NASA Contractor Report 4612

On the Various Forms of the Energy Equationfor a Dilute, Monatomic Mixture of NonreactingGases

Christopher A. Ken nedy

(NASA-CR-461Z) ON THE VARIOUS

FORMS OF THE ENERGY EQUATION FOR A

DILUTE, MONATOMIC MIXTURE OF

NONREACTING GASES Final Report

(California Univ.) 19 P

N94-35864

Uncl as

HI/02 0016064

Grant NAGI-1193

Prepared for Langley Research Center

July 1994

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NASA Contractor Report 4612

On the Various Forms of the Energy Equationfor a Dilute, Monatomic Mixture of NonreactingGases

Christopher A. Kennedy

University of California at San Diego • La Jolla, California

National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001

Prepared for Langley Research Centerunder Grant NAGl-1193

July 1994

ON THE VARIOUS FORMS OF THE ENERGY EQUATIONFOR A DILUTE, MONATOMIC MIXTURE

OF NONREACTING GASES

Christopher A. Kennedy

Department of Applied Mechanics and Engineering Sciences

University of California, San Diego, La Jolla, California 92093

Introduction

Choosing the appropriate form of the governing equation set to solve in

fluid dynamics is the first step in the solution of a particular problem. In

the case of gas mixtures, the governing equations become rather formidable

and a complete listing of the equations in their various forms and methods

to evaluate the transport coefficients is difficult to find. This paper seeks to

compile common as well as less well known results in a single document.

Forms for the various quantities involved in multicomponent hydrodynam-

ics may be derived either phenomenologically 6, is, 10 or by methods of kinetic

theoryal, 11. Ideally, the full limitations of the derivation need to be stated a

priori. For this reason, we choose the kinetic theory approach. In addition,

kinetic theory allows us to explicitly calculate the relevant transport coeffi-

cients. This paper will focus on the issue of complications to the governing

equations because of multiple species but will not concern itself with issue of in-

ternal degrees of freedom of the molecules leading to relaxation phenomena a4

or antisymmetric stress 2r' 24 exothermic heat release that may cause high-

energy "tails" to the distribution function of particle velocities 2° and differing

temperatures of certain species 23, or the effects of ternary, quartenary, etc.

collisions found in dense gases 14 and liquids. Having made those severe restric-

tions, we also limit discussion to small perturbations from equilibrium where

one may safely assume that the first nonequilibrium expression derived from

the Chapman-Enskog-Burnett a method of approximating the solution to the

Boltzmann equation is valid. In terms of classical irreversible thermodynamics,

"local" equilibrium prevails. This means that although the flow is not in equi-

librium, locally, equilibrium thermodynamic variables are still defined. Results

using the Chapman-Enskog-Burnett method are not valid for severe spatial

gradients and for sufficiently high frequency phenomena. This amounts to no

variation in quantities whose characteristic lengths and times are of the order

of the mean free path or the time between collisions. Collision times are also

presumed to be much smaller than the time between collisions. In cases where

more severe phenomena occur, local equilibrium does not exist and unique def-

initions for the nonequilibrium temperature and entropy are not forthcoming 2.

This is the topic of "Extended Irreversible Thermodynamics 26, 12,, Techniques

such as Grad's moment method 9, which allow for terms such as the heat flux

vector, stress tensor, and species flux to become independent variables, are

likely to lead to more accurate approximations to the Boltzmann equation in

situations far from equilibrium 29. Although the governing equations derived

by Grad's method are more complicated than those traditionally confronted,

they are likely to be essential in describing the finer points of polyatomic and

reacting dilute gases a3 as well as rarefied gases as where the Knudsen number

may no longer be considered a small expansion parameter.

It is the goal of this paper to summarize the various relationships between

equations describing conservation of energy for a dilute, monatomic, nonreact-

ing gas in local equilibrium. The gas is treated as nonrelativistic, not subject

to magnetic or electric fields, or radiative effects.

