ENGINEERINGr DESIGN, AND EQUIPMENT a number of illustrative examples of interest to chemical engineers. Its purpose is to familiarize a wider circle of chemical engineers with the field of matrix algebra, which is becoming more and more useful in many branches of applied mathematics, and to demonstrate the power of this new approach by examining some typical chemical engineering problems. It is felt that the use of matrix methods will develop rapidly in chemical engineering problems since i t seems to be an ideal tool for application to multicomponent and multistage systems. Nomenclature
A A-’ A’
square matrix inverse matrix transposed matrix element in i t h row and j t h column of A = cofactor of a,j = adjoint matrix of A = det A = determinant of A = column matrix or vector = unit matrix = null matrix = concentration of i t h substance in n t h reactor = reaction velocity constants = vapor liquid equilibrium constant of i t h component in liquid mixture leaving nth stage = liquid holdup on n t h plate, moles = liquid leaving nth plate, moles/unit time = reciprocal relative volatility of ith component = reflux ratio = time = vapor leaving nth plate, moles/unit time = mole fraction of i t h component in overhead = mole fraction of ith component in feed = mole fraction of ith component in liquid stream leaving feed tray = mole fraction of ith component in liquid leaving n t h tray = mole fraction of i t h component in vapor leaving n t h tray = a determinant = a characteristic root of A
= = = =
literature cited (1) Amundson, N. R., Trans. Am. Inst. Chem. Engrs., 42,939 (1946). (2) Arnoldi, N. E., Quart. A p p l . Math., 9, 17 (1951). (3) Boole, G., “Treatise on Calculus of Finite Differences,” Macmillan, London, 1880. (4) Courant, R., and Hilbert, D., “Methods of Mathematical Physics,” Interscience, New York, 1953. (5) Dwyer, P. S., “Linear Computations,” Wiley, Mew York, 1951. (6) Ferrar, W. L., “Algebra,” Oxford University Press, Oxford, 1941. (7) Ferrar, W. L., “Finite Matrices,” Oxford Univ. Press, Oxford, 1951. ( 8 ) Fettis, H. E., Quart. A p p l . Math., 8 , 206 (1950). (9) Frazer, R. A., Duncan, W. J., and Collar, A. R., “Elementary
Matrices,” Cambridge Univ. Press, Cambridge, England. 1938.
Fry, T. C., Quart. A p p l . Math., 3, 89 (1945). Hicks, B. L., J. Chem. Phys., 8 , 569 (1940). Jahn, H. A., Quart. J . Mech. and A p p l . Math., 1 , 131 (1948). Jordan, Ch., “Calculus of Finite Differences,” Chelsea, New York, 1949. (14) Kincaid, W. M., Quart. A p p l . Math., 5, 320 (1947). (15) Lapidus, L., and Amundson, N. R., IND.ENG.CHEM.,42, 1071
(10) (11) (12) (13)
(1950). (16) LlacDuffee, C. C., “Theory of Matrices,” Chelsea, New York, 1950. (17) Mason, D. R., and Piret, E. L., IND.ENG.CHEM.,43,1210 (1951). (18) Michal, A. D., “Matrix and Tensor Calculus,” Wiley, New
York, 1950. (19) Milne-Thompson, L. M., “Calculus of Finite Differences,” Macmillan, London, 1951. (20) Murdoch. P. G.. Chern. Eno. Proar.. 44. 855 (1948). Norlund, N. E., “Vorlesungen uder Differenzenrechnungen,” J. Springer, Berlin, 1924. (22) Perlis, S., “Theory of Matrices,” Sddison-Wesley Press, Boston,
(21)
1952. (23) Robinson, C. S., and Gilliland, E. R., “Elements of Fractional Distillation,” McGraw-Hill, New York, 1951. (24) Tiller, F. M., Chem. Eng. Progr., 44, 299 (1948). (25) Tiller, F. M., and Tour, P. S., Trans. Am. Inst. Chem. Engrs., 40, 319 (1944). (26) Underwood, A. J. V., Chem. Eng. Progr., 44, 603 (1948). (27) Wallenberg, G., and Guldberg, A., “Theorie der Linearen Differenzengleichungen,” B. G. Teubner, Berlin, 1911. (28) Wayland, H., Quart. A p p l . Math., 2, 277 (1945). RECEIVED for review September 7, 1954.
ACCEPTED M a r c h 1 , 1955
A Test for the Validity WALTER P. REID U. S. Naval Ordnance Ted S f a f i o n , China Lake, Calif.
