If the first q components are infinitely dilute, Equation 4 gives n
(Df - D,+I) 6i VCf
=
0
(6)
f+1
so
D f= D,+I;
e = q
+ 1, q + 2, . . . n + 1
Literature Cited (1) .Ark, Rutherford. IND. ENG. CHEM. FUNDAMENTALS 3, 28 (1964). (2)’ Bird, R. B., Stewart, LV. E., Lightfoot, E. N., “Transport Phenomena,” Lt‘iley, New York, 1960. (3) Burchard, J . K., Toor, H. L., J . Phys. Chem. 6 6 , 2015 (1962). (4) Pao, Y.-H., Chem. Eng. Sct. 19, 694 (1964). 15) Shuck. F. 0..Toor. H. L.. J . Phi’s. Chem. 67. 540 (1963). (6) Toor, H. L.. A.1.Ch.E. J . 8, 561 11962). (7) Zbzd., 10, 448 (1964). (8) \vei, James, J . Catalysis 1, 526 (1962). I
(7)
while the D f , i = 1, 2 . . . q > 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, Pa.
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. T h e amount p
364
l&EC FUNDAMENTALS
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
constant, K , Lshich is determined by the outer stream conditions. Assuming a linear velocity profile (3) where u t is the tangential velocity of the liquid a t the gas-liquid interface, one may write Equation 1 with Equations 3 and 4 , and bu,ldx = K in the following form:
Solving this equation for clo yields
(5) The assumption of linear profiles for velocity and concentration is justifiable as long as the viscosity of the liquid is reasonably small (2). and only for such cases is diffusion of practical importance. Assuming equilibrium between the surface layer and the bulk phase adjacent to it. the concentration, cm, immediately below the surface will now cause a surface concentration. r’, which is determined by Gibbs’ law:
With Equation 5, the last equation may be written
The conservation law for the total mass is ~ 3 .Eliminating
rizi =
Kp6
(8)
mi in Equati.on 7 and using Gibbs’ law yields (9)
where ri0is now the equilibrium surface concentration, corresponding to the bulk concentration, co, a t the interface temperature. Equation 9 may be called Gibbs’ law for ablating liquids. As expected, I” is always smaller than T r o ; when m + mi-that is, when total evaporation occurs-r’ will approach rZo.The same is true for 6 + 0. Thus both limiting cases give plausible results. Gibbs’ law for ablating liquids has two important applications-the stabilization of liquid layers and the improvement of ablation effectiveness. Under certain conditions ablating liquid films become unstable and the coherent film agglomerates into droplets. This results in excessive: mass loss caused by the roll-off of droplets from the surface under the action of aerodynamic shear forces. The addition of a detergent or a surface active agent suppresses droplet formation. However, to be effective the surface active agent must be maintained a t a sufficiently high surface concentration even under ablating conditions. Equation 9 determines this surface concentration as a function of m,6, and the surface temperature, which is necessary to calculate Tie. These parameters can be determined from the mass and heat transfer conditions ( 3 ) . The work described also has some significance in the area of improving the effective heat of ablation by adding a surface active compound. The effective heat of ablation determines the quality of an ablating material and is defined by the
amount of heat absorption per unit mass loss. In liquid ablation there are generally t\vo loss mechanisms-flow to the back, and evaporation. In a liquid ablator, the larger the mass ratio of evaporated material to total ablated material, the higher the heat of ablation, as expressed by the following formula ( 7 ) : h,if =
hb
+ (Lti+ 0.68M3.26 h,)
mi,’&
(10)
where hb is the enthalpy absorbed by the body including the heat of fusion, L , is the heat of evaporation, M is the ratio of molecular weights of air and injected vapor, mlis the rate of mass loss by evaporation, m is the total rate of mass loss, and h, is the enthalpy difference between stagnation and interface conditions. Since ablation is in many cases the transport of mass through surfaces and interfaces, their constitution must have an influence on the mass and heat transfer conditions. The physicochemical constitution of the surface can most easily be changed by the addition of surface active agents. Even with a small concentration in the bulk of the liquid, the surface active agent may have a high concentration in the surface and may thus control the vapor pressure of the liquid and its rate of evaporation. A surface active agent will usually not influence the bulk properties of the liquid to any degree, because of its generally minute bulk concentrations. The viscosity and heat conductivity, which control the flow of the liquid layer to the back, will not be altered by any considerable amount as long as the temperature remains constant. However, if the addition of a surface active agent produces a higher total vapor pressure, the increased cooling action of the injected vapor \vi11 decrease the surface temperature. The resu!t will be an increase in the viscosity and a reduction in the amount of liquid which flows to the back under the action of aerodynamic shear forces. Under suitable conditions the presence of a surface active agent will therefore increase, to a considerable degree, the ratio mtJm. However, a prerequisite for obtaining this effect is to find a system in which the surface active agent increases the total vapor pressure above its ideal value. Surface activity and the increase of the total vapor pressure in binary systems above the ideal value are closely related. Both phenomena are caused by the heterophobic behavior betkveen solvent and solute. T\vo species of molecules which repel each other in solution, favor each other’s transfer into the vapor phase. In a very similar way this heterophobic behavior is responsible for surface activity, where one species of
4
G A S BOUNDARY LAYER
DIRECTION O F FLIGHT Figure 1 .
Mass conservation of surface a c t i v e a g e n t VOL. 4
NO. 3
AUGUST
1965
365
molecules is forced out of the bulk into the surface layer. Surface activity and the increase of total vapor pressure in binary systems are therefore generally parallel phenomena. Based on this concept, a general theorem may therefore be stated: Any liquid ablation material can be further improved by adding a proper surface active agent. This theorem is valid also for film cooling. In order to estimate the relative improvement that can be made, Equation 9 is of fundamental importance ; however, more information is needed on the heats of evaporation, boiling points, and activity coefficients of the chemical components.
Literature Cited
(1) Bethe, H. A,, Adarns, M. C., J . AerospaceScz. 26,321 (1959). (2) Roberts, L., Natl. Aeron. Space Admin., NASA Rept. R-10 (1959). (3) Steverding, B., “Ablation of Glasses Containing a Surface Active Compound,” to be published in J . Appl. Mech. (4) Steverding, B., J. Am. Inst. Aeron. Astron. 2, 549 (1964).
BERNARD STEVERDING Army Missile Command Redstone Arsenal, Ala.
RECEIVED for review September 21, 1964 ACCEPTEDApril 12, 1965
CORRESPONDENCE BATCH AND CONTINUOUS THICKENING. PREDICTION OF BATCH SETTLING BEHAVIOR FROM INITIAL RATE DATA WITH RESULTS FOR RIGID SPHERES SIR: In a recent article by Shannon, De Haas, Stroupe, and Tory (2) on settling of suspensions, the principal theme was a discussion of the batch settling behavior of suspensions whose solids flux LIS. concentration (S-C) curve is “doubly concave”Le., has a shape similar to curve a of Figure 1 as against the more usual shape (curve 6). While this may be an interesting and perhaps useful exercise, it is certain that the particular experimental slurry chosen for discussion is not of this class. The slurry in question was made up of closely sized glass spheres of mean diameter 66.9 microns in water. The 5’-C curve was established by experiments in which the initial rate of fall of the upper boundary of a slurry of initially uniform, known concentration was measured ( 3 ) . This measurement was made for a number of concentrations up to, but not including, C,, the ultimate concentration of a fully settled slurry. for which the settling rate was taken to be zero. This was the sole value establishing the “double concavity.” If this concentration had been included in the program, the slurry would, indeed, have refused to settle and the experiment would have indicated that the settling rate for this concentration is zero. This result. however, indicates only a fault in the logic of the method. The siurry does not settle, not because its possible setting rate is zero, but because it is on the bottom of the vessel. Settling in such a situation requires that the concentration should increase towards the bottom. .4s the slurry is already a t its ultimate concentration, this is clearly impossible and no settling occurs. The fact that the settling rate is not in general zero a t the ultimate concentration can be seen from consideration cf the correspondence between settling and fluidization. The two are, of course, relatively speaking, the same. I n settling, a suspension moves through a substantially stationary fluid ; in fluidization the fluid moves through a substantially stationary bed. In each case the whole immersed weight of the solids is supported by the fluid. The only case in tvhich the relative velocity is zeco is the physically unreal one of C = I-Le., the slurry is a solid piston. For this very simple case of a suspension composed of spheres of nearly uniform size, the correspondence makes it possible to estimate the settling rate Lvith some accuracy. For the very low Reynolds number concerned, Ergun’s 366
I&EC FUNDAMENTALS
equation for pressure drop through packed beds (7) takes the simple form:
and when the bed is fluidized this must equal the buoyant weight per unit area of the bed APlZ =
.*.
Us
(Ps
d P S
=
~
- P)(l - 4 - P)
g
l50p
-E
€3
DP21
the corresponding reduced solids flux S / p , equals Us(l velocity.
- e) - a
giving for the ultimate concentration and properties specified by Shannon et al. a value of 0.22 X lo-* cm. per second. The curve of the above equation is shown with the experimental
C Figure 1 .
Form of flux-concentration curve a.
Shannon et of.
b. Usual form
