Dynamic simulation of a distillation column with multiple liquid phases

Compurers them. Engng Vol. 19, Suppl., pp. S393-S398,1995 Copyright @J1995 Elsevier Science Ltd 0098-1354(95)00047-X Printed in Great Britain. All rights reserved 009s1354J9S $9.50 t 0.00

Pergamon

DYNAMIC

SIMULATION OF A DISTILLATION WITH MULTIPLE LIQUID PHASES E. ECKERT

Department

COLUMN

and M. KUBICEK

of Chemical Engineering and Department of Mathematics Prague Institute of Chemical Technology 166 28 Prague 6, Czech Republic

ABSTRACT A rigorous mathematical model of dynamical behaviour of a distillation column with a possible occurrence of several (e.g. three ) liquid phases on individual trays is described. The generalized model is based on the concept of the equilibrium tray. By solving the algebraic part of DAE system, the phase situation is tested and the system is changing adaptively in the course of the integration. Examples of a distillation of mixtures with possible existence of two and three liquid phases are presented. KEYWORDS Distillation; splitting.

equilibrium

stage column;

dynamic

model; multiple

liquid-vapour

equilibrium;

phase

INTRODUCTION Dynamic modelling of rectification columns where splitting of a liquid phase may occur has recently appeared in the literature, e.g. Rovaglio and Doherty (1990), Wong et al. (1991) and Widagdo et al. (1992). The possible splitting of the liquid phase requires a more complicated mathematical model and a sophisticated solution algorithm is necessary both for stationary and dynamic simulation. Wong et al. (1991) use the tangent plane analysis proposed by Michelsen (1982) for a decision whether the splitting of the overall liquid phase on two liquid phases occurs. Their dynamic model considers only the overall liquid phase and the vapour hold-up is neglected. Widagdo et al. (1992) which use a similar approach suggest standard software for the numerical integration of differential algebraic equations (DAE) which appeared recently (Brenan et al., 1989). We present here a generalized rigorous model for the dynamic behaviour of a rectification column, where p liquid phases can coexist on individual trays. The model is based on a concept of the equilibrium flash calculation which has been described in our recent paper (Eckert and KubiEek, 1993). Following paper (Eckert and KubiEek, 1994) was devoted to dynamic modelling of single tray of this kind. The present paper is a generalization to the rectification column. Examples of simulation in cases where two and three liquid phases can coexist are presented. MODEL

OF THE RECTIFICATION

COLUMN

Consider a rectification column composed from s equilibrium trays (including condenser and reboiler). Such an equilibrium tray is schematically drawn in Fig. 1. Several possible feed streams and liquid side withdrawals are considered. Heat exchange stream, Qj, is important particularly for j = 1 (condenser) and j = s (reboiler). The dynamic operation of the column can be then described by the following set of equations (j = 1,. . . , s): Material

balances:

9 dt

=

‘~j+l~;,j+l

t

~ L~_,Z~,j_l I;=1

-

GjYi,j - f:

LSz~j(R~ + 1) +

k=l

f +

c

F;x&,

i.=l ,...,I&,

k=l s393

(1)

European Symposium on Computer Aided Process Engineering-S

s394

1 P Lj-lLj,l

j

... +

,fi,

i

Gj+l

Lj

p

Lj

Fig. 1. Equilibrium

Energy

stage j.

balances:

dEj = Gj+l Hj+lt f:Ljk-1 h:_l - GjHj - f: dt

+

L;ht(R$

1) + Qj +

k=l

k=l

f

Equilibrium relations for p possible reference liquid phase): 0 =

yi,j

-

O=K~.z?._Kk.z~. %I

Summation

*J

liquid

*,J

phases and a vapour

i = l,...,n

Ki:jZ~,j,

l,...,

i=

W’

phase (r is the index of a chosen

n,

k=

(3) l,...)

p,k#r

(4)

equations: 0 = 1-

eYi,j

(5)

i=l

O=

or

l-~~e

0 = $$,

k = l,...,p

(6a,b)

i=l

Component

hold-ups: i=

O=Ni,j-(~N,j)((l-~I0:)y;,jt~~~~e), s-1

Internal

energy

Volume balances:

hold-ups:

k=l

k=l

l,...,n-1

(7)

s39.5

European Symposium on Computer Aided Process Ertgineering-5

Pressure

drop equation,

e.g.: (10)