T
HE person performing a sedimentation experiment wishes to know how large a concentration he may safely use. I n the
literature one can find many statements to the effect that interactions between particles may be neglected provided that the concentration of sedimentary material is below about 1 to 5%. T h e analysis that follows indicates that one must also take into account the shape of the particle size distribution curve. The sedimentation method of determining particle size distributions is based on the assumption that the particles fall independently of each other. For this assumption to be completely valid, i t would be necessary, among other things, for the particle,. to pass freely through each other. This, of course, they cannot do. Xevertheless, it is possible to assume that the particles can pass through each other and that is what is done in this paper. -4ny attempt to take into account the intricacies of particle interactions during sedimentation would lead t o a very difficult problem. However, by assuming that the particles fall independently one obtains a simple problem which is easily handled matheAugust 1955
matically. The results obtained in this way serve to help one guess what is happening during sedimentation. Throughout this paper it is assumed that the particles fall independently, and hence that they pass freely through each other. The expected number of times, h, that a particle OC random initial position passes through other particles, and the expected number of times, H , that other particles pass through it are calculated. These results are g k e n in Equations 9 and 6. Both h and H depend on the particle size distribution, g ( r ) , and each is proportional, not to the concentration, but t o the amount of sedimentary material per unit of cross-sectional area of the (cylindrical) settling vessel. T h e expected fraction, f, of its settling time that a given size particle spends passing through other particles is given in Equation 14. The expected fraction, F , of its settling time during which other particles are passing through it is given in Equation 12. Bothf and F depend on the particle size distribution, and each is directly proportional t o the concentration.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1541
ENGINEERING, DESIGN, AND EQUIPMENT The g ( r ) from a particle size distribution curve obtained experimentally may be used in Equations 12 and 14 to find F and f by numerical integration. If the basic assumption of essentially unhindered settling is to be satisfied, then it seems necessary, though not sufficient, that the calculated values of F and f should not be large. T h a t is, if one assumes independent fall to obtain a g(r) experimentally, then finds that with this g ( r ) particles must spend an appreciable fraction of their settling time passing through each other, then the assumption of independent fall is not correct, and the Q ( T ) is probably in error. Thus Equations 12 and 14 give a means for testing the validity of a particle size distribution curve obtained experimentally. I n this paper no guess will be made as to how large a value for F or f would be too large.
spends passing through other particles is obtained by multiplying the ordinates of Figure 3 by the concentration, v, of the solution. The expected distribution of sizes among the particles that do the hitting is shown for one case in Figure 4. When considering the results for h and H it should be borne in mind that a near-grazing impact occurs considerably more frequently than a nearly direct impact. Or, stated otherwise, if p denotes the projection on a horizontal plane of the line joining the centers of the two particles, then the probability of occurrence of a collision with p lying between p and p d p is directly proportional to the area of the circular ring between two concentric circles of radii p and p d p , respectively. For d p sufficiently small, then, this probability is directly proportional to p .
+
+
“Collisions” occur during sedimentation
Consider a depth L of liquid containing sedimentary material initially thoroughly mixed throughout its volume. Take the sedimentary particles all to be spheres. Starting a t time t = 0, assume that each particle begins settling to the bottom of the liquid with a constant speed u given by Stokes’ law:
where T is the radius of the particle, and X is a constant. Cylindrical coordinates p, 8, z will be used to designate the initial location of the center of any particular particle, p , where z = 0 and z = L are the bottom and top of the liquid, respectively. Single out one of the particles and call it P. Denote its z coordinate by Z, and its settling speed and radius by U and R. Choose the origin of coordinates directly below this particle so that its coordinates are 0, 0, Z. Let the distribution of radii, r, among the spheres exclusive of P be given by g ( r ) , where
2
0
Figure 1.
0.04
4
6
kr
8
C f g ( r ) d r = average number of particles with centers in (2) unit volume of the original mixture which have radii between r1 and r2, where r2 > r1 The constant C, here, is t o be determined by the fact, that the volume, v, of sedimentary material per unit volume of the original solution is considered to be known. Thus
Size distribution curve assumed for particles in example
It will be assumed throughout that all spheres have radii which are small compared t o L, and that they pass freely through one another. A hit will be counted when one particle overtakes and overlaps another. Particles p and P will reach the bottom of the liquid in times z/u and Z / U , respectively. Hence if z > Z and z / u < Z/U,-~i.e., z/r2 Z/RZ-then particle p will overtake particle P before the latter reaches the bottom of the liquid. If in addition p < T R, then particle p will overlap particle P. This will be counted as a hit. The expected number of particles that will hit particle P before it reaches the bottom of the liquid is
-
henegative of expression 10. Three integrations may then be performed, giving finally for the expected sum of transit times (13).
