Nornographs for Mean Driving EDWARD G. SCHEIBEL AND DONALD F. OTHMER Polytechnic Institute, Brooklyn, iY.1 .
HE use of mass transfer coeificients and the related values of transfer units which are necessary in the design of diffusional equipment For distillation, gas absorption, extraction, etc., depends on the mean driving force which causes diffusion to take place. Tliii driving force is the difference between the concentration (or partial pressure) of the diffusing material in one phase and the Concentration (or Courtesy, iM. W. Kellogg Company \VA.TER COOLIYG T O W E R I N PALESTINE, WHICH, BY HUVIDIPIC4TION OF AIR, ILLUSTR4TE3 partial pressure) of the same maTHE SIMPLE DIFFUSIONAL O P E R ~ T I OOF N DESORPTION terial in equilibrium with the other phase. The solution of diffusional problems requires the knowledge of the effective mean throughout the system of this driving force, geometrically the distance between the assuming that the K values of the components vary linearlgequilibrium line and the operating line. The theory mill he with 2.) applied first to gas absorption. It is important to note that the integrated mean driving force, which is the true mean driving force in systems where Gas Absorption the concentrations of the gas and liquid vary continuously The mean value of this driving force is obtained easily if with respect to one another, as in packed columns, is given by the equilibrium line and the operating line are each assumed the formula: to be straight. Then the driving force varies linearly with either the ordinate (or the abscissa) on an x, 21 diitgram; and A Y M = Yl - Yz the integration of the differential equation of mass transfer obtained with this assumption results in a log mean of the This form is preferable to the same expression, as usually terminal driving forces as the effective over-all driving force. given with N T on the left-hand side, because it seems that the A straight equilibrium line is rare, however; and in gas abmean driving force is a more fundamentul physical concept sorption problems (which may be considered as a first exthan is the value for the number of transfer units. ample) even in dilute solutions where Henry’s law applies. The formula derived by the authors was: the heat effects of the diffusional operations are such that the equilibrium line is curved. N r = 2.3 (Y - Y2) log u4-s In the study of the curvature of the equilibrium line for the s u-s case of a simple absorption problem, the authors arrived a t a formula (6) which applied to absorption problems where the Combining these two equations, change in liquid temperature in the column is approximately s proportional to the amount absorbed. (This is the usual asAY>lf = U+S sumption in calculating the equilibrium line when exact data 2.3 IO^ T/ - s are unknown.) The formula was based on a curvature of the equilibrium line such that the value of slope m of the line, between a point on the equilibrium curve and the origin, varies linearly with 5. The formula, therefore, was extended to the calculation of other types of diffusional operations such as Similarly, for the case where the values under the square distillation and extraction, and gave an expression for the root sign are negative, the following formula applies : number of transfer units in the system as a function of the 8’ total increment of the change in concentration in one of the AYM = (4) S’ phases. (The definition of m = v*/z is the same as the K 2 tan-l U value which is so commonly used in multicomponent distillation. With a two-component system this amounts to
T
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Forces in Diffusional Problems and for the particular case where S = 0, AYM = U/2
Nomographs have been developed to determine the mean driving force involved in different diffusional operations to make possible the design of industrial equipment. The use of the nomographs requires the preliminary calculation of two values, U and S, which are functions of the terminal given concentrations and equilibrium conditions of the system under consideration. The choice of the proper nomograph depends upon the relative values of these two functions. The use of the nomographs simplifies to a great extent the application of the formula recently proposed (6) for use in the design of equipment in such diffusional operations as gas absorption, liquid extraction, and distillation.
(5)
This last expression also holds with an error of less than 3 per cent if 8 is less than 0.3U. Using the original formulas, it was found difficult to calculate an accurate answer if U was nearly equal to S, since the denominator of the log term became very small. So a form which could be more accurately applied to slide rule calculations was derived for this limiting case, and from that form there results:
These formulas all refer to the operating line above the equilibrium line; however, the opposite form can be derived in which all values of y1 and %XI as well as y2 and mlxz are interchanged. While these formulas appear somewhat cumbersome. most oDerations consist of mecia1 cases which
'
sorption or extraction is carried out with solute-free feed, xa = 0, and several terms drop out; and in distillation xz = gz in the stripping section or x1 = y1 in the fractionating section to give a slight simplification also. The formulas 8s given above apply to the mean driving force based on gas concentrations; but if for any reason it was desirable, the analogous derivation could be carried out to obtain A X M ,in which case the formulas would be of the same form with all of the y and z values interchanged and m = x*/y instead of g*/x. Similarly, the formulas for molar ratios or weight fractions may be used if desired.
