Lecture 2 Constitutive Relations

Lecture 2 Constitutive Relations Conservation equations Dv = @ + r v =0 @t rp + r + g DY i + r ( Y iv i )= i i =1, 2,...,N { De = pr v + r q Dh + r q Z Y i p = R h = Y i h o i + c p d o W i We already have expressions for the viscous stress tensor and the dissipation function. We need constitutive relations for the heat flux vector q, the di usion velocities V i and the rates of consumption/production i. University of Illinois at Urbana- Champaign 1

Heat flux vector he total energy flux q is the sum of the separate energy fluxes, including the fluxes due to di usion and the radiant flux q R, but neglecting the Dufour heat flux (due to di erence in di usion velocities) q = r + Y i h i V i + q R Dh N + +r ( r ) r X Y i V i h i the energy transfer is not just by conduction, but also through di usion fluxes of the di erent species Radiative heat transfer will also be removed and added only when necessary. Dh N + +r ( r ) r X Y i V i h i r q R h = Y i h o i + c p d o Dh N + +r ( r ) r X Y i V i h i D c p d + DY i ho r Y i i V i [h o i + c pi d ] o o r Y i V i h o i + r Y i V i c pi d o D c p d = Dp + +r ( r ) X N DY i + r Y iv i h o i r Y i V i c pi d o o i (I) D o c p d = Dp N + +r ( r ) r X Y i V i c pi d o i h o i University of Illinois at Urbana- Champaign 2

Alternative form of the energy equation h = h i Y i ) dh = dh i Y i + h i dy i = c pi Y i d + h i dy i = c p d + h i dy i ) Dh = c D p + N X h i DY i Dh N + +r ( r ) r X Y i V i h i D N + X h i DY i N + +r ( r ) r X Y i V i h i D N + +r ( r ) X Y i V i rh i N X h i apple DY i + r ( Y iv i ) i D N + +r ( r ) X Y i V i rh i N X h i i (II) h i = h o i + c pi d o ) rh i = c pi r D N + +r ( r ) X i Y i V i r h i i and it can be shown that the two versions are equivalent. Moreover, when all the c pi are the same, i.e., c pi = c p for all i, the energy equation reduces to D N + +r ( r ) X h o i i University of Illinois at Urbana- Champaign 3

Di usion Fick s law of binary di usion In a binary mixture Fick s law of di usion states that the di usion mass flux of a given species is proportional to its concentration gradient Y 1 V 1 = D 12 ry 1 Y 2 V 2 = D 21 ry 2 he constrains Y 1 V 1 + Y 2 V 2 = 0, and Y 1 + Y 2 = 1, imply that D 12 = D 21 D Y i V i = DrY i DY i + r ( Y iv i )= i i =1, 2 DY i + r ( DrY i)= i i =1, 2 For D = const. ) DY i + Dr2 Y i = i i =1, 2 For multicomponent di usion in gases (Curtiss & Hirschfelder 1949) X i X j rx i = (V j V i ) +(Y i D ij di erence in di usion velocities X i ) rp p pressure gradient rx i = X i X j + D ij Dj Y j Di Y i r temperature gradient Soret e ect X i X j D ij (V j V i ) Stefan-Maxwell equations + p Y i Y j (f i f j ) di erence in body force he e ects of pressure gradients, and temperature gradients (known as Soret e ect) on di usion are typically small. If, in addition, there are no significant unequal forces acting on the species (often relevant for ions and electrons where f is due to an electrical field), then where D ij are the binary di usivities, whichhavebeenproventobesymmetric, i.e., D ij = D ji. For a dilute ideal gas mixture they are independent of composition, and determined by a simple two-components experiment where a species in low concentration di uses through a second species that is in abundance. University of Illinois at Urbana- Champaign 4

