A new implementation for analytical derivatives of thermodynamic properties and its beneficial application on dynamic simulation

Computers them. Engng, Vol. 11, No. 516, pp. 441450,

0098-1354f93 $6.00 + 0.00 Copyright 0 1993 Pergamon Press Ltd

1993

Printed in Great Britain. All rights reserved

A NEW IMPLEMENTATION FOR ANALYTICAL DERIVATIVES OF THERMODYNAMIC PROPERTIES AND ITS BENEFICIAL APPLICATION ON DYNAMIC SIMULATION A. KILAKOS’ and B. KALI~VENTZ&~~~ ‘LASSC, Universite de Liege, Sart Tilman, Bit. B6, B-4000 *BELSIM

s.a.,

Part

Industriel

(Received

du Sart Tilman,

30 July

All&e des Noisetiers

f992; received for publication

Liege,

1, B-4031

30 September

Belgium

Angleur

Liege,

Belgium

1992)

Abstract-A

formulation to obtain analytical partial thermodynamic derivatives (PTHDs) of thermodynamic functions with respect to selected state variables for liquid, vapour or liquid-vapour systems, is presented. The objective of this paper is first to illustrate the advantages using analytical PTHD information and secondly to present a systematic procedure, based on an implicit system of nonlinear equations, which allows the user to choose his own state variables and to obtain analytical derivaties of any property (also chosen by the user) with respect to these state variables. The procedure is based on few PTHDs with respect to a basic set of state variables, coded explicitly and an algorithm that allows to compute all the other PTHDs needed, with respect to any set of state variables. The algorithm presented is completely independent of the mathematical solution procedure and independent of the thermodynamic model used. It therefore does not modify the structure of the system describing the plant units. It is now implemented in BELSIM-EPIC thermodynamic package. Its flexibility and contribution to solution accuracy are illustrated. Computer time savings using the algorithm, are presented and analyzed in the case of liquid-vapour equilibria calculations using different thermodynamic models and also in dynamic simulation of multicomponent distillation units in refinery plants.

INTRODUCTION Until now the general idea is that in practical modelling and design applications were equation-oriented strategies are applied using quasi-Newton type solution procedure, complete analytical partial thermodynamic derivative information is either unavailable or expensive to calculate. Although previous studied mention that analytical computation of partial thermodynamic derivatives (P’THDs) leads to substantial savings of computation time, especially in cases where physical property calculations dominate the simulation time, like multicomponent distillation, very few studies have been reported on that subject. All of them deal with steady-state simulation and/or optimization although in dynamic simulation CPU time savings is a quite crucial subject. The approaches used to tackle this problem are:

2.

3.

4.

1. Numerical computation of the PTHDs, (Kvalues and enthalpy) with finite forward differences. The step is adjusted by rules based on “experience” these tTo

whom

derivatives

or

arbitrarily. using

all correspondence

finite should

Approximating differences be addressed.

is in

5.

most cases expensive (many calls to the physical property subroutines) and also the values obtained depend on the differentiation step chosen. Modifications of the forward difference method in a noisy environment for process optimization based on the use of second derivatives and an estimate of the level of noise in the system (Kaijaluoto and Morton, 1987). A method for approximating quantities that involve physical property derivatives in equation-oriented process design, using a hybrid algorithm that makes combined use of Newton’s method and the Schubert update (Lucia and Macchietto, 1983). In several cases the sparsity of the system is destroyed. Analytical expressions for the derivatives of the activity and fugacity coefficient are proposed by Michelsen and Mollerup (1986) for LV calculations and some computation time results are presented for four thermodynamic models. Individual case of practical application of analytical expressions of PTHDs in steady-state simulation like the Naphtali-Sandholm distillation by Christiansen et al. (1979).

A.

KILAKCB

and B. K~~rrvrmrzsss

Table 1. Equations for an adiabatic single-stage flash No. of of equat&ms

Equation z,-y,-x,=0 &..(K P.Y.) -JkL.Ar. h” - &.(C p, Y,) = 0 hl. - &(T, P, &) = 0

Equation No.

