Ideal Binary Distillation Module 10

liquidFIGlJH:E !VItO. I Closed system \vithliquid and vapor in cquiJibrililn.

I)istillation is based on the separation of components of a liquid mixture by virtueor the difTcrcnccs in boiling points of thccomponcnts. For illustration purposes, \\le willbase our discussion on the separation of liquid streams containing two cOinponents (I)i

nary mixture). We will refer to the pure component that boils at a lower temperature asthe light component and the pure component that boils at a higher temperature as theheavy component. For example, in a mixture of benzene and toluene, benzene is the lightcOlnponcnl and toluene is the heavy component.

A saturated liquid mixture of two components at a given concentration is in equilibrium with a vapor phase that has a higher concentration of the light component than theliquid phase. Let x represcnt the mole fraction of the light component in the liquid phaseand y represent the mole fraction of thc light component in the vapor phase. Consider r·'igme rvIlO. I as a conceptual representation of phase (vapor/liquid) equilibrium. 'fhe salu~

rated liquid is in equilibrium with a saturalcd vapor. The concentration of the Iighl COtll~

pOllcnt will be larger in the vapor phase than the liquid phase.Figure M 10.2 is an example of an equilibrium diagram that rcprescnts the relation

ship bctvvecll the liquid and vapor phase cOITlpositions (mole fraction). For example, if theliquid composition is 0.5 mole fraction of the light component, we rind frOID FigureM! 0.2 that the vapor composition is 0.7.

O.B

w00

'" 0.6cQ

"0,

'" OA>:';;

02

0.5

x, liqUid phase light componentFIGURE i\:IlO.2 Vapor/liquidequilibrium diagram.

Sec. Ml0.2 Conceptual Description of Distillation 599

For ideal mixtures, it is cOlllmon to model the phase equilibriulll relationship basedon constant relative volatility

o:x}'=. I + (o:-I)x (MIO.I)

where (~is known as the relative volatility. FigurcMIO.2 was generated based 011 equation (M 10.1) with (X ~ 2.5.

M10.2 CONCEPTUAL DESCRIPTION OF DISTILLATION

Thcft.JlIowing is a conceptual description of the operation of a binary (two-component)distillation colunm. The feed typically enters close to the rniddlc of the column (above thefeed stage), as shown in f'igurcMIO.3. Vapor flows from stage to stage lip the colullm,while liquid flows from stage to stage down the column. The vapor from the top tray iscondensed to liquid in the overhead condenser and a portion of that liquid is returned asrcflux. Thc rest of that vapor is withdrawn as the overhead product stream; this overheadrroduct stream cOlltains a concentrated amount of the light componcnt A portion of thcliquidatthc bottom of thc column is withdrawn as a bol.loms product (containing a con-

Condenser

(Heat removed from condenser)

Overhead receiver

FeedStream -----1>1 NF

NS

_...-'::=r=::.-=-----...Reflux Distillate Product

Vapor Boil-up

(Heat addedto Reboller)

Bottoms Product

FIGURE MlO.3 Schematic diagram for a distillation column.

600 Ideal Binary Distillation Module 10

WE8c,oo

FIGUIU: 1\'110.4 Schematic diagraolfor it distillation C()lumn tray.

centratcd amount of the heavy component), while the rest is vaporized in the rcboiler andreturned to the column.

'file liquid from one tray goes over a weir and cascades down to the next traythrough a downcomcr. As the liquid moves across a tray, it comes in contact with thevapor from the tray below. The schematic diagram for a sieve tray is shown in FigureMUlA.

Generally, as the vapor from the tray below comeS in contact with the liquid, turbulellt mixing is promoted. Assuming that the mixing is perfect, allows one (0 IlHldcl thestage as a lumped parameter system, as shown in !·'igurc M 10.5. Notice that the vaporfronl stage i is modeled as a single stream with molar flowrate Vi and light componentvapor composition (mole fraction) Yi' The liquid leaving stage i through the downcomer ismodeled as a single stream with molar fJowrate Li and light component liquid composi

tion (mole haction) xi'

The conceptual diagram for the feed stage is shown in Figure MIO.6. It differs fromFigure M10.5 in that an additional input to the stage is from the feed to the column.

Li__ l Xil ViYi

FIGURE 1\'110.5 Conceptual materialLixi Vi-t-l Yill balance diagram for a typical stage.

