G . NARSIMHAN
1476 15
14
13
PAg.
12
11
-1
-+
10
Vol. 60 9
8
7
6
Na’
-2
c
z
v
0
0
-3
La3+
-4 ,e--
2
1
3
4
I
I
5
6
7
8
9
10
PI. Fig. 5.-The
influence of PI u on the coagulation values Na+, Ba2+ and Laa+ and upon the negative activity limit
of stability in the AgI system. &dinate: logarithm of the concentration of the neutral coagulating cation. Abscissae: PI or pAg of the cell: “Saturated calomel electrode-Ag-AgI electrode.” Concentration of AgI: 0.0005 M , PI 1 to 5 ; 0.0001 M , P I 5 to 10.
the concentration of the neutral electrolytes approaches their constant coagulation values. The activity limit seems also to be independent of the valency of the counterions as well as their size in low concentrations of the neutral electrolytes. This was checked with Cd(NO& and Ba(NO& with no other cation present in the concentration 0.0001 N . Figure 5 shows t,hat no smooth and continuous transition from the function of the activity limit (vertical part) into the function of coagulation val-
ues (horizontal part) occurs and that a very well expressed discontinuity can be observed. This may be considered as a confirmation of the opinion that a sharp distinction between the mechanism of the isoelectric coagulation and the mechanism of coagulation with neutral electrolytes should be made. Previously1 this was concluded from the difference of the adsorption capacities measured on suspensions which were obtained by both coagnlnmtions.
. ON THE EQUATION OF MOTION OF A SPHERICAL PARTICLE IN A CENTRIFUGAL FIELD *
BY G. NARSIMHAN
Department of Chemical Engineering, Laminarayan Institute of Technology,Nagpur University, Nagpur, India Received February 14, 1066
The equations of motion of a particle settling in a centrifugal field, under ultimate streamline and eddying conditions have been completely solved. Subsidiary expressions have been derived to compute the o timum feed rate t o continuous centrifuges. I n deriving the analytical ex ression for the velocity of a article settling ugimately under eddying conditions, an appropriate boundary condition has teen chosen, recognizing the E c t that the particle has successively passed through the streamline and buffer zones involving different force fields before encountering fully developed eddying resistance. This important fact appears to have been overlooked by previous investigators.
The critical mathematical relationship governing by Stokes, as the law that bears his name.l Amthe velocity of migration of a particle of a dispersed (1) G . G. stokes, “Mathernatioal and Physical Papers,” Trans. phase in a continuous medium was first developed Cambridge Phil. SOL voi. 9,1901,Part 11.
b
Nov., 1950
EQUATION OF
MOTION OB A PARTICLE IN A CENTRIFUGAL FIELD
bler2 analyzed the mathematical aspect of continuous centrifuges for predicting the performance of industrial equipment. There have been relatively few fundamental studies on the dynamics of particles in a centrifugal field, when compared with the availability of a large amount of data for performance characteristics of industrial machines. Coulson and Richardson3 have indicated the general equation of motion of suspended particles in a centrifugal field and given the approximate solution. The present paper will deal with the derivation of a rigorous analytical solution of the general equation of motion with the help of appropriate boundary conditions. The rate of phase separation in a centrifuge is considerably greater than that obtained with gravitational settling but the particles do not attain a constant terminal velocity because the accelerating force increases as the particles approach the walls of the basket. Figure l a represents a section of a tubular centrifuge of radius zw. Under the action of centrifugal force developed when the basket is rotated a t high speed, the surface of the liquid contained in the basket takes on the shape of a paraboloid, The equation of a section of the surface through the center of rotation is At high speeds of rotation, the slope of the liquid surface is very high and therefore the liquid surface is almost vertical. The paraboloid is not complete NOMENCLATURE constant B constant d particle diameter, ft. acceleration due to gravity, ft./sec.2 height of the centrifuge, ft. pi k , kl etc. constants a function P vol. rate of flow, ft:a/sec. Q time, sec. t N velocity, ft./sec. A
Cartesian coordinates Greek symbols
2’?