Construction and Use of Noniographs To facilitate the application of the formulas to design work, nomographs were drawn which solve the several formulas graphically. They eliminate a major part of the mathematical work because, while it is still necessary to determine the values of factors U and S, the operations involved in the calculation of these values are chiefly addition and subtraction which can be done more rapidly arithmetically than graphically. Figure 1 is the nomograph for solving Equation 3. The reference line (pivot line) is drawn horizontally and uniformly EXTRACTION BATTERY IN A CHBMICAL PLANT, THB DESION OF WHICHMAY BE ASSISTED OB Trim PERFORMANCE CHECKED BY rnE ACCOMPANYING NOMOGRAPHB Couriesy, Vulcan Copper & S u p p l y Company
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calibrated from 1 to 19. (In this, as in Figures 2 and 3, the calibrations on the reference line are of more or less use in construction of the nomographs. These calibrations indicate values a t some step in the final solution, but these values do not have t o be read to obtain the desired answer. It is felt, however, that since the reference line is a “pivot” line, the calibrations may well be retained as an aid in the use of the nomograph.) The U scale is drawn perpendicular to this line a t unity, and the S scale is drawn paraliel to the U scale and two units distant to the right. This construction gives values of the relation ( U S)/( U - S) by the intersection of a line through the values on the L7 and S scales with the reference line. The U and S scales are both reciprocally calibrated over a convenient range. The reference line is temporarily calibrated logarithmically and the S and A Y v scales are then obtained by actual construction, by fixing two starting points on the AI-?{scale and calculating intermediate points and corresponding values on the S scale, according t o the original formula. In using Figure 1, the line between the desired values of U and S on the vertical scales is dravin and extended to cut the reference or pivot line. Through the point so located, a second line is projected through the value of S on the other S curve to cut the AY,ii curve a t the bottom. (Because of the manner in which S appears twice in this particular formula, it was necessary to have two S scales, on each of which the same value of X is taken.) The value on the last scale is the desired value of AYM, which is then substituted in Equation 1 to give the number of transfer units. Any values of S and U may be used in this nomograph, provided the proper scales are multiplied by the same scale factor. Thus the two vertical scales of U and S must be multiplied by the same scale factor (powers of 10 are, of course, the simplest) t o determine the point on the reference line, and then the value of AYu can be obtained by applying the same scale factor to the other S scale and the A Y Mscale. (This factor does not necessarily have to be the same as the previous one, but it will be the same if powers of 10 are used.) Thus if S = 0.8 and U = 1.2, the line through S = 0.008 and C = 0.012 gives the desired point on the reference line, and the line thiough 0.008 on the other S scale gives a reading of 0.0050 on the AY,wscale, so A Y M = 0.50. The insert sketch of Figure 1 demonstrates the use of this nomograph in the solution of part B of the example in gas absorption, given later. Figure 2 presents the nomograph for Equation 4. The reference line is first drawn and uniformly calibrated from 0 t o a or an even number beyond. With a radius of n / 2 the two S’scales are drawn as shown, having a center at n/2 on the line. Similarly the positive and negative U scales are drawn with centers a t 0 and T and the same radius. The U and S’scales are calibrated from the relation S’/U tangent of the angle in radians indicated on the reference line by the intersection of a straight line between values of S’ and U . Thus in Figure 1, the value U = 0.1 was located as a starting point directly above the value of n/4. From this point, lines were drawn through the calibrated points on the reference line and extended to the S’ scale; and these intersections were calibrated so that the tangent of the angle in radians on the reference line n‘as equal t o S’/U. T h e n the S’ scale had been calibrated, the positive U scale was calibrated by projecting all the calibrations through point a/4 on the reference line; and the other S’ scale was obtained by projecting the same calibrations through the point a/2. The negative U scale \vas then obtained by projecting the calibrations of the second S’scale through point 3n/4. The A Y Mscale and the upper horizontal S‘ scale are drawn so as to constitute the usual type of nomograph for multiplication and division with the scale on the reference line.
