Unsteady state fugacity model by a dynamic control system. - Portal Embrapa

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A O PUED H A n t H V n f A l /KCDHU ISC

ELSEVIER A p p lied M a th em atic al M odelling 22 (1998) 4 8 5 -4 9 4

Unsteady state fugacity model by a dynamic control system

R a f a e l B ru a’*, J o s é M a r ia C a r ra s c o b, L o u r iv a l C o s ta P a r a íb a a,c

Departament de M atemática Aplicada, E T SIA , Universität Politécnica de Valência, Cami de Vera, sin, 46022 Valência, Spain

b Departament de Biotecnologia, E TSE A , Universität Politécnica de Valência, Valência, Spain

Centro Nacional de Pesquisa de Monitoramento e Avaliação de Impacto Ambiental, C N PM A, E M B R A P A , Jaguariúna, SP, Brazil

R eceived 11 N o v em b er 1997; received in revised form 16 A p ril 1998; accep ted 2 Ju n e 1998

A bstract

A co n tin u o u s tim e d y n am ic system o f an u n steady state fugacity m o d el is p resen ted . P ro p erties o f this m odel as stability are studied. In o rd e r to evaluate num erical results a discretization preserving th e stab ility an d yielding the positivity p ro p e rty o f th e m o d el is used. Finally, algorithm s to determ ine th e values o f th e fugacities, th e c o n cen tratio n s an d th e d issip atio n tim e a re given. T h e above stu d y is illu strated w ith num erical resu lts in a th ree co m p a rtm e n ta l en v iro n m en tal system . © 1998 Elsevier Science Inc. All rights reserved.

Keywords: F u g acity m odel; M u ltip h ase environm ental m odel; U n ste a d y state m odel; D y n am ic co n tro l systems; N o n n eg ativ e c o n tro l system s; F e n itro th io n

1. Introduction

The therm odynam ic concept o f fugacity was introduced in 1910 fo r Lewis [1] in o rd er to ex plain the behaviour o f the real gases with respect to th at o f ideal gases, in the study o f free energy corresponding to an expansion process, isotherm , reversible a n d infinitesim al.

The fugacity is a therm odynam ic m agnitude related to the chemical p o ten tia l and characterized by the leak trend o f a substance in a com partm ent [2,3]. T he fugacity express the chem istry ac tivity o f a substance and has been applied m ainly in th erm odynam ic problem s im plicating equilibrium am ong phases, especially in com putations encountered in chem ical separation p ro cesses such as liquid extraction, distillation and adsorption.

M athem atical m odels based on the therm odynam ic theory o f the fugacity are outlined fre quently by a linear system o f equations describing the bu lk balance o f a chem ical substance in an ecosystem constituted by com partm ents. Thus, when all fugacities are equal an d co n stan t in all com partm ents the concentrations are evaluated directly, this case corresponds to the well-known ‘Level I Fugacity M odel’ or ‘Level II Fugacity M odel’ if in ad d itio n there are reactions and advections (see [4]). ‘Level III Fugacity M odel’ supposes th a t the d istrib u tio n o f the substance is not in equilibrium and th a t each fugacity can have different values, w hich are determ ined by a

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486 R. Bru el al. / Appl. Math. Modelling 22 (1998) 485-494

linear system o f equations when there are reactions, advections, em issions and transfers o f the substance am ong com partm ents in stationary state [5,6]. In ad d ition, there is a n o th e r m odel describing the unsteady state behaviour o f a substance in the environm ent, which perm its to observe substances w hose emissions vary w ith the tim e an d to determ ine the tim e in w hich the system reaches the steady state. This last m odel, know n as ‘Level IV F ugacity M odel’, usually is described by a system o f differential equations (3) (see [7]).

In this w ork, we will present a proposal o f the m ultiphase fugacity environm ental ‘Level IV ’ m odel fo r a ecosystem constituted by n com partm ents where the fugacities change w ith the tim e in response to m emissions and they are determ ined by a continuous tim e dynam ic control system describing the to tal bu lk balance o f the substance.