(s=l,...,Ns) (s=l,...,Ns) :

Moap babmcea Equilibrium cqwMioIls Vapour molar cnthalpy Liquid molar cntbalpy

::; (3) (4)

1

Total enthalpy

(5)

a -*~,Y*/,~,z*=o

1

vapow

(6)

vv - VAT, P, Y.) = 0 OL- V,(T,P,+=O

1 1

Vapour molar volume Liquid molar volume

1

Total volume

sv-WT,J’,Y,)=O g, - G,(T, P, x.) = 0

1 1

Vapour molar free energy Liquid molar free energy

(10) (11)

C=Iasv+(1-~l~~l~~,z.=O

1

Total free energy

(12)

H-[ah,+(l

vt -[av,+(l

-a)&]

-a)vJ

pv x.) = 0 .v* c 2,-o r-1

E z, = 0 s-1

PTHDs with respect to selected state variables, for LV systems are of crucial importance in modern modelling. Accuracy of the solution and computer time savings are aspects directly related with PTHDs computation. Analytical PTHDs improve both aspects. In this work a procedure to obtain accurate values for these derivatives in a systematic way and also to reduce the computation time spent, is presented. The procedure is based on few PTHDs with respect to a basic set of state variables, coded explicitly and an algorithm that allows to compute all the other PTHDs needed, with respect to any set of state variables.

fraction

Although multicomponent liquid-vapour equilibria (LV) is described by complex state equations, the basic principles can be formulated using a rather simple model. The equations required to describe an adiabatic single-stage flash where NS components are involved, are presented in Table 1. Equations (l-6), (2Ns + 4) deflne the state of the system with 3Ns + 6 variables, leaving NS + 2 degrees of freedom. To this set, more equations can be added in order to include more physical properties. In dynamic simulation the density variations are considerable, so equations (7-9) can be added, including the computations, the molar volume (m’ kmol-I). In cases were free energy calculations are required, equations (10-12) can be added. Formulation of the problem The systems we are interested in are mixtures consisting of Ns components where an LV equilibrium only can occur. All quantities involved are

(9)

classified into two categories: the state variables (independent variables that determine the state of the system, as for instance the partial flowrates, the temperature and the pressure for a process stream) and the dependent variables (like the mixture mean density, the vapour fraction, the molar entropy, etc.) derived from the state variables by, in most cases, explicit formulas. Symbolizing the state variables array by X (containing NET entries) and the dependent variables array by Y (containing N entries), the system of equations presented in Table 1, can be written using matrix notation as follows: for, Y) = 0,

THE ALGORITHM

LV equilibrium equations

:z

(13)

with f=‘[fi

. ..fk. ..&I,

x = ‘[X, . . 1x,

Y=‘[Y,.

. . r;.. . . Y”],

. . . XNET],

where N is the dimension of f and Y and NET the dimension of X. The purpose is to obtain the derivatives (a?/~%,) wherej=l,..., Nandi=l,..., NET,basedonthe implicit form of the system f, since the model describing the equilibrium is not known explicitly. Differentiating equation (13) we get:

wherek=l,.... Let: A = g(N

i

x NET matrix)

N,

(14)

A. KIL.AKOSand B. KALITVENTZ~PF

444

calculated:

(ah$//ah,), (%i/;laP,),

(&‘;/3P,).

The

new procedure to compute them is presented here below, in comparison with the finite differences approximation: finite dilierences

-It is independent of the state variables chosen. -Flexibility. Equations can be added or subtracted adding or removing quantities without important modifications. (See Table 1.)

procedure

new anaIytIcaI

-Given hl and P, compute the entropy s, (PI, h,) --Given sI (PI, h,) and Pz compute h F(s, , Pz) Perturb hl to h, + &I -Given h, + 6h and P, compute the entropy s’l (PI, h, + 6h) Given s;(P,, hl + 6h) and Pz compute hk (s; , Pz) -Approximate the derivative with the difference:

-Given

(ah:/as,)