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Thus, a value on the upper horizontal S’ scale divided by the value on the reference line tan-’ (S’/U) gives the desired value on the A Y Mscale, since the factor 2 in the denominator is taken care of in the calibrations. If U in Figure 2 is positive, the line between the calculated value of U and S‘ on the arcs a t the left-hand side is drawn to cut the reference line. Then a line is projected through the point so located and the proper value of S’ on the upper horizontal S‘ scale; the value of AY,Mis read by the intersection of this line with the diagonal scale. (Here again two scales are used for the same value of S’ because of the manner in which S‘ is involved in the equation.) If U’ is negative, the point on the reference line is located by connecting the proper values between U and S on the arcs a t the right-hand side of Figure 2. This nomograph may also be used for all values of U and S’, the requirements being the same as for Figure 1-namely, that the circular scales of U and 8’ be used with the same scale factor, and that the horizontal S’scale and the AYM scale be used with the same scale factor. Thus if S’ = 0.8 and U = 1.2, the point on the reference line is located by the intersection with the line between 0.08 and 0.12 on the circular S’ and U scales; and the reading on the A Y M scale is made by projecting a line from the reference point through S’ = 0.08 to 0.068. Thus the value of A Y M = 0.68. The insert sketch of Figure 2 demonstrates the use of this nomograph in the solution of part B of the problem in gas absorption. Figure 3 represents Equation 6. To construct this nomograph, which is of the set-square type (using the two legs of a right angle as connecting lines) combined with a three-scale type (using a single straight connecting line), the reference line is first drawn and calibrated uniformly from 0 to 200. The (yl m2x1 V ) scale is drawn perpendicular to the reference line a t the zero point; and this scale is calibrated so that the distance equal to 10 on the reference scale corresponds to 0.01. The V scale is then drawn parallel to the reference scale and a convenient distance below it (in this construction a distance equal to 10 on the reference scale was chosen). The V scale is calibrated with the zero point below the center of the reference scale and a distance of 10 on the reference scale equal to 0.001. The (yp - mlxz)scale is drawn parallel to the V scale and a distance equal to 10 on the reference scale below it. This scale is calibrated the same as the V scale. The values on the reference line correspond to the fraction (y1 - m2x1 V)/(yz - m1x2- V ) . The S and A Y Mscales are obtained by actual construction: two points are fixed on the A Y M scale, and the S scale is calibrated along with the rest of the A Y Mscale by the relation given in Equation 6 . I n using this nomograph a right triangle is laid on the diagram so that one leg passes through the given values of (y2 - mlx2)and of V on the corresponding lower horizontal scales. The triangle is adjusted so that the other leg passes through the proper value of (yl - mzxl V ) on the vertical scale a t the right. The point where this second leg cuts the reference line is noted; and a line is projected from this reference point through the proper value on the S scale to give the desired value of A Y M on the upper scale. The value of (y1 - m2xr V ) is represented by a single scale, because in a great many cases the value of V is negligible with respect t o (yl - mgc,) so the scale becomes simply (y1 - mfll). In problems where the operating line is below the equilibrium line, this last or approximation Equation 6 becomes
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FIQURE 2. NOMOGRAPH FOR DETERMINATION OF MEANDRIVING FORCE WHENVALUEOF
,\here
,~
( w 2
- XIYI)(nl' - ))h)
-
~ Z X I m1x2
+
- VI
and the values of (m2sl - yl +'V), (m1z2- yz), and V must be used on the (yl - mpzl VI, ( 2 ~ 9 - mlzz), and V scales, respectively.
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Is NBQATIW
Figure 3 ako lends itself t o the determination of A Y M for the entire range of values. The conditions to be satisfied are: (a) The same scale factor must be used on all three scales in the primary set-square nomograph to determine the point on the reference line; and (b) the value of A Y M is determined, applying the same scale factor on both the R and A Y v scales.
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INDUSTRIAL AND ENGINEERING CHEMISTRY
In the few instahces where A Y M falls outside the range of values on the scale, a scale factor of 2 may be readily applied to bring the value of AYMon the given scale. The insert sketch of Figure 3 demonstrates the use of this nomograph in the solution of the problem in liquid extraction.