FIGlJRE i\-1l0.6 Conceptual rnatcrialLnfxnf VIlf11 Ynf-t-l balance diagram for the feed stage.

Sec. Ml0.3 Dynamic Material Balances 601

M10.3 DYNAMIC MATERIAL BALANCES

M10.3.1 All Stages Except Feed, Condenser, and Reboiler

The component balance for the liquid phase of a lypical stage as shown in FigureMIO.S is:

accumulation liquid frollltray above

vapor from liquid vaportray below leaving leaving

dNlrti

dl(M 10.2)

where Mi is the liquid molar holdup on stage i.[.'01' this simple binary distillation model, we will make the COllllllon assumption of

cquimolal ovcrllow (King, 1980). For any stage except the feed stage, we assume that thevapor tlowratc from one stage is equal to the vapor molar fJowratc of the stage below:

(MI0.3)

and that the liquid leaving the stage is equal to the liquid flowing from olle stage ahove:

(MIOA)

M10.3.2 Feed Stage

Let {{F' represent the quality of the fccdstrcam. If the feed is a saturated liquid, thcn qF =: (,

while 'iF ~ 0 fbr a saturated vnpor. The vapor molar f10wratc leaVing the feed stage is(where NF =: number of the feed stage)

(M 10.5)

Similarly, the liquid molar nowrate of the slrcanl leaving the feed stage is:

(M I0.6)

M10.3.3 Condenser

A total condenser removes cnergy from the overhead vapor, resulling in a saturated liquid. Assuming a constant molar holdup in the distillate receiver, the total liquid nowratefrom the distillate receiver (reflux + distillate flows) is equal to the nowrate of the vaporfrotH the top tray:

V2 (MI0.7)

where Lv and D represent thc reflux and distillate molar nowratcs, respectively.

M10.3.5 Summary of the Modeling Equations

where Vreboilcr is the reboiler molar llowrate and B is the bottoms product molar flow rate.

The rectifying section (top section of column, above the feed stage) liquid molar nowratesarc:

(MIO.8)

Module 10Ideal Binary Distillation

Ii = LNS _ 1 - Vrehoi1er

A total material balance around the rcboilcr yields:

M10.3.4 Reboiler

602

L R = L{) (MIO.9)

The stripping section (bottom section of column, below the feed stage) liquid molarflowratcs are

(MIO.IO)

The stripping section vapor molar flow rates are:

The recti fying section vapor molar flowrates arc:

Vii = V, + FCI - 'lr)

In the following ~e assume a constant liquid phase molar holdup (dM/dt) = 0).The overhead receiver component balance is:

(MIO.II)

(MIO.12)

(MIO.l3)

The rectifying section component balance is (from i = 2 to NF~l):

dX j ="I (MIOI4)

The feed stage balance is:

dXNF

dl (MIO.IS)

The rectifying section component balance is (from i = NF+l to NS-I):

"Xi 1 [= Lxdt M r 'S i

And the rehoiIer component baLance is:

(MIO.16)

dxNS' =

"I1

M [L\"NS_l - BxNS - V\YNS]Ii

(MJO.17)

c

Sec. M10.4 Solving the Steady-State Equations 603

EQUILIBRIUM RELATIONSHIP

It is assumed that the vapor leaving a stage is in equilibrium with the liquid on the stage.The relationship between the liquid and vapor phase concentrations on a particular stagecan be calculated using the constant relative volatility expression:

O'Xiy-=, I + (IX - I )x,

M10.4 SOLVING THE STEADY-STATE EQUATIONS

(MIO.18)

To obtain the steady-state concentrations we must solve the system of equations, f(.\') :;:: O.From the overhead recci vcr component balance:

I, ~ y,~X, ~ 0

}<rol11 the rectifying section component balance (i = 2 to NF~]):

f~ = Ll\};:i t + VRYi+ 1 ~" LW'Ci ~ VRYi = 0

From thefced stage balance:

j~'F = LgxN/' -J + VSYNr+ I + FZ r ~ LS'''-NF - VUYNI '-co 0

From the stripping section component balance (i ::::: NF+ I to NS-l):

f; :0'::' LSXi _1 + V.SYill - LSxi - VSYi = 0

And from the reboiler component balance:

f:vs = LSX NS _ 1 - BxNS - VSYNS = ()

(M 10.19)

(M 10.20)

(M 10.21)

(MIO.22)

(MI023)

where B = L, - V,.We must realize that (M 10.19)--(M I 0.23) constitute a set of nonlinear algebraic

equations, since the relative volatility relationship (M I0.18) is nonlinear in the state variable. Equations (M IO.19)~(M 10.2:1) arc N.S' equations in NS' unknowns. A Newton-basedtechnique will be used to solve the equations.

