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 F e b r u a r y 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 = apn2 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.
Vol. 20, No. 9
INDUSTRIAL AA-D EA'GINEERI.VG CHEMISTRY
900
Write
pnl = a p l l Then, by (4), an =
For the second tank, formula (2) gives directly
+ als + a2s2+ . . . + a n - - l s n - l ) + + + ai + 1
whence it can a t once be proved-by induction (from n to n + 1) that a, = Z n - l for any n. Thus Pn,= a p ( 1 s 2 9 . . . + 2s-2sn-1 1 or
+ + +
4
s)%-1
(5) vanishes for n = a. Thus the required limiting concentration in the first tank becomes
whence, for n =
a , the
p, =
a2pja1
=
+ azs + . . . + ans"-'l
with the previous coefficients, only shifted by one place. Thus, if we write fins = a3@ { 1 bis . . . bn-ls"-') (12) the coefficients b will be seen to satisfy the relation
+ +
bn
=
2P 1 - 2s
+
+ an+,
bn-1
(13)
This enables us to find the limit value of pn3 without ever considering the numerical values of these coefficients (apart from the first which is bl = 3). I n fact, subtract from the series (12) its product into s and the product of p,, into a 2 / s and pass a t once to n = . Then owing to the relation (13) the result will be (1 - s)P3 - ?Pi = a3fi11
limiting concentration
(11)
ffP)
limiting concentration
+
(13)
For the second tank we have, directly by ( 2 ) ,
a ,the
Xext, for the third tank, formula (3) gives 9.3 = orfin: s 9 n - i 3 . Kow, by (11) and (7),
p,, which holds for any stage (n). Kow, since s 5 I-, ( 2
-
$Pn+1,1
and thus also, for n =
an-,
an-i
1
Pnz =
+
(61
:I
- 1 - U J S - ai - -
and since b, = 3, a1 = 1, a2 = 2, the required limiting concentration, after substitution of P1from (I4), will become
a
Finally, for the third tank, by (3'),rpn3= apn2, and thus also for n = a ,Ps = aP2 or
(1114)
Finally, for the fourth tank, by (3'), P4 = ffP3. (Iv4) FOUR-TANK SYSTEM-In this case (3') is p,, = qn3, FIVE-TANK SYsTmf-In this case (3') becomes pn5= ap,, and a repeated application of (2), (3) leads to the formula and a repeated application of (2) and (3) gives the recurrence formula e n 1 = CUP sPn-1,i aiS'Pn-2, I *. . ~ n - & ' - ~ P u (14) reducing the nth stage concentration in the first tank to all in which the coefficients a turn out to be identical with those the preceding concentrations in the same tank. If we write appearing in the first series for the four-tank system-viz., 1, 2, 5, 13, 34, etc., satisfying the relation (9). If we write pn1 = cYp(aO a,s a2s2 . . . an-,sn-') (7) then = a1 = 1 and, as follows readily from (6), all succeedPnl = a p ( c o CIS . . . C,-,S"-') (15) ing coefficients will be given by the new coefficients will be determined by the recurrence an = a n - 1 1un4 2a,-, . . . 2n-*ao. (8) formula Thus the second and the following coefficients are 2, 5, 13, 34, Cn-I = Cn-z U1Cn-a a&%-4 f . f an-zCo (16) 89, etc. Now, these show the remarkably simple property, and the obvious value co = 1. This gives for c1 and the followan + = 3a, - an-, (9) ing coefficients 1, 2 , 5, 14, 41, etc., and these numbers have and that this property holds generally can be proved at once the simple property by the method of induction with the aid of (8). By (9) the C n + 1 = 36, - 1 (17) successive coefficients could be rapidly written down, but since whose general validity is easily proved by induction, with we are ultimately interested only in the limit of the series the aid of (16) and (9). A repeated application of (17) leads (7) for n = a,they may be left on one side. I n virtue of a t once to (9) that series can be summed a t once. In fact, subtract Cn = (3n-1 1) (17') from (7) its own value multiplied by 3s and add its value multiplied by s2. Then, by (9), all but the first and the last The series (15) is thus reduced to an aggregate of geometries1 two terms will cancel and the result will be progressions which are readily summed, giving, for any n,
+
+ +
+
+
+
+
+
+
+
+
+
+
=
ffP
1
1 X = -(3
On the other hand, s
+
- 2s - a,s" + U , , - , S " - ~ 1 - 3s + S'
Now, the ratio of the successive coefficients a, +l/a, steadily increases from 2 up to a finite limit X given, according to (9), 1 by X = 3 - -, i. e., X 2
+
+
+
;
Pn1
+
+ d'5)+ 2.6180
1
-.
Thus, for n
4
= m
- 3s + s2
=
a ,since
t
both s" and (3s)" tend to zero,
- 2s)
(Is) ffP !;I + (1 - s ) ( l - 3s) This being the limiting concentration in the first tank, that in the second tank will be, again by (2)j =
the products
a#, a, -ls"- 1 vanish, and the required limiting concentration in the first tank is P1 = ap(1 - 2s)
1
and for n
while for any n, by (2)and (E), pnl = a2P(c1 c2s c8s2 . . . c , s " - ~ ) (19) which will be utilized presently. The coefficients are as in p n l , only shifted by one place.
