It is shown that, for the multivelocity model, the resulting in- dependent variables are the densities and velocities of the components and the entropy per unit volume

PRODUÇÃO TECNICO CIENTÍFICA DO IPEN

DEVOLVER NO BALCAO DE EMPRÉSTUVIO

= Ost, — P + EA(1)p(i) (3)

i=1

The time derivative of the internal energy per unit vol- ume can be written as:

auv _at = (i) 7--opo) t + 0 ^asv at (4)

An analog description is adopted for the internal energy per unit volume corresponding to the ith-component uti):

u1,0 uta)(s„, (1(1), ..., p(0) (5) The following potentials are defined:

A New Approach for Modelin.g Multicomponent Solutions

Presented at IAPWS EC Meeting, Buenos Aim, Argentina, July 2002

Balizio, J. L. (jlbalinodnet.ipen.br ) Instituto de Pesquisas Energéticas e Nucleares,

Av. Prof. Lineu Prestes, 2242 - Cidade Universitária, São Paulo - SP - CEP 05508-000, Brazil

Abstract,— This paper shows an application to Multicom- ponent Solutions of a new, Bond-Graph based formalism for Computational Fluid Dynamics (CFD) problems. It is shown that, for the multivelocity model, the resulting in- dependent variables are the densities and velocities of the components and the entropy per unit volume. The state equations are derived, showing the potentials and constitu- tive relations needed to describe a multicomponent system.

Based on the muitivelocity model, the diffusion approxima- tion is presented. This model differs from the multivelocity model In the way the kinetic coenergy of the components is taken into account, reducing the number of state variables from multiple velocities to a mean velocity. In the diffu- sion model, the dynamics of a multicomponent solution la described in terms of the average (center of mass) velocity of the mixture and the mass flux of each component rela- tive to the average velocity. The relative fluxes are assumed to be dependent on the entropy per unit volume and the component densities. This functional dependence allows to deal with ordinary (concentration driven) diffusion, pressure diffusion, forced diffusion and thermal diffusion. Based on the contribution of the diffusion fluxes to the kinetic coen- ergy, different potentials associated to the entropy per unit volume and component densities are deflned. It is shown that these potentials modify the generalized effort variables appearing at the inertial, rnass and thermal ports of the /C- fleld representing the system total energy. Besides, these potentials introduce linkages between these ports.

Keywords: Bond Graphs, Computational Fluid Dynamics, CFD, Multicomponent Solutions, Diffusion.

I. INTRODUCTION

In recent works [1] [2] a theoretical development of a gen- eral Bond-Graph approach for CFI) was presented. This new methodology, which was called BG-CFD [3], is a re- sult of the right combination of Bond-Graph concepts [4]

with elements of numerical methods. In this paper, the methodology described above is extended to multicompo- nent solution systems.

A classical mbcture, or solution, is a material in which the components are not physically distinct, that is, the mbdng is at molecular level. In this case, when described using continuum theory, all the components of the solutions are able to occupy the same region of space at the same time [5] and can be assumed to be in thermodynamic eqUilib- ritun. In a solution, each component has its own velocity, density and internal energy. The balance principles for the constituents resemble those for a single component, except that the constituents are allowed to interact with one an- other.

Concerning the nomenclature used in this paper, bold

—4i) letters will be, used to define first order tensors ( v • , pti),etc.). Column vectors associated to nodal values will be denoted by single tmderscored plain or bold type (m(i),

cp_e_(1), etc.) while multidimensional matrices will be identified by double underscored plain type ()UM, SYpi), etc.). Second order tensors will be denoted by bold, dou- ble underscored type (721, I, etc.). Einstein convention of summation over repeated in—dices is not used.

II. INDEPENDENT VARIABLES AND POTENTIALS

A. Internal Energy per Unit Volume

For a multicomponent solution with r components, the internal energy per unit volume ut, can be written as a func- tion of the entropy per unit volume st, and the component densities p(i):

= p('), •••, P(r)) (1)

The following potentials are defined:

' 14" ap .)