EXAlV1PLE MIO.l Stead'y~StateOperation of a 41-Stage Column

Consider a 41-stage column with the overhead condenser as stage 1, the feed tray as stage 21and the reboiler as stage 41. The following parameters and inputs apply

ex 1.5F ::: 1 mol/min

zr 0.5 mole fraction of light componentR 2.706 mol/minD "'" 0.5 mol/minqF 1 (sat'd liquid feed)


r:rom an overall material balance, the hottoms product f10wratc is;

B = F - D = 1 --- 0,5 mol/min

the stripping section flowr:ltc is:

Ls = R -j Fqr = 2.706 + I = 3.706 mol/min

and a balance around the reboiler yields:

V\' ,."'" Ls -- B ,= 3.706 -- 0.5 c=. 3.206 mol/min

'[ he ll1-tJle d 1st C,b, Tn (shown in the Appendix) is used to sol ve for the steady-state composi

tions.

x fsolve( 'di~-oL S,3' ,xO)

The resulting compositions are shown in Figure M 10.7. Notice the strong sensitivity to rellux

flowrate.

alpha 0 1.5. R 0 2.706. V 0 3206

08

0.0.6E8:c£>04

0.2

[_-J

nominal

_~__:_ ~11 O~O,, _

o ____1... _

5 10 15

__-------'__ _ .-----L.-

20 25stage

l.,_ ... _

30 35-~

40

FIGURE MJO.7 Liquid phase composition (mol fraction of light component)as a function of stage number. Solid := nominal reflux, dashed;::: +1II(, reflux,dOlled:::o. --1% rellux.

'Che overhead composition (stage I) is 0.90 and the bolJOllls composition (stage 41) is 0.0 I for

tile nominal rcllux rate (2.706mollmin).

Steady-State Input-Output Relationships

The scnsitvity to rdlux rate is also shown by the plot in Figure MIO.X. The steady-stale gain(change in oUlput/dmnge in input) for distillatc composition is large whcn reflux is less than 2.7,

Sec. Ml0.5 Solving the Nonlinear Dynamic Equations 605

but small when the reflux is greater than 2.71 1l1OlImin. 'T'hc opposite relatiollship holds for bottoms composition, where the gain is small when reflux is Jess than 2.7 mol/min, but large for n.>

flux greater than 2.71 mol/min. This sensitivity has important nnnificati{)HS for control systemdesign.

~ 0.95

2.66 2.68 2.7reflux

2.72 2.74

0.06

~ 0.04

0.02

o ::--::.----- _~J:_~_'::-

2.66 2.68 2.7reflux

1_________ _----L _

2.72 2.74

FIGURE i\i110.8 Steady-state input (reflux)- -olltput (distillate or boUomscomposilion) relationship.

M10.5 SOLVING THE NONLINEAR DYNAMIC EQUATIONS

Equations (M IO.I3)--(M 10.17) arc a set of initial value ordinary equations, which can hesolved using numerical integration techniques. 'rile next example L1ses the variable stepsize MATLAB routine ode4 5 to perform the intcgratiOlL

EXAMPLE lV110.2 Dynamic Response

Consider now the previolls problem, with the initial condiLions of Lhe stage compositions equalto the steady-state solution or Example M1O. L The ~\dditional parameters needed for the dymunic simulation are the molar holdups on each stage. Here we use the following parameters:

MJ=Motv}] =M3 :::: 5

overhead receiver molar holdup

feed tray molar holduphottoms (reboiler) molar holdup

:::::;5 mol

= 0.5 mol=5 mol


To illustrate the nonlinear behavior we compan.' the results of J-, I(j{ step changes in the refluxrate at time t:;:: 5 minutes.

[t,x]o::ode45( 'dist_dyn' IO,400,xO)

Note that the currenl version of ode4 5 docs 1101 allow model p,lnullClerS to be passed throughthe argument list, so global parameters afC defined in the m-filc dis L_dyn. m shown in the Appendix.