Solution of a Problem in Gas Absorption PARTA. I n the work of Dwyer and Dodge (8) on the absorption of ammonia in water, the number of transfer units in run 73 was calculated according to the basic assumption in the derivation of the formula; that is, the value of m (which is directly dependent upon temperature) varies linearly with the amount of ammonia absorbed. The data give y1 = 0.01851, y2 = 0,00120, x1 = 0.00502, and z2 = 0, and the equilibrium data for ammonia in water give ml = 1.092 and ma = 1.138; hence the calculated values of U and S are 0.01400 and 0.01155, respectively. On the nomograph of Figure 1, the value on the reference line is 10.5 and AYM = 0.00498. Thus N T = 3.48. The calculated value of NT is 3.51 and the slight difference may be attributed to manipulation in using the nomograph. PARTB. In the same work run 64 was calculated by the same assumption. The data give the values of yl = 0.01977.
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= 0.00803, 21 = 0.01535, and rz = 0; and from the equilibrium datami = 0.978and m2 = 1.511. Thus U = 0.00460 and S’= 0.01159. In Figure 2 the value on the reference line is found to be 1.205, and AYM = 0.00482. Thus N T = 2.44, compared with a calculated value of 2.42 from the original formula. 1/2
Solution of a Problem in Liquid axtraction The case of liquid-liquid extraction rarely presents a straight operating line since the presence of the solute, soluble in both solvents, usually causes them to be mutually soluble in each other to a greater extent than when each solvent is present in the pure state. The theory for adopting the fundamental formula to these problems was fully discussed in the previous paper (8) together with the special methods for obtaining accurate results in this case. One of these methods is illustrated in the following problem in conjunction with the application of the nomograph to extraction. A 5 per cent aqueous acetone solution by weight is to be extracted with methyl isobutyl ketone down to a concentration of 0.07 per cent acetone by weight. A solvent-solution ratio 50 per cent greater than the theoretical minimum is to be used. In this problem the equilibrium data of Othmer, White, and Trueger (6) were employed. The theoretical minimum solvent-solution ratio was found to be 0.518, so the actual ratio used was 0.776. The operating data give the following coordinates for the operating line; 21 = 0.0613, y1 = 0.05, z2 = 0, and y2 = 0.0007, where y refers to the water solution and x refers to the solvent phase. The equilibrium data give y: = 0.0287 and the slope of the equilibrium line, dy*/dx = 0.392 a t 2 = 0. In this problem, since AY1 is more than five times AYz, the point-slope method previously proposed (6)was used. The formula for obtaining a value of na~, corrected for straightening the operating line so that the formula may apply, is:
ml =
= 0.468 21
From these values U = 0.0256 and S = 0.0242, so that Figure 3 must be used. The values of (y2 mlzd, V, and (VI - m2x1 V ) are 0.00070, 0.00010, and 0.0250, respectively. These values applied to the set-square nomograph give 43 on the reference line; and from this the value of AYM is found to be 0.0065 compared to a calculated value of 0.00649, Thus N T is 7.59. In ‘this problem weight units were used; and to give the final answer o n a m o l e b a s i s , t h e value of the correction terms derived by Colburn (1) becomes
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Courtesy, Vutcan Copper & Supply Company
DISTLLLING-COLUMNS FOR PRODVCTIOX OF ETHANOL BY VAPOR RE-USE, AN ADVANCEDUSEOF THE DIFFUSIONAL OPERATIONOF RECTIFICATION
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ii
>= a
FIGURE 3. NOMOGRAPH FOR
DETERMINATION OF MEANDRIVING FORCE WHEN VALUE OF TEE DISCRIMINANT, S, IS GREATER Tn.4~0.9 u
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+0.014 which in this case is practically negligible. The number of transfer units calculated either by the nomograph or by the algebraic formula is 7.59; and the number determined by graphical integ r a t i o n is 7.39. Thus t h e error is 2 . 7 per cent.