„(,)

ouv) out, (.)

where 0 is the temperature and p,(i) is the ith-component chemical potential per unit mass. The pressure P can be obta.ined from the Euler equation [6]:

(2)

7 ^{3- 62 02}

COPIE C21.Lt0,0i. OR ."

a (i)

(g) (uuv (ii) _ (--v

; – —

– 88„ _p(i) 00)

a,„ p(k0

where r(i) and ti(ii) can be regarded as ith-component contributions to the temperature and chemical potentials.

From the Euler equation:

uti) r(i)sv _ p(i) E ^{A(0) pU)} ₍₇₎

i=1

where P(i) is the ith-component contribution to the pres- sure. Since:

Euto

the following relations are verified:

Er(i) ; ; p p(i)

i=1 ₁₌₁

As an example, the potentials associated to the entropic representation of the ith-component internal energy are cal- culated in Section VII for the case of a mixture of ideal gases.

B. Kinetic Coenergy per Unit Volume

The kinetic energy per unit volume t: can be written as a function of the component densities and velocities V(i):

t; = r 1 - p(i) V(i)2 i=i 2

The following potentials are defined:

,,(i) at;',

kap(%) p000,vo ²

v(i)2

(i) atz, ₍₁₎

v(a) Pv =

FV-71) p(1) , V(k9id) = p

where K(i) and pl,i) are correspondingly the kinetic co- energy per unit mass and linear momentum per unit ma.ss for the ith-component. The time derivative of the kinetic coenergy per unit volume can be written as:

jt'

U6 = E E ^{Pv .0}

_(i) (i) ,v(i)

(13) C. Total Energy per Unit Volurne

The total energy per unit volume e: includes the internal energy and the kinetic coenergy:

e: = + t: (14)

The time derivative of the total energy per unit volume can be written as:

aez = ⁹ _j_co + ^as E [(0) ^{,(0) p(i)}

4.1 at ^v at

(15) Since the internal energy and kinetic coenergy are contin- uous functions of the independent variables, the potentials multiplying the time derivatives of the independent vari- ables satisfy both constitutive and Maxwell relations from Thermodynamics [6].

III. BALANCE EQUATIONS

The balance equations are power equations correspond- ing to ea.ch one of the terms that contributes to the time derivative of the total energy per unit volume. For multi- component solutions, the balance equations can be derived from the mass, momentum and energy conservation equa- tion corresponding to each component [5]:

(i)

= –V . (p(i) V(i)) + C(i) ₍₁₆₎

(i) avw

— –0) V(4) .V V(i) + v .

+p(i) di) 4_ f (i) C1i) V(i) (17)

°UV)

= -V. (14,i) Vi)) - f ^{C(i) _1} vii)2

+la vv(i) p(i) Ib(‘) V.q(i) e(i) 2 08) where C(i), f (i) and e(i) are correspondingly the mass, momentum and energy interaction terms (per unit vol- ume), cI3(i) is the heat power source per unit mass, di) is the body force, q(i) is the heat flux and T(i) is the stress state for the ith-component.

Since there are no distributional sources, it is postulated that the sum of the interactions of mass, momenttun and energy vanish, that is:

t. = o ; Ef (i) = 0 ; Ed) ^{= 0} (19)

i=1 1=1

and viscous components r(i) as:

The stress state can be expressed in terms of the pressure

= _p(i)1+7.(1) ₍₂₀₎

Taldrig into account the conservation equations and Eqs.

(4) and (7), the balance equations result:

(1.4(4) K(i)) aotP(i) _v. {p(i) (14(i) +. no))

+c(i) K(1)) + (JO V(i).Vp(i) v(i).vnta) (21) avo)

140 — = V . (r(i) .V(i)) – p(i) et .V tc(i) – .V PO)

+ f (i) .V(i) – V V(i) – 2 C(i) n(i) + p(i) G(i).V(i) (22) (6)

(8)

(9)

(10)

(12)

at

.,. 11 3

0 -- as. Lif—v ^.[q(i) (ir(i) sv _ Ao) po)

at _i=1

+ E pu) + V(i) .V Po)

.1=1

—f (1) .17(i) + La: VV(i) + 2 di) n(i) _c(i) (AO) tc(i)) _ p(i) VO).VA(i) ±p(i) CVO

(23) According to the balance equations it can be seen that it is necessary to know, for each component, the potentials coming from the entropic representation of the internal en- ergy and from the kinetic coenergy, as well as the mass and momentum interaction terms, the heat flux, the heat power source and the viscous stress.