The following results arc shown in l;'igurc MIO.9. A positive lr;::- step change in the reflux

rate yields a snlal! increase in the distillate composition; tbis makes since because the maximumpossible increase in distillate purity is (J.()J (the composition cannot he greater than I mole haclion) while it call decrease much more than that. A negative J (J{ step change ill reflux causes alarger change in the distillate purity. 'fhe opposite effects arc observed l~)r bottoms composition,where a positive reflux change yields a large bottoms composition change. A negative refluxchange yields a small bottoms composition change.

0.98

~0.96

0.94o

0.06

0.04

?0.02

oo

50

50

100

100

150 200 250time, min

150 200 250time, min

300

300

350

350

400

400

FIGURE M10.9 Illustration in Ilonlinear response or distillate and bottomscompositions to :'>tcp changes in reflux. Solid line c::: +l(k, Dashed line;:::: -19().

M10.6 STATE-SPACE LINEAR DISTILLATION MODELS

Linear state space models arc useful for slability analysis and control system design. Herewe develop models of the (orm:

x ,- A x' t n u'

y' ;::::: C x'

(M 10.24)

(MI0.25)

Sec. M10.6 State-Space Linear Distillation Models 607

where! is used to represent the deviation variables. Xl :;:;: X -- x~, u' ;:;;: u -- u.\ (the subscript S

indicates the steady-state values). Defining

av 0'.I( = .' =

, ilx; (I+«I-I)x)'

and linearizing the dynamic equations (M IO.13)--(M IO.I?)

(MIO.26)

(M 1(27)

.;01' i = 2 to NF-] :

ill;JXi 1

ill;Au I I = -)

(-t'i+ 1

F'or the feed stage:

V/JCA11)

M[

VnJ(; I

I'vf/

(M 1(28)

(M 10.29)

(M 10.30)

(MI031)

ANF,NF I

A,V/<Nr+ I

Fori=NF+1 toNS-I:

A iJI

and for the rcboiler (stage iV,,')

illy/

JXA'V_I

ill"()xNP

ill,,·dX,vF' I

(If;ax; 1

ill;()Xii I

V)<',/

M,

M[

V/<i-+JiVf/

(MI032)

(MIO.33)

(M I 0.34)

(ivlIO.35)

(M 10.36)

(ivlI0.37)

608

at;A NS,NS-l ."C:_-

O.li_ I

ANS,NS

Ideal Binary Distillation

Mil

_(13 + V,KN ,)

MlJ

Module 10

(M 10.38)

(M 10.39)

Now, for the derivatives with respect to the inputs; ttl ;;;;: LR ;;;;: L! and Ill:::: Vii'::::: VrehoJler:

For i = I to NS-I:

anel for the bottom stage:

Mr

B,.2aj,

0au?

aj; Yil -B'.2 .

OllZ M r

(M IOAO)

(MIOAI)

aj;B .....N.'), I ~-- au] M NS

ill,!!"\2 ~ '. ~1'_,. aU

2

- YvsMNS

(M I OA2)

If the output variables arc the overhead and bottoms compositions, tIlcn:

C u " 1, while eli .c. 0 for i j. 1

C?NS = 1, while C'!.i -::0' 0 for i ;f!

M10.6.1 Transforming the State Space Linear Modelsto Transfer Function Form

The matrix transfer function is:

G(s) ~ (,(sf-A)'!!

(M I 0.43)

(M IOA4)

(M IOA5)

It is easy to generate MATLAB m-files to calculate each of the state~sIxlcc matrices(A,B,C) for a particular set of parameters (and steady-state comrJositions). For a colunm ofreasonable sile (say the 41-stagc example) the denominator polynomial ill (M 10.45)would be quite large (say 4pt order). \Vhat is often more llseful is to be able to directlycalculate the steady-state gain matrix, as shown below.

The steady-state gain matrix is:

(M IOA6)

where, again, it is easy to generate a MATLAB m-file to perform this calculation.

M10.7 MULTIPLICITY BEHAVIOR

Even simple ideal binary distillation columns have been shown recently to have interesting steady~state and dynamic behavior, including multiple steady-states. Nice cxalnplesare shown by Jacobsen and Skogestad (1991, 1994). The key assumption that must bemade for this behavior to occur is that mass flows, rather than molar flows, arc manipu-

References and Further Reading 609

lated. The reader is encouraged to read these papers and modify the MArLAB m-filespresented in this chapter to illustrate the behavior shown by Jacobsen and Skogestad.