(Po is given pressure Relations

on the output

of the condenser)

for liquid flows from the tray, e.g. Francis formula: 0=

$ Ni,j $ $;v: -

Aj

(Wjt Cj( f: Lj”Vj”( Ri t 1))““)

(11)

k=l

Mixing of liquid phases: o=

Ljk(R; + 1)

$

k=l

(12)

,...,p,k#*

If no liquid phase flows out from the plate j, i.e. if

i=l

k=l

then p equations o= replace

k = l,...,p

L$,

(13)

Eqs (11) and (12). If the k-th liquid phase does not exist, i.e. +j = 0, then obviously 0 = L$

If the vapour

(14)

phase does not flow out from the tray j, i.e. if Pj-1 2 Pj

then equation 0 = Gj is considered

instead

of Eq.(lO).

(15)

This case can occur for an overcooled

plate (condenser).

Equations ( l)-(5), (7)-( 9) and (10) when Pj_1 < Pj are always present in the model independently of the phase and hydraulics situation (i.e., of the number and the volume hold-up of coexisting liquid phases) on the stage j. Remaining equations are chosen from Eqs (Ga,b), (ll)-(13) depending on the phase and hydraulics situation, cf. Tab. 1 for generic cases and p = 3. In the model described above we assumed that individual stages are in thermodynamic equilibrium. Eq. (11) depends on the construction of the plate. Ideal dispersion of the liquid phases is assumed in Eq. (12). Extension of the model by including vapour side streams, heat capacity of the plate etc. does not bring any complications. THE 1NTEC:RATION The above formulated

equations

ALC:ORITHM form a system of differential dX dt 0

- algebraic

equations

(DAE):

=

F(X,Y)

(16)

=

G(X,Y)

(17)

where the variables Ni,j and Ej form the vector X, dim X = s(n+ l), and variables Z~,j, yi,j,L$‘,$t, Gj, Tj, Pj form the vector Y, dim Y = 8((p+l)(rL+2)41). Eqs (16) represent s(n+ 1) differential

European Symposium on Computer Aided Process Engineering-5

S396

equations (1) and (2), Eqs (17) consist of altogether s((p + l)(n + 2) + 1) algebraic equations with variable structure depending on the phase situation on individual trays. This structure can change in the course of integration and, therefore, difficulties arise (Widagdo et al., 1992) when using a standard software for DAE. Therefore we used a simple Euler method for integration of Eqs (16) and an adaptive iteration algorithm (Eckert and KubiEek, 1993), which is based on the Newton method, for solution of decomposed system (17), i.e. we solve this system successively for j = 1,2,... , s. The switching between phase situations on individual stages has been made according to the algorithm presented in our recent work (Eckert and KubiEek, 1994). As the system (16)-(17) is more or less stiff, the step size used for integraton has to be chosen small enough. Table

1: Choice of equations in dependence

;I

on the phase situation

on the stage j for p=3.

((6a),k=l), ((6b)(W, k=W), (11) ((64, k=V), ((6b)(l3), k=% (111,(12)

EXAMPLES Calculation of values of Zr’,h, H, w and V,is described in detail in our recent paper (1994). There are presented also all necessary stage parameters. We used the same values for each stage here. Application of the algorithm presented above is illustrated on two examples. In the first one two liquid phases can coexist in equilibrium with the vapour phase (p=2). In the second example three liquid phases can arise (p=3). Example

1

Consider a 5-stage column, where the single feed to third stage is a vapour mixture of n-butanol, water and n-propanol (F,’ = 7 mol/s, x:,~ =(0.15; 0.8;0.05), T~,F = 370 K). Such a mixture can form two liquid phases (p=2, i.e. the phase situation of type V-L-L), as shown by Block and Hegner (1976). Reflux ratios in the condenser (j = 1) are Ri = Rq = 1 and heat supplies are Ql = -350kW and Qs = 59.1 kW. No side withdrawals are considered, PO = 0.1 MPa. The dynamic The steady composition values. The the number Example

response of the column to a change in the feed composition is illustrated in Figs 2a-d. state solution has been used as the initial condition for t=O s. For t E (0,25) the feed was changed to x i ,F=(0.05; 0.65; 0.3) and for t > 25 s it was returned to its original step size At = 0.005 s was used for integration. It can be seen from the figures that of liquid phases in the condenser is reduced to one for t E (36.6; 83.5).