For a binary mixture rx 1 = X 1X 2 D 12 (V 2 V 1 ) rx 2 = X 2X 1 D 21 (V 1 V 2 ) ) r(x 1 + X 2 ) 1 1 =(V 2 V 1 ) =0 X 1 X 2 D 12 D 21 D 12 = D 21 D and the Stefan-Maxwell equations reduce to Fick s law of di usion Y 1 V 1 = DrY 1 Y 2 V 2 = DrY 2 More generally, Fick s law is recovered if one assumes that the binary di usion coe cients are all equal, i.e., D ij = D. Y i V i = DrY i for i =1,...,N If the binary di usion coe cients are all equal, i.e.,d ij = D, we again cover Fick s law. rx i = X j (V j X j V j X i X j (V j V i )= X i D ij D V i )= X i D X i D V i D rx i X i = X j V j it can be shown, after some algebraic manipulations that V i P N X jv j = D(rW/W ) D rx i X i = D rw W V i and, since X i = WY i /W i ) rx i X i ryi D + rw Y i W = ry i Y i = D rw W + rw W V i Y i V i = DrY i University of Illinois at Urbana- Champaign 5

he use of the Stefan-Maxwell equations is quite complicated because the diffusion velocities V i are not expressed explicitly in terms of the concentration gradients. Generalized Fick equations he coe cients D ij, which don t have to be necessarily positive, satisfy D ij = D ji, for all i, j Y i D ij = 0 for all j V i = D ij rx j would be more convenient for the evaluation of the di usion term in the species equations, but the Fick di usivities D ij are not simply related to the binary di usivities D ij, and they are concentration-dependent. Note that there are many definitions for multicomponent di usivities; the one quoted here (with the D ij symmetric) follow Curtiss & Bird (1999). One can find the relation between the multi-component di usivities D ij and the binary di usivities D ij. For a binary mixture Stefan-Maxwell rx 1 = X 1X 2 D 12 (V 2 V 1 ) rx 2 = X 1X 2 (V 1 V 2 ) D 21 Multi-component di usion D 12 = D 21 V 1 = D 11 rx 1 + D 12 rx 2 V 2 = D 21 rx 1 + D 22 rx 2 D 12 = D 21 D 11 Y 1 + D 12 Y 2 =0 D 12 Y 1 + D 22 Y 2 =0 ) D 11 = D 22 = 2 Y 2 D 12 X 1 X 2 2 Y 1 D 12 X 1 X 2 D 12 = D 21 = Y 1Y 2 X 1 X 2 D 12 University of Illinois at Urbana- Champaign 6

Expressions for ternary and quaternary mixtures are available, but are more complicated. For example, for a ternary mixture D 11 = D 12 = (Y 2+Y 3) 2 X 1D 23 + Y2 2 X 2D 13 + Y3 2 X 3D 12 X 1 D 12D 13 + X2 D 12D 23 + X3 D 13D 23 Y 1(Y 2+Y 3) X 1D 23 + Y2(Y1+Y3) 2 Y 3 X 2D 13 X 3D 12 X 1 D 12D 13 + X2 D 12D 23 + X3 D 13D 23 with the additional entries generated by cyclic permutation of the indices. For the general case, one can write a relation between the matrix of Fick di usivities D ij and that of binary (Stefan-Maxwell) di usivities D ij ;seecurtiss& Bird Ind. Eng. Chem. Res. 38:2515-2522 (1999). We are interested in rewriting the generalized Fick equations in terms of the mass fraction V i = ˆD ij ry j j6=i which requires expressing rx i in terms of ry i, or using the di erential relation (Bird, Stewart & Lightfoot) rx i = W 2 apple 1 1 W i W + Y 1 i ry j. W j W i his leads, in general to complicated expressions for the ˆD ij. For a binary mixture, rx 1 = X 1X 2 Y 1 Y 2 ry 1 rx 2 = X 2X 1 Y 2 Y 1 ry 2 ˆD 11 = Y 2 Y 1 D ˆD22 = Y 1 Y 2 D ˆD12 = ˆD 21 = D Y 1 V 1 = DY 2 ry 1 + DY 1 ry 2 = D(Y 2 + Y 1 )ry 1 = DrY 1 and we recover Fick s law, as we should. University of Illinois at Urbana- Champaign 7

Dilute mixture here is an abundant species in the mixture, denoted by N, while all other species may be treated as trace species (for combustion in air, nitrogen is abundant) rx i = Y i 1 for i =1,,N 1, Y N 1 X i 1 for i =1,,N 1, X N 1 Y i V i =0, ) V N 1 W = X i W i W N X i X j D i,j (V j V i ) X iv i D i,n for i =1, 2,,N 1 ry i Y iv i D in for i =1, 2,,N 1 Y i V i = D i ry i for i =1, 2,,N 1 Fick s law may be used as an approximation with the di usion coe cient interpreted as the binary di usion coe cient with respect to the abundant species, i.e., D i = D in. he P equation for N is still complicated, but is not needed N 1 because Y N =1 Y i A useful approximation is the e ective binary di usivity for the di usion of species i in the mixture, obtained by assuming that V j 0 for all the species, except for V i 6= 0. rx i = Y i V i = D i,e ry i or X i V i = D i,e rx i X i X j D i,j (V j V i ) note, however, that this can be only used for N from the constrain P Y i V i = 0. X i V i D i,e X i V i ( X i D i,e + D i,e = 1 X i NP X j /D ij X i X j D i,j V i = X i V i ( X i D i,e + j6=i j6=i X j D ij ) j6=i X j D ij ) 1 species, with the last obtained University of Illinois at Urbana- Champaign 8