The balance equations show one of the advantages of the

BG-CFD methodology, that is, the representation of the power structure of the system. In the balance equations there can be identified three type of terms: divergence, source and coupling terms. The divergence terms take into account the power introduced in the system through the boundary conditions. The source terms constitute the dif- ferent power sources, external to the system. Finally, the coupling terms represent power transfer between the ve- locity, mass and entropy balance equations; these coupling terms appear, with opposite signs, in pairs of equations.

"Diking into account Eq. (15) it verifies that coupling terms cancel out when the balance equations are added, resulting:

— = aet --r {—V. [(4i) P(i) p(i) nii)) Vii)1 1=¹

+V . (7(1) .V(i)) — V .q(i) + ^di) .V(i) +p(1) (DM

(24)

The cancellation of the coupling terms means that they influence the power distribution among the different ports but not the total power in the system.

IV. DISCRETIZATION

The independent variables are discretized, in the domain volume f2, in terms of time-dependent nodal values (4) , VLi) and st,i) and interpolation (shape) functions (corre- spondingly ço(pii), , o(vi),,,, ^{and (psi):}

n(pi)

P(i) (r, ^t) = ^{Ep(ki) (t)} v(i) (r) ^{(i) T (1)}

2_— ^•Wp

k=1 Pk

n(i)

V(i) (r, t) = E v;ii.,)(t) (40,7, (r) = V(')T .ep(i) (26)

m=1 n.

8t, (r, t) = (t) (psi (r) = suT .cp„ (27)

1=1

For any position r E ft, the shape functions have the following properties:

Ecpp(a2 (r) -- 1 ; E (r) = 1 ; E (pai (r) = 1

n(i) v . n.

k=1 rn=1 1=1

(28)

For simplicity in the treatment of the boundary condi- tions, we also require for the interpolation functions to have the value one for the reference node position, be monotoni- cally decreasing with respect to the distance from the refer- ence node and be zero for the rest of the nodes. Since this is the only discretization restriction, it is possible to work with any kind of grids. Notice that it is possible a priori to have different densification in the nodalization, this is, the ntunber of nodes 4) and 4) can be different for each component; this is important, for instance, in boundary layer problems.

Nodal vectors are defined as Bond-Graph state variables, namely mass and 'velocity for the ith-component and en- tropy. The mass and entropy vectors are obtained by inte- grating the corresponding nodal independent variables in the support of the shape functions:

m(i) p(i) ; ,E na. sv P r.__

The diagonal matrices fl(pi) and f2. are defmed as:

P p kn ^{rpk kn}

o(i) if111) f ^da

a n

= t 1 = f ^{dn (31)}

where bij is the Kronecker's delta (15ii = 1 if i = j, ^{6ii := 0}

otherwise). The system mass for the ith-component and entropy are related to the integrated variables as follows:

n(;)

M(i) = p(i) dEl = Em(i) (32)

k=1

S ^{= f St; =} Es, n. (33)

1.1

The system total energy E* is defined as the surn of the internal energy U and the kinetic coenergy T*:

E* = UCE, m(1) , rt_2_1) + T.(73311 723L, v(1)

(34) where:

E* = e: cif/ ; U ⁼ f ^{at, d12 ; T* = clil (}35) fl

From Eq. (35), it can be easily shown that the system kinetic coenergy can be expressed as the following bilinear form:

(25)

(29)

(30)

COTE'

m(1) p•(1)

N MTF N

po)

Fig. 1. Modulated itli-eomponent inertial transformer.

r T

= E__t=1 2 ^{VW . M(°}

F")

v")

According to Eqs. (38) to (40), the nodal vectors a, 22

and EL can be regarded as system volume averages of the corresponding local values, weighted by the interpolation functions. The time derivative of the system total energy can be written as:

= 9,7%1+ E [(2.(1.4_ K(0) T ^{+ E(IT} VW]

i=1

(36⁾ where M(i) is the inertia matrix corresponding to the

ith-component:

Al(i) = {Mal = P") (P(147)Trt dfl

We define the following potentials:

a= (211-) = 519-1. [f cp„ di-21

(i) ⁽³⁸⁾

(i) ⁽ au _{do] (39)}

mu#0, ^•

=A-1(0-' [.1 (0 (0 I

\anz(i) mu ^{0,, v,k, _p__ • It (22._cift}

(40)

ffr*

p(i) - = v(i) p") (i)

OV(1) (i) 17"a

(41)

where a, tt(0, K(') and F22. are correspondingly nodal vectors of temperature and ith-component chemical poten- tial per unit mass, kinetic coenergy per unit mass and linear momentum. It is important to notice that Eq. (41) defines, in the Bond-Graph terminology, a modulated multibond

transformer relating the nodal vectors of velocity and lin- ear momentum, as shown in Fig. 1; in this and in the following figures, it is drawn the causality resulting from the Bond-Graph causality assignment procedure [4]. Ac- cording to the power conservation across the transformer, the generalized effort is given by:

F(i) m(i) ¡Ai)

According to Eq. (41), the nodal vector of ith-component

linear momentum can be regarded as a system volume in-

tegal of the local values weighted by the velocity inter- polation function. It can be easily shown that the system

ith-component linear momentum can be obtained as:

P(i) f ^{a./ = Ev}

m=1

n(4)

(43)

(44) It can also be shown that the voltune integrals of the left side terms of Eqs. (21) to (23) can be calculated as:

(ii(i) (i)) op(i) ds-1 = T

(45)

(46) (47)

The system constitutive relations are:

= mo), ..., m(0) + K(i) (v(0) (48) 2(1 E,(/ ^(7-n(i), v(i)) = m(i) v(i) (49)

fi = a ^{(1, M(1) M(r)) (50)}

The Maxwell relations corresponding to the system total energy arise from the equality of the mixed partial deriva- tives of the system total energy expressed as a function of the independent variables a, ^{and V(').} These vari- ables are regarded as the state variables for the BG-CFD

methodology:

The constitutive relations (Eqs. (48) to (50)) and the Maxwell relations (Eqs. (51) to (53)) define, in the Bond- Graph terminology, a multibond /C-field associated to the system total energy, as shown in Fig. 2. This field has

r inertial ports (the velocity ports) and r + 1 capacitive ports (the entropy port and the r ^MRSS ports). The gener- alized effort variables associated to these ports are ir(i),

and (21+ K(i)), while the generalized flow variables are correspondingly el, ,¡ and 621.

For the sake of convenience, we also define the following diagonal matrices, whose elements are the components of the corresponding vectors of nodal potentials:

(42)

f Pta) • $914: 61S-1 =

(37)

f1

f 8÷; c/f2 =

fl

(97-Ti(i) =

„ (,))T 1942

Tn(ioi) mu)

= (— ^op(a)

T

(an As ;3) 0.2 (i) y22. = -

t ap±\ (8K(.0\

v,,,= UT:EL)

C gF ^L74

Fp(i) (12.1.fi) ,p(12dr

(65) (64)

(68)

F(i) — - Z, • (0 v (i) V' v dn V. SYSTEM STATE EQUATIONS

cfirm ^{r 7 7-}

•••

Fig. 2. System /C field reprmonting micro storage.

ri2(4) - (AM + K(i)) . f ^{w(i) /1(0} V(i).77A(') cift (62)

71. (i) _ ( \— 0) + KM) -1 _e_ w(i) di) ( (in

(63) Th(i)

K ^(±). C__

KM) -1 Id) p(i) v(i).vn(i) di)

O.= {el.} ⁼ et bin

= {4;:,;} = AL') sk.

K (0 = {14,2 = bk.