(ah:/ah,)= --(ah~/asl)(ah,/as,)-l (ah$/aP,)= -(ah:/ah,)(ah,/a~,)

ables may be chosen since the procedure is completely independent of the mathematical approach, in order to compute intermediate PTHDs that are convenient for the calculations that follow. Here, molar entropy and pressure have been used as “intermediate” state variables although the purpose was to calculate PTHDs with respect to pressure and molar enthalpy.

described iter-

Table 2. The main steps of the algorithm. for LV systems I. Consider a set of basic sate variables (Ns + 2) (ea.. K P, z) that define the state of the system described by equations (l-6). 2. Perform a Hash calculation (only if the system has not been initialized). In most modelling softwares this step and the previous one are part of the initialization phase and so they can be skipped. 3. Choose a set of state variables fex. . H. P. zl. This choice will be used only in Step 8. 4. Compute the matrix A = d/,/dX,, k = 1, IV, i = 1, NET. 5. Compute the matrix B = df,JdY,, k = I, N, j = I, N. 6. Inversion of matrix B. i = I, NET. 7. Compute D,= --B-l A,, 8. If the state variables are not the basic ones, then proceed to the ‘change of state variables procedure’, applying equation (22). 9. Exit with the derivatives. I,,

2)

-The two other derivatives are calculated using the formulas:

applied. In an equation-oriented environment, molar enthalpy and pressure may be the state variables for the mathematical solver. In order to apply the alogrithm proposed, another set of “intermediate” state vari-

-Simple mathematics--direct calculations-no ation procedure. -It is independent of the solution procedure.

the

(ah, iaP, ) -Using Pz and s, as state variables, the algorithm (see Table 2) returns the derivatives: (ah $//aP,),

It is clear that the number of calculations increases significantly if tinite differences approximation is

of the procedure

hl and PI compute

entropy sI (PI, 4 1 -Using P, and s, as state variables, the algorithm (see Table returns the derivatives: (dh, /as,),

-Repeat the previous steps perturbing P, and P2, respectively

Advantages

procedure

-It

removes the “headache” of the differentiation step. -It leads to substantial computer time savings, especially in multicomponent cases.

APPLICATTON

In this work, we demonstrate the computer timesaving advantages using analytical expressions of PTHDs in steady-state LV-equilibrium systems, and computer time-saving and accuracy in the solution advantages in dynamic multicomponent distillation. The thermodynamic models used for the applications are among those in EPICLB of the BELSIM-EPIC package (EPIC Manual, 1988). Ail simulation experiments have been performed using a Hewlett-Packard 9000 Workstation, Series 425, with HP-UX (release 7.05, Vs B) operating system.

Derivatives

of LV mixtures

(steady state)

Results of CPU time comparison for PTHDs computation of two steady-state LV systems are presented in Fig. 2. The benefit increases as the number of components in the mixture increases, for all properties except liquid fugacity. This incoherence is due to the complexity of the thermodynamic model used for the liquid phase property prediction. In the second LV system, the Peng-Robinson equation of state was used for both phases and this implies the solution of a cubic equation, while the GraysonStreed model uses the Hildebrand equation, based on a regular solution model, for the activity coefficient prediction. A special implementation for cubic equations of state (see a brief description in “Dynamic

New implementation for derivatives f, =iI,-

1

-Is+

f2 =Pp-

i,-

445 w,

7 P,=

fa q?__-;i?-

= Q tJ

‘, h

+1=

i rl

0

f, = li’; - ii: ( 8 1, Pz ) = 0

Fig. 1. Simplified compressor model.

results analysis”) makes it computationaly more intensive than the direct formula of the Grayson-Streed model. Application into dynamic simulation Dynamic simulation has been chosen to demonstrate time-saving advantages because although memory requirements remain modest, computer time needed raises significantly compared with that of steady-state simulation. For testing purposes, a dynamic simulator developed in LASSC, based on a predictor-corrector integration procedure has been used. All cases tested had an index of not greater than one. Cases. Four cases from refinery distillation have been used for testing purposes: (A) A TBP characterization experimental batch column startup, from the point where total reflux steady-state has been achieved. Simulation has been performed using different thermodynamic models and mixtures with a different number of pseudocomponents. Convergence has been achieved with one Jacobian matrix inversion and three residuals computation, for each time-step. (B) A column for the stabilization of the gasoline, operating around a nominal steady-state. Thermodynamic model used:

K-values: Enthalpy: Molar volume: I

Convergence has been achieved with one Jacobian matrix inversion and two residuals computation, in each time-step. (C) Two thermally coupled distillation units for a new deisopentanizer scheme, in a refinery plant; the reboiler of the first column is the condenser of the second. The Teng-Robinson equation of state was used for equilibrium calculations. Convergence has been achieved with one Jacobian matrix inversion and three residuals computation, for each time-step. (D) A distillation column for the Cl, C2 separation from a mixture of ClC4. This column operates in low temperatures and pressures around 26 bar. Top pressure is controlled via the condenser’s heat load. The column undergoes disturbances coming from sudden changes in reboiler’s heat load. These perturbations lead to sudden changes in top composition since the top mixture is a narrow-boiling system, and also to equilibrium conditions like pressure. The whole system is very sensitive in equilibrium calculation accuracy. The Peng-Robinson equation of state has been used for equilibrium calculations. Cases A, B and C are used to demonstrate timesaving advantages while case D is used to demonstrate the contribution to solution accuracy.

Vapour phase Redlich-Kwong Soave-R-K Redlich-Kwong

s9 AnalyticalpTHDs

FiniteDifferenceforPTHDs computation 9 components

0.6

Liquid phase Grayson-Streed Soave-R-K Redlich-Kwong

42 comtxments

1

6-

E $

a4

a ” 0.2 0-l

Fugacity Enthalpy M.VdumG Vapour Fraction

qacity _ Enthalpy . M.Voium6 Vapour Fkction property

Fig. 2. Steady-state LV equilibrium results: CPU time comparison between finite difference approximation and analytical F’THDs computation.

A. KILAKC~

446

Table 3. Dvnamic

and B.

distillation

KALITVENTZEFF

column

model assumptions

case -Vapour and liquid leaving each theoretical stage are in equilibrium established without delays. Murphrce plate el?iciency equal to one was used -Vapour and liquid on each tray and in each downcomer are perfectly mixed -Vapour hold-up is negligible. -Pressure profile was constant across the column -Thermal inertia of metal walls was neglected -Heat exchanger dynamics were negkctcd -Transportation delays between trays were neglected -Number of theoretical stages (reboiler, condenser included) -Number of equations -Number of components -Condenser fT = total. P = aartiall

The system of differential/algebraic equations describing the duty of an individual column plate is presented in the Appendix, while the assumptions made for each case are presented in Table 3. Integration procedure. The integration procedure used was a predictor-corrector, first-order BDF implicit procedure. The correction step was performed using a modified Powell’s dogleg method (Chen and Stadtherr, 1981), which is very efficient for treating badly initialized cases. The reason for not using a higher order BDF was that the interest of this work has been focused on the computer time gain per integration step for cases where the solver behaves more or less the same way (one Jacobian matrix computation and two to four residuals calculation). A higher order procedure would reduce the effect of the computer time gain using analytically computed PTHDs, in cases where it contributes to avoid a new Jacobian matrix computation. The combined effect of higher order and analytical computation of PTHDs depends strongly on the case studied. In cases where discontinuities exist, the predicted value by a higher order method may be far from good and the solver

A

B

C

D

Yes

Yes

Yes

Yes

Yes

Yes

Yes

Yes

No

Yes

Yes

Yes

YCS

Yes

No No Yes Yes Yes 41 1200

Yes Yes 19

V=Y Vary T

Yes Yes Yes 47 2071 18 T

Yes No 3z,z 2898

will step back, adjusting the order and integration step. In this latter case more than one Jacobian matrix computation is needed and the contribution of PTHDs may be crucial. In cases where the prediction is very close to the solution, the contribution of the higher order method to computer time savings may be important, but it is quite difficult to isolate the effect of the analytical derivatives contribution. By choosing a fixed order with a fixed integration step BDF method, the PTHDs contribution can be isolated more easily. A special procedure to take discontinuous actions when an event has been detected, implemented in LASSCs dynamic simulator allowed us to simulate the batch distillation of case A. Results. The results of A are presented in Table 4. The same model was used for liquid and vapour physical property prediction. The gain factors reported, correspond to mean values calculated over a great number of integration steps and having a standard deviation not greater than the system routine accuracy in calculating the CPU time.