Introducing the di usion law Y i V i = D i ry i in the energy equation D o c p d = Dp N + +r ( r ) r X Y i V i X N c pi d i h o i o D c p d = Dp + +r ( r )+r X N Z D i ry i c pi d o {z o } energy transfer due to di usion fluxes caused by imbalances in the di usivities and specific heats. i h o i If D i = D for all i D c p d = Dp + +r ( r )+r D X N Z ry i c pi d o {z o } i h o i ry i Z c pi d = r Y i o = r Y i c pi d o Z c pi d o Y ir c pi d o z } { Y i c pi r = r Y i c pi d c p r = r c p d c p r o o D D Z c p d = Dp + +r ( r )+r D X N Z ry i c pi d o {z o } o c p d r " Dr o c p d + r D r c p d o # Dcp 1 r =0when D = / c p r = Dp + i h o i N X i h o i When all the c pi are also the same, c pi =c p =c p ( ) for all i, the energy equation reduces to D r ( r )= Dp + N X i h o i University of Illinois at Urbana- Champaign 9

Governing Equations with Fickian di usion (dilute mixture) and equal c pi @ + r v =0 @t Dv = rp + r µ[(rv)+(rv) ]+r(apple 2 3 µ)(r v)+ g DY i D r ( D i ry i )= i, i =1, 2,...,N 1 r ( r )= Dp + N X i h o i ransport Coe cients he bulk viscosity is negligible in combustion processes; apple 0. = µ[(rv)+(rv) ] 2 3 µ(r v)i = 1 2 µ @vi + @v 2 j 2 @x j @x i 3 µ(r v)2 Although the transport coe cients, µ in a multicomponent mixture depend in general on the concentrations, this dependence is generally ignored in theoretical studies and average quantities are used; they remain dependent on pressure and temperature. Similarly W is taken as constant, so that the equation of state reduces to p = R/W. University of Illinois at Urbana- Champaign 10

he dependence of the binary di usivities D i on pressure and temperature is D i /p 3/2 apple apple 2 ypical values at p = 1 atm are in the range 0.01 10 cm 2 /s. he kinematic viscosity µ/ has the same units and the same pressure and temperature dependence as the molecular di usivities µ/ /p 3/2 apple apple 2 ypical values at p = 1 atm are in the range 0.1 he thermal conductivity depends mostly on temperature, and behaves as /p with 1/2 apple apple 1. More relevant, however, is the thermal di usivity / which has the same units and the same temperature and pressure dependence as the molecular di usivities and the kinematic viscosity. In particular / /p 3/2 apple apple 2 ypical values at p = 1 atm are in the range 0.1 1cm 2 /s. 1cm 2 /s. Since / c p,µ/, D i have the same dependence on temperature, their ratios are nearly constant. / c p = Le i Lewis number D i µ/ / c p =Pr Prandtl number µ/ D i =Sc i Schmidt number We can write D i = Le 1 i ( /c p ) and µ =Pr( /c p )with = ( ). A good approximation (Smooke & Giovangigli, 1992) for the latter is =2.58 10 4 c p 0.7 298K g cm s he Prandtl and Lewis numbers are nearly constant; Pr = 0.75 and for common combustible mixtures Le varies in the range 0.2 1.8. University of Illinois at Urbana- Champaign 11

If the transport coe cients µ,, D i assume constant values (an average value, say) the equations simplify to Dv = t + v =0 p + µ 2 v + 1 3 ( v) + g DY i D D i r 2 Y i = i, i =1...N 1 r 2 = Dp + p = R/W N X i h o i there are N +6 variables (,p,,v,y i ) and N +5 equations, supplemented with the constrain P N Y i = 1. University of Illinois at Urbana- Champaign 12