Solution of a Problem in Distillation The following problem demonstrates the use of the nomographs in the solution of a more difficult diffusional problem. The distillation of the acetonewater system was chosen because of its peculiar equilibrium curve; it illustrates the use of all Courtesy, M . W . Kellogg Company three nomographs, and also DISTILLING TOWERS FOR SEPARATION OF PETROLEUM PRODUCTS IN PALESTINE, ILLUSthe special case where a TRATING THE DIFFUSIONAL OPERATIONOF RECTIFICATION nomograph is not necessary, in the solution of a single problem. The problem is difficult and interesting because the unusual shape of the equilibrium curve prevents a complete solution by previous mathematical methods; the only other the value of x = 0.20. The equilibrium data and the given means for evaluating the system accurately is the graphical conditions of the problem are summarized as follows: integration method. As shown in the previous paper, such a Reference problem may be broken u'p into its several component parts Line No. X Y m for solution of each individually. 2 - 0 977 -0.711 0.563 It is desired to recover 99 per cent of the acetone present 3 -0.800 -0.584 0.310 4 -0.500 -0.370 0.400 at a concentration of 10 per cent by weight in a water solution, 5 -0.045 -0.046 0.750 as an acetone product 99 per cent pure by weight, using a NOT^. In the application of the formula to the fractionating section, the origin is taken a8 (1.0.1.0) and the values are used algebraically b y oonsiderreflux ratio of 2.5. The number of transfer units in the ing the signs of the terms. stripping section and in the fractionating section are to be calculated. Using the values corresponding to 4 and 5, the following In order to give a straight operating line on the x, y diavalues of U and 8' were calculated for fractionating section A: gram (i. e., to accommodate for the difference of the molar U = 0.022, S2 = -0.00717 or S' = 0.0847. Figure 2 gives latent heats of acetone and of water), a fictitious molecular AYM = 0.0318 for the values of U and S'; thus N A = 10.2. weight of 84 was used for acetone. The equilibrium data of Similarly, calculation of U and S for fractionating section York and Holmes (7) were used; and the value of m a t z = B, using values corresponding to 3 and 4 in the above table, 0 was extrapolated to give a value of 33, which corresponded gives U = 0.479 and S = 0.0313. I n this case S is less than to the value obtained by Levy (4) in the extrapolation of the 30 per cent of U; thus the nomograph is not necessary, and original data of Bergstrom given by Hausbrand (3). This AYM = U/2 = 0.239 or N B = 0.90. and other parts of the problem are indicated in Figure 4. The values of U and S for fractionating section C are calThe stripping or exhaustion section of the column, S , has a culated from the values corresponding to 2 and 3 in the above normal-shaped equilibrium curve; therefore the mean driving table and U = 0.550, S = 0.263. For these values Figure 1 force and the number of transfer units were calculated by gives AYM = 0.27 or N O = 0.47. considering this section as a whole. The given conditions of The number of transfer units in the system was determined the problem show that the coordinates of the ends of the by application of the formulas algebraically and by the operating line are 2 1 = 0.000232, y1 = 0.000232, 22 = 0.0232, following agreement: and yz = 0.289; and the equilibrium data give ml and m2 equal to 33 and 19.8. Thus the calculations give U = 0.481, Read from Calcd. Graphiqal S = 0.476, m2zl - yl = 0.477, mls2-yz = 0.00437, and V = Section Nomograph8 Algebraically Integration 0.00172. From Figure 3, AYM = 0.090, whence N E = 3.2. Reotifying (A + B + C) 11.57 11.50 10.8 Stripping (S) 3.20 3.15 3.2 The upper or rectifying section is divided into three parts, Total 14.77 14.65 13.5 A, B, and C. The first dividing line is approximately a t the point of inflection of the equilibrium curve-i. e., x = 0.50. The second dividing line is approximately a t the point where Thus the error in the use of the nomograph over the value the tangent to the equilibrium curve passes through the point calculated algebraically by the formula is less than 1 per cent; (1.0, 1.0) as shown in Figure 4. This occurs approximately at and the exact application of the formula gives an error of 8.5
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If S is less than 0.9 bi but more than 0.3 U , determine A Y Mfrom Figure 1. (The restrictions thus imposed on this nomograph are such that it should not be used if the value on the reference line is less than 1.9 or more than 19.) If S’must be used, determine AYu from Figure 2. In this case it is possible for U to be negative, but care must be taken that the formula has been chosen for the proper relation between the equilibrium curve and the operating line. If U is Less than 10 per cent greater than S, the value of V must be calculated as follows and A Y M determined from Figure 3: If operating line is above equilibrium curve:
Y
=
(WZ
- w J ( m - mz) - mzxi
YL - yz 4-m1x2
If operating line is below equilibrium curve: V =
DATAFOR ACETONE-WATER SYSTEX 4. EQUILIBRIUM CALCULATED ox BASISOF FICTITIOUS MOLECULAR WEIGHT FOR ACETONE OF €2 (PROBLEM C)
Summary
tem. 2 . The values of U and S or S’ are calculated as follows (extraction may be either case, depending upon choice of 2 and y) : a. If operating line is above equilibrium curve (absorption is most common example) : U = yl
+ y2 -+wwct -- m2Xi)’
8’ = (YI - g2 b.
