INDUSTRIAL A N D ENGINEERING CHEXISTRY
September, 1928
895
Computation Tables Test Pressure, kg. per sq. c.m. T , hours 0 , liters per sq. m. 01, liters per sq. m. Rq
02 QIZ
RnQz Q I ~ - RnQz T I ,hours RqT Ti-RqT WP Po
' U
STARCH 22328 0.703(10) 0.1 186.2 450 2,415 34,700 202,500 83,800 118,700 0.45 0.2415 0.2085 569,000 0.7361 773,000 C = 0.87
22328A 1.406(20) 0.1 201.5 546 2.68 40,600 298,000 108,800 189,200 n.. 4~. 5 0.268 0.182 1,040,000 1.345 773,000
22428 2.812(40) 0.1 328.5 639 1,944 107,800 408,200 209,600 198,700 0.30 0,1944 0.1056 1,880,000 2.402 773,000
KIZSELGUHR Test 31527B Pressure, kg. per sq. cm. 0.321 (4.57) T , hours 0.1 0, liters per sq. m. 86.9 Qi. liters per sq. m. 236.0 2,715 Rq 7,550 55,700 Kq02 20.770 34,930 Q12-RqQP 0.50 Ti, hours 0.2715 RQT Ti - RqT 0.2286 W P 153,300 P c0.3936 W 389,500
%
Test Pressure, kg. per s q . cm. T . hours Q, liters per sq. m. Q I , liters per sq. m. Rq
52526
3527 0 . i03(10)
0.1 136.5 306.0 2.24 18,650 93,600 41,800 51,800 0.40 0,274 0.176 294,000 0.7477 392,500 C = 0.82
3527A
3527B
1.406!20) 0.1
2 812(40)
0 1
195.5
174.0
391.0 2.00 38,200 152,700 76,400 76,300 0.35 0.20 0.15 509,000 1.322 384,500
385 0 2 21 30,300 148,200 66,900 81,300 0 2 0 1105 0 0895 909,000 2 333 390,000
Q12- RnQa T i , hours RaT Ti - RgT WP Pc W
0.703(10) 0.1
40.3
55.6
80.6 2.0
108.5 1.95
RnQ2
012-Rn02 T I ,hours RqT T I - RnT U'P Pc
iv
Test Pressure, per sq. T , hours Q , liters sa. m. 01, 'liters sq. m.
RqQ2
1.406(20) 0.1
2 812(40) 0 1
Ti, hours RqT T I- RqT WP PO
71.3 158.1 2.218
rv
FULLER'S EARTH 32527A 42027
32527 kg. cm.
0 1 4
22526C
62.00
51027 5927 1 406(20) 0 7 0 3 ~ ') 0 02 0 05 117 8 96.1 296 0 260 5 2 51 2 71 13,850 9230 87,600 67,820 34,800 25,000 52,800 42,820 0 225 0 25 0 1265 0 1355 0 0985 0 1145 536,000 374,000 1 196 0 831 448.200 450,000 c = 0,525
Q2 QiZ
52526B
130 2 2.1
3845 16,950 7880 9070 0.4 0.21 0.19 47,700 1.27 37.540
5080 25,030 11,260 13,770 0.4 0.2218 0.1782 77,200 2,062 37.950
MASONS' LIME
012-
0.322(4.67) 0.1
3090 11,670 5940 5730 0.4 0.195 0.205 27,900 0 783 35.700 C = 0.7
-
Test Pressure, kg. per sq. cm. T , hours Q , liters per sq. m. 01, liters per sq. m. Rq
;:
R E D COLOR 525268
1623 6490 3246 3253 n~ ". 0.2 0.2 16,260 0.4496 36,200
kiQ2
RgOl
per per
0.703(10) 0 1
1.406(20) 0.1
2.109(30) 0.1
130.2
155
297 2.28 16,950 88,200 38,660 49,650 0.4 0.228 0.172 288,000 0.9605 301,000
331.5 301 2.14 2.068 24,100 21,200 109,900 90,600 51,600 43,800 58,300 46,800 0.4 0.35 0,214 0,2068 0.186 0.1432 313,500 326,500 1.04 1.09 301,500 298,200 c = 0.115
145.6
5927.'' 2 812(401 0.05 151.8 390 5 2 575 23,000 152,300 59,200 93,100 0 25 0 1288 0 1212 768 000 1 721 449,000
42227 2.812 (401 0.1 148.9 341 2 29 22,000 116,200 50,400 65,800 0.425 0.229 0.196 33 1,000 1.126 294,000
Abrasion Tests of Rubber Stocks Containing Various Types of Carbon Black' W. B. Plummer and D. J. Beaver COXBVST:OXDTILITIES CORPORATIOK,
HIS paper presents certain interesting results obtained during the course of a general study on the abrasion resistance of rubber stocks compounded with various types of carbon black. All tests have been carried out using the Grasselli abrasion tester as described by Williarn~,~ since this method measures the actual power consunied per unit of material abraded. and hence eliminates some of the uncertainties inherent in other abrasion test devices. Vogt3 has tested seven redaim stocks by this method as well as with four other devices, all designed to measure or take account of the power consumed during abrasion. Averaging the results of three of his methods, the fourth being definitely discordant with the others, the relative abrasion resistances for the various stocks were 100-95-91-83-80-74-71, whereas with the Grasselli abrasion machine the results were 100-9189-83-77-72-67. The concordance between these results is very close. In his second paper Vogt discusses certain differences between the results of his "angle abrasion" machine
T
L
Presented before the Division of Rubber Chemistry a t the 75th Meeting of the American Chemical Society, S t . Louis, Mo., April 16 t o 19, 1028. IND. ENG.CHRM.,19, 674 (1927). * Zbid., 20, 140, 302 (1928).
LONGISLAND CITY,
S . E'.