Table 4. Results of dynamic case A. Integration over one time-step consists of one Jacobian matrix inversion and four nsiduals computation Equation of state

No. of comuonents

No. of eauations

CPU time eain factor’

CPU time gain factor/time-steab

Ideal

40 30 20 IO

1590 1230 870 510

19.7 15.6 10.7 6.2

I .83 1.70 1.53 I .21

Peng-Robinson

40 30 20 IO

1590 1230 870 510

9.1 8.0 6.8 4.3

1.80 1.57 1.41 1.38

Redlich-Kwong

40 30 20 IO

1590 1230 870 510

7.0 6.3 5.2 3.8

1.70 1.52 1.36 I .27

*CPU time gain factor = time for PTHDs computation using finite difference approximation/time for analytical PTHDs computation. %ZPlJ gain/time-step (Jacobian inversion and residuals computation included) = t,,.,,,/t.,. where: t.., = mean CPU time for one time-step with finite difference approximation of PTHDs, t,, = mean CPU time for one time-step with analytical PTHDs computation.

New implementation for derivatives

447

20

10

c

case

-

Fig. 3. Dynamic distillation cases results: CPU time comparison between finite difference approximation and analytical PTHDs computation for the three cases.

The CPU time gain factor (column 4 of Table 4) grows almost linearly with the number of components in the mixture. The slope is greater for the Ideal model and less for the Peng-Robinson and Redlich-Kwong. This is due to the increased complexity of computations using cubic models like the S-R-K or Peng-Robinson instead of the Ideal. If a cubic equation of state is used for an LV equilibrium, in order to construct matrix A, the compressibility factor is needed as an intermediate variable. The last one is calculated solving a third-order polynomial. implemented in BELSIM-EPIC An algorithm

package (EPIC A4anual, 1988) allows to solve a cubic equation of state and make sure that the root obtained can be associated with a specified phase (Gosset et al., 1986). The algorithm has been optimized with respect to computer time, but remains a supplementary computational effort, nonnegligible compared with that of the Ideal model CPU time comparison between finite difference approximation and analytical PTHDs computation, for cases A, B and C is presented in Fig. 3. The CPU time values reported correspond to mean values.

0.000161 0.00016

0.000159

G 0.000158

‘=; 1 0.000157 $ 0.000156 E 2 0.000155 VI 5 0.000154 ~0.000153 0.000152 0.000151 0.00015

-100

0

100

200

300 Time (8)

400

500

600

Fig. 4. Condenser’s top CH, flowrate evolution with the time. Finite difference approximation was used for the PTHDs calculation.

700

A.

448

KILAKOS

and B.

The benefit for cases B and C is less compared with that of case A. This is due to the complexity of the thermodynamic calculations (cubic equations instead of the Ideal model) and also to the fact that mixtures of the last two cases consisted of less components (see Table 3). The CPU time gain factor/time-step (column 5 of Table 4) is less than the gain factor concerning the PTHDs computation (column 4 of Table 4) because the time step is dominated by the Jacobian matrix inversion procedure. These gain factors are related to one Jacobian matrix evaluation, since in all cases studied here, one Jacobian computation was sufficient to obtain convergence. In cases where more than one Jacobian evaluation is needed, the total gain will be equal to the gain factors presented multiplied by the number of Jacobian evaluations per time step. In case D, the solver does not achieve convergence if finite difference approximation is used for PTHDs calculation, although initialization has been performed using accurate steady-state simulation results. The source of this fail has been localized in the condenser. The condenser was isolated and simulated alone since the solver fails to obtain a solution for the column. A small step change to the condenser’s heat load has been imposed in order to assimilate the top controller’s action, when the top pressure is moved