SUMMARY

In this chapter wc have dcveloped modeling equations to dcscribe tile steady-state and dynamic behavior of ideal, binary distillation COIUlllllS. The 41-stagc column example showsthat stcady-statc distillate and bottom compositions arc a nonlinear function of the manipulated inputs (distillate and vapor boil~up flows). Also, the dynarnic responses or these COll1

positions depends on the magnitude <lnd direction of changes in the manipulated inputs.

REFERENCES AND FURTHER READING

The following undergraduate chelnical engineering texts develop the steady-state modeling equations ror ideal binary distillation:

King, C,J. (1980). 5,'cparatiolls Processes. 2nd cd. New York: McCiraw-l-lill.

McCabe, W.L., & J.e. Smith. (1976). Un;f Operations (?F Chemlt.-ol EJlgineering,3rd cd. New York: McGraw-Hill.

The dynamic lnodcling equations for distillation are prescIlted by:

L.uyben, W.L. (J 990). Process A1odding, Simulation 01/(1 COlltrol.!iJr Chemical h,'l1gineers, 2nd cd. New York: McGraw-HilI.

More advanced treatments of stcady-st::lte and dynamic distillation models arc prescnted by:

Holland, C.D. (1981). FUlldamenfals (~l Multicompoflent Distillation. New York:McGraw-Hili.

Holland, CD., & A.l. Liapis. (1983). Computer Methods'for Solving Dynamic .)'ep

aratioll Problems. Ncw York: McGraw-HilI.

The panlllleters for Example M I0.1 are presented in the following two references:

Skogestad, S., & M. Morari. (1988). Understanding the dynamic hehavior of disljl~

lation columns. /ild. 6·lIg. Choll. Res., 27( 10): 1848-1862.

Morari,M., & E. Zafirioll. (1988). Robust Process Control. Englewood Cliffs, NJ:Prentice-Hall.

The possibility of multiplc steady-state behavior in ideal binary distillation is presented by:

Jacobsen, E.W., & S. Skogestad. (199 J). Multiple steady~stales in ideal two~product

distillation. A/ChE.I., 37(4): 499-511.

Jacobsen, E.W., & S. Skogcstad. (1994). Instability or distillation columns. A/ChE.1.,40(9): 1466-1478

610 Ideal Binary Distillation

STUDENT EXERCISES

Module 10

1. Consider a simple I tray (3 stage) column with the overhead condenser as stage I,the feed tray as stage 2 and the reboilcr as stage 3. Usc the following parameters andinputs:

R =ifF =FlJ =Zp =

53 mol/minI1 mol/min0.5 mol/min0.5 mole fraction of light component

Find the bottoms product Jlowratc, the stripping section flowratc and the vapor boilup rate (stripping section vapor Ilowrate). Usc fsolve and dist_ss.m to findthe resulting compositions:

[

0.703] [ distillate composition ]x = 0.486 = composition of stagc 2 (the feed tray)

0.297 bottoms product composition

Consider now the dynamic behavior, with the initial conditions of the stage COITlPO~

sitions equal to the slcady~state solution. The additional paramcters needed for thedynamic simulation are the molar holdups on each stage. Usc:

M, kIn overhead receiver molar holdup 5 molM2 = feed tray molar holdup 0.5 mol.M, = 5 = bottoms (reboilcr) molar holdup = 5 11101

At lime zero, the reflux is changed from 3.0 mol/min to 3.2 mol/min. Usc ode45and dist_dyn. m to simulate the dynamic behavior shown in the figures below.

0.78

0.76

~ 0.74

0.72

0.7

time

0.08

(J) 0.06~·x.:- 004~

0.02

S 10 istime

The Reflux is step changed from 3.0 to 3.2 at 1 :::: I minute.

Appendix

APPENDIX

function f ;::: dist_ss(x);%% solve for the steady-state stage compositons in an ideal% binary distillation column using fsolve.%% (c) 1993 B. Wayne Bequette - 21 june 93% revised 31 Dec 96%% All flowrates are molar quantities. Stages are numbered% from the top down. A total condenser is assumed.% ThE~ overhead receiver is stage .1. The partial reboiler% is stage ns (the number of equilibrium "trays" is then% Ds-I). The column parameters should be specified in the% DIST_PAR array.%% to use this function, enter the following in the con@and% window, or from a script tile (after defining parameters% in the DIST_PAR array:%% x ;::: fsolve (' dist_ss' I xO)%~~ wher(~ xO is a vector of initial guesses for the liquid% phase stage compositions (length(xO) :c: DS)