w P P
9
roots of the eq. 3 constants angular velocity, see.-’ density lbJft.3 viscosity lb./ft. see. function, defined by eq. 23
and yo is negative. This condition is represented in Fig. lb. Consider a fluid element a t a distance y from the bottom of the basket, rotating with the angular velocity w. Let the radius of the inner surface of the liquid be xi. For a spherical particle of diameter d, density ps, settling under streamline conditions, the equation of motion of the particle in the radial direction is
1477
where the first term of LHS represents the centrifugal force, the second, the drag and the third, the accelerating force. ?rd3 Dividing throughout by T ( p s - p ) , equation 2 may be rewritten as The primitive of the above equation may be written as x = A exp(d)
+ B exp(a2t)
(4)
where aland a2are the roots of the equation 3
Let the following boundary conditions be imposed on the general solution Whenx = xi, t = 0
I n introducing the second boundary condition an assumption has to be made. It is clear from inspection of eq. 2 that the particle does not attain a terminal velocity because the accelerating force increases as the particle travels toward the centrifuge wall. However, if this free term can be neglected, as being small in comparison with the others, then the particle velocity a t the centrifuge wall can be calculated as equal to Xw2/xw. Equation 4 can be differentiated with reference t o t and the differential set equal to the above quantity. This would provide the second boundary condition: when t = t,, (dzldt), = Xw2/x, where
and tm represents the minimum time the particle stays in the centrifugal field. Introducing the first boundary condition in eq. 4 (7)
xi=A+B
Introducing the second boundary condition
$
=
x2wxw= A
cy1
exp
(allm)
+ Baz exp(azt,)
(8)
Solving 7 and 8
B =
-
[Adz, exp( -
c&,)
- ~l~zi] ~
4 1
1
- P)
where ff1
exp [(a2-
cyl)tml
(10)
The particular solution of 3 is given as x =
1
ffdl - 8)
-
{ [Xw2xW exp( -
[xw2xW exp(
-
cult,)
- ~ P x i exp l (cylt) - aIxJ I exp(a2t)) (11)
orllm)
An expression relating the velocity of the particle t o its radial position can be derived from the above equation on differentiation. (2) C. M. Ambler, Chsrn. Eng. Prog., 48, 150 (1952). (3) J. M. Coulson and J. E. Richardson, “Cliemical Engineering,” VoI. 11, Pergamon Press, Ltd., London, 1955, p . 462.
G. NARSIMHAN
1478
Vol. 60
Z I
/L
i i
i iL.-
C
x
/
Y
Y
! I
i n I
I
A
8
Fig. l.-Liquid
surface during centrifugal separation: a, low speed, yo
The time a particle takes to reach the wall is tm. This may be determined from 10 after substituting the terminal conditions. Therefore
+ ve; b, high speed, yo - ve;
c, particle path.
turbulent conditions is ?rds
-g- ( p a -
p)w2
x
x
- 0.22p 4 d4
?rd8
( P . - P ) dtz = 0 (17) 1-p [hw2xs, - exp (a2 - cy,) tml - xi ( B exp (alt,) - exp ( a z t m ) ) ] (13) where the second term denotes the drag force under eddying conditions. Dividing throughout On the assumption that p can be neglected as by (ad8/6)(p, - and rewriting
xw =
small, it can be shown that tm N
1log,
(1
012
-
$)]
(14)
Equation 14 is helpful in determining the optimum feed rate t o the centrifuge to give a clear discharge. One need not assume a constant settling velocity to be operative at all times during the motion of the particle. If the height of the basket is H, the volume retained in the basket a t any time is r(xWz- Xi2)H. If the volumetric rate of flow is Q ftes/sec., the residence time tR
= ,(x,2
- Xi2)H
Q
(15)
For clear discharge, it is obvious that t~ 2 tm. The condition t R = t, for the smallest particle gives the optimum feed rate, Qo. Qo =
T(x,'
- xia)H t,
(16)
The equation of motion of the particle under
d2x 0.33~ dt2 f d(ps - p ) (E)'
- wzx = 0
(18)
Substituting
dx
dT = p;
dZx @
dp
= P.ax
Equation 18 may be transformed as p ddPT + k l p z - k 2 x = 0
(19)
where
Equation 19 can be rewritten after substituting p2 = 9. Accordingly
% !dx
+ 2kl+ - 2kzx = 0
(20)
The integrating factor is exp( f 2kldx) = exp(2klx)
Therefore +exp(2klx) = f 2k3x exp(2klx)dx
(21)
+ AI
(22)
EQUATION OF MOTION OF A PARTICLE IN A CENTRIFUGAL FIELD
Nov., 1956
or dz
ai=
[1z - 2klkz + AI exp ( -2k,z)]"* 2-
(24)
I n applying a suitable boundary condition to evaluate the constant AI, in eq. 24, it must be remembered that the general equation of motion that eq. 17 represents is for point condition and only for a particle settling under eddying resistance. I n as much as the particle settling from rest passes through the streamline and buffer regions before encountering fully developed eddying resistance, the appropriate velocity t,erm, necessary to evaluate AI, should either be the lower or upper critical velocity corresponding to Reynolds numbers 0.1 and 800, respectively. The eddying resistance becomes fully developed for Reynolds number greater than 800. For simplicity, it will be assumed that beyond the lower critical velocity, v", the general equation of motion will be governed by eq. 17. Let the distance traversed by the particle, in the streamline region, to attain this critical velocity be x". On applying the boundary condition, x = x*, when dx/dt = v*
