Density profile for two-fluid single phase gas mixture under body force from uniformly mixed upstream conditio

Kao, T. W.

English

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NASA CONTRA ACTOR-. 
REPORT 
ON THE ESTABLISHMENT OF 
DENSITY PROFILE FOR 
THE FLOW OF A TWO-FLUID 
SINGLE PHASE GAS MIXTURE 
by Timothy W. K&o 
Prepared under Grant No. NsG-586 by 
CATHOLIC UNIVEKSITY OF AMERICA 
Washington, D. C. 
for Lewis Research Center 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION l WASHINGTON, D. C. l OCTOBER 1965 
https://ntrs.nasa.gov/search.jsp?R=19650026362 2020-03-16T22:04:35+00:00Z
- 
TECH’ LIBRARY KAFB, NM 
lInIllIIlllnllmIIl~IIIIHBIII 
bO=iqbbL 
NASA CR-330 
ON THE ESTABLISHMENT OF DENSITY PROFILE FOR THE FLOW 
OF A TWO-FLUID SINGLE PHASE GAS MIXTURE 
By Timothy W. Kao 
Distribution of this report is provided in the interest of 
infor mation exchange. Responsibility for the contents 
resides in the author or organization that prepared it. 
Prepared under Grant No. NsG-586 by 
CATHOLIC UNIVERSITY OF AMERICA 
Washington, D. C. 
for 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
For sole by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - Price $1.00 

FOREWORD 
The research described herein, which was conducted at the Catholic 
University of America, was performed under NASA Research Grant NsG-586 
with Mr. Robert G. Ragsdale, Nuclear Reactor Division, NASA-Lewis Re- 
search Center, as Technical Manager. 
iii 