(54) f p(i) V te(i) so(vi) c/SI (66)

(55) "n

(56) ^k2pil f ^{(vp(i) _ f (0) (p(i)} dn (67)

The system state equations are obtained by systemati- cally volume integrating the balance equations correspond- ing to each port of the /C-field representing the total sys- tem energy. The expressions for the system state equa.tions are:

Fg) _ f ^di) vi) ^cis/ (69)

Fg) f /3(0 di) ,p(vi)

(70) (57)

4) ,_ ta ^- 1 . 1 _ f w. t. [q(i) + (Ir(i) s. _ 0) p(i) (58)

+

r ^i--.1

E pU))V0)]. fidr}

(59) i=1 ⁽⁷¹⁾

The different terms in the system state equations (57) to (59) arise from integrations over the domain volume _SZ or

the domain boundary F. Their definitions are: ^{

I ^{Via, . E [q(i) +} r ^{(r(i) sv _ 0) p(i)} i=i

0-1

mw • (NO _ (p(i) K(i)) — f _e_ w(i) p(i) (A(i) (0) Vi). Jiff]

K(i)) p(i)

+Pc(i)) V(i).Vw(i) ____P__.

+K

+ + +4,9K +ThZ)

ir(i) = M(') . (FP(i) - FZ) -

- Fg)+F(,i))

4r) + ‘.12 + k-JJ+ L'CK +

+E p.(q) pU)) /=1

(72)

(60)

(61)

= Er ; = '4/ ;

i=1

(E p(i) Vi)) ds1

'11 (73)

= E ⁽⁷⁴⁾

(75)

EL

i=1 ₌ 1

two

F (i)

e \I 1

FrX° + Fr

where:

[ /. w (VP(i) — f (a)) .V(i)cifti 'n

43) = 0-1. [ f tv. (V V(i) : r(I)) c/SZ 'n

4;!) = g- i . - f [ ^{w. 2 CO) tc(i)dft} 'n

[

4) . a-i. ^{ftv,p(i) v(i) Nip) cin}

a

[

21 ,..._ fr 1 . f lilt co) (11(0 + n(0) do 'n

:5(0

—P—. (76)

(77)

(78)

(79)

(80)

1 0 / ISf

Fig. 4. 0-Junction representing the balance equation at the entropy port.

Although the complete Bond Graph is not shown here, it can be said that the state equations (57) and (59) are represented, in the Bond-Graph terminology, by multibond 0-junctions, in which correspondingly the ith-component

mass rate nodal vectors and the entropy rate nodal vector are added (see Figs. 3 and 4). Eq. (58) is represented, in the Bond-Graph terminology, by a multibond I-junction, in which the forces are added (see Fig. 5). A multibond 0-junction is also used to represent Eqs. (74) and (75).

Fig. 3. 0-junction representing the balance equation tit the ith- component mass port..

The convective (upwind) nature of the fluid equations is handled through the definition of density and entropy weight functions, namely w(1,i) and w., which are introduced to satisfy the power interchanged by the system through the boundary conditions, as well as to share the importance of different power terms among neighboring nodes. In the

discretization procedure, all the terms of the ith-component

mass balance equation and entropy balance equation were multiplied correspondingly by w(i) and w.; although this procedure has the advantage that the steady-state balance equations are satisfied locally for the different nodes, other

discretization strategies are possible and should be investi- gated. This concept was successfully applied to convection-

Fig. II. 1-junction representing the balance equation at the ith- component velocity port.

diffusion problems [7] [8]. It is very interesting to notice that no weight functions result for the velocity state equations.

As in Ill, all kind of boundary conditions can be handled consistently through the terms representing surface lute-

(„1,(r)(i) v(r)(i) 6(r)‘

grals _''"(4 and can be represented, in the Bond-Graph terminology, either as generalized modu- lated effort sources at the inertial ports or modulated flow sources at the capacitive ports, as shown in Figs. 3 to 5.

The discretized representation of the power coupling ap- pearing in the balance equations per unit volume is per- formed through the coupling matrices, which relate gener- alized variable,s whose product gives rise to power terms ap- pearing in more than one port. Depending on the variables being related, these matrices define, in the Bond-Graph ter- minology, power conserving two-port elements (modulated transformers or modulated gyrators), as shown in Figs. 6

to 8.

It can be shown that the relationships corresponding to Figs. 6 to 8 are:

P' a

(87)

g")+K")

MTF

MK")

FP m(i⁾

FP + + F g) M

e

MTF

+

Fig. 6. Power coupling between the ith-component velocity and entropy ports.

F(i) + F(i) + F(i) __I2- 4,(i) 440

4) + mr,), r,(i) + K(i))

th») = mMT a

Ad, . KT)

• T .