and the Williams machine, to which further reference nil1 be made later. In the present tests with the Grasselli machine the standard procedure outlined by Killiams has been used except that round blocks are used instead of square ones. This simplifies the construction of molds very greatly and permits test blocks to be made by rolling up narrow strips of the raw stock, which completely eliminates any possible effect of calendar grain on the test results and in general gives a more satisfactory test block. Check determinations have shown no difference in results between round or square blocks. The average deviation of test results for a given stock by this method has been found in general to be less than 3 per cent, although occasionally a given test will give obviously discordant results, ordinarily in the direction of OW abrasion loss. A priori this might result from changes in the motor speed (line voltage), in the size of the abrasive, or in the condition of the abrading or abraded surface. KO detectable differences in speed haie been observed during tests,and although the abrasive siae is important, as discussed later, it cannot account for the occasional erratic results. Such results are attributable only to the condition of the surface involved, and hence in routine testing special atten-
896
I N D CSTRIAL AATDESGISEERING CHEMISTRY
tion should be paid to cleaning the surface of the test blocks with a cloth moistened with benzene, to insuring the absence of entrained oil in the air blast used for keeping the paper clean, and to avoiding the presence of accidental oily or greasy impurities on the surface of the abrasive paper. - Effect of Size of Abrasive
Vol. 20, No. 9
near to constant for given conditions of load on the test pieces, speed of the abrasive surface, etc., regardless of the actual abrasion loss of the stock under test. Vogt has found this to hold very closely with his angle abrasion machine, but finds less constancy in his tests with the Williams machine. Of the tests reported herein, for example, the actual rate of work during the tests of the various stocks of Table I lay be-
watts similarly. With the d i p h e n y l g u a n i d i n e , 25 h i g h l y compounded stock black, the cure a t 140” C. The abrasion loss of rubber stocks containing differ“C,” black “G,” the power being 120 minutes for black ent types of carbon black was determined by means consumption was 10.7 to ttM,, and 8o minutes for of the Williams abrasion apparatus. It was found that 10.5 watts. black “G.” C o m p o u n d the abrasion loss per unit of work increased as the “c,,is a highly compounded As pointed out by Vogt, size of the abrasive increased up to 0.45 mm. and then stock containing 100 pale this relative constancy of decreased quite rapidly. No adequate explanation crepe, 5 zinc oxide, 5 sulfur, p o w e r consumption indifor these results has been found. It was shown further l . 5 D. p. G,, Bo black, cure cates that only a small perthat the ratio of the abrasion loss of unaged stocks at 1400 c. for black i l M , , 75 centage of the power input containing various types of carbon black to these minutes, for black 45 goes to do true abrasive stocks after aging in the Geer oven was not constant, minutes. The figures near work. He supports this by since it was found that some types of carbon black the arrows at the bottom of calculations showing that increase the rate of deterioration of rubber much more the graph refer to the size the power consumption of than Other types. number of standard emery abrasive test machines of the Damr used. T h e Doints 1I ‘I present types represents 35 marked “Saw” refer to a t o 120 times the-energy inspecial abrasive surface made by coiling tightly a 0.47-em. put per cubic centimeter of rubber removed as would be neces(3/1,-inch) metal-cutting band saw and soldering it t o a brass sary to disintegrate 1 cc. by stretching. This value he deterplate. The average particle size of the various papers was mines based on the conception that a piece of rubber “superdetermined by microscopic examination, using an ocular mi- stretched” above its normal break until its energy of resilience crometer and measuring a large number of particles. is 50 per cent greater than normally possible will disintegrate It will be observed that up to a particle size of about 0.45 to a powder on rupture instead of a single break. This calmm. the results fall accurately upon lines having the slope culation, however, neglects the