%

%% DIST~PAR is a vector of distillation column parameters% used by both dist sS.m and dist~dyn.m

%

611

if 1engthIDIST_PAR) < 8;disp ( I not enough parametlC':rs given in DIST~PAR')

disp (' ')disp (' check to SQe that global DIS'I' PAR ha;" bCE"n defined ')return

end%

alpha DIST_PAR(l) ; 96 relative volatility 12. OJ)

ns DIST_PAR(2) ; % total number of stages (3 )nf DIST~PAR(3) ; % feed stage (2 )feed DIS'r_PAR (4) ; % feed flowrate (1 )zfeed ~ DIST_PAR('j) ; % feed cc)mposition, Liqht comp (0.5)qf IJIST_PAR I 6) ; % teed quality (l sat'd liqd,

% 0 = sal'd vapor) (1)reflux DIST PAR(7) ; % reflux flowrate (3 )vapor - D151' PAR(8) ; ~) rE'-"boiler vapor flowrat.e ( 3 . 'j )..

%% DIS'I'_PAR(9:19) used by dist_dyn.m (distillation dynamics)% dist = distillate product flowrate


~t, f (i)°6 Ibot% II'% Is% VI'

% VS

% x(i)% y(i)%

ith comp mat bal equationbottoms product flowrateliquid flow in rectifying section (top)liquid flow in stripping section (bottom)vapor flow ~ rectifying sec (::::: vapor + feed* (l-qf)

~ vapor flow ~ stripping section (~ vapor)::0:- mole frae light component on stage i, liq

mole frae light component OD stage i, yap

% rectifying and stripping section liquid flowrates%

%

lrIs

reflux;refl_ux + feed*qf;

% rectifying and stripping section vapor flowrates%

%

vsvr

vapor;vs + feed*(l-qf);

% distillate and bottoms rates%

dist_Ibot

VI' - reflux;Is - vs;

%if dist < 0

disp('error in specifications, distillate flow < 0')return

endif Ibot < 0

disp{'error in specifications, stripping section ')disp(' ')disp ( I liquid flowrate is negative')return

end1;% zero the function vector%

f :::;: zeros(ns,l) i

%% calculate the equilibrium vapor compositions%

for i=l:nsiy(i)=(alpha*x(i))/(l.+(alpha-l.l*x(i));

end9"% material balances%% overhead receiver%

f (1) = (vr*y(2) - (dist+xeflux) *x(I));%% rectifying (top) section

Appendix

'r,for i";2:nf-l;

flil=lr*xli-ll+vr*yli+ll-1r*x(il-vr*yli) ;end

%% feed stage

613

f Infl

%

lr*x{nf-l)+vs*y{nf+ll-ls*x(nflvr*y(nf)+feed*zfeed;

% stripping (bottom) section~;

for i=n£+1:n5-1;flil ls*x(i-l)+vs*Yli+l)-ls*xli)-vs*y(il;

end%% reboiler%

f(ns)=(ls*xlns-l)-lbot*x(ns)-vs*y(ns));

dist dyn.m

function xdot. == dist~ __dyn (t J x) ;oo

% solve for the transient ;-Jtage compositions in an ideal% binary (listillat~on column using ode45.%% (e) 1997 B. Wayne Bequette - 24 Jan 1997% revised 31 Dec 96%~t, All flowrates art::" molar quant~ities. Stages are numbered% from the top down. A total condenser is assumed.% The overhead receiver if3 stage 1. The partial reboiler% is stage DS (tIle number of equilibrium "trays" is then~f, n8-1). Th(~ column parametE~rs should be specifi(~d in the% DISrr PAR array.

% to use this function, enter the fol1ow,ing in the command% window, or from a script file (after defining parameters% in the DIST_PAR array:%96 [t, xl ode451 'dist_dyn' ,to,tf,xO)

% where xO is a vector of initial values for the liquid% phase stage compositions (length(xO) ~ DS)

%7; DIST _PAR is a vector of distillation column parameters% used by both dist 88.m and dist_dyn.m


if 1ength(DISTPAR) < 11;di~3P ( I not enough parameters given in DTST~PAH_')

disp (' ')disp ( 'check to sec that global DIST_PAR has been defined')return

end%

alpha DIST_PAR(I) ; %n;:; DIS'r__ PAH (2) ; ').