The expected ratio of collision time t o settling time is obtained by dividing this expression by Z / U
This is the expected fraction of its settling time during which a particle of radius R is passing through other particles. Both f and F are seen to be proportional to C and hence to t h e concentration of the solution. If for a given case F or f is greater than unity, this means that the simultaneous passage of two or
INDUSTRIAL AND ENGINEERING CHEMISTRY
1543
ENGINEERING, DESIGN. AND EQUIPMENT more different particles through ,P. or P through two or more different particles, has become a significant factor. Example
I n this example the distribution of sizes amon@;particles will be assumed to be g ( r ) = k4r4e-k~
This distribution is illustrated in ~ value of g ( r ) in Equation 3
be the expected number of particles of radius r(r1 < r < r z ) that will hit a particle of radius R and random initial position. With d r ) and c given by Equations 15 and 16, one obtains from Equation 6 13440 9 (Icr,kR) = k V ( r R)3(r - f 2 ) e - k ’ (17i
+
(*”
i 1. L~
6720sC = v k 4
Figure 4 shows how 4 ( k r , k R ) varies with kr for the case when k = 2. T h a t is, it shows the espected distribution of size? T using ~ ~ this ,~ ~~ R ~ among the particles that do the hitting. The total area under is equal to the ordinate of the curve from kr = 2 t o kr = the H / ( v k L ) curve in Figure 2 a t kEz = 2. (16)
After substituting these values for y ( ~ and ) c in Equations 6, 9, 12, and 14, one obtains for H / ( o k L ) , h / ( v k L ) , F / v and ,f/v> the results shown in Figures 2 and 3. The expected Eize distribution among the particles which hit a particular size particle may be obtained, if desired. For example, consider the particles hitting particles of radius R. Let
Acknowledgment
This study of collisions during sedimentation was suggeBted bgr Henry Seaman. T h e author wishes to thank John W. Odle and R. T. KnaPP for constructive criticisms. RECBIVEDfor review March 8, 1964
ACCEPTED Ianuary 11. 1853
Barrier Systems in Thermal Diffusion Columns J. C. TREACYl
AND
R. E . RICH
Departmenf of Chernicol Engineering, University o f Nofre Dame, Nofre Dome, Ind.
A
T PRESENT, the Clusius column is the only gas separation
apparatus in industrial use that utilizes the thermal diffusion effect. Clusius columns consist of long, cooled outer containers with heated wires or other surfaces placed along the column axis. The thermal diffusion effect operates horizontally, tending (with few exceptions) to concentrate the lighter molecules at thg hot surface. Thermal siphoning moves the partially separated gases to the column ends. Light gas can be removed from the top and heavy gas from the bottom of the column. I n order to be effective, Clusius columns must be long, have small hot-cold surface clearance, and utilize large temperature differences. With equipment 20 feet or more in length, built with only fractions of an inch of hot-cold surface clearance, considerable care must be exercised in construction, power requirements are high, and gas must spend a considerable time in moving through the very considerable length. The work described in this article presents an attempt to utilize barrier systems between hot and cold surfaces instead of small clearances between hot and cold surfaces. Several of the objectionable features of thermal diffusion separation are thus eliminated. Brewer and Bramley ( 1 )suggested the value of baffles attached to the hot surface in enhancing separation effected by thermal diffusion columns. Donaldson and Watson ( 3 ) reported that spacers used for positioning of hot wires had the effect of increasing separations. This effect was attributed to a possible advantageous effect of controlled turbulence. Barriers (horizontal and vertical sections suspended from the ends of the column into the gas space, but not attached to either hot or cold surface) are considered to be of more value than baffles, inasmuch as they allow flow at both hot and cold surfaces. Thus, they do not require molecules to move long distances while diffusing and do not limit separation rates as seriously as do baffles. I
Deceased.
1544
Apparatus i s designed to test requirements for precisely constructed column
This nork was partially a t e d of barrier effectiveness with respect to reducing constructional requirements. Thus, in construction of the apparatus precise requirements were intentionally abandoned. Apparatus. The column was constructed from 2-inch water pipe closed at the ends with couplers and plugs. The column was 23 inches long and was cooled by a n open spray of tap water at 17” f 1” C. The hot surface was made from l/c-inch water pipe with a borosilicate glass tube insert containing a coil of 20 feet of Sichrome wire (1.75 ohms per foot). One end of the Nichrome wire was grounded while the other was led from the equipment through rubber insulation t o a rheostat and the 110-volt alternating current line. Spacer plates used in conjunction with the end-plug shape served to center the hot element when it was introduced into the column. Taps cloped with stopcocks were provided a t the ends of the column for removal of samples and for connection to vacuum and to a manometer. A mid-column t a p served for introduction of feed taken from tankage or a premixing vessel. Runs were made with the open column and with the following types of barrier systems placed between hot and cold surfaces:
I . Horizontal barriers, consisting of 1/64-inch metal plates (for certain runs asbestos was used), 1 T / 8 inches in outside diameter, with a 3s/6,-inch hole bored in the center. These were suspended from the ends of t h e column and were not attached t o the hot or cold walls. An annulus of area 0.06 square inch was between the horizontal barrier and the hot surface, and an annulus of area 0.60 square inch remained between the barrier and cold wall. The 1/4-inch pipe was inserted through the 39/@-inch holes.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47 No. 8