+ +
+
+
IXDUSTRIAL A S D ESGINEERISG CHEMISTRY
September, 1928
For the third tank the relations are particularly simple. In fact. by (2) and a repeated application of (3) and (3') one finds without trouble p n 3 = ffpn2 Spn-1 3 s2p7Z-Z 3 . . . Sn-'P13 (20) nhere p13 = a 3 p . Substituting p,, from (19) and writing pn3= a3p{1 e,s e,s2 . . . en-lsn-l ) it nil1 be readily seen that e,,-l = 1 e, + e , . . . en-a ci Thiq, together with (17') and (20). which gives pL7 = a 3 p il 35), i. e., el = 3, leads at once to e .. , = 3n 1 - (3s)" Thus pna = a3P 1 - 3s (21)
+
+
+
+
+
+
+
for the four-tank system, ap(1 - 2s) P1
+
+
+
+
+
901
=
PB
1 - 3s =
&(l
+ s2' p* = 1
- ;;p+
-
- 3s
s)
+ s2
s2)P4 = aPl
and for the five-tank system, P1 = ap(1 - 3s s2) P, =
+
(1 - s ) ( l - 3s)'
a2p(1 - 2s) (1 - s ) ( l - 3s)
+
Application
~
~
and for n
=
05,
the limit concentration
For the fourth tank a single application of (3) and (3') gives PPn-I,, = apn3 sPn-1,r Pn4 = aPna so that, all pn3being already known, pn4 is directly reduced to its predecessor in the same tank. A repelition of this process leads a t once t o pn, = 4Pn, spn-1 3 . . . sn-?p23) P-lPlr and since = a 4 p , nhile any p13 is given by (21)) n e have, after simde reductions. p n, = 2 % - 3 (3" - 1 p , (22) 1 - 3 s ( l - s 2 ! which holds for any n. Whence, for n = 03, the limiting concentration ff4P P4 = (IVK)
+
+
+
+ +
(1
+
- s)(l - 3s)
Finally, for the fifth tank, by (3'), pn5 = also, for:n:= a, P K =
Qpn4,
and thus (VK)
&C
To sum up, the limiting values of the concentrations are: For the three-tank system, p1 = aP(l-),p - ff2P p 3 = -a 3P 1-2s
*--
1 - 2s'
1 -- 2s
To illustrate these formulas by a practical example, consider a mass of material containing 37A pounds of adhering caustic soda solution of 40 per cent strength. Let this mass he leached by the countercurrent process in a series of tanks. each wash consisting of 700 pounds of liquid from the next tank. It is required to find the limit concentration of wash liquid in the last tank. The constants are 375 io0 -, s = 0.22iO p = 40 per cent, CY = 1075' - p = 1075 Thus, if three tanks only are used, the required concentration of the liquor to be discarded is ff3p - 3.1 per cent. 1 - 2s
+ s2 = 1.60
With the four-tank system it will be 1 - 3s per cent, and with five tanks
ff
5P
(1 - s ) ( l
- 3s)
= 0.84
per cent. The addition of a fourth tank results, therefore, in the recovery of 2.1 per cent, that is, of 7,875 pounds more caustic soda, and the addition of a fifth tank saves a further 3.0 pounds. With the same data, the concentration of the strong Rash liquid leaving the first tank amounts, in these three cases, by (I3), (I4), (I5),to 19.8, 20.8, and 21.0 per cent, respectively.
Inflammability of Automobile Exhaust Gas'sz G. W. Jones PITTSBURGH
EXPERIMENT STATION,
u. s. BUREAUOF
Composition of Exhaust Gas UTOMOBILE exhaust gas consists mainly of carbon dioxide, oxygen, carbon monoxide, hydrogen, methane, nitrogen, and water vapor. Gasoline vapor and unsaturated hydrocarbons exist to an appreciable extent only under abnormal operating conditions-for example, through faulty ignition or too rich carburetor adjustment. For normal conditions a t ordinary temperatures it may be assumed that the combustibles in exhaust gas are hydrogen, carbon monoxide, methane, and the inert gases, nitrogen and carbon dioxide. From 75 to 98 per cent of the exhaust gas consists of nitrogen and carbon dioxide, or 3.5 volumes or more of inerts for each volume of combustible. The combustibles carbon monoxide, hydrogen, and methane depend largely upon the carburetor adjustment or the air-fuel ratio of the mixture exploded in the engine. From
A
1 Presented b y G. D c'. Jones a n d G. St J. Perrott before t h e Division of Gas a n d Fuel Chemistry a t t h e 75th Meeting of t h e American Chemical Society, St. Louis, Mo., April 16 t o 19, 1928. 2 Published b y permission of t h e Director, U. S. Bureau of Mines. ( S o t subject t o copyright )
MINES, PWfSBURGH, P A .
another problem on the ventilation of vehicular tunnels3 a large amount of information was obtained on the composition of automobile exhaust gas with reference to carburetor adjustment. Table I gives typical analyses of exhaust gases collected during road tests in that investigation. The mixtures were chosen to cover a wide range of air-fuel rat'ios. As would be expected, the proportions of combustibles decreased with increasing air-fuel ratios. The proportions of inerts (air-free analysis) varied from 76.9 to 98.5. The small percentages of oxygen shown in the analyses are typical for samples of exhaust gas collected on road tests. Application of Law of Mixtu,res t o Inflammability of Gases One of the laws used by chemists and engineers for determining certain properties of complex gases is the so-called 8 Appendix ICo. 3, Tunnel Gas Investigations, Amount a n d Composition of Automobile Exhaust Gases, b y Fieldner, Straub, and Jones. Report of
New York S t a t e Bridge a n d Tunnel Commission, 1921, p. 91; Fieldner, Straub, and Jones, J. IND. EXG.CHEM.,13, 51 (1921); J . SOC.Automotive E n g , , 8 , 295 (1921); Bur. Mines, Repls. Inwesligalions 2226 (March, 1921); Fieldner and Jones, J . F r a n k l i n Insl., 194, 613 (1922); Fieldner and Jones, Bur. Mines, Repts. Investigations 2487 (1923).