KALITVENTZEFF

from its setpoint

(due to a change in the rehoiler’s

heat load). In Fig. 4, the top CH4 flowrate response in this perturbation is presented using finite difference approximation for the PTHDs computation. The controller was not included in the simulation. The same variable was observed under the same conditions using analytical expressions for the PTHDs computation, and its evolution is presented in Fig. 5. In this case the whole column can be simulated with the top control loop since the solver achieves convergence. The results of the whole column simulation are not presented here because they cannot be compared with those using finite difference approximation since the last ones do not exist. The behaviour of the system can be interpreted as follows: the top mixture is a narrow-boiling system, and so the PTHDs are very sensitive to the differentiation step. The noise generated by the finite difference approximation at the top composition is transferred inside the column. The scaling procedure makes all variables having the same level of significance and as a result the noise corrupts other variables as well. Using finite difference approximation, the correction procedure of the solver fails to meet the specifications imposed since certain affected variables behave arbitrarily and finally gives up. The procedure for PTHDs computation affects the solution.

0.000159

0.000158

?i0.000157 . ';: 3 0.000156 J : 0.000155 g G: 2 0.000154 0 8 B 0.000153

0.000152

-100

0

100

200

300

400

500

600

Time (8) Fig. 5. Condenser’s top CH,

flowrate evolution with the time. Analytical computation

of F’THDs.

700

New

implementation

CONCLUSIONS

A systematic procedure for calculating analytical partial thermodynamic derivatives for modelling purposes, where LV systems are involved, has been presented. This procedure is based on a few PTHDs with respect to a basic set of state variables, coded explicitly and an algorithm that allows us to compute all the other PTHDs needed, with respect to any set of state variables. This approach does not depend on the solution procedure and therefore does not modify the mathematical structure of the plant unit models and can be used as an independent thermodynamic tool for all modelling-simulation-optimization purposes. The user can choose his own state variables and obtain the derivatives needed. In dynamic simulation of multicomponent distillation, where a large number of components are present, analytica derivatives can lead to really substantial computer time savings. The procedure described becomes interesting for refinery plant simulation. In certain cases where narrow-boiling systems are present in the process, the solver may fail to achieve convergence if finite difference approximation is used for the F’THDs calculation. The procedure proposed allows to overcome such problems_ Advantages in process optimization and three-phase equilibrium calculations will be assessed in future work. NOMENCLATURE e = Murphree

F= f = fg = g = h= H = HL = NY = HVL = HVV = K = L = N = NET= NSI = Ns = P = SL = SV = s= sx = sy = T= V= Vt = o = W = X = x = Y = y =

plate efficiency Total inlet molar flowrate. kmol s-l Vector of equation residuals Fugacity Gibbs free energy, kJ kmol-’ Molar enthalpy, kJ kmol-’ Total enthalpy, kJ s-’ Liquid holdup, kmol Vapour holdup, kmol Liquid volumetric holdup, m3 Vapour volumetric holdup, m3 Equilibrium constant Liquid molar flowrate, kmol s-l Number of dependent variables Number of state variables Number of stage inlet streams Number of components in the mixture Pressure, bar Liquid drawoff molar flowrate, kmol s-l Vapour drawoff molar flowrate, kmol s-l Molar entropy, kJ kmol-* K-l Veotor of species liquid drawoff flowrate. kmol s-l Vector of species vapour drawoff flowrate, kmol s-l Temperature, K Vapour molar flowrate, kmol s-I Total volume, m3 Molar volume, m3 kmol-’ Mechanical power, kW Vector of state variables Vector of species liquid flowrate, kmol s-l Vector of dependent variables Vector of species vapour flowrate, kmol s-l

449

for derivatives 2 = Vector of kmol s-l

species

inlet

partial

molar

flowrate,

Greek symbols a= AP = q= r = B=

Vapour fraction, kmol kmol-’ Pressure drop, bar Isentropic efficiency Compression ratio Differentiation step

