INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
2844
OTHERREGENERANTS
Because sodium hydrosulfite is a relatively expensive chemical, considerable effort was expended in a search for cheaper regenerants. However, potential agents which are relatively cheap, such as alkaline sodium sulfite and alkaline formaldehyde, are either very inefficient in the reduction of the copper-amine resin complex or do not work a t all. The only additional reducing agents found so far which work efficiently are sodium hydrosulfite and sodium formaldehyde sulfoxalate. The latter, prepared by the method of Wood ( S ) , is of interest in that the cupric ion-resin complex is reduced only to the cuprous state.
Vol. 41, No. 12
flow rates and larger beds would increase efficiency and reduce the 4.8-cent charge appreciably. The 4.8-cent figure is also somewhat inflated because excessive regenerant unquestionably was used. However, it is seen that the method is not competitive on a cost basis with sodium sulfite dosing nor is it competitive with mechanical deaeration of boiler feed water at high dissolved-oxygen concentrations. I t is anticipated, nevertheless, that there will be numerous commercial applications for this novel method of removing dissolved oxygen from aqueous solutions.
ECONOMICS OF THE OXYGEN-REMOVAL PROCESS
The cost of oxygen removal by this new method is a matter of prime interest. The largest item of cost in this method is the cost of regenerating the oxygen-removing resin. Other costs, such as the regeneration of the cupric ion trap and the occasional reimpregnation of the anion exchange resin with cupric salts, are not considered here but would be relatively small. Overhead and amortization items, although appreciable, cannot be calculated with any exactness here. The calculated cost of this treatment on the basis of a 25cent per pound price for sodium hydrosulfite and 4 cents per pound for sodium hydroxide is 4.8 cents per 1,000 gallons of water containing 1 p.p.m. dissolved oxygen. Data listed for sample 2, Table I, are employed in arriving a t this figure. Slower
ACKNOWLEDGMENT
hluch of the work here reported was carried out under contract with the Bureau of Ships, Navy Department, Washington, D. C. Permission to publish this information has been granted. LITERATURE CITED
(1) Milla, G. F., patent applied for. (2) Scott, W. W., “Standard Methods of Chemical Analysis,” 5tb ed., Vol. 11,p. 2079, New York, D. Van Nostrand Co., Inc.. 1939. (3) Wood, H., Chem. Age (London),38,85(1938).
RECEIVEDM a y 10, 1949. Presented before the Division of Industrial and Engineering Chemistry a t the 115th Meeting of the ~ M E R I C A NCHEXICAI, SOCIETY, San Francisco, Calif.
ractional Distillation of Multicomponent Mixtures NUMBER OF TRANSFER UNITS A. J. V. UNDERWOOD 38 Victoria S t . , London, S . W . I , England
Equations are presented for calculating the number of transfer units required for the fractionation of multicomponent mixtures. Constant relative volatility and constant molaI reflux are assumed.
Considering, for example, a ternary mixture and applying Equation 2 to each of the three components, then
THE
number of transfer units required to effect a given separation in a packed column is defined by Chilton and Colburn (1)as d&I dl\7 = 7:)
The same integral occurs in the expression given by Thormann (6) for the height equivalent to a theoretical plate in a packed column and in the expression given by Hausen ( 4 ) as a proposed conception of a theoretical plate. For binary mixtures matbematical solutions of this integral have been given by Chilton and Colburn ( 1 ) for the case of total reflux and by Hausen ( d ) , Dodge and Huffman (S),Colburn ( 2 ) and by the author (8,9) for the case of partial reflux. In all these cases constant relative volatility and constant molal reflux were assumed. For multicomponent mixtures, when constant relative volatility and constant molal reflux are assumed, the equations corresponding to Equation 1 can be integrated readily by making use of a mathematical transformation that has been employed by the author (‘7, 10-18) for calculating the number of theoretical plates. Differentiating Equation 1 gives
-
(3c3
Yl
where the subscripts 1, 2, and 3 denote the components.
