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chloride to sulfate (the latter is especially nonhygroscopic in combination with urea). 3. Specific formulas containing large amounts of an active material can be developed which remain hygroscopically inactive for long periods and are commercially feasible in certain territories where suitable raw materials are available and marketing practices satisfactory. The formulation of mixed fertilizers will perhaps remain an empirical matter for several years, in so far as control of hygroscopicity and drillability is concerned. It is hoped, however, that the moisture absorption rate patterns as obtained by the method here presented will be of immediate practical use in comparative evaluations and possibly lead to better understanding of the fundamental principles.
Acknowledgment The author wishes to express his grateful appreciation of the assistance rendered by E. F. Harford in the design of this apparatus.
Bibliography (1) Adams and Merz, IND. ENQ.CHEM.,21,305 (1929). (2) Beaumont and Mooney, Ibid., 17,635 (1925).
VOL. 31, NO. 7
Ber. Ajdeel. Handelsmuseum Koninkl. Ver. Kolon. Inst., No. 27 (1926). Berlin et al., J . Chem. I n d . (U.S . S. R.),13,24 (1936). Damiens, Compt. rend., 198,1233 (1934); Bull. soc. chim., [ 5 ] 2, 1893 (1935). Demougin, Chimie & industrie, 34,517 (1935). Diesnis, Bull. soc. chim., [5] 2, 1901 (1935); Ann. chim., 7, 6 (1937). Edgar and Swann, J . Am. Chem. SOC.,44,570 (1922). Feiltizen and Lugner, Chem.-Ztg., 35,985 (1911). Gorshtein and Dizhevskii, J . Chem. Ind. (TJ. S . S.R.),13, 1413 (1936). Guyer and Schiitze, 2.angew. Chem., 46,763 (1933). Hals, C h e m d t g . , 35,1130 (1911). Lenglen and Milhiet, Chimie & industrie, Special No., p. 826, March, 1932; Superphosphate, 5,49 (1932). Merz, Fry, Hardesty, and Adams, IND.ENQ.CHERI.,25, 136 (1933). Peddle, J . Chem. SOC.,105, 1025 (1914). Pestor and Glazova, J . Chem. I n d . (TJ. S . S . R.),13,868 (1936). Prideaux, J.SOC.Chem. Ind., 39, 182T (1920). Robert, Industrie chimigue, 22, 887 (1935). Toknoka, Bull. Agr. Chem. SOC.J a p a n , 10,148 (1934). Werner, Nature, 139,512 (1937). Whittaker, Adams, and Jacob, IND.ENG.CHEM.,29, 1144 (1937). PRESBNTBD before the Division of Fertilizer Chemistry a t the 96th Meeting of the American Chemical Society, Milwaukee. Wis.
SEPARATION PROCESSES Point Transformations of y vs. x Diagrams’ MERLE RANDALL AND BRUCE LONGTIN University of California, Berkeley, Calif.
T
HE previous paper (8) showed that the y us. x diagram could be obtained in an exact form by a contact transformation of the H us. N diagram. The two types of diagram supplemented each other. However, in a number of cases, the y US. z diagram is somewhat more convenient than the H us. N diagram. In any case of relative volatilities near unity, the ordinary y us. x diagram of the McCabe and Thiele type (4) becomes difficult to use. The equilibrium curve lies close to the diagonal, y = x, and the stepwise construction cannot be easily carried into the corners of the diagram. These difficulties become important in any case of relative volatilities lying in the range from 0.7 to 1.5. I n studying the performance of fractionating columns intended to strip heavy forms of water from ordinary water (9) (in which case the relative volatilities are in the range 1.01 to 1.05), we were led to consider various possibilities of exaggerating the usual y vs. x diagram in such a way as to increase the distance between the equilibrium curve and the reflux or operating line.
General Mathematical Transformations The y us. x diagram is useful particularly because certain graphical stepwise constructions may be made with straight lines. The only point transformations of the diagram which This is the fifth paper in this series. 1939 (6-9). 1
The first four appeared in 1938 and
Graphical stepwise calculations of the McCabe and Thiele type may be carried out on a diagram in which (y - x ) is plotted as ordinate against x as abscissa. This transformation of the usual y VS. x diagram has the advantage of economy of space. Furthermore a magnification of the vertical scale will serve to spread the region between the equilibrium curve and the diagonal, y = x . Such an exaggeration is useful for relative volatilities near unity. Other transformations of the y VS. x diagram are discussed. will be useful are those preserving straight lines. Mathematically all such transformations belong to the. groups known as projective transformations (2). They may be obtained by placing a projection plane a t any angle to the given plane figure and then projecting the figure orthogonally, obliquely, or perspectively onto the .projection plane. Let us consider only those obtained by parallel (i. e., not perspective) projection. They may also be considered as combinations of rotation, shear, and stretching. If x’ and y’ are the transformed coordinate, the general equations of this group of transformations are : 2’
= A(r
+ ay)
(W
Y‘ = B(r 3- by)
(IB)
+
Graphically the transformation is obtained by plotting (z by) to a scale of B units = 1, against (z ay) plotted to a scale of A units = 1. (If the number 1 is plotted as B units, by) units. Hence then (z by) is to be plotted as B(z Of units = is the same as y’ (’ by) plotted to plotted to a scale of 1 unit = 1.)