Som e n o n -statio n ary m odels for the study o f the kinetic o f a substance in the environm ent have been pro p o sed for the analysis o f the concentrations changing w ith the tim e [8-10]. But, it seems th a t no n e o f them presents a m athem atical form alism th ro u g h c o n tro l theory.

T he m athem atical unsteady state fugacity m odel o f a substance in an ecosystem for continuous tim e c o n tro l system perm its to analyse the asym ptotic stability and the non-negativity o f the corresponding discrete system. M ore precisely, we will see th a t the fugacity m odel can be solved as a positive discrete tim e control system obtained by discretizing the continuous tim e control system. F u rth e r, we will prove th a t the continuous an d the discrete tim e c o n tro l systems are b o th asym ptotically stable a n d th a t the discrete system is non-negative. T he num erical algorithm s presented indicate th a t this control m odel turns o u t to be easy to com pute. Finally, we illustrate o u r m odel by a num erical exam ple studying the insecticide fen itro th io n in the fugacity m ultiphase m odel constituted by air, w ater and b o tto m sedim ent.

2. Notations and definitions

The distribution o f a small qu an tity o f a substance betw een tw o com p artm en ts denoted by the indices i an d j , respectively, u n d er co n stan t tem perature an d pressure, yields constant concen tra tio n ratio s between these two com partm ents. The p a rtitio n coefficient ky o f the substance betw een tw o com partm ents is then defined as the quotient CJCj o f the concentrations o f the substance in each one o f the com partm ents.

T he relationship betw een the fugacity an d the co n cen tratio n is given by C = Z f where C is the co n cen tratio n in m ol m “3, / i s the fugacity given in Pascal (Pa) an d the c o n sta n t o f p ro p o rtionality Z is the capacity o f fugacity in m ol m -3 P a -1. The estim ate o f the capacity o f fugacity Z, o f a substance in a com partm ent i depends on the n a tu re o f the c o m p a rtm e n t and o f the partitio n coefficient o f the substance in this com partm ent.

F o r exam ple, in the atm osphere the fugacity o f a substance is equal to the p artial pressure, the one th a t can be expressed in term s o f the concentration in the air Ca a n d by the equation o f the ideal gases given by

/a = CaR T °. (1)

where ^ = 8.314 x Pa m 3 m ol-1 is the gas constant and 7° is the absolute tem perature. By ex pression (1), the air capacity o f fugacity is Z a = 1 ART0.

T he fugacity o f a substance dissolved in w ater is ap p ro x im ated by its p a rtial v ap o u r pressure, the one th a t is p ro p o rtio n al to its concentration in w ater, th a t is,

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R Bru et al. / Appl. Math. Modelling 22 (1998) 485-494 487

where H is the H en ry ’s c o n stan t in Pa m 3 m ol 1. C onsequently, the capacity o f fugacity o f w ater is Z w = \/H .

F o r the oth er com partm ents, the capacity o f fugacity is expressed as the p ro d u ct o f the ca pacity o f fugacity in w ater or in air, by using the density o f the co m p artm en t and the ad sorption coefficient [6,11]. F o r exam ple, fo r the b o tto m sedim ent the capacity o f fugacity is evaluated by expression Z s = k dp sZ w, where k d is the ad so rp tio n coefficient and ps in kg m “3 is the density o f the dry soil, consequently, the fugacity or concentration in this co m p artm en t can be calculated by equation

Cs = Ps

Zv„/s-In an unsteady state, the fugacities are functions o f tim e and they can be determ ined as a system o f o rdinary differential equations, describing the to tal bulk balance during an infinitesim al tim e interval d t. In this balance, the storage capacity o f a substance in a co m p artm en t i is given by the p ro d u ct o f its volum e Vt and by its capacity o f fugacity Z,.

The biological and chem istry reactions are supposed to be a first o rder process where the rate o f reaction is rt in m in -1 and the reaction com ponent for each co m p artm en t is r ^ Z , .