"n£ DIST PAR(3 ) "; -,:i

[cedi DIS'I" PAR (4) ; %zfeedi - DIS'T' PAR ( ~) ) ; %--

qf DIST PAR(6) ; %~~

rc;£luxl DIST.. PAR(7) ; 90

vapori DIST_PAH(S) ; %

mel DIST_PAR(9) ; ~6

mb DIS'I' PAR (10) ; %ml- DIS'l' PAR (] 1) ; 'x

%

relative ~olatility (1.5)total number 0 f s Laq(-'-;s (41)feed stage' (21)initial feed flowrate (1)initiaJ feed composiLiol1, light comp(0.5)feed quality (1 = sat:' d liqd,

o ~ saL'd vapor) (1)initial reflux flowrate (2.706)initial reboiler vapor £lowrate(3.206 )d.1.stillatE' molar hold~up (5)bottoms molar hold-up (5)sLaqe molar hold-up (D.5)

if length (DIST_Pt\R) == 19;stepr DIST_PAH(12); ~~-; magnit_udn step in reflux (O)U:,:;tepr DIST__ PAR (13); % t~ime of reflux step change (0)st~cpv rnS'l'. PAR(14); % magnitude stc':'};) in vapor (0)tstepv DIS'1' YAR(15) i % time of vapor step change (0)stepzf DIST_PAR(16) i 96 magnitude of fc(~d comp change (0)tstepzf DIST_PAR(17)i % time of feed camp change (0)stepf DIST]AR (IS); ';, magnitude of feed flow change (0)Lst(~pf IHST PAR(19) i % time of foed flow change (0)

elsestepr 0; tstepr = 0 i stepv = 0; tstepv = 0 ist:epzf 0 i tstepzf = 0; stept = 0; tst.epf = 0 i

end%

%

%%

%

:(; DI~_::T_PAR{9:19) used by di~3t_dyn.m (cJistillat:ion dynamics)dist disLillate product flO1.,vrateIbot bottoms product flowrateLI' liquid flow in rectifying section (top)Is liquid flow in stripping section (bottom)vr vapor flow - rectifyin9 sec {= vapor + feed* (l-qf)vs vapor flow - stripping section (= vapor)x (i) mole frac light component on stage i I liqxdot(i) light component ith stage mat bal equationy (i) molc.:> frac light componenl- on stagei, vap

~6

Vs check disturbanc0f; in rt-'lflux, vapor boil-up, feed composit::.ion<J(, and feed flowrate%

if t < tstepr;

Appendix 615

reflux reflux,i;else

refLux refluxi + sLepr;('Ond

if t < tstepv;vapor vapori;

elseViJ.por

endvapori sLepv;

if t: < tsLepzf;zfeed. - zf(:,-cdi;

elsezfeed - zfeedi + stepzf;

end%

if t <feed

elsefeed

end

tstepf;feec1i;

feec1.i t- SLOpr

(I, rectifyinglr1s

and strippi.nq sectionreflux;reflux! feed*qf;

liquid flowrates

9,5 rectifying and st~.ripp_ing section vapor flowrates%

vsVI'

vapor;vs + feed*(l-qfl;

% distil.late and bottoms rates%

dist.Ibc) t:

vr -- reflux;Is - V~;;

if di;:3t: < 0disp('error in specifications, distillate flow < D')return

endif lbat < 0

disp{'error in specifications, strippin9 section ')disp (' 'Idisp ( 'liquid flowrate is Df:~gative ')return

end

% zero the function vector%

zeros (ns, 1) ;

%


% calculate the equilibrium vapor compositions%

for i==l:ns;y(i)=(aIpha*x(i)I/(I.+(alpha-I.)*x(i)l;

end%% material balances%% overhead receiver%

xdot(II=(I/md) * (vr*y(2) -(dist+reUux) *x(II);%% rectifying (top) section%

for i=2:nf-l;xdot (i I = (I/mt) * (Ir*x (i-l I +vr*y(itl) -lr*x (i) -vr-'y( i) ) ;

end%% feed stage%

xdot (nf) (lImt) * (Ir*x (nf-l I +vs *y (nf+ 1) ·-ls'x (nf 1-r*y(nf)+feed*zfeed);

%% stripping (bottom) section%

for i==0£+1:os-1;xdot(il=(l/mt)*(ls*x(i-ll+vs*y(i+l)-ls*x(il-vs*y(i)) ;

end%% reboiler%

xdot (ns 1= (l/mb) * (18 *x (ns-l) -Ibot *x (ns) -vs *y (ns) I ;

bequette module ideal binary distillation

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