fact that a tremendously of 0.67. This shows definitely that with varying abrasive greater amount of new surface is produced in the first case, size the volume loss per unit of work done is proportional which, as shown by the present tests (Figure I),is an important to the new surface generated per volume removed, since factor. On the other hand, if we assume that the ordinary for spherical particles surface is a function of the ‘/3 power tensile rupture of 1 cc. of rubber produces 1 sq. em. of new of volume. It is rather surprising that this relation should surface, and if we therefore multiply his value for the energy hold so accurately, since the shape of the particles removed of rupture by the amount of surface generated in abrading by abrasion is shown by microscopic examination to be long 1 cc. to. for example, 0.2 mm. cubical particles, we find that and stringy, rather than in any way spherical. The results this new figure is much greater than the total measured energy observed for abrasive sizes over 0.45 mm. are still more sur- consumption during abrasion. This may be taken to indiprising, particularly with respect to the suddenness of the cate that removing small particles by abrasion is more of a break in the curves. Apparently, with the larger abrasive tearing than a tensile phenomenon, since resistance t o tear is sizes new and unknown factors come into play or a new balance always lower than the ultimate strength in pure stretching. is set up between effective abrasive work and work lost in To consider another aspect of this matter briefly, it is friction. The data shown hare, however, been confirmed difficult to conceive of any major points of consumption of by numerous check determinations which are not shown. energy in the Williams machine save actual abrasive work and It is also interesting to note the results obtained using the heat generated by friction between the rubber and abrasive side of an ordinary 30- to 40-mesh carborundum grinding or as result of hysteresis of the specimen. If we assume that wheel in place of the abrasive paper, the loss for blacks “G” 90 per cent of the total heat generated is dissipated into the and “M” (compound “A”) being, respectively, 260 and 208 testing machine or the air, and take Williams’ figure of 7 ” cc. per kilowatt-hour. Using no abrasive but merely running C. as the final temperature rise of the (10-gram) test blocks the blocks against the machined surface of the brass support- a t the end of a typical 20-min. test, the total energy liberated ing disk, a small but appreciable loss was obtained, respec- as heat throughout the test would not be more than 10 to tively, 9.3 and 7.4 cc. per kilowatt-hour. 20 per cent of the observed input. On the other hand, when running test blocks of compound “A” (blacks “M” and Energy Consumption in Abrasion Tests “G”) against the bare machined brass wheel of the Williams The total power consumption during abrasion tests, as machine, the tests showed an abrasion loss of only 7 to 9 measured in a machine of the present type, is surprisingly cc. per kilowatt-hour, but the power consumption was 17.0
ILVDUSTRIAL A N D E-VGIXEERISG CHEMISTRY
September, 1928
to 17.1 watts. This is actually much higher than for normal abrasion tests, and shows the magnitude which it is possible for thermal losses in a machine of this sort to attain under certain circumstances. Unfortunately, however, the consideration of the foregoing facts leads mainly to the conclusion that there is much to be learned about the mechanism of abrasion before the testing machines and methods will be entirely understood and satisfactory. I
I
AVEeRGE
I
,
,
I
’
1
PAeTlCLE f1.7.5 OF A B U S I V E PAPEP. R m
Figure 1-Abrasion
Resistance us. Abrasion Size
Abrasion Resistance of Stocks Containing Various Carbon Blacks