+
+
+ +
JULY, 1939
INDUSTRIAL AND ENGINEERING CHEMISTRY
909
V'
x = Mol fraction of benzene in liquid
FIGURE 1. STANDARD MCCABEAND THIELE DIAGRAM Separation of 50 mole per cent benzene-toluene mixture into 90 mole per cent toluene fractions with a reflux ratio, Rt = 2/3. in the enriching'seotion.
FIGURE2. TEE (y
- x) vs. x DIAGRAM, WHEREa = 0 and b = -1
A (left), B = -A;
Y'
B ( ~ i g h t )B,
-2A
Y'
K
FY
+ y) DIAGRAN, WHEREa = 1 A (left), E = - A ; B B =
~IGU1w3. THE (Y -
5 ) us. ( 5
(right),
-2A
The stepwise construction of the McCabe and Thiele (4) and other y us. x diagrams is made up of horizontal and vertical lines, together with the operating line. The vertical lines are lines for which 2 is constant, x = xl. These lines become the oblique straight lines:
+
y' = (bB/aA) X' (1 - b/a) BZI (2) Similarly, the vertical lines, for which y = yl, become: 2/'
= (B/A)x'
+ ( b - a ) By1
(3)
In particular the two sides, 2 = 0, and y = 1, of the square are transformed into the straight lines, y' = (bB/uA)z',and
AND
b = -1
-
-
FIGURE 4. THE(y - 5) us. y DIAGRAM A
0, Aa
=
1, E
=
-1, and b
-1
+
y' = ( B / A ) d ( b - a)B, respectively. The vertical steps of the construction were parallel to the vertical side 2 = 0, and become parallel to the line y' = (bB/uA)z', which is the transformation of the originally vertical side. The horizontal steps, originally parallel to the side y = 1, become parallel to its transformation. These two transformed sides may be taken as reference lines parallel to which the steps are constructed. When the McCabe and Thiele assumptions are valid, the reflux line is expressed by an equation,
y = h + R x
(4)
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VOL. 31, NO. 7
In Figure 2A the diagram has undergone simple shear in the vertical direction in order to bring the diagonal, y = x, down onto the horizontal axis. In 2B the sheared diagram of 2A is plotted to an increased vertical scale in order to spread the region between y = x and the equilibrium curve. In Figure 3A the standard diagram has been rotated to bring the diagonal into coincidence with the horizontal axis. In Figure 3B the vertical scale has been increased to obtain the desired exaggeration. In Figure 4 the original diagram was rotated through 90 O before undergoing shear to bring the diagonal into a horizontal position. The exaggeration may now be obtained by changing the vertical scale. In Figure 5 the original diagram has undergone shear in both directions. Figure 6 shows the effect of a change of vertical scale on the untransformed diagram, and Figure 7 the effect of a slight vertical shear.
-
FIGURE 5 . THE(y -
Z) US. ( 2 X 9) DIBGRAM A = 2, A a = -1, B = - 1 , b = -1
in which R has the value Ot/V in the top section and Ob/V in the bottom section of the column, while h has the values (P/V)x, and (W/V)xW, respectively. I n the transformed coordinates the reflux line becomes the straight line : B(l
+ ~ R ) x ‘= A ( l + aR)y’ 4 AB(u - b)h
(5)
In general, the reflux line is most easily constructed by locating the point a t which it crosses the line y = x, together with the intercept a t either x‘ = 0 or a t x’ = A . The intercepts for x’ = 0 and x’ = A are: 2’ =
0 : y‘ = hB(b a)/
4 aR1
-
(6-4)
Particular Forms of the Transformation The properties of the transformation have b e e n s t u d i e d for a number of different choices of the constants a, b, A , and B. The principal features of each type are tabulated in the accompanying diagrams. Figure l shows the standard FIGTJRE 6. THE y VS. x DIAQRAM diagram of which all WITH 2 TO 1 VERTICALSCALE the others are trans(Little is gained in exaggeration of the formations. The 00working area.) ordinate network of this diagram is shown in its transformed shape in each of the other diagrams to make clear what has happened to the diagram in the transformation.