Xow
ya = 7?1X3 i- a3
and
* Ys -
ff3x3 a323
f apx2
+
01151
mith similar expressions for the other two components and Equations 3a, b, and c become
INDUSTRIAL AND ENGINEERING CHEMISTRY
December 1949
2845
or
+
+ +
For an enriching column m = R / ( R 1)and a3 = X ~ . L J / ( R 1 ) s etc. For a stripping column m = (S l)/S and a3 = - x ~ , w / S , etc. Let 4 be a quantity defined by the equation a2az
a3011
a 3 - + + F 3
dl
+aloll=l
(5)
a1-4
Multiplying Equations 4a, b, and c by a I / ( a 3 - 4), ~ / ( 0 1 2 - +), and O I I / ( L Y I - 6,)respectively, adding them and substituting from Equation 5
d
013x3
+
1 012x2
+
011x1
(
a3
I
-4
or
There are two similar equations, one involving 42 and 41 and the other involving +I and +8. There are, however, only two independent equations in all. Equation 11 or one of those similar to it gives the number of transfer units required to effect fraction& tion from the composition denoted by z' to that denoted by 5". The method of integration has been given for a. ternary mixture, It can obviously be applied in exactly the same way to a mixture of any number of components. The number of equations similar to Equations 3a, b, and c and 4a, b, and c will be equal to the number of components and there will be appropriate additional terms in Equation 5 and in Expression 7 for E ( x , 4 ) . There will finally be obtained equations exactly similar to Equations 10 and 11 involving as many values of + as there are components. The method of calculation may be illustrated by reference to a ternary mixture for which a3 = 4, a2 = 2, and a1 = 1. With a reflux ratio of 3, a mixture of composition x3 = 0.3, x2 = 0.3, and $1 = 0.4 is to be concentrated in an enriching column to give a product, x 3 , D = 0.999, X ~ , D= 0.001, and X ~ , D ,being very small and unspecified. The original composition is denoted by x' and the product composition by 5". For an enriching column, Equation 5 becomes
or
4
x
0.999
4-4 Equation 5 is of the third degree and is satisfied by three values of + which will be denoted by $11, +z, and +a (in ascending order of magnitude).
-
E ( z , 4 ) CEZ 01353 013
-4
%XI
01222
+
012
-++
0
1
7
. .
. .
x 0.001 = 2 - 6
from which $13 = 3.0015 and 62 = 1.999. (There is also a solution of this equation 41which is very nearly equal to 1 and,which corresponds to the term X ~ , Dwhich has been neglected aslbeing very small.)
(7)
While E(x, $I), E ( x , +2), , denotes this expression when 4 = 4 = 42, * * For the three values of 4 there are three equations corresponding to Equation 6-namely,
$1,
2 +
E(x'' ")
4
4 X 0.3 2 X 0.3 - 3.0015 + 2 - 3.0015 '1
4 -0'3.0015
= 0.403
Similarly
E(x', 42) = 600.2 When x: = ~ 3 , etc., ~ , are substituted in Expression 7 for E ( q - 4 ) it will be seen from Equation 12 that E ( x , $) is then equal to ( R 1) for all values of 4. Thus
+
E(z", +3) = E(x", $2)
i=
4
Insertingnumerical values in Equation 11 then gives N = 19.56. The number of theoretical plates, n, required for this separation can, for comparison, be calculated from the appropriate equation as given elsewhere (7, l0-16)-namely,
Integrating between the limits represented by x' and x"
from which n = 17.99. CASE OP TOTAL REFLUX. For total reflux, R = m and the solutions of Equation 12 are 4 3 = 013, +2 = 012, and 41= cy1. The corresponding equation for the stripping column has the same solutions, For these values of 4 all the terms E(x, 4) in Equation 10 or 11become infinite. However,
2846
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Adding these threc equations ttnd noting that
Thus, for total reflux, Equations 10 become SO
that, eliminating dyl/dL
and Equation 11becomes
~ r d integrating, :
These equations can also be derived directly from Equations 4a, b, and c. For t,otal reflux, m = 1 amd aa = a = ai = 0 giving
