INDUSTRIAL AND ESGIiVEERING CHE.VISTRY
960
This same apparatus was employed for studying the absorption of hydrocyanic acid from the mixtures of that gas with carbon dioxide obtained, as described in the previous article, by passing carbon dioxidq over moistened sodium cyanide. For this purpose the flask F was removed and a U-tube, equipped with stirrer and containing sodium cyanide, was connected in its place.
Vol. 17, S o . 9
Conclusion
As a result of these and former experiments, it is seen that it is practicable to decompose impure sodium or calcium cyanide and by working in a closed cycle to obtain thereby high-grade sodium cyanide with a high degree of safety and with very low temperature requirements.
Graphical Rectifying Column Calculations' By E. V. M u r p h r e e THESOLVAY PROCESS Co., SYRACUSE, N. Y .
Graphical m e t h o d s of m a k i n g rectifying c o l u m n calculations for binary mixtures i n b o t h plate a n d packed columns based o n m a t e r i a l transfer e q u a t i o n s have been outlined a n d examples given of their use. Methods of calculation for packed c o l u m n s w h e n m o r e than two c o m p o n e n t s are present have been discussed.
R
ECTIFYIKG columns may in general be separated into two classes-plate columns and packed columns. ' In the former the rectification may be conceived as occurring in a series of finite steps, whereas in the latter the rectification is continuous. As the phenomena are somewhat different in the two cases, they will be considered separately. A-Plate
Columns
The discussion of plate columns will be limited to binary mixtures and the column will be assumed to be isothermal. This assumption may be made when the mols of vapor condensed for heating are small compared with the total mols of vapor. I n this case average quantities of liquor and vapor should be used and the temperature should be taken as the column average. It will also be assumed that an artificial molecular weight has been assigned to one of the components so that the molal heats of vaporization are equal. The mols of vapor going up the column, the mols of overflow above the feed plate, and the number of mols of overflow below the feed plate are therefore constant. One mol of product will be used as the basis of calculation. NOMENCLITURE
0 = mols of overflow above feed plate per mol of product V = 0 1 = mols of vapor per mol of product F = mols of feed per mol of product 0' = 0 F = mols of overflow per mol of product below feed plate W = 0' - V = mols of waste per mol of product x n = mol fraction of component A in liquor on nth plate yn = mol fraction of component A in vapor above nth plate xc = mol fraction of component A in product xu = mol fraction of component A in waste yrL*= mol fraction of component A in vapor which is in equilibrium with the liquor on the nth plate x o = mol fraction of component A in liquor a t bottom of packed column yo = niol fraction of component A in vapor a t bottom of packed column x1 = mol fraction of component A in liquor a t point of introduction of feed C T ,= ~ concentration of component A in liquor on nth plate Cc = concentration of component A in liquor which is in equilibrium with vapor P = total pressure a t any height L in column K , = over-all conductivity coefficient for component A when driving force is partial pressure difference
+ +
K , = over-all conductivity for component A when driving force is concentration difference
L = height of any point in packed column LI = height of point a t which feed is introduced Li = total height of column U = area of contact per mol of vapor a t any height L in packed column 'J = linear velocity of vapor a t point L in column M , M', R, and R' are constants determined from column tests
In a previous article2the author has shown that under the above conditions, when the driving force of the material may be represented as a partial pressure difference, that Y* = yn* - M (Y,* - yn-1) (1) When the driving force of the material transfer is represented by a concentration difference, the relation between y,, and yn-l is given by
For cases where it is possible to make approximate integrations by means of straight line relationships between y and C, the above equation becomes yn
Received June 1, 1925
- 8-S.R (Y"*- rn-1)
(3)
where S, is the slope of the straight line relating y and C uqed for the nthplate. Thiele and McCabe3 have recently shown how graphical methods may be used in calculations involving the rectification of binary mixtures under isothermal conditions when the concept of the theoretical plate is used. Their method may be readily modified to agree with Equations 1, 2, or 3, instead of using the concept of the theoretical plate. Consider the sketch in Figure 1. By equating input to output in the dotted section above the feed plate the following equation is obtained: Yn =
%+lo+ X e Y
Similarly, for the dotted section belolv the feed plate: &+I 0'- x, w Ym =
V
On substituting the analogous equations for yR-l and in ( I ) , the results are, respectively: yn = (1 2
1
yn*
3
M - Aq rn* + 7 (xnO
THISJOURNAL, 17, 747 (1925). I b z d . , 17, 605 (1925).
+
2,)
~ ~ " - 1
(6)
I,l'DL'STRIAL A,VD ELVGINEERING CHEMISTRY
September, 1925
I n the remainder of this section it will be assumed that Equation 1holds. The discussion given below may, however, be readily adapted to the use of Equations 2 or 3. By combining Equations 4 and 6 the following equation is obtained:
961
which the waste will not contain more than 1 mol per,cent of component A . The value of 1M from tests on similar columns rectifying these two substances has been found to be 0.4. The mols of overflow per mol of product will be arbitrarily taken as 4. The y x equilibrium curve for this binary mixture is Curve 1 of Figure 2.
-
The closer the values X , ~ + I and zn,the less, other things being equal, is the rectification. When xn+1 and xn are equal there is no rectification. In order for there to be rectification it is therefore necessary that (1
- .M)ytr* >(1 -
0
-M) - xn V
- nf + 1V
Xe
By rearranging this inequality and remembering that V 1, the following inequality is obtained:
0
+
O>-
xc
=
- Y"* - xn
yn*
This shows that for every plate above the feed plate upon which rectification is occurring, (8) must hold. The quantity Xe
- h*is the theoretical minimum mols of overflow per Xn
yn* -
mol of product for any value of x,,. Identical quantities are obtained3 for the minimum overflow when the concept of the theoretical plate is used. Equations 4, 5, 6, and 7 express the value of y above any plate as a function of the value of 5 on that plate or on the next plate above. These equations mav be used as the basis OF a graphical method of making rectifying column calculations. Equations 4 and 5 are linear equations connecting y and x and hence will be represented by straight lines on a plot of y against z. The value of 2 for any plate above the feed plate corresponding to any value of y on the plate below must be on the line of Equation 4. The same applies for the line of Equation 5 for the feed plate or any plate below the feed plate. Equations 6 and 7 are in general curves on a y - z plot. For any plate above the feed plate the value of y above any plate corresponding to the value of z on that plate must be on the curve of Equation 6. Similar conclusions apply to Equation 7