W hen there is a gradient o f fugacity betw een two co m partm ents i a n d j it results a flow o f the substance in the direction o f this gradient. The flow o f the substance is the p ro d u ct o f the dif ference (ft - f j ) by the transfer dy betw een these two com partm ents. T he coefficients o f tra n s ference dy a n d djt are equal an d positive. The differences (/j- - f ) are positive or negative depending on the direction o f the transfer determ ined by the relative values o f f and f .

T he advection in a co m partm ent i can be introduced in the m odel as a first o rder process. In fact, the advection can be considered as a constant speed defined as the algebraic sum betw een the entry flow G,CBi and the exit flow G,Cj o r in term s o f fugacity as G i Z f , w here G, is the m atte r flow in m 3 m h r 1 entering into co m partm ent i with concentration CBi and leaving this com partm ent w ith concentration Q .

The emissions in com partm ent i, as a function o f the tim e in m ol m in -1, are denoted by

Ej = Ei(t). In this position we are supposing th at there is n o t any effect o f dilution, th a t is, the

volum e Vj o f each one o f the com partm ents is constant.

3. Fugacity model of a continuous and discrete time system

G iven a co m p artm en t i and a co m p artm en t j there exist the follow ing excluding possibilities for transferring substance betw een these tw o com partm ents: (a) there is a co n tact area between co m partm ent i and com p artm en t j , in this case, there is a positive g rad ien t while the equilibrium o f the fugacities will n o t be reached am ong these two com partm ents; (b) there is no contact area betw een co m p artm en t i and co m p artm en t j , consequently, there is n o t any possibility o f a direct transfer o f the substance am ong these two com partm ents. These tw o possibilities are fundam ental in the position and in the analysis o f the fugacity m odel for a con tin u o u s tim e control system.

Let / , — { / € N: the set o f indices i ^ j o f com partm ents in w hich there exists a contact area w ith co m p artm en t

/}-T he v ariatio n rate o f the fugacity in the tim e, for each one o f the com p artm en t, is calculated using the to ta l bulk balance, w hich is described by the follow ing system o f ordinary differential equations:

VtZt ~ = E, + GiCm + Y d f i f j ~

(

G‘Z‘

+

r‘ V‘z >

+

Y f l j

J

f" ^

j€J, \ jeJ: J

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488 R. Bru et al. I Appl. Math. Modelling 22 (1998) 485-494

T he system expressed by Eq. (3) can be w ritten as the follow ing continuous tim e co n tro l system:

f = A f + Bu,

w here f = f(i) is the fugacity derivative vector o f com ponents [/}] = d f / d t .

GiZj

+

riVjZi

+

Y2jeJi dij

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A — [a,y] —

&ki '

_Vlzl

if k = i,

&ki

_v,z,

dki

_{in e J l'}

. a/a = 0 if k £ J i

is the n x n state m atrix, f = f(f) is the state vector o f fugacity o f com ponents [/]} = f ( t ) , B = I„ is the identity m atrix o f o rder n and u = u(7) is the co n tro l vector o f com ponents

[«,] =

(Ei

+

GiCxyViZi.

3.1. E xam ple

F o r a simple ecosystem constituted by air ( / = 1), w ater (i = 2) and b o tto m sedim ent (z' = 3), system (3) is determ ined by the system o f ordinary differential equations:

V\Z\ = E\ + G\CB\ + d2\ f i — (d\2 + G\Z\ + r\ V \Z \)f\ ,

VlZl = Eo G2CB2 + d \lf\ + <^3 2 /3 {d2\ + ^23 + G1Z2 + t 2 V2Z2) f 2,

P3Z3 = £3 + G^Cb3 + d jifl —(^ 32 + G3Z3 + ^3^ Z ^ fs

w ith the initial conditions /i( 0 ) = / f \ f 2{0) = / 2° a n d f 3(0) = / 3°, or by the continuous time co n tro l system (4) w here the m atrices A and I3, are respectively,