Energy Equations

Using the first nonequilibrium approximation according to the Chapman-

Enskog-Burnett theory in the kinetic theory of gases TM s (the Navier-Stokes

level),

nO3 nC$

q_ = p_(h_Y_Q_)-)_0vor-p__,Dr_d_ (1)i=1 i=1

_cs

= p__,(hiYiVii,_)+q_(red) (2)i=1

fLC$

_ _ M DTV_(ln T)= -- Dij dj_ - (3)j=l

p__ nC8d,_ = V_X, + (Xi- Y/)V_(ln p) + [fi_ - _, Yjfj_,] (4)

j=l

aZ_ = -pS_ z + A.05_ + 2ttD_ z (5)

where i and j are species indicies, a and /3 are spatial indicies, ncs is the

number of chemical species, q_ is the heat flux vector, q_(red) is the reduced

heat flux vector, I)i_ is the diffusion velocity, _r_ is the stress tensor, dic, is the

diffusion driving force vector, D T is the thermal diffusion coefficient, Di M is the

multicomponent diffusion coefficient, )_0 is the partial thermal conductivity, ,_,

is the second coefficient of viscosity, # is the shear viscosity, Xi is the mole

fraction, Y, is the mass fraction, hi is the partial enthalpy per unit mass, p

is the pressure, T is the temperature, p is the density, fi_ is the body force

per unit mass, 0 is the dilatation, 5_ z is the kronicker delta, and D_Z is therate of deformation tensor. Several comments about these terms are in order.

There are two different definitions in the literature for the multicomponent

diffusion and thermal diffusion coefficients, those of Waldmann a° and those

of Curtiss et. al. 4' 16. It is essential that readers be aware of which one they

are confronting at any given moment. We chose those of Waldmann for their

simplicity22, s and their straightforward correspondence to coefficients found in

linear irreversible thermodynamics 11' is. For the second coefficient of viscosity,

2/,Stokes hypothesis holds rigorously and we may use/_R = ()_, + 5 ) = 0, where

#B is the bulk viscosity,. Note that the partial thermal conductivity is not the

conductivity that is measured in experiments because thermal diffusion effects

can not be completely isolated. We also note that all diffusion velocities and

diffusion driving force vectors are not independent but are related through

72C8 rig8

_-_Y,._=O ; _-_di,_=O (6)i=1 i=1

By defining the velocity gradient tensor as

L_ = V_u_, (7)

the rate of deformation tensor and its deviator are given by

D_O= _(L_ 9+L9_ ) ; D_o=D_a- 05_ (8)

where uo is the barycentric, hydrodynamic, or center-of-mass velocity of the

gas. It will also be useful to write out the equations of state (9), species

continuity (10), overall continuity (11), and momentum conservation (12) for

future reference;

p-_- + V_. (pY//l_) = dai (10)

Dp

D--7+ pO = 0 (11)no8

Du_ _ VZ. (_r_) + p__, Yifi_ (12)P Dt _=1

where the reaction rate, d;i, is presumed to vanish and n and k are the number

of atoms per unit volume and Boltzmann's constant. The conservative forms

of equations (10) and (12) may be obtained by simply adding zero to both

equations in the form of Eq. (11) times }4 and u_, respectively,. Use of the

following relations has been made in order to transform the equation of state;_C8 nC8

n = _i=1 hi, Pi = nirni, p = F.i=l fli, where rai being the mass per atom,Ro RO ,Yi = p-o, Xi = n,n, R = W' = kl\a, Wi = Natal, where l_i, W, R °, N_ are

the atomic weight of species i, the average atomic weight of the mixture, the

universal gas constant, and Avogadro's number. Clearly _i_] Xi = _i_l Yi =

1.

The mole and mass fractions may be related by,

L XiW;WI- = (13)

(Ek=l XkWk)

and the average molecular weight, W, is given by

nc$

w- = (XkWk)nc8

k=l Wk k=l

These may be related through

(14)

Equations for the heat flux vector and diffusion vector appear in many

different, but equivalent forms. Beginning with the expression 2s' 17, 16

ncs XiXj .DME -B-g-[ " - D_] = 6;,- Y/ (16)j----1 u o

where D B is the binary diffusion coefficient, we may rewrite the diffusion'3

velocity equation (3) as the Stefan-Maxwell equation (17) using X:i_] kT = 0

and ncs_i=1 dic_ : 0 as

j=l --ij

= d,_ + kTV_(ln T) (17)

nc8

= dis -t- E XiXjaTv_( In T) (18)j=l

where the thermal diffusion ratios, k/, and antisymmetrical thermal diffusion

factors, c_iT, are defined by

nc8

E M T D T .Dij kj =j=l

nc8

kr = E XiXja_ (19)j=l

or, equivalently, using (16),

.c_ XiXj T DT- DT (20)k T=_-_iB (Di -D T) ; a T- D B

j=l

Because there is no explicit mole fraction dependence in the thermal diffusion

factors, they are sometimes the preferred term to describe thermal diffusion

effects. There are [ncs(ncs + 1)/2] independent multicomponent diffusion co-

etticients and thermal (ncs - 1) independent thermal diffusion coefficients, ascan be seen in the relations