2

Consider a 2-stage column with three feed streams of temperature 295.2 K to both stages (F,’ = Fi =1.05 mol/s of nitromethane, F: = l?,” = 1.05 mol/s of dodecanol, F;” = F; = 1.4 mol/s of ethylene glycol). Such a mixture can form three liquid phases (p = 3) as shown by Null (1970). A transient behaviour of the column in the case when the feed flows are changed is illustrated in Figs 3a-b . The steady state solution has been used as the initial condition for t = 0 s, i.e. the solution of the type V-L-L-L. For t E (0; 70) values of F,’ = Fi and FT = l?f are changed to 2.2 mol/s and 0.25 mol/s, respectively. After 58.8 and 62 seconds the third liquid phase disappears on the second and first stage, respectively. At time t = 70 s the feed flows are changed to their original values. At the time t = 70.3 s the third phase reappears on both stages and the system moves to the original steady state. The step size At = 0.002 s was used for integration. CONCLUSIONS In this paper the algorithm for dynamic simulation of a distillation colu~nn capable of predicting the appearance and disappearance of heterogeneous liquid phases on individual trays was developed. Thus,the changes in the liquid phase pattern and the resulting effects on the dynamics of the colu~m could be followed. The dynamic responses of the column to upsets in feed flowrate (composition) were simulated for mixtures which can lead to a coexistence of two or three liquid phases with

s397

European Symposium on Computer Aided Process Engineering-5

2

G1 /l

a

L1 0.1

b

0.11 .-

a6

0.10 . .

0.5

0.09 .

0.4 0.08

0.3 0.07

Q2

1 0.06

a1

t

0.05

0.0 0

20

40

&.I

WI

loo

t

J 0

20

40

60

a0

loo t

‘2 L5

1

L1

d

4.0 3.0 3.8 2s 3.6 20

15

3.2 0

20

40

Fig. 2. Example

60

a0

1. Response

t

lal

of the column

0

20

40

60

SO

loo

to the change in feed composition.

260

a

2 L2 zis

b

2.45

”

20

40

Fig. 3. Example

60

80

2. Response

loo

t

of the column

0

20

40

60

so

to the change in the feed flows.

t

European.Symposium

S398

on Computer Aided Process Engineering-S

vapour phase. The algorithm is relatively for integration gives satisfactory results.

robust

and it is shown that the simple Euler method

used

NOTATION

c

effective cross sectional area internal energy holdup number of feeds molar enthalpy of vapour phase vapour/hquid distribution coefficient molar holdup mol pressure heat supply volume of stage time molar volume of liquid phase liquid mole fraction

height over weir coefficient molar feed flow rate molar flow rate of vapour phase molar enthalpy of liquid phase (or feed) molar flow rate of liquid phase number of components number of possible liquid phases side withdrawal ratio number of stages molar volume of vapour phase weir height vapour mole fraction

F G h L n P R s V W Y

Greek letters $ molar fraction of a liquid phase < pressure drop parameter Subscripts i component ,P component stage j j

Superscripts k, m liquid phases k, 111 T reference liquid phase

i s

REFERENCES Block U. and B. Hegner (1976). Development distillation. AIChE Jl22, 582-589.

and application

of a simulation

model for three-phase

Brenan K. E., S. L. Campbell and L. R. Petzold (1989).Nutnerr’caI Solution of Initial Value Problems in Diflerential - Algebraic Equations. North - Holland, New York. Eckert E. and M. KubiEek (1993). Multiple liquid-vapour approach. Computers them. Engng 11, 879-884.

equilibrium

Eckert E. and M. KubiEek (1994). Modelling stage. Cl&em. Eng. Sci. 49, 1783-1788.

for multiple

Michelsen

M. L. (1982). The isothermal

Null H. R. (1970).

of dynamics

flash problem

flash calculation

liquid-vapour

by global

equilibrium

I, II. Fluid Phase Equil. 9, 1,21.

Phase equilibrium in process design. Wiley , New York .

Rovaglio M. and M. F. Doherty (1990). Dynamics A IChE Jl& 39-.52.

of heterogeneous

Widagdo S., W. D. Seider and D. H. Sebastian (1992). azeotropic distillation. AIChE JIB, 1229 -1242.

azeotropic

Dynamic

distillation

analysis

columns.

of heterogeneous

Wang D. S. H., S. S. Jang and C. F. Chang (1991). Simulation of dynamics and phase pattern changes for an azeotropic distillation column. Computers them. Engng Irj, 325-335.