A -— (G\Z\ + d\i + r\ V\Z\) V\Z\ d\2 W 2 0 a 21 V & — {G2Z2 + Ai -I- d2i + r2 V2Z2) V2Z2 ^23 W i 0 i/32 V&2 — ( G3Z3 + d^2 + r3V3Z 3) '1 0 O' I 3 = 0 1 0 _0 0 1

The vectors f = f(i), f = f(i), u = u(i), fo = f(0) are, respectively,

f = d /i d/ 2 d/ 3

d t di d t f « = [ / H 0 / 2 ( 0 / 3 « ] T,

u(0 = [{El + G\CB\)/V \Z \ (E2 + G2CB2)/V 2Z2 ( £ 3 + G2CBi)/V 3Z3]T, and f(0) = [f? . In this

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R Bru et al. I Appl. Math. Modelling 22 (1998) 485-494 489

N ow we retu rn to Eq. (4) w hich adm its the follow ing integral solution in [0; t\ (see [12]):

t

f(f) = eA'f(0 ) + J e A^ \ u ( t ) d r. (5) o

Let m be the n u m b er o f emissions a n d let T be the tim e interval betw een each emission. Discretizing Eq. (5), see [12] for details, for each t = k T , w here k = 0 , 1 , 2 , . . . , (m — 1) we have

f((* + l ) r ) = W( T ) f { k T ) + H (T )u (k T ), (6)

where

T

w (T) = eAT, H( T ) = J e Ml n dx, x = T - t .

o

W hen the tim e interval T betw een each em ission is fixed, the m atrices W(7~) an d H ( I ) are constants. W e are supposing th a t the emissions E l = E ^ t) an d consequently the vectors u = u(/) are co n stan t in a given interval betw een any tw o consecutive in stants o f em issions, th a t is, u = u(£7), for k T ^ t < k T + T.

Thus, if the emissions E ^ k T ) fo r each com partm ent i are know n, for the corresponding times

k = 0 , 1 , 2 , . . . , (m — 1), and consequently the control vectors u(kT ), then, it is possible to com pute

the fugacities an d the respective concentrations in these times for each one o f the com partm ents

i = 1 , 2 , . . . ,n, using Eq. (6). N o te th a t u(i) = 0, for all t ^ m T , an d in this case system (4) is

determ ined by

f((Jfc + 1)T) = W ( T ) f ( k T ) fo r k ^ m . (7)

We will see below th a t the discretization (6) has the p ro p erty o f m aintaining the stability o f Eq. (5).

In the m odel given by Eq. (4) and the corresponding solution (5), o r alternatively the dis cretized m odel (6) we suppose that: (i) the volum e o f each co m p artm en t is co n stan t, (ii) the emissions are know n fo r each tim e k T in the tim e interval [0; (m - 1)7] an d (iii) for the times outside o f th a t interval the em issions are null.

4. Analysis of the model

In this section we deal w ith som e properties o f o u r m odel. F irst we study the stability o f the continuous tim e control system described by the m atrix eq u atio n (4). L ater, we will prove th a t the discrete m odel (6) is asym ptotically stable. In ad d itio n , since we are m odelling fugacities, it is im p o rta n t to know the non-negativity o f the discrete tim e c o n tro l system given by m atrix equation (6) (i.e. all m atrices o f (6) has only non-negative entries). In the follow ing theorem s we will use the term inology, corresponding to non-negative m atrices used by Berm an and Plem m ons [13],

Theorem 1. The discrete control system defined by the pair (W( 7), H(7")) is a positive system, that is,

the components o f the matrices W(7~) and H(T) are non-negatives.

Proof. T he m atrix A is essentially non-negative since there exists a c o n stan t <5 > 0 sufficiently

large, for exam ple 5 > m axi=12 „[a,-,-], such th a t (A+<5I) > 0. T hen, by T heorem (3.12) o f [13],

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490 R. Bru et al. / Appl. Math. Modelling 22 (1998) 485—494

As a practical consequence o f this theorem one can deduce th at if we apply non-negative controls and the initial conditions are non-negative then we always ob tain non-negative state vectors th a t in our problem are fugacities. In fact, this is the definition o f positive discrete tim e control systems, and this is equivalent to the m atrices W (T) an d H (7) are non-negatives.