_C8 nC8

EED, =o; ESDT =o (21)i=l i=1

XiX

Each of the terms, DiM, DiS, and Rii = _ are symmetrical, the last

term being referred to as the impedance matrix. In the special case of a

w,_ 5 (15)W X_

binary mixture, the multicomponent diffusion coefficients are related to the

binary diffusion coefficient as

D_- XIX2DM _ XIX2DM _ XlX2 M (22)YI Y2 Y22 Y12 D22

Solving the Stefan-Maxwell equation (17) for dis and then substituting the

result into the original expression for the heat flux vector (1) gives

i=1 i=1

,_c,,_, (XiXjD_(_ _ 1)/_)) (23)- pEEi=1 j=l _k ij

An experimenter measuring thermal conductivity is likely to measure it

when all diffusion velocities vanish at steady state. For this reason the thermal

conductivity is defined in terms of the partial thermal conductivity as

p nO8

A=A0- _kTD T (24)i=1

so that in the absence of a diffusion velocity, the coefficient in front of V'_T is

the value which is measured. With this we may write four equivalent forms of

the heat flux vector (1),

'_ (-_nikT T)q_, = -AV_T + y_ 5 + pk (/ii_ (25)i----1

rico

= -aVoT+E (ph,v,+ pk ) (26)i=l

= -AV_T +pE Xi + k Qio (27)i=1

'_ '_" '_ (XiXjD_ (_ _ _/_)) (28)=-AV"T+pEhiYi_"_-PEE_ Dgi=1 i=1 j=l

where }kT is the enthalpy per atom, or with the use of the equation of state,

sp and hi is the partial enthalpy per unit mass. The relation between the2n _

thermal and partial thermal conductivities may also be written in terms of thethermal diffusion coefficients instead of the thermal diffusion ratios as

A = A0-_EE D/_ (D T-D r ) (20)i=1 j=l

ncs ncs XiXj 2

= A0- _-_.= J=' Dg

The energy equation may be found in the literature in many forms. It mayderived as a' IS, 10

5

TOTAL ENERGY EQUATION (eo)

nc8

Deo =_V .q_+Vz.(crZ .u_)+py_yifi .(Q_+u_ ) (31)P D---T _=1

Dotting the momentum equation (12) with the center-of-mass velocity, u_,

yields

MECHANICAL ENERGY EQUATION (½u_u_)

_C8

D (_) _ u_Vz. (a;3_) + p E Y_f'_ .u, (32)P Dt i=1

and upon subtracting (32) from (31), we get

P-5-(- p Dt - pDt (33)

or

INTERNAL ENERGY EQUATION (e)

nc8

De_ V .q.+az.:L.o+p__yif, ._ (34)P Dt i=a

With the thermodynamic relation h = e + °a, one may obtain

Dh _ De + 1_Dp_ p__Dp (35)Dt Dt p Dt p2 Dt

and hence, with the aid of the continuity equation (11),

ENTHALPY EQUATION (h)

Dh

P Dt

p rl, C8

i=1

Adding the mechanical energy equation to this gives

DhoP---D-t = P--D-[ + P Dt

or

TOTAL ENTHALPY EQUATION (h0)

(36)

(37)

6

D ho Op ,_c,P Dt --V_.q_+V_.(r_,_.u_)+-_+py_Yifi_.(_+u_) (38)

i=1

The viscous stress tensor, r_, is given by

o

rz_ = A,05_Z + 2#D,_ = #B05_,Z + 2/t D,Z (39)

or

a_ =-pS_z + r_ (40)

and with it, we may define the viscous dissipation function, _, as

(I) = r,_ :L_ = A,02 + 2#D_ :D_ (41)o o

= #B02 + 2/t D_t_:D_t_ (42)

By considering h = h(Yk,T,p) and e = e(Yk, T,p)in flows where local

equilibrium exists, we may write the total differentials of the enthalpy and

internal energy as 32

dh = __,k=l T,p

de =k=l T,O

dYk + _ Yk,P Yk,T

dYk + -_ Yk,P rk,T

OhThese may be simplified using (gf)Yk,p - Cp, the isobaric heat capacity,

Oh

(_T)yk, ° = Cv, the isochoric heat capacity, (b--_k)r,, = hk, the partial enthalpy