• (i) m(t) v(s) MV •

where the rectangular matrices MV, (nv(i) rows, n.

columns), Mrf)s (n. rows, n(pi) columns) and MiTt, (n(vi) rows, 4,i) columns) are defined as:

{m(01 r,

çvfmi = -6; / RvP(') — ^C(i)

+T(i) ' V(P(Vind Wai

e ^ACs 0+K")

MG Yi

Sq0 + ;50)

Ihu + riiCK(i)

u

Fig. 7. Power coupling between the ith-coinponeut entropy and WW1

ports.

m.(;) 1

(i) MSf 01 ALS) KV et [P

+C(i) (AM + KM)] u ridtv(pii), df2 (88) M(i) ^mv = ⁽ⁱ⁾ 1 ^ii.(,) P(i) VK(i) (P'(tif)m (in

Ak k '11

Since the coupling matrices relate nodal vectors which may have different sizes, they are rectangular and may be not inversible, setting a restriction in the allowable causal- ities. For instance, from Fig. 6 and Eqs. (81) and (82) it can be seen that the input variables to the ports of the modulated transformer must be a and V(i), while the out- put variables result correspondingly the nodal forces and the nodal entropy rates for the ith-component.

Finally, initial conditions are needed for the nodal vec- tors of state variables. If initial conditions are given as continuous functions, these nodal vectors are determined in such a way that the ith-component total mass and mo- mentum, as well as the total entropy, is kept constant after the cliscretization.

VI. DIFFUSION APPROXIMATION A. Introduction and some definitions

It is common in the literature [9] to describe the dynam- ics of a multicomponent solution in terms of the average (center of mass) velocity of the mixture and the mass flux of each component relative to the average velocity. These relative mass fluxes are modeled using diffusion theory.

The diffusive rnass fluxes consist of different contribu- tions associated with the driving forces (mechanical or ther- mal) existing in the system [10i.

In ordinary diffusion, the mass flux depends in a com- plicated way on the concentration gradients of the compo- nents present; in most of the problems, this is the most important contribution.

The pressure diffusion indicates that there may be a dif- ferential net movement of a component in the mixture if there is a pressure gradient imposed to the system; this ef- fect is important in centrifuge separation, in which tremen- dous pressure gradients are established.

The forced diffusion appears when the components are under different external forces, as in the case of ionic sys-

tems in presence of electric fields.

Finally, the thermal diffusion describes the tendency for the components to separate under the influence of a tem- perature gradient. Although this effect is small, it can be enhanced by producing very steep temperature gradients.

Before presenting the diffusion approximation, some def- initions of parameters associated to the multicomponent solutions are introduced. The density of the mixture p is defined as:

m(i) .0 m(i) T v(i)

(81) (82) (8.3) (84) (85) (86)

(89)

Fig. 8. Power coupling between the WI-component velocity and illaRA

ports.

— (i)

P — P (90)

frk, rfi-3 fi

(19 ^{k 5} ' 4.,,A

Other quantities associated with the mixture can be de- fined, in order to regard the motion of the solution as a single body. For the case of additive functions, the follow- ing definitions are adopted:

PG = EP(i)di) ; P4'=EP(i) i=i

r = ^; q = q(i) (92)

i=1 1=1

where G is the body force, tD is the heat power source per unit mass, T is the stress and g is the heat flux corre- sponding to the mixture.

The average (center of mass) velocity of the mixture V is defined as:

v — E p(i) P i=i

(93) The velocity of the ith-component can be expressed as:

V(i) = V + v(i) (94)

where v(i) is the ith-component velocity deviation with respect to the mean velocity. The relative mass flux corre- sponding to the ith-component J(i) is defmed as:

j(0 )9(0 v(0 From the definition, it is verified that:

e2

j(19 Em(i) mu) vii) x(i) mu) v(i) 1 vp

pRO .1=1 MU) p

(100)

_2 r

JO) \--"‘ MO) mU) D(ii) x(i) MU) (GU) pRO 4—■

i=1

(101) j(i) _D(i) V °

T T ⁽¹⁰²⁾

In these equations, R is the gas constant, /1)(i) and v(i) are respectively the pa.rtial molal Gibbs free energy and volume, /1/(i) is the molecular weight and x(i) is the mole fraction for the ith-component. The mole fraction is calcu- lated as:

C (0

X" — —

c ⁽¹⁰³⁾

where c(i) is the molar concentration for the ith- compone,nt and c is the total molar concentration, defined as:

(104) (95) c (0 m(i) ; c P(i) = E co)

i=i

EJ(^') = o

i=1

The D(q) are multicomponent diffusion coefficients and the D(i) are thermal diffusion coefficients, with the follow- (96)

ing properties:

In the diffusion approximation, it is assumed that the rel- ative fluxes can be expressed as a function of the thermo- dynamic state of the system, this is:

( i) = ( i) ⁽s v p (1) p(r)

(97)

V") = 0 (105)

E op) mu) Doi) _ mo) 11/1(k) D(ik)) = 0 (106) (107)

E Do) _T = o

B. Kinetic Coenergy per Unit Volume

The main difference between the multivelocity and the diffusion model is the way the kinetic coenergy of the components is taken into account. Since the velocity devi- (98) ations are functions of the thermodynamic state, the set of state variables corresponding to the inertial ports reduces to the mean velocity. Taking into account Eqs. (94) to (96), the kinetic energy per unit volume can be written as:

= (V , 8 t, p(1), p(r)) = p ^{v2 4.} It ^j(i)2

) 2 (99)

i=i 2 P(i) (108) The following potentials are defined:

•

J(4) = +4,) + +4)

The formulas for these flux contributions are [14 j(i) C2 M(i) MU) D(ii) yrU)

C pRO

0.0))

vx(k) x (a),(k)

k.i 0, P,

k#i

This functional dependence allows to deal with ordinary (concentration driven) diffusion, pressure diffusion, forced diffusion (with forces dependent on the thermodynamic state) and thermal diffusion; the corresponding mass films are J(ci.) , J (19 , J g) and J , resulting:

y(i)2

v _ r(i) sv _ iu(i) U)

2 p(i) p(i)

f=1

+TT; Jt) + V.VP + VV j(02

( etZ,`

+ e(4. = r ) flu _oi)

( at: ) Pv — p V

8., p(1)

ju) aju) c at; \

v,p(i) pu) • as.

where:

tc= v- 1 (112)

ei) j(i)2 JU) op)

2 P (02 2-• P ) U a P(s) •

From the definition, the potential e(i) is the contribution of the diffusion fluxes to the ith-component kinetic coen- ergy per unit mass, while C can be regarded as a linkage between the mechanical an thermal ports. Notice that pu results the linear momentum of the mixture. The time derivative of the kinetic coenergy per unit volume can be written as:

et; ay as„ r

at C -Ft + E ^(it + 0)) aP(i)

at = -

• at

i=i c. Total Energy per Unit Volume

The total energy per unit volume e* includes the internal energy and the kinetic coenergy:

e* = ut, + (115)

The representation of the internal energy is the same as in Section II-A. The time derivative of the total energy per unit volume can be written as:

av es r

= Pv.-0-F + (0 + ()--a-tt z (,_t=i ^{u(i) 6(1 8P(i)} _at

(116) The potentials defined for the diffusion approximation also satisfy both constitutive and Maxwell relations.

D. Diffusion Balance Equations

Starting from the conservation equations for each compo- nent, the mass, momentum and energy conservation equa- tions corresponding to the mixture can be expressed as:

8 p = (P

e V = t. V. (p( V(i) V(i) at _{V (p V} V)

—pV.VV —VP +Z(V.-r(i) p(i) di)) (118)

aUv = [-V. (//1,i) V(i)) C(i) IA° - P(i) V.V(i) t=1. 2

–f (i) .170) + : V V(i) +p(i) (I)(i) – V.q(`)] (119) It can be seen that the non-linear terms appearing in Eqs. (118) and (119) give raise to additional terms in the conservation equations, when compared to the ones corre- sponding to a single component. As a consequence, the conservation equations for the mixture are not the same as the equations for a single continuum. This conclusion disagrees with many textbooks 15] [10j.