nz122
1)2iL::
$- 4(X1?/2
A Y x = C/’2
Nomenclature m = y*/x Np=
s
- XzY~)(ffli-
=
number of transfer units In system, other subscripts refer to sections over which the transfer units are determined
v‘
(yI
- yz + mlxz - r n z x ~+~4 h y z - xJvi)(ml - md
or l/(muxl S’ =
+ 4(xlyz - x2yd(m - ma) - xlz/L)(ml - mz) + ya - yi)l
- m1x2+ y, -
4 4(Xz1Ji - xly2)(rnl - r n -~ (yl - y2 + mix%- m d 2
or l/4(x-vl U = y, y2 - mlxa
+
(m2zl
-
( X ~ ? / Z myI)(ml
2
~ ~ 1 x 2
- m2x1or WLL~XIf mlxa - y~ - ?/z (z1y.2 - X?Y,)(rnI - mi) yl - y~ + mlxl - m,x1
V’
+
- na2)
or m2x1 - mix2 Y L - Y, = concentration in one phase (liquid phase in distillation and abwrpt ion)
conrentration in other phase (vapor phase in distillation and absorption) x* = conrentration in equilibrium with y y* = concentmtion in equilibrium with 3c AYM = mcan driving force Subscripts 1, 2 = terminal value of section under consideration y
1. From the equilibrium data and material balances, the values of m,2, and y are determined at both ends of the sys-
+ -
The calculation of B is simple because both the denominator and the numerator have been evaluated in the fkst and second terms €or the calculation of S. If S should be zero or small with respect to U (less than three tenths of the value of U ) , the value of AYM is given by the formula:
FIGURN
per cent in the total number of transfer units. Since most of the transfer units are concentrated in the upper part of the fractionating section where the equilibrium line approaches a pinch, a slight variation in the equilibrium data produces a large change in the number of transfer units determined, so it is doubtful whether the equilibrium data is exact enough to justify greater accuracy in this case. However, when equilibrium data are known with a high degree of accuracy, any of the three refinements mentioned in the original paper (6) may be used with these nomographs to give an answer practically within the limits of error of manipulation possible in the graphical integration method, which is presumed to be exact. In the case of an equilibrium curve of more regular shape, it would not be necessary to break down the problem into so many segments as in the given problem. However, this is one of the three methods previously mentioned which can be applied to any equilibrium curve to give a high degree of precision.
-
(xiyz - x~yl)(ml mu) mzx? - mlx2 YZ yl
=
Acknowledgment The authors wish to evpress their appreciation to Frederick G. Sawyer for drafting the nomographs in final form.
Literature Cited m2)
If S2is negative, S’is calculated as (S’)* = S2 If operating line is below equilibrium curve (desorption and distillation):
If P i s negative, S’is calculated as (SI)* = S2 3 . Then AY.w is determined from nornographs as follows:
( I ) Colhnm. A. P.. TND. ENG.CHEM., 33, 459 (1941). (2) Dwyw. 0.E.,and Dodge, B. F., Ibad., 33, 485 (1941): private communication. (.3,) Haushrancl. E.. “Pnnriulrs and Practice of Inductrial Distillation”, p 215,New York, John Wilcy & Sons, 1926. (4) Levy, R. M.. IND. EN^. CHFIY., 33, 998 (1941). (5) O t h i r i ~ r ,D. P.. Khite, R. E., and Trueger, E., Ibid., 33, 1240 ( 1 941). (6) Schpihcl. E. G . , and Othmer, D. F., Trans. Am. Inst. Cfiem. Enurn., 38.33R (1942). (7) York, R ,, and Holmes, EL C , IND.EKG.CHEM., 34, 345 (1942).