( 0e ) = hk _ where x is coefficient of isothermalper unit mass, _ T,p pYk_'

compressibility and a is coefficient of thermal expansion at

{Oh\ (1-oT) [Oe'_ 1 [ _el). constant pressure,_-_P)Yk,T _--- O ' _'_P)Yk,T = _ _P -- In addition, Cp - C. = TT_-_. For a

1 and x = 1, reducing equations (43) and (44) toperfect gas, a =t

ncs

dh = __, hi dY_ + Cv dT (45)i=1

he8 I pX_ _de = _ hi dYi + C,, dT (46)

i=1 P]/// ]

From equations (45), (46) and (35) we may write

Dh DT '_( DYi_p--_-[ = pCv--ff_- + P_I= ',h_ Dt ]

p-_ = pC_--_ + p y_ hii=l pY,. ] Dt J

Dh De Dp

P-5-i= P-5-i+ -5-i+ ppo

(47)

(48)

(49)

which enablesuse to derive

Dp - po+pE (pX, (50)DT

Dt i=1 \ PYi Dt + pR-_

With equations (36) and (47), one form of the temperature equation appearsaS 21

DT Dp

PCp Dt - V, . q_ + r_o : L_ + D---(nC8 nC3

+ PY_ (h. DY_\ ' Dt J +P_Y'f'_ 9,_ (51)i=1 i=1

By eliminating the material derivative of the pressure from equation (51)

with equation (50) we have

TEMPERATURE EQUATION

_C8

pC_ DT - -V_ . qo + a_ : L_ + p _ Y_fi_ . _._Dt _=1

_C8 I D Y_ ]

+ pE [(pX, h, (52)i=1 [ \ pYi Dt J

Dividing equation (50) by the perfect gas relation RU= = (7 - 1) results in

DT pO p_p_ _ (pXi DYi) 1 Dp (53)flCv---_ - (__ 1) (7 1) i=1 _pYi Dt + (7- 1) Dt

and with this we find

PRESSURE EQUATION

1 Dp _ _ pO _c_

(7- 1) Dt (7- 1) - V_.q_ +'rz_: LoZ+ p_,Yifi_" _i=1

'_ (h DYi_ p _ (pXi DY,_ (54)- P_-_\ i--_-] + (7-1) \pY_ D-t]

i=1 i=1

The expression for the time rate of change of entropy per unit mass may

be written as

Ds- V_.J:+a s. (55)P Dt

where J_ is the flux of entropy and is written in the context of linear irreversible

thermodynamics10, 8, is, with the use of #i = hi - Tsi, as

J_ =T

nc$

= p Y_ siYi_ + %(red) (56)i=1 T

8

where #i is the chemical potential per unit mass and si is the partial entropy

p'er unit mass. Therefore,

nc_

J: = pZ s,y,_ _o p ncsT rot - T E O_d,o (57)i=li=1

and

V_ • (ncs) = V_. p_-_siYiQ_ -4- (V_T).(V_T) - 1TV_.(AOV.T)i=l

nC$ _C8

+ _-_:(V_T). (_-_ DTd,_) - 1_V_ . (p E DT d'.) (58)i=1 i=1

= V_- p_s,YiQ. + V_- (59)i=1

Equations (58) and (59) represent the reversible change of entropy. The ir-

reversible component of the change in entropy is derived by noting that the

entropy generation is of the bilinear form

a s = 'y_ JX

= _ LXX (60)

where J = LX, X, and L are the respective thermodynamic fluxes, thermo-

dynamic forces, and Onsager coefficients. Summation is considered over all

forces and fluxes of the same tensorial order. The four thermodynamic forces

are the diffusion force (X!_)), the thermal force (X_)), and the scalar (X (_°))cy(v2)

and tensor viscous forces t,,_Z ) given by 7

Xlg ) - P (61)pTy:d'o

X_ ) 1= -vV_T (62)

1 (63)X(_O) _ TO

X(V2) _ 1 f)o,_' (64)a_ -- -- T

Similarly, the six Onsager coefficients are given by

L(e,a) 1 M (65)ij = P2TYiYjDij

L} i'd)= L} e't) = pTYin T (66)

L (t't) = .XoT 2 (67)

L (_°'_°) = FtsT (68)

L (':'_2) = 2ttT (69)

and hence

where

j(,)j(,,o)

nc8

L(d'd)xd L(t'd)Xt PYi_= _]-/j _+-, --a=

j=l #?L(,_,OX d= L(t't)Xt 'l.I'-i ia = qa(red)