Subscriprs F L s V P

= = = = =

Feed stream Liquid stream Species index Vapour stream Column plate index

Superscripis sl = Side liquid drawoff sv = Side vapour drawoff is = Isentropic

REFERENCES

Chen H.-S. and M. A. Stadtherr, A modification of Powell’s dogleg for solving systems of nonlinear - - method equations. Compulers &em. &gni 5, 143-150 (1981). Christiansen L. J.. M. L. Michelsen and A. Fredenslund. Naphtali-Sandholm distillation calculations for NGi mixtures near the critical region. Computers hem. Engng 3, 535-542 (1979). EPIC, Vs. 7.0 EPIC Manual. Belsim s.a., Part Industriel du Sart Tilman, All&e des Noisetiers, Li&ge (1988). Gosset R., G. Heyen and B. Kakitventzeff, an efficient algorithm to solve cubic equations of state. Fluid Phase Equif. 25, 5144 (1986). Kaiialuoto S. and W. Morton, Numerical calculation of dkrivatives in process optimization. Proc. CEF87, Giardini Naxos. Italy, DD. 779-783 (19871. Kilakos A. and C6. .dGarlier, Analiticai derivatives computation of an implicit system of equations for LV equilibrium. LASSC Technical Report (1990). Lucia A. and S. Macchietto, New approach to approximation of quantities involving physical properties derivatives in equation-oriented process design. AIChE JI 29, 705-712 (1983). Michelsen M. L. and J. Mollerup, Partial derivatives of thermodynamic properties. AIChE Jl 32, 1389-1392 (1986).

APPENDIX Equations for a dynamic equilibrium stage p (starting counting from the top and without considering condenser, reflux drum and n&oiler) having one liquid and one vapour phase:

Material balances:

-

Ys,, -

xs.p- SYS, - S&p s=l,...,

9

Ns

(Al)

Energy balamz

+ Vp+lh”*p+l + Lp -&h,,

- SV,hy

I h,,p

-

- SL,h:

I -

Vph”*p 642)

450

A. KIUKOS

Liquid-vapour

Y*p+l(l

Feed streams:

phase equilibrium:

- ep) + epKp

VP+l

($J-($0, s=l,...,

F~=~L~~, s Ns.

K-values: &, Thermal

= K..,(Tv.~,

J’v.,,. yp, J$,),

Mechanical

. . , Ns.

Molar

Tv.p= TL,~

(AS)

T”” = T V.P3 P T: = TL,p.

(‘46)

pv.p =

(448)

hv, = 61 CT,,

h&, = HF(Tb.p, P&

. pv.p>yp ),

=

.

p v.p

Pi = P,,

3

.

UL.,, =

(A101

v;

“L (TL,I,

vV(T;,

=

v,” = q(T$ Volumic

H=,

(Al21

Side streams: (Al3) (Al4)

(‘416) (Al7) (AIS) (A19) G-8

Pv,p- Y,),

e PL,,,

9 xph

(A211 G-2)

Pr, sy,).

(A23)

P;, q,).

(~24)

holdups:

streams:

SL, = 1 sxs,p. 1

P;, sy,),

~v,~ = VY(TV+ 1

streams:

=p =cx,p.

(Al%

volumes:

(A9)

(All)

i = 1, . . , NSI,,,

H,(T&~$sx,).

h;l=

(A7)

of liowrates:

vp = CY..,. ,

PL+. xp), z$,),

h; = H,(T;,

Molar PL.p

NSI,.

enthalpies:

(A4)

equilibrium:

Summation

i=l,...,

h,.,, = &U-w

p"

Liquid

s = 1,.

(A3)

equilibrium: 1

Vapour

and B. KAL~VENTZEFF

“L.P

HV,v,, Hydraulic

=wvL,p,

(4425)

= HVV,.

6-6)

correlations: P L.p = %,pLp = L(HL,,

AP, = AP(L,,

I + APp.9

(~27)

plate p characteristics),

(A=)

V,, plate p characteristics).

(~29)