Taking into account the conservation equations pre- sented in Section III, the mixture conservations equations, the constitutive relations and the diffusion approximation, the balance equations result:

ap(i)

(11(i) K 6(i)) = -V. KAM (p(i) V + J(4))]

+ (p(i) V + J(i)) .V + (p(i) V + J(i)) .V p.(i) +C(i) (ti(i) + K 0)) - 6(i) V. (p(i) V + J(i)) (120)

OV

pi,. = ^V. [r. v — ^{E (J(0 j(i)]}

i=i P

– V .V P – T : VV - Ep(i) V.VK + p G.V (121)

+ () — =V .1-0 s, V – es, q [-1 T(I) j(i)

(i) — i=1 P

– [(p(i) V + ./(i)) .C440) – 6(i) V. (p(i) V + .0)) i=i

C(i) (0) E di)..1(4) p (I) (122) i=-.1

As in the multivelocity model, the balance equations for the diffusion model represent the power structure of the system. Concerning the constitutive and closure laws needed to model a multicomponent solution, it can be ob- served that in the diffusion model it is not necessary to know the momentum interaction terms between compo- nents. It is aliso interesting to notice the way ale diver- gence and source terms split among the different ports.

For instance, the power term corresponding to the stress state splits in two: a term considering the total stress and the mean velocity (influencing the velocity port) and power (113)

(114)

C ^{; ,rt}

L

uto = L.(:{; p(i) .14 cv) _{(e') de'}

On

[ (01)

= E ^8;,v, + ^{p(i) cv} 0, dlY + ^p(i) R In 2(itcp

t'

(124) (123)

= (uvo

Po . c(i) ^v (0') dO)

(i) 9

0

p(i) cti) [s(vid fe ^{Cti) (01}

d01

+R ln (II)) —

pC7) (126)

p(i) p FP) cl,i) p(i) cti))

where the total pressure results:

P = ^{(Ê p(0)} RO

i=3

From Eqs. (125) and (127), it can be seen that the con- tributions to the total temperature and pressure from the ith-component are weighted by the product of the density and the specific heat.

p(k) elk)) Po 00 k.1

(127) terms considering the stress state and the velocity devia- Lions for the different components (influencing the entropy port). A similar behavior can be found for the power term corresponding to the body force and the kinetic coenergy.

The discretization of the balance equations a,nd the re- sulting Bond Graph for the diffusion model is not shown here, but it is performed in a similar way as in the multi- velocity model.

VII. MIXTURE OF IDEAL GASES

As an application example, the potentials defined in Section II-A are calculated for a multicomponent solution (mbcture) of ideal gases. The entropic representation of the internal energy for an ideal gas is [61:

where R is a universal constant, ^po and ⁰⁰ are cor- respondingly the density and temperature at a reference state, tcy(i( and sy(it.; are the internal energy per unit vol- ume and entropy per unit volume for the ith-component at the reference state and cli) is the ith-component specific heat at constant volume (function of temperature only).

Eqs. (123) and (124) are a representation of Eq. (5) in parametric form, being the parameter the temperature O.

From this representation, the potentials result:

ir(i) 0 p(i) cti)

(t PU) eV))

.1=1

VIII. CONCLUSIONS

In this paper, the BG-CFD methodology was extended to Multicomponent Solutions. A multivelocity and a diffu- sion model were presented. Based on the total energy per unit volume, the BG-CFD methodology allowed to define a set of independent variables, potentials and constitutive re- lations needed to de,scribe a multicomponent system. The state equations were obtained by systematically integrat- ing a set of power balance equations. The resulting Bond Graph represents the power structure of the system, show- ing energy storage, power interchange through the bound- ary conditions, power source:3 and power couplings between the different ports. The author believes that the BG-CFD methodology is the foundation of a bridge between Bond Graphs and Computational Fluid Dynamics, two fields that have been following almost separate paths until now. It is hoped that the fmdings of this paper encourage other re- searchers to use this formalism in other problems.

ACKNOWLEDGMENTS

The author wishes to thank Conselho Na.cional de De- senvolvimento Cientifico e Tecnologico (CNPq, Brazil) for the financial support as Visiting Researcher at IPEN. The finantial support of Fundação de Amparo A Pesquisa do Estado de Si° Paulo (FAPESP, Brazil) in the travelling expenses for this Meeting is deeply acknowledged.

REFERENCES

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