= L(vo,vo)x vd''= -#BOo

T (v2,v2)yv2.... a_ = -21_ Daz

(70)

(71)

(72)

(73)

.(v2) = (74)J(V°)6a_ + "a_ -v_

The irreversible generation of entropy, a 8, is then given by

rtC8 TIC8 nC8 nC$

_" = _ _ --.L!d'd)X!d)--,a• --,ax(d)+ _ -/L("_)X(')--a•x!da)+ _ -,L!d")X!d)"--,a X_)i=1 j=l i=1 i=1

• r.(v2,_2)y(v2) y(v2) (75)+ L(t't)x(a t) X(at) + L(_°'_°)X(v°)X(V°) + "_ _'_ : _"a_nc$

• • "w("2)X(% 2) (76)= E J!2 x!2 +J_) x_)+ J(v°)x(°°)+.an :i=1

or

a" = TA--_(VaT) • (VAT) +

+ __ (d_a.d_.)O/_ + (Valnr).(_O_rd_.) (77)i=1 j=l i=1

= _ - (VAT). (qa(red)) "4- vf_a: La_ - p_-_(Qa " d/a) (78)i=l

Because the second law of thermodynamics requires that entropy must increase

until it is maximized at equilibrium, it is important to verify that a. > 0. By

manipulating the Stefan-Maxwell equation (17), using (20), and the definition

of the impedance matrix, we may write the diffusion driving force vector as

nCS nCS

d,a = )-_[R/j(Vja - Qa)]- Y'_,_,[R/j(D T - DT)]Va(ln T) (79)j----1 j=l

and taking the reduced form of the heat flux vector (28), we have

_1C8 nO8

qa(_¢d) = -AVaT - pE Y_ [DTRij(_a - Qa)] (80)i=1 j=l

Upon substitution of (79) and (80) into (78), all occurances of the thermal

diffusion coefficients cancel, leaving

2# o o

a' - T2 (VAT)-(VAT) + --_--(D_:Oaz)

p _CS _C$

- E - (Sl)'= j=l

10

wherethe vanishingbulk viscosityhasbeendropped,and may be rewritten as

0 -8

A 2# ° °- T2(VaT). (VAT)+ -_--(D_z:De`;_ )

p _c__ [X_Xj _ ]i=1 j=l

(82)

It is now clear that entropy production is never negative because A,/_ and Dg _>

0 and all other terms occur in quadratic form (thermal diffusion coefficients

may be positive or negative).

Putting the irreversible entropy generation (77) and the entropy flux (58)

into (55) yields

ENTROPY EQUATION

,_c8 2 02+ TVe`. (p}2s,v,G) = -g, + 2_(De`_: D_)i=1

no8

+ V_.(AoVe`T)+V_.(p__,Dr_&_)i=1

nC8 nO8 tiC8

+ p(V_lnT).(y_D_dio)+p_-_}-_ [(dio.dj_)Dia]]i=1 i=1 j=l

no8

= rfi_: Le`fi - _'e` "qcqred)- PE(Vie`" die,)

i=1

(83)

(84)

or

_cs

+ Tve`. (pE _,v,.G)i=1

D(_ nCS IIC$

= pO + p-_[ + E [(pYiifi_ - pdi_)" _,] + V_. (py_ hiY/ld/o) (85)i=1 i=1

Each of the four fundamental transport coefficients appearing the equations

(It,)_0, D_, D/) may calculated to any desired accuracy, and in particular

those occuring in (16). Formally, the transport coefficients are calculated by

evaluating the following matrix expressions

11

r/(_ + 1) =

HOO HOl H02 H03 ... HO_ X T

H 10 H 11 H 12 H 13 ... H 1_ 0

H 20 H 21 H 22 H 23 ... H 2_ 0

H 30 H 31 H 32 H 33 ... H 3_ 0

: : : : ".. : :

H _° H _1 H _2 H _3 .. • H_ 0

X 0 0 0 ... 0 0

HOO H ol HO_ H 03 ... HO_

HlO Hll H12 H13 ... HI_

H20 H21 H22 H23 ... H2_

H3O H31 H32 H33 ... H3_

: : : : ".. :

H _° H _1 H _2 H _3 ... H_

-

L °° L °l L °2 L 03 ... L °_ 0

LlO Lll L12 L13 ... LI_ X T

L 2° L 21 L 22 L 23 ... L 25 0

L 30 L 31 L 32 L 33 ... L 3¢ 0

: : : : .. : :

