Nomenclature
SUBSCRIPTS
-1, a
G L
= = = =
a‘ ~
il
DC -
= =
fG
= =
i?C
H Kc
= =
KL
=
k are unrestricted. Hence if the matrix is diagonal, the nondilute components have like Fickian diffusion coefficients while the dilute components may have unlike coefficients. If there are n dilute solutes in a solvent, all the n independent diffusion coefficients may be unlike. Under certain conditions a nondilute system with small but nonzero cross terms may be usefully approximated by neglecting the cross coefficients and using unequal diagonal coefficients. However, since Equation 4 will not be satisfied, the resulting diffusion equations will not be self-consistent.
\
,
H. L. TOOR K . R. ARNOLD Carnegie Institute of Technology Pittsburgh, P a .
RECEIVED for review November 12, 1964 ACCEPTED MAY10, 1965 Research supported by a grant from the Petroleum Research Fund administered by the American Chemical Society. Grateful acknowledgment is hereby made to the donors of the said fund.
COMMUNI CAT1ON
GIBBS’ LAW FOR ABLATING LIQUIDS Gibbs’ law for determining the concentration of a surface active agent in an ablating liquid is derived. The result is obtained by assuming a linear velocity and concentration profile in the ablating liquid film. These assumptions are valid only for liquids with small viscosities and therefore substances such as glasses, polymers, etc., are excluded from the analysis. This is not a serious restriction, since diffusion in high viscosity liquids is too slow to b e of any practical importance for the discussed purposes. Two applications of Gibbs’ law for ablating liquids are given: the stabilization of films and the enhancement of ablation by adding surface active compounds. SURFACE
active agents and detergents have found considerable application in controlling the evaporation of lakes in hot climates, but relatively little attention has been paid to the possibility of utilizing these substances for ablation cooling. One reason for this neglect is certainly the fact that most of the studies of surface active agents have concerned aqueous solutions; relatively little is known about surface phenomena in the materials which are of interest to ablative cooling. According to Gibbs’ law a surface active agent of bulk concentration c, will have, under ideal conditions, the surface concentration, r,, in thermal equilibrium:
siders only diffusion transport and assumes that the liquid contains no gas bubbles. T h e mass balance for the surface active agent can best be described by considering the transport in a small control volume of the thickness of the liquid layer, 6 , unit width, and length, dx, as shown in Figure 1. T h e x-coordinate is along the ablating body contour and the y-direction is perpendicular thereto. T h e amount of surface active substance melting per unit time and unit area is c,m, where m is the total rate of ablation. From this amount, the quantity c,mi - pD bc/by is transported by flow and diffusion perpendicular to the solid surface contours and enters the upper surface of the control volume, where mi is the ablative loss by evaporation per unit time and area, and ,G is the bulk concentration just below the
where u is the surface tension and R is the gas constant. In an ablating or rapidly evaporating liquid the equilibrium concentration, I’,, will be disturbed because of the steady loss of material by flow and evaporation. T h e surface layer of the liquid containing the surface active agent is removed at the fastest rate, and in order to be effective in controlling evaporation the depleted surface must be replenished steadily by some transport provision. T h e decrease of r0 \vi11 cause a concentration gradient in the liquid layer, so that diffusion transport is automatically coupled with ablative loss. I n those cases where gases are generated during the ablation process by boiling, degassing, or thermal decomposition of the substrate. the surface active agent may also be transported to the liquid surface by means of rising gas bubbles This transport mechanism and its utilization for ablation cooling were treated by Steverding ( 3 ) HoLvever, the following discussion con-
surface.
364
l&EC FUNDAMENTALS
T h e amount p
l
c
bu/bx dy is the net flow through
the side faces. T h e following equation may then be written to express the mass conservation of the surface active species : mc, =
c,mi
- pD
bc
.4t given ablation rates the equation can be solved for 6, if the concentration and velocity gradients and bu/bx are known. Assuming a linear concentration profile, we obtain for the concentration c = co
-
(6,
- c,)y/S
(2)
In the vicinity of the stagnation point there is a very simple expression for buibx. Therefore, the following analysis will be carried out for stagnation conditions where bu/bx is a