Various carbon blacks have been compared by two methods with respect to their abrasion characteristics when compounded in rubber. In the first series of tests Grasselerator 808 was used as accelerator, the amount of which was varied t o compensate for the different curing characteristics of the blacks, while in the second series a constant amount of accelerator (diphenylguanidine) was used in all compounds. The blacks used included “ink-making” black, as prepared from natural gas by the channel process as regulated with view to the requirements of the ink trades; black ‘W,” a standard gas black as prepared by the channel p r o c e s s for the rubber trade; black “G,” a black prepared by the combustion of natural gas in a special type of app a r a t u s . Acetylene black, lampblack, and Thermatomic carbon are products requiring no further identification for present purposes. Series 1-The reFigure 2-Compound “A” Aged 4 Days sults for this series a t 70° C. are shown in Table I. The adjustment of the accelerator was such as to give stressstrain curves of the same shape, or in other words with maximum cure a t 40 to 50 minutes for all blacks except the first one tabulated, which is not used in rubber compounding but is given as of general interest. The compound with this black was obviously greatly oyeraccelerated in order t o cure it a t all. The results for the other blacks are in line with general knowledge, but are nevertheless interesting as a summary comparison. It will be noted that in general the:blacks having the lower initial abrasion loss show the greatest percentage degradation on aging and vice versa,
897
although acetylene black is somewhat of an exception to this rule. Table I-Abrasion Resistance Data. Series 1 100 pale crepe, 5 zinc oxide, 3 sulfur, 1 stearic acid, 25 black, accelerator as noted MODU- ULTIABRASION GRASSEL-CURE LUS MATE Loss RATIO TYPEOF ERATOR AT AT TEN- Before Aged Aged/ BLACK 808 14OOC. 500% SILE aginrr 4davsa Normal M i n . Kg.>sq. cm. -Ccr/kw-hr.Ink-making black 2 . 0 80 41 239 306 447 0.68 Gas b1ack“M” 1.5 40 197 310 254 362 0.70 Gas black “G” 0.5 40 183 253 296 392 0.76 Lampblack 1.6 40 180 221 430 487 0.88 Acetylene black 0.5 40 184 250 329 366 0.90 Thermatomic carbon 0.5 40 101 205 430 465 0.92 a Aged in Geer oven at i o o C. and the abrasion test blocks run 10 minutes under standard conditions with the removal of approximately 2 mm. of surface stock, as distinct from the tests shown in Figures 2 to 7 , which were r u n for 2 minutes only with the removal of approximately 0.4 mm. of stock.
Formula:
Series 2. The formula used and the re‘*. sults for the s e c o n d series of compounds are ehown i n T a b l e 11. The stock containing “ink-making” black is again obviously out of 2 line, being greatly un- s dercured even a t 140 m i n u t e s . Differences in the r e e n f o r c i n g power of the blacks are m o r e noticeable here than in Table I. The fast-curing (non-retardpk ing) blacks, “G” and the Figure 3-Compound “A” Aged 6 Days a t 70’ C. highest stiffening effect, a l t h o u g h black “M” definitely has the highest ultimate strength. The abrasion results are in the same general_order-,asin ‘the first series of tests. Table 11-Abrasion Resistance7Data. Series 2 CURE AT MODUI,USAT ULTIMATE TEN- ABRASION 140OC. 500% SILE LOSS Kg./ Lbs./ Kg./ Lbs./ Cc./ Min. sq. cm. sq. rn. sq. cm. sq. wi. kw-hr. Ink-making black 140 70 1000 95 1330 810 G a s hlack“h1” 100 183 2600 302 4300 299 Gas b1ack“G” 80 211 3000 243 3450 380 Lampblack 80 95 1320 184 2610 576 Acetylene black 80 218 3100 244 3470 416 Thermatomic carbon 80 98 1400 228 3280 464 TYPEOF BLACK
Effect of Aging on Abrasion
The use of the Grasselli abrasion machine permits the study of the abrasion resistance of aged stocks during the removal of progressive small increments of material from the surface inwards. The test is carried out under standard conditions, except that a large number of two-minute runs are made in series instead of one longer run, the blocks being always replaced in the same position with respect to the direction of abrasion as shown by the friction marks. If the abrasion loss for a series of such tests is plotted us. the total cumulative depth of abrasion as calculated from the cumulative total weight removed, the results are interesting. For the test stock compound “A” these results are shown in Figures 2 to 4, where the normal loss for the unaged stocks with blacks “M” and “G” is, respectively, 290 and 375 cc. per kilowatt-hour. It will be observed that for all aging periods this relation is reversed for the first millimeter of stock from the surface, black “M” now showing the higher abrasion loss. With an aging period of 8 days, but more particularly
I S D L-STRIAL A N D EXGINEERING CHEMIXTRY
898
Vol. 20, No. 9
600
s
2
500
6
f
T I M
JM
D+fLd Abmsm
Deys
Depth of Abrasi‘
mm.
Figure &Compound “A” Aged 10 Days a t 70° C.
& of w-c
mm’
Figure 6-Compound “B” Aged 4 Days a t 70° C.
Figure &-Surface Abrasion Loss us. Aging Time-Compound “A”
a t 10 days, the deg- a normal resistance (unaged) of 375 cc. per kilowatt-hour, radation of the rub- after 4 days a t 70” C.the abrasion loss in the middle of the ber is so complete that block (2.5 to 4.0 mm.) is only 300 cc. per kilowatt-hour, showthe differences in the ing that there has been considerable improvement due to s u r f a c e layer, owing aftercuring. The cures used in all these abrasion tests were to the type of carbon 10 to 15 minutes below the maximum cure as indicated by black used, are much the modulus a t 500 per cent elongation. The Geer oven less pronounced than test a t 70” C. has been compared unfavorably by some to a t an aging period of the oxygen-bomb test, being claimed to give results more 4 days. I n Figure 5 influenced by purely thermal deterioration from the long for compound “A” the exposure a t an elevated temperature, rather than resulting abrasion loss for the from the actual reaction of oxygen on the rubber. The data surface layer has been of the present tests show conclusively the deterioration to be plotted against aging progressive from the surface inwards, which obviously can time. I n order to give be done only to the action of oxygen, since there is no apprea c o n s t a n t basis of ciable thermal deterioration in the interior of the block, comparison the abra- but rather as noted above, a definite improvement due to sion loss has been read aftercuring. It should perhaps be emphasized that the from the various in- stiffening of stock due to overcuring will not improve the d i v i d u a l curves as abrasion results on a testing machine of the present type, Figure 7-Compound “B” Aged 6 Days that loss indicated at which measures abrasion per unit of work done. The fact at 70° C. a cumulative total of is, however, that the curves as obtained represent the re0.5 mm. depth of abrasion. Here it will be seen that after 2 sultant effect of two changes-first, a certain actual improvedays’ aging the relative abrasion loss of the compounds with ment due to aftercuring; and second, the deterioration from the two blacks is reversed, although a t and after 8 days the the surface inwards due to oxygen penetration, which pregeneral surface degradation is so great that the results are cludes their giving any accurate data as to the rate of such practically indistinguishable. There is no particular evi- penetration. dence from this curve of the existence of any induction period Conclusion in aging such as has been found by various authors, although it might be perceptible a t shorter aging periods. With respect to the results noted herein on the relation That the effect of the type of carbon black on aging is of abrasion loss to size of abrasive material, it is regretted not restricted to one particular compound or accelerator is that as yet no results have been obtained which cast much shown by Figures 6 and 7 for compound “B,”whose formula light on the reasons for the break in the curves of Figure 1. was 100 pale crepe, 5 zinc oxide, 3 sulfur, 1 stearic acid (25 It is possible, if not probable, that the unexpected phenomena black “M,” 1.5 No. 808) or (25 black “G,” 0.5 No. S08) both whose existence is indicated by these results may be responsicured 40 minutes a t 140” C. The normal abrasion of the ble for the commonly observed discrepancies between laboraunaged stocks of compound “B” for blacks “M” and “G,” tory abrasion tests and road tests. It is hoped that further respectively, was 254 and 296 cc. per kilowatt-hour. It studies may give more useful information on this relation. will be seen that here also, as with compound “A,” the relative The abrasion results on aged stocks present several imporabrasion resistance in the surface layer of the compounds tant points. It is, of course, obvious and fully realized that with the two blacks is reversed after aging. abrasion in use takes place only on the surface layer of a tire It might be expected that curves such as the foregoing tread, but attention has not before been called to the very would give a definite index of the rate of penetration of oxy- considerable effect which the aging characteristics of the gen into the stock on aging. Consideration of the curves, carbon black used in the stock may have on the abrasion rehowever, shows that the reactions occurring are too complex sistance of this surface layer. The present results show that, t o permit this. For example, in Figure 2 black “G,” having with two stocks differing only in their carbon black content,
INDUSTRIAL d S D ESGINEERING CHE.?IISTRY
September, 1928
the relative abrasion resistance of the surface layers after aging may be in reversed ratio to those of the unaged stocks. This evidently will be a decisive factor in the relative useful life of the tread if the rate of surface aging is greater than the rate of tread wear. Surface conditions or, in other words, the time of storage before use, the daily mileage, existing road and climatic conditions, etc., will determine this relation.