W ith the following theorem we can assure th at system (4) is stable because the m atrix A has eigenvalues w ith negative real p arts (see [14]).

Theorem 2. L e t A be the m a trix o f system (4), - A is positively stable, that is, its eigenvalues have

positive real part.

Proof. F irst, by the definition o f the p roblem all param eters dtj are positive. F u rth erm o re, it is

observed th a t

then - A is strictly diagonal dom inant. Since - A has positive diagonal elem ents then the Ge- rschgorin’s disks are in the positive sem iplane. F rom b o th properties we deduce th a t the eigen values o f - A have positive real p a rt an d consequently - A is positively stable. Then A has eigenvalues w ith negative real p a rt. In ad d itio n A is n o t singular. □

The discretization (6) m aintains the stability o f the original continuous time co n tro l system, then the po in t where all fugacities vanish is asym ptotically stable for system (6). By com pleteness, we give the short p ro o f o f this result fo r o u r particular m atrices.

Theorem 3. The discrete positive control system (W (7), H (7 )) is asym ptotically stable, that is, the

point where all fugacities are null is an equilibrium stable point.

Proof. I f a, are the eigenvalues o f A then, e'-,r are the eigenvalues o f W (7). By T heorem 2 the

eigenvalues A,- have negative real p a rt, then we have |e/-, r | < 1 and therefore, | | W ( —» 0, when

p —> +oo. H ence the system (W (7), H (7 )) is asym ptotically stable. □

5. Algorithm to compute the settling time

To determ ine the needed tim e to get low concentrations in all com partm ents, we have to com pute the time for w hich the fugacities vanish, after the emissions finish. This tim e is called fugacity settling time. T he fu g a city settling time is defined as the tim e required for the fugacity values stay w ithin a range o f the final value. The settling tim e can be conceived as the time re quired so th a t the system arrives and stays within a range o f the equilibrium point. It can be com puted by the follow ing algorithm whose convergence is guaran teed by T heorem 3.

The com p u tatio n o f the m atrices W (7) and H(T) w hen the m atrix A is know n can be ac com plished by the m athem atical scientific package M A T L A B [15], w hich calculates the expo nential o f a m atrix th ro u g h P a d e ’s approxim ations with the function expm using the com m and

[W,H] = c2d (A,I„,7).

Algorithm 1 (A pseudo-code algorithm to com pute the settling time).

Begin

C om pute m atrices W(T) and H(T) by: [W,H] = c2d (A ,I„,7) F o r k = 0 until (m - 1) calculate

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R. Bru et al. / Appl. Math. Modelling 22 (1998) 485-494 491

f((k + l)T ) = W( T ) f ( k T ) + U (T )u(kT )

fo r i = 1 until n com pute

Q (k T ) = Z jfi(kT )

L et tol > 0 be a tolerance o f the origin. F o r k > m com pute

i((k + l)T ) = W (T )f(k T ) + H (7)u (J)

until [|f(k + 1)71| ^ tol fo r i = 1 until n com pute

Ci{ { k + \) T = Z if i{(k+ l)T )

End;

T he first value o f k s = k , fo r w hich the inequality o f A lgorithm 1 is satisfied, determ ines the settling tim e ts = k sT for the tolerance tol. The value o f tolerance can be established as a function o f the lim its o f c h ro m ato g rap h y detection o f the concentrations o f the substance in the com partm ents.

6. Algorithm to compute the dissipation time

By Eq. (7), for each co m partm ent, it is possible to com pute the in sta n t tim e for w hich the value o f the concen tratio n re d u c e d p % o f the first value o f the co n cen tratio n after the em issions finish. This tim e is called the dissipation time in the co m p artm en t i, d enoted by fp a n d calculated by the follow ing procedure:

Algorithm 2 (A pseudo-code alg o rith m to com pute the d issipation time).