ON THE ESTABLISHMENT OF DENSITY PROFILE FOR THE FLOW 
OF A TWO-FLUID SINGLE PHASE GAS MIXTURE 
by Timothy W. Kao 
Abstract. The establishment of density profile for a two-fluid single phase gas mixture 
under a body force from a uniformly mixed upstream condition is analysed. The flow is 
assumed to be two-dimensional and confined between two parallel walls. An inviscid 
h ydrodynami ca I mode I is adopted. A perturbation procedure is used to obtain a closed 
from zeroth order solution. The interplay between Fickian and baro-diffusion is brought 
out. The problem bears on cavity type gaceous nuclear reactor propulsion device where 
the critical concentration of the nuclear fuel in a fuel-propellant mixture is important. 
V 
(1) Introduction 
For many industrial purposes it is desirable to know the establishment of stratificatification 
for a two-fluid single phase fluid system in the presence of strong body forces. In particular, it is 
very often necessary to know the concentration of the heavier species at various points downstream 
of the ,inlet where the two fluids are uniformly mixed. This knowledge is needed, for example, in 
gaseous nuclear propulsion device where the heavier fluid is uranium, and the critical concentration 
for the onset of reaction is of paramount importance. In most problems of this nature the body force 
is usally a centrifugal force, and for high flow velocities the effect of viscosity is generally negligible. 
The dominant effect is one of mass diffusion. 
In this paper an inviscid, incompressible, hydrodynamical theory is proposed. Strictly 
speaking the thermodynamics of the system has to be considered together with the mechanical 
equations in order to obtain a complete set of equations (see for example Landau and Lifshitz (1)). 
However, when the change of density of the fluid mixture taken as a whole is assumed to be 
proportional to the change in concentration of the heavier fluid, a purely mechanical consideration 
suffices and thermodynamics can be left out of the analysis. This of course results in a major 
simplification of the problem. A perturbation scheme is then used to solve the problem, which is here 
considered as a two-dimensional flow in a horizontal duct with a body force in the vertical direction. 
2. The Governina Eauations 
The equation of continuity for the total mass of fluid is 
(1) 
(1) L. D. Landau and E. M. Lifshitz, Fluid Mechanics.Addisonj-Wesley Publishing Go., Inc. 
Reading, Mass. pp 219-227. 
where p is the total density of the fluid and 2 denotes the velocity. 
We note that velocity is here understood as the total momentum per unit mass of fluid, 
and the equations of motion are the Euler’s equations 
where 
t 
is the pressure, 
4 
$ 
is the body force and 
is the substantial derivative. 
(2) 
#.. z ( +k +zv) 
If we denote c to be the mass concentration of the heavier fluid, the equation ‘of 
continity for that species is 
DC bt =-v.;z 
-W 
where i is the mass flux of that species. 
The mass flux is made up of three parts 
I L = -DCVc + ( > P T VT 
where D is the diffusion coefficient or mass transfer coefficient 
%T is the thermal diffusion ratio 
5 
is the barodiffusion ratio 
and T is the absolute temperature. 
(3) 
(4) 
-CT + and are determined by thermodynamic properties alone. For the purpose of this analysis 
it can be shown (see Landau and Lifshitz (1))that 
-ct 
is negative. 
In the present analysis we shall assume a uniform temperature distribution so that 
where k z - k is positive. From the above equation we can conclude at once that equilibrium 
P 
is reached when the flux due to mass concentration is balanced by the pressure flux. 
Substitution of (5) into (3) yields 
(l), (2), and (6) are five equations for the six unknowns, c, 
P IPI 
;u’ 
system we assume 
( p-p*‘> = p cc - c9, 
(6) 
To complete the 
where P and C.* are reference quantities and 0 is a constant of propationality. 
We shall now formulate our problem in terms of a duct flow. The flow is assumed 
to be steady two-dimensional and bounded by two parallel walls at y = 0 and y = L as shown 
in Figure I. The body force $ is taken to be in the negative #- direction. If we denote 
b I v ) to be the components of v’ in the (x, y ) - directions, then for an incompressible 
flow the equation of continuity is 
au. 
ar 
+3x so . 
“;) 
(8) 
3 
The equations of motion are 
= -%, 
and (6) becomes 
where v=i&+;$ and 
Hence 
02) 
(13) 
The boundary conditions ore 
3. Solution of the Boundary Value Problem and Discussion 
The boundary value problem is now to be solved by a perturbation procedure, In 
general g may be assumed to be a function of x. Now D and k are known functions 
of P and p. We let D  = f; w  PL 
We now assume a perturbation series of the form 
u = U,l$ + 44, + fur + 1 . - 
v-= 0 + v-l 4 G* 4 . . . 
t = +bq> t.-p 3 fa +.a- 
e = pq) + p, -t y z  t  8 l # 
where terms of 1st and higher orders are << than the zeroth order quantities. It then follows that 
fl = J&p ++y, ++, + ij $4 +p;%ppJi + ‘a ’ 
Substitution ot the above into the equatioNand collecting zeroth order terms we have 
Utilization of the 1st two we then have 
Zcoax # 
(14) 
(17) 