899
Acknowledgment The authors wish to express their appreciation to the Combustion Utilities Corporation for permission to publish these results, and t o C. J. Wright, chief technologist of this company, whose interest and support have made it possible to carry on this work. The cooperation of Ira Williams, of the Grasselli Chemical Company, is also gratefully acknowledged.
Quantitative Relations of the Countercurrent Washing Process1" Ludwik Silberstein E A 5 T M A V R O D A K COMPANY, ROCHESTER, pu'.
A
PROBLEM frequently encountered in chemical manufacture is that in which an insoluble solid is to be freed as completely as possible of a soluble substance accompanying it in such a way that the smallest quantity of washing liquid practicable is employed, with the minimum amount of equipment. This is accomplished by countercurrent washing exemplified in industry by the Rogers wet machine for washing wood pulp, and represented diagrammatically in Figure 1. The washing is carried out in a series of tanks, three being shown in the diagram, although any number may be used. In tank I is placed a definite quantity of insoluble solid which we shall assume to be wet with a definite quantity, a, of liquid. This wet solid is mixed with a definite quantity, b, of wash liquid taken from tank 11. When equilibrium has been established the resulting solution, amounting to b in weight, is removed from the system, and the insoluble solid, wet with a quantity a of liquid having the same concentration of solute as the liquor removed from tank I, is transferred to tank II?where it is mixed with a quantity b of liquor from tank 111. A fresh charge of insoluble solid, wet with the quantity a of the liquid containing the original concentration of solute, is placed SOLID + SOLUTE
p &
i
n
i
r-
It is desired to construct formulas from which to calculate the concentration of solute in the liquid adhering to the solid removed from the system a t the last tank. Derivation of Formulas Just as in actual practice, it is necessary to start the system with pure water in all the tanks, and to calculate the limiting value to which the concentration in each tank tends. The meaning of a and b being as above, put for brevity
-
b a f b p = 1, and therefore s = a0 5 a
+
fin,
=
(1) The concentration (p) of the solute in the liquid leaving the system from tank I. (2) The amount of liquid a, which adheres to the solid on transferring this from tank to tank. (3) The amount b of liquor transferred a t each washing. (4) The number of tanks m in the system. 1
2
ffp
+
Received February 2 5 , 1928. Communication No. 340 from the Kodak Research Laboratories.
n 2 2
PPn-,,,
(2)
For any further tank except the last Pnn
= aPn,m--i
+
n 2 2, m'
PPn-I,m+~
>m
2 2
(3)
and finally, for the last tank, Pnm'
in tank I and there mixed with the quantity h of wash liquid from tank 11. This process is continued systematically, the solid in the last tank I11 being treated with a quantity h of pure water. We thus have a series of unit quantities of solid, wet with quantity a of liquid, passing out of the system a t one end and a series of unit quantities b of a solution having definite concentration of solute passing out of the system a t the other end. We have as quantities a t our disposal:
(1)
amp
Pin
For the first tank, a t any stage after the first,
SOLUTE + WATEP 1
p=-
= a + '
so that cy which will be seen to insure the convergency of the series occurring in the sequel. Further, p being, as before, the given concentration of the liquid adhering to the solid on entering the system, let pnm be the concentration of the liquid in the mth tank in the nth stage and rn' the total number of tanks. Then the process described above can be formulated as follows : For any tank m, in the first stage
WATFD
n i--l
Y
= aPn, m'
-i
(3'1
Such being the recurrence formulas of the system, it is required to express in terms of the given numbers p , a , p any concentration p,, and especially to evaluate, for each tank, the limit corresponding to n = 03, which will be written Pam
= Pm
As even the form of these expressions depends materially upon the total number, m', of tanks employed, it is preferable to treat the case of each m' separately. To illustrate the procedure, as well as to cover the needs actually met with in practice, it will be enough to consider here the cases m' = 3, 4, and 5. THREE-TAKK SYSTEX-A repeated application of equations ( 3 ) , (3), and (3'), which in the present case becomes pn3 = a p n z gives for the nth concentration in the first tank the recurrence formula, P", = ffp spn-1.1 s*pn-_?1 , .. Sn-IlPll (4, where s = a 3 and, by ( l ) , pll = ap. Thus
+
p21 =
ap(1
+
+ SI,
p3l =
+ + a p ( 1 + s + 2s2), etc.