The (y - x) us. x Diagram Of all the different tvwes the easiest to construct graphically are the two in which (y - x) is plotted to any desired scale as ordinate against either x or y as abscissa (Figures 2 and 4). Of these two, the (y - s) vs. x diagram is a bit more readily understood. The authors have found in three years of experience in using the diagrams that this particular transformation is of great utility. However, since the (y - x) vs. y diagram (Figure 4) exaggerates the slope of the equilibrium curve more in the right-hand corner than does the (y - x) vs. x (Figure 2A), this other form may be useful in particular cases. Rodebush (IO)proposed a plot of (y - x) os. x as a basis of fractionating column design, but the later simplified methods of McCabe and Thiele have not until now been adapted to this diagram. It has certain obvious advantages. In o r d i n a r y design problems the region outside the area bounded by the equilibrium curve and the diagonal, y = x , i s rarely used. The (y - z) vs. x diagram may be plotted on a long narrow strip of coordinate paper with n o need of a n y l waste area. Furthermore, the distance between the equilibrium curve and line y = x m a y easily be magnified b y a FIGURE7. T H E (y - Z) us. (y - x/2) DIAGRAM change in the vera = 2, A = -1/2, b = 1, B = -1 tical scale. If it is desired to obtain a comparable magnification of the y vs. x diagram, it is necessary to use an extremely large sheet of coordinate paper, of which the major portion is unused, or to piece a number of sheets together diagonally. Neither procedure is convenient. A comparison of the (y - x) us. x diagram with the y os. x and with the (y - 2) vs. (22~- x) diagrams is shown in Figure 8. These diagrams are all con-
INDUSTRIAL AND ENGINEERING CHEMISTRY
JULY. 1939
!OOOO
39995
b
P
911
With these limitations, all features of the standard McCabe and Thiele diagram may be represented in the transformed diagram, including the plate efficiency (3) constructions of Murphree (6),Baker and Stockhardt ( I ) , and Warden ( f l ) . Furthermore, a contact transformation from the H us. N to the (y - x) us. x diagram may easily be deduced, by which the correct curved reflux lines may be obtained.
E
0 29990 p
Acltnowledgment
x
.6 -E
29985 .c oa
2 _‘I
49980
The testing and evaluation of the various types of diagrams were done by students in chemical technology, in particular by Harold Cowdrey, H.h i t c h , H. Finch, W. F. Schindler, x) diagram and Paul D. Williams. The (u - x) us. (y was originally suggested and studied by E. J. Haven. The clerical assistance of the Works Progress Administration is gratefully acknowledged. Bruce Longtin, the junior author, is the Shell Research Fellow in Chemistry a t the University of California.
+
3.9975
Nomenclature = mole fraction of more volatile component in liquid = mole fraction of more volatile component of vapor =
new variable plotted on horizontal axis
=
composition of distillate
= =
composition of residue value of 2’ along line z = y, corresponding to z = xa and r = x,,,, respectively constants determining size of transformed diagram constants determining shape of transformed diagram moles of liquid overflow per unit time at point in question moles of distillate withdrawn as product per unit time reflux ratio at point in question moles of vapor arising per unit time at point in question moles of residue withdrawn per unit time
= new variable plotted on vertical axis = constant values of x and y = composition of feed
=
= = =
= =
I
W
=
FIGURE 8. STRIPPING COLUMNFOR ORDINARY WATERTO OBTAIN 0.00227 MOLEFRACTION OF HDO, USINGTOTAL REFLUX; TWENTY-EIGHT THEORETICAL PLATES REQUIRED a. b. c.
Standard McCabe-Thiele diagram The form (y 5 ) us. z with a vertioal scale of 5 t o 1 2) with a vertical scale of 5 t o 1 The form (y - z ) 1s. (2y
-
-
Literature Cited (1) (2)
Baker and Stockhardt, IND.ENO.CHEM.,22, 376 (1930). Hewes and Seward, “Design of Diagrams for Engineering Formulas”, Appendix B, New York, McGraw-Hill Book Co., (1923.)
structed from the same data, obtained by Randall and Webb (9)in their work on the fractionation of isotopic forms of water. It is apparent that the two transformed diagrams give much more useful constructions than the standard diagram, Neither of the transformed diagrams is much better than the other from the point of view of ease and accuracy of the stepwise construction, but the (y - 2) us. 2 is more readily plotted. I n the (y - 5) vs. 2 diagram the vertical lines of the original diagram remain vertical. The horizontal lines, however, become displaced to make an angle of -45” with the horizontal. If the vertical scale is magnified, they have a slope numerically (- l),but the angle with the horizontal is greater than 45”. Geometrically the slope will be (-1) multiplied by the scale factor. The stepwise construction is most readily obtained by plotting the reference line, y = I, or some other line parallel to it. The steps are then constructed with sides alternately vertical and parallel to the reference line. This is easily done with the aid of a straight edge and triangle. If the vertical scale is magnified too much, the corners of the steps will become too sharp, and there will be no further gain in accuracy of the construction. Hence in practice the magnification is limited to five- or tenfold a t most.
(3) Lewis, J. IND.ENQ.CHEM.,1, 522 (1909). (4) McCabe and Thiele, Ibid., 17, 605 (1925). (6) Murphree, Ibid., 17, 747, 960 (1925). (6) Randall and Longtin, Ibid., 30, 1063 (1938). (7) Ibid., 30, 1188 (1938). (8) Ibid., 30, 1311 (1938). (9) Randall and Webb, Ibid., 31, 227 (1939). (10) Rodebush, Ibid., 14, 103.6 (1922). (11) Warden, J. SOC.Chem. Ind., 51,405 (1932).