Begin

C om pute m atrices W (T) a n d H (7 ) by: [W,H] = c l d (A,I„,7) F o r k ^ m com pute

f((&+1) 7) = W ( T )f(kT )

Ci(kT) = Z f ( k T )

until C,(kT) {(100 -/? )/1 0 0 ■ Q ((m - 1 )T j End;

The first value o f k p — k , fo r w hich the inequality is satisfied, determ ines the dissipation tim e at level p % , fv — kpT fo r the co m p artm en t i.

7. Numerical simulation

W e have selected the insecticide fenitrothion to verify o u r m odel. F e n itro th io n is an insecticide frequently used in crops an d forest protection. F o r th a t, we have considered a hypothetical three com partm ents environm ental system consisting o f air, w ater an d b o tto m sedim ent w ith volum es o f 1.0 x 106, 1.0 x 104 a n d 1.5 x HP m 3, respectively. T he sedim ent density and co n te n t volu m etric o f carb o n are, respectively, 1.5 x 103 kg m ~3 an d 4.0%. In this system , we apply in the air one m ole o f the insecticide fen itro th io n during 60 m in, th a t is, E \(k ) — 1/60 and E2(k) = E^ik) = 0 fo r k = 0 , 1 , 2 , . . . , 59. A ccording to Z itko and M cLeese [16], the c o n sta n t rate o f disappearance in m inutes o f the fen itro th io n in air, w ater and b o tto m sedim ent are respectively 4.76 x 10-4, 3.80 x 10“4 and 2.85 x 10-5. H e n ry ’s co n stan t for the fen itro th io n (see [17]) is 6.65 x 10-2 P a m 3 m o l-1. T he coefficient o f a d so rp tio n k A for this substance in this b o tto m sedim ent is 27 x 10-3 m 3

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492 R. Bru et al. I Appl. Math. Modelling 22 (1998) 485-494 1.0E-HH l,0E+00 1.0E-02 l,0E-a3 1.0E-M 1.0E-05 M>E-06 l,0E -07 1,0EMB 1.0E-09 M>E-10 1,0E-11 0 30 60 7200 14400 21600 28800 36000 43200 50400 57600 64800 72000 79200 86400 93600100800 Time (m inutes)

-•-A ir + W ater - k Sedim ent

Fig. 1. F ugacities o f fe n itro th io n in a ir, w a te r an d b o tto m sedim ent b efo re a n d a fte r th e em issions, o b ta in e d by A l g o rith m 1.

k g -1. T he value o f T an d tol are 1 m in and 1.0 x 10_u Pa, respectively. T he exit advection flow in air, w ater and b o tto m sedim ent are, respectively 6.87 x 102, 5.49 and 6.20 x 10-2 m 3 m in -1.

W ith these d a ta we wish to evaluate, during and after the em issions, the values o f the con cen tratio n a n d the fugacity o f fen itro th io n in the air, w ater an d b o tto m sedim ent an d then to determ ine the needed tim e to stabilize the fugacity, th a t is the fugacity settling time. In this ex am ple the air is the only one co m p artm en t receiving em issions a n d we can consider th a t the advections are null fo r all com partm ents.

1,OE+Ol 1,OE+00 1,0E-01 l,0E-02 i L,0E-fl3 e l,0Er04 c g l,0E -05 a o 1,0E~06 o o vj l,0E -07 I,0E-08 l,0E-09 1,0E-10 1,0E-U Time (m inutes) ■♦•Air “h W ater rh Sedim ent

Fig. 2. C o n c e n tra tio n s o f fe n itro th io n in air, w ater a n d b o tto m sed im en t b efore a n d a fte r th e em issions, o b ta in e d by A lg o rith m 1.

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494 R. Bru et al. I Appl. Math. Modelling 22 (1998) 485—494

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