1479
By substituting the value of v* in eq. 12 and then inserting the calculated value of t" in eq. 11, z* may be obtained; v* is approximately equal to 0.1(P/dP). Equation 30 is useful in determining the optimum feed rate to the centrifuge. The optimum feed rate is obtained as T[x,'
Qo = '
- xi2)H &a
(31)
The maximum velocity of the particle can be calculated from eq. 17 as
A more rigorous procedure in analyzing the dynamics of a particle in a force field would be to divide the region into streamline, buffer and turbulent zones and then solve the general equation of motion, choosing the appropriate term t o define the resisting force. The transition from streamline to buffer zone and from buffer zone to turbulent zone occurs at Reynolds numbers 0.1 and 800, respectively. This would help satisfy the boundary conditions and, depending upon the ultimate settling conditions, the time of migration can be computed by adding the residence periods for the particle in appropriate zones. The general equation of motion for a particle settling under such conditions that the Reynolds number is between 0.1 and 800 may be written as
Equation 24 may be rewritten as
- z*))I
exp ( -2k1(x
(26)
Equation 26 defines precisely the magnitude of the particle velocity in the turbulent region as a function of its radial position beyond x*. This equation is difficult to solve to provide a relationship between distance traversed and time. However, on the assumption that the magnitude of the term in the exponent is large, eq. 26 may be simplified as 2 = [,*z + z)mz (% - x*)I1I8 (27) dt
Xw
The general solution of the above equation is 2xw [,*2 Vm2
+ .xuwrn2 (z - z*)
Applying the boundary condition that t = 1, when x = xw,B can be evaluated. The particular solution of eq. 27 may hence be given as
{u*'
+ xw Ht.1" (Z - $*)
The above equation defines the distance traversed by the particle as a function of time under ultimate eddying conditions. The time taken by the particle to reach the centrifuge wall may be evaluated from eq. 29 by specifying any set of values for t and x. The condition that t = t* when x = x", gives
A complete solution of the above differential equation has not been attempted here, because eq. 33 is itself rather approximate, due to the doubtful validity of the resisting force made up of the second and third terms. Discussion The general equation of motion of a particle in a centrifugal field, under streamline conditions, has been solved completely to provide eqs. 11 and 12. On the assumption that, in the case of a particle settling ultimately under eddying conditions, only two zones need be considered, namely, the streamline and turbulent, eqs. 26 and 29 have been derived. I n calculating the optimum feed rate in industrial centrifuges a constant velocity is usually assumed to be operative for all the time the particle takes to reach the centrifuge wall and hence provide B means for comparing centrifuge performance.2 But it is now possible to calculate with more accuracy the time taken for the smallest particle to reach the wall. Equations 16 and 31 are more reliable to calculate optimum feed rate than existing equations. Finally, it mag be mentioned that in assessing the centrifuge performance, one usually assumes t h a t the angular velocity is constant throughout. But there might be a slight decrease in its magnitude from the centrifuge wall t o the inner surface consequent on the shear transferred to the central air space by the rapidly revolving inner liquid
1480
C. N. SPALARIS
surface. I n such cases there may be a gradient of angular velocity in the direction perpendicular to the axis of rotation, causing the settling particle to assume a curved path instead of a straight one
Vol. 60
(Fig. IC). The shear transferred may be evaluated if the system is considered as one where the air is blown a t equivalent velocity over a plane surface. Investigation into this effect is desirable.
THE MICROPORE STRUCTURE OF ARTIFICIAL GRAPHITE1 BY C. N. SPALARIS~ Engineering Department, General Electric Company, Richland, Washington
r
ReceQed February 67, 1066
The surface characteristics of artificial graphite were investigated by means of nitrogen adsorption-desorption isotherms. Changes in surface area, pore size distribution and density of graphite were studied as a function of oxidation. It was concluded that the binding materials used in the manufacture of graphite are largely responsible for the majority of the pores present. The variation in the pore size distribution with increasing oxidation must play an important role in the rate and/or mechanism of graphite oxidation.
I n the study of heterogenous reactions between artificial graphite and gases it was decided that information concerning the surface characteristics of graphite was necessary. I n the past, several attempts have been made to correlate the rates of carbon dioxide-graphite oxidation with its surface characteristics, but the data reported are only fragmentary.3-6 Moreover, the artificial graphite used in the experiment reported here is of a different type from that used by other investigators. The surface Characteristics of five kinds of artificial graphite were investigated. These studies include the determination of surface area, pore size distribution, density and the changes of these properties taking place upon successively higher oxidation. Methods and Materials A standard B.E.T. all-glass vacuum apparatus was used to obtain the data for the surface characteristics. Nitrogen gas was used as an adsorbent to obtain the adsorptiondesorption isotherms. The surface area was determined from the analysis of the adsorption isotherms using the B.E.T. method.6 The pore size distribution curves were obtained from the analysis of the nitrogen desorption isotherms using the B.J.H. method.’ Some of these isotherms were analyzed by the modified B.J.H. method develope: by C. .Pierce.* Density studies were made a t 25.0 f0.1 by helium immersion using a container whose volume was determined to the nearest 0.02 cm.3. The volume difference before and after the graphite sample was placed in the container was the volume of the solid sample. The density was then calculated using the volume and the weight of the sample. The COZ-graphite oxidation studies were made in a horizontal combustion furnace a t a controlled temperature of 1000 f.3 ” . The temperature was recorded with a chromelalumel thermocouple, its welded junction being imbedded in the center of the graphite sample. The Graphite Samples.-All of the graphite samples used in this experiment were artificial, high purity, Acheson
type. The types of samples used and some of their physical properties are listed in the Table I. SOME
TABLE I GRAPHITEUSED IN THIS EXPERIMENT
PHYSICAL PROPERTIES OF
Typea
Graphitization temp.,
X-Ray Co spacing,
Ash content,
OC.
A.
%
CSF 2800 6.71 0,037 to TS-GBF 2500 6.74 0.084 CS-GBF 2500 6.72 KC 2800 6.71 WSF 2700 6.72 a The abbreviations used in the above table for the graphite samples are as follows: CSF, Cleves Coke (Petroleum Coke), standard pitch used for binder, purified by the “F” process; CS-GBF, Cleves Coke, standard pitch, gas baked, purified by the “F” process; TS-GBF, same as CS-GBF except Texas Coke (a petroleum coke) is used instead of Cleves Coke; KC, Kendall coke and Chicago pitch; WSF, Whiting coke, standard pitch, purified by the “F” process. The purification process “F” is achieved by circulating “Freon-12’’ around the graphite bars, which are held at a temperature of about 2500”. Commercially obtained gases (Ns and He) used for the surface characteristics were purified by passing them through a tube filled with copper turnings heated to 450°, then through a bubbler filled with concentrated sulfuric acid and finally through a liquid nitrogen trap. Liquid nitrogen of good purity was obtained commercially.
Results and Discussion The surface area of graphite, calculated by using the B.E.T. method, was found to be between 0.30 and 1.00 m.2/g. for formed cylindrical samples of 1 cm. in diameter and 10 cm. long. The surface area values depend greatly upon the temperature of outgassing and the type of graphite sample. The surface area of all graphites used was found to increase with increased outgassing temperature, (1) Presented before the Division of Colloid Chemistry, 128th A.C.S. particularly in the range of 25 to 500” (Fig. 4). National Meetins, September 11, 1956, Minneapolis, Minnesota. The increase in surface area is very small beyond (2) General Electric Co. APED, San Joee, California. the outgassing temperature of 500”. All of the (3) E. A. Gulbransen and K. F. Andrew, Ind. Eng. Chem., 44, 1039 (1952). outgassing in this experiment was made between (4) P. L. Walker, Jr., R. J. Foresti and C. C. Wright, ibid., 46, 1703 450 and 550”. (1953). All types of artificial graphite examined possess (5) P.L. Walker, Jr., F. Rusinko, Jr., and E. Raata, THISJ O U R ~ L L , a structure consisting of pores with radii smaller 59, 245 (1958). (F) 8. Brunauer, P. H. Emmett and E. Teller, J . Am. Chem. Soc., than 200 A. termed the micropore structure. BO, 309 (1938). These micropores seem to be an intrinsic property (7) E. P . Barrett, L. G. Joyner and P. P. Halenda, ibid., ‘73, 303 of the manufactured graphite. It can be seen in (1951). Fig. 1 that artificial graphite has a large number of (8) C. Pierce, THIS JOURNAL, 6’7, 149 (1953).
r