The system consisting of the above equation together with (14) and (15) is the non-linear zeroth 
order equations. Bearing in mind that $, and 5, are really diffusion coefficients we could 
assume(as leading terms in an ordinary iterative procedure) $, and 4, to be constants and 
denote 4, by -k, , and 5, by k2. Hence we have 
5 
or 
The boundary conditions are now 
(18) 
The velocity u. is determined by the inlet velocity distribution. For any given U, at the 
inlet and g equation (18) together with the above boundary can be readily solved 4 finite 
differences. For the important case when UO is constant =A and g is constant an analytical 
solution is presented. 
We first cast the equations into non-dimensional form by using L as the reference 
length, A the reference velocity and 
c 
as the reference density, we have using the 
same symbols to denote now the corresponding dimensionless quantities, the following: 
where 
if we denote -&2 and (t3 x F-*) by y 
I 
we have 
(19) 
By separation of variables, the solution to the above problem is 
where are eigen-functions of the Sturm- 
Liouville system from separation of variables and are orthogonal with respect to the weight function 
2% in the interval (0,l). Q, are the Fourier coefficients given by 
= vl.Jr jf [(-I)“” eb/’ + \ -J / c (g,’ + m f , 
and 
a = li /(I - e9 . 
Note that eqn. (19) with its associated boundary conditions automatically ensures that 
J ‘jw/~l+f = 1 as it should be. This last result has been utilized to obtain a. 0 
From the solution it is immediately clear that as x -+ 00, p6 3 (Ye8d)/(\-e”>. 
It is also seen that the important parameter of the problem is the dimensionless number y. 
For y ----I 0, fi is constant throughout as to be expected. If y is very large then 
the heavier gas sinks to the bottom. For some physicaily realistic value there would of course 
be a balance between baro-diffusion and mass diffusion and the asymptotic form of /a 
above indicates the equilibrium distribution. It has an exponential behavior as one would expect. 
The approach to the equilibrium position is also exponential. It is seen that the series converges rather 
rapidly for all ~>a . 
7 
Figures (2) and (3) are two typical cases (for fixed o< , and zf ) of density profiles as a function 
of depth for various values of distance from the inlet, The exponential profile is reached 
asymptotically for larpx,Fig (4) shows for fixed & and ‘d the depth along the channel 
where an increase in density can be expected. For given values of oc and y, the above figures 
thus yields the intormation needed for determining the critical concentration for example. 
Figures (5) shows the depth of where density increase can be expected for various values of. y . 
It shows that as y increases, the depth of the region of increasing density decreases. 
Figures (6) shows the distance downstream from the inlet OS a function of y for the density profile 
to reach 80 per cent of its asymptotic value. The graphs are plotted for constant values of d . 
It exhibits a maximum of y approximately e8mL to 5. The curves goes to zero as y = 0. 
Figure (7) is a typical plot of the density as o function of depth with natural scales. 
Appendix 
Solution of the boundary value problem 
By separtion of variables we assume 
Equation (19) and boundary condition become 
From (Al) we have, on using the condition at X = d, 
x = A $-zaF-d-d)+ 
and from (A2),we have 
W  
(A3) 
where LO= 
J 
Substitution of (A3) into the boundary conditions following (A2) we obtain the secul& equation 
9 
or 
u=nfl, uL= 0, I, 2,3,. - l 
that is 
$n’ 4AZ-f = 
I 
4 > or L = my+ TF . 
A particular solution, not an ekgen solution, also comes from 
‘1( 2 
a2=- 7 l ) 
4x’--f T2 
or - L--, 4 4 > uv A--O 
For A=O, ? = earq , and g=a, a constant 
For 4: , hence X satisfies the 
condition at X = 4 . Indeed X + 0, OS z4oa. The corresponding e.lgen 
functions are then 
We note that the eigen functions ‘7, (1’ are orthogonel with respect to the weight function 
9”;s in the interval (0,l) and form a complete basis for expansion of L2 - functions. 
Thus we write 
At & =o, p.cx+j> =l, at x. -06, pdoq) -, a &‘6”6 
and from the conservation of mass we have 
10 
At r = 0, then 
The series converges by virture of the Sturm-Liouvi Ile Theorem. 
11 
Y’ 
/ / / / ////////////////////////////// 
E L 
b L w ///‘-//////////////////////////////// 
X 
Figure 1. - Definition sketch. 
- 
IC 
e . 
IC 
5 . 
P I 
c . i- 
0.1 ,- 
5- 
a=2.0 
Y= 5.0 
0 0.1 0.2 03 0.4 0.5 0.6 0.7 Cl8 0.9 I.0 I.1 1.2 1.3 I4 
Y 
Figure 2. - Variation of density with depth for different distances down- 
stream from the inlet Case for y = 5, a = 2. 
13 
0.0 I I I I I I I I I I I 
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 
1; lil 
1.2 1.3 I.4 
Y 
Figure 3. - Variation of density with depth for different distances down- 
stream from the inlet. Case for y = 15, a = 0.5. 
24
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Figure 7. - Natural scale of typical plot of variation of density with depth. 
18 NASA-LandeY, 1965 


On the establishment of density profile for the flow of a two-fluid single phase gas mixture

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