Equation 21 must hold good for all points in the colurnri , Then fixing the value of y3 will fix the value of (and also the value o! yl). Thus, whatever the length of the column and the reflux ratio, there would be, for a specified vdue of ya, only one corn. position that could be fmctionated to give the required product For a mixture of any number of components there would, nocord. ing to this analysis, be one degree of freedom less when a. frao. tionation is carried out in a packed column than when it. is car. ried out in a plate column. There is no evidence which would support this conclusion. It must, therefore, be concluded that the coefficients MI, iM2,etc., in Equations 17a, b, and c are the same for all components so that Equations 18 and 19 are identical Equations 17a, b, and o then become the same as Equations 3tt, b, and c. The equality of the coefficients M implies equality aQ tjhemass transfer coefficients for all components. The integration of Equation 20 to give Equation 21 asfiurrm t,hat MI, &I*,and M a are constants. This was also assumed by Xlurphree in Equation 16. It might be conceived that the n~am transfer coefficients are different for different coniporients and t,hat the values of M 1 , M z 9 and Ma vary with composition, If this were the case, the terms dN in Equations 3a, b, and c w o d d not be identical. This would mean that it would not be possible to state ~l definite number of transfer units required to effect a given separstion. Instead there would have to be stated a different nurnbej of transfer units for each different component. I n such circumstances the conception of a transfer unit, as the measure of the difficulty of effecting a given separation, would come to have little meaning. The conception of a transfer unit is largely a question of defrnition and it appears legitimate to use equations such as Equations 3a, b, and c with identical terms d N as a logical extension to multicomponent mixtures of the conception of transfer unite that has been accepted for binary mixtures. It might well be the case that experiment,alinvestigations designed to examine the relationships expressed by Equations 20 and 21 would vieldi fruitful results.
oiaza
+
1
+
~ZZZ
~
z
which, on integration, gives Equations 14. The number of theoretical plates required with total reflux tor the separation of a multicomponent mixture is the same as for a binary mixture if the ratio of the key components in the top product and the ratio in the bottom product is the same for both mixtures. A similar relation does not hold for a packed rolumn. LVriting Equation 15 in the form
i t will be seen that the number of transfer units is not determmed solely by the ratios of the key components in the top and bottom products. Equations -la, b, and c with total reflux are the same as thr Rayleigh equation for differential distillation if dx/dN is replaced by w .dx/dW. Thus the integrated equations for differential distillation of a multicomponent mixture are obtained by ITplacing ,?' by In w"/w' in Equations 14 and 15. XASSTRANSFER COEFFICIENTS.Murphree (6) has presented equations for a packed columr, which, for earh component, are nf the form
where L is the length of column required to effect the specified separation and $1 is a coefficient depending on vapor velocity, area of contact, and transfer coefficient. L is obviously proportional to the number of transfer units. Murphree assumes that the coefficient iM is different for each component, Thus, for a ternary mixture, differentiating the equation corresponding to Equation 16 for each coinponent gives: (174
t
NOM ElVCIATURE
= intercept of operating line m = slope: of operating line n = number of theoretical plate8 z = mole fraction of component, in liquid y = mole fraction of component,in vapor LC
December 1949
INDUSTRIAL AND ENGINEERING CHEMISTRY
y * = mole fraction of component in vapor in equilibrium with
liquid of composition L = length of column M = coefficient as defined in Equations 17a, b, and c N = number of transfer units R = refluxratio s' = reboilratio E(z, 4 ) = function as defined by Expression 7 LY = relative volatility 4 = parameter defined by Equation 6 Subscripts 1, 2, 3!, etc., are used to identify a component. Superscripts, as ' and , are used to denote the values of a component, a t different points in the column. LITERATURE CITED s
ip
(1) Chilton, T.H.,and Colburn, A. P., IND. ENG.CEEM.,27,255-60 (1935).
2847
(2) Colburn, A.P.,Ibid., 33,45%67 (1941). (3) Dodge, B. F.,and Huffman, J. R., Ibid., 29, 1434-6 (1937). (4) Hausen, H.,"Der Chemie-Ingenieur," Vol. I, Part 111, Leiprig, Akad. Verlags., 1933. (5) Murphree, E. V.,IND. ENG.CHEM.,17, 960-4 (1925). (6) Thormann, K.,"Destillieren und Rektifizieren," Leipzig, Spamer, 1928. (7) Underwood, A. J. V., Chem. Eng. Progress, 44, 603-14 (1948). (8) Underwood, A. J. V . , J . Inst. Petroleum, 29, 147-56 (1943). (9)Ibid., 30,225-42(1944). (10)Ibid., 31,111-18 (1945). (11) Ibid., 32,598-613 (1946). (12)Ibid., pp. 614-26. RECEIVEDApril 1 , 1949.