L ¢° L _1 L _2 L _3 ... L_¢ 0

0 X 0 0 ... 0 0

LOO L 01 LO2 LO3 ... LO¢

LlO Lll L12 513 ... LI_

L2o L21 522 L23 ... L2_

L3o L31 L32 L33 ... L3_

: : : : ".. :

L_O L _ L_2 L¢3 ... L_¢

2TDT(_) = -5P

L °° L °1 L °2 L 03 ... L °¢ 0

L lo L n L 12 L13 ... LI_ X T

L 2° L 21 L 22 L 23 ... L 2¢ 0

L 30 L 31 L 32 L 33 ... L 3_ 0

: : : : ".. : :

L _0 L _1 L 52 L _3 ... L _5 0

_ 0 0 0 ... 0 0

LOO LOl LO2 Lo3 ... LO_

L 10 L n LO2 L13 ... L_

L20 L21 L22 L23 ... L2_

L30 L31 L32 L33 ... L3_

: : : : ".. :

L_O L_ L_2 L _3 ... L_

12

M 4T= - 25p

LOO L m LO2 LO3 ... LO_ ATLlO Lll L12 L13 ... LI_ iL 2° L 21 L 22 L 23 ... L 2_ 0

L 3° L 31 L 32 L 33 ... L 3_ 0

: : : : ".. : :

L _° L_1 L _2 L _3 • • • L _ 0

6i 0 0 0 ... 0 0

LOO LOl LO2 Lo3 ... Lo(

L lo L u Lo2 L13 ... LI_

L2o L21 L22 L23 ... L2_

L30 L31 L32 L33 ... L3_

: : : : ".. :

L _° L _1 L _2 L _3 ... L_

Formal expressions for the (ncs x ncs) matricies H pq (p,q < 2) and L pq

(p,q < 3) are given in the literature 1' 19, 13 The variable _ denotes the or-

der of the approximation and the vectors X, 6_, and Ai are given by X =

{Xl,X2,.. .,Xncs}, (_i = {_il,_i2,''',_incs}, /_i = {((_il-Yl),(_i2-Yl),...,(_incs-

Y,_c_)}. The superscript T denotes transpose. Note that the first approxima-

tion to both the partial thermal conductivity and thermal diffusion coefficients

vanish identically.

Acknowledgements

The author acknowledges the support of NASA Langley Research Center's

Theoretical Flow Physics Branch while in residence under NASA grant NAG-

1-1193. Assistance from the staff at the NASA Langley Library is appreciated.

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16

Form ApprovedREPORT DOCUMENTATION PAGE OMBNo. 0704-0188

Public reporting burdan fo_this colloction of Informalion _1uttmltlKI to iv_rsge 1 hour per rl_ponle, including the time for rewew_g knstruct_onl, s_vchlng exiling dais sources,gethedng and milnlatnklg the data needed, and completing and t'evlewir_ the cofllKltorlof informalk_l. S_rld oomrclents mgording 1hisburclan N,tin_tte Or any Other_ of th_coflectlorlof intomlatk011,including |uggGstionl for reducingth_"buKlan, to Washington Heado_orlan; Services, Dimolorale for Inforrratlon O¢_rlaionl and R_, 1215 Jetflnon DavisHighway, Suite 1204, Arlinglon.VA 2220_4302, and to the Office of Managetrt_nt and Budge(, PaperworkRiK_JctionPro_c't{0704-0188). Washington, DC 20503.

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July 1994 Contractor Report4. TITLE AND SUBTITLE 5. FUNDING NUMBERS

On the Various Forms of the Energy Equation for a Dilute, MonatomicMixture of Nonreacting Gases

6. AUTHOR(S)

Christopher A. Kennedy

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University of California at San DiegoDepartment of Applied Mechanics and Engineering SciencesLa Jolla, CA 92093

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National Aeronautics and Space AdministrationLangley Research CenterHampton, VA 23681-0001

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13. ABSTRACT (Maximum 200 words)

In the case of gas mixtures, the governing equations become rather formidable and a complete listing ofthe equations in their various forms and methods to evaluate the transport coefficients is difficult tofind. This paper seeks to compile common, as well as less well known, results in a single document.Various relationships between equations describing conservation of energy for a dilute, monatomic,nonreacting gas in local equilibrium are provided. The gas is treated as nonrelativistic, not subject tomagnetic or electric fields, or radiative effects.

14. SUBJECTTERMS

energy equations, linear irreversible thermodynamics, Navier-Stokes

equation, dilute gas, multicomponent diffusion, kinetic theory, perfect gas

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