Viscosity of Pulping Waste Liquors KENNETH A. KOBE' AND EDWARD J. McCORMACK2 University of Washington, Seattle, Wash.
In the wood pulp industry the sulfite, sulfate, or soda process may be used for the pulping process to give a cellulose fiber. In the alkaline processes it is usual practice to concentrate the liquor and burn it to recover the soda base. I n the sulfite process it is becoming necessary to carry out some processing prior to disposal. For all of these waste liquors a knowledge of their viscosities is necessary for calculations involving pumping costs, pressure drop in fluid flow, heat transfer coefficients, and cost of concentrating the liquor. Because the viscosity of the waste liquor is largely due to the dissolved sugars and colloidal lignin molecule i t was hoped that some general relation might exist among these three liquors so that the viscosity of a l l could be represented by one set of curves with an accuracy sufficient for most engineering calculations, as was previously found to be the case with the specific heats of these liquors (2,3).
s
w
v
ULFITE waste liquor, sulfate black liquor, and soda black liquor were obtained from pulp mills of the state. The sulfite waste liquor was from the pulping of western hemlock a t the Soundview Pulp Company, Everett, Wash. The sulfate black liquor was from the pulping of western hemlock with a small amount of Douglas fir a t the St. Regis Kraft Company, Tacoma, Wash. The soda black liquor was from the pulping of cottonwood with some hemlock a t the Everett Pulp and Paper Company, Everett, Wash. The liquor was taken directly from the digester. Before use the liquor was strained to remove any fibers or particles that might clog the capillary of the viscometer. Part of each sample was evaporated to approximately three quarters, one half, and one quarter of its original volume, This evaporation was conducted on the stream plate at approximately 80" C. by allowing 1 liter of the liquor to evaporate, portions being removed a t each of the three desired concentrations, Each time care was taken to decant the liquor from any objectionable solid matter that might be present due to surface film formation. Total solids was determined on each sample by pipetting a 5-ml. sample onto 10 grams of sand in an evaporating dish. The sample was placed on a steam plate for 24 hours and in an oven a t 105" C. for another 24 hours, An Ostwald viscometer was calibrated against glycerol-water 1 Present address, Department of Chemical Engineering, University of Texas, Austin, Tex. 9 Deceased.
solutions over the range of temperatures and viscosities found in the waste liquors (6). The viscometer was placed in a small thermostat, the temperature of which could be varied from 0 " to 100"C. at approximately 20" intervals. It was not possible to cover the entire temperature range with the most concentrated samples because the liquor would not flow through the capillary a t the lower temperatures. Even the unconcentrated solution possessed a certain amount of gel-like properties below 20" C. The specific gravity of each sample was determined by means of a hydrometer graduated to 0.002 unit. The experimental values of density and viscosity for the various concentrations of solids over the temperature range used are recorded in Tables I, 11, and
111. Various methods of plotting were attempted to produce a straight line for viscosity and temperature, such as the reciprocal of viscosity against temperature ( O K.) or reciprocal of temperature, or the logarithm of viscosity against the logarithm of temperature. All of these methods gave a certain amount of curvature in the lines a t lower temperatures. An Othmer plot (6) of the logarithm of the viscosity of the waste liquor plotted against, the logarithm of the viscosity of water a t the same temperature
TABLE I. VISCOSITY OF SULFITE WASTELIQUOR Total
Solids, %
Temp.,
11.2
0.2 20.8 38.6 62.8 65.8 74.8 81.0 96.8
c.
Density arams/cb. 1.065 1.057 1.049 1.041 1.040 1.036 1.033 1.027
Viscosity,
CP. 3.072 1.620 1.083 0.712 0.673 0.605 0.657 0.468
15.0
26.6
0.3 20.0 42.8 60.4 77.7 94.8
1.133
1:ii4 1.103 1,094 1.083
9.36 4.200 2.462 1.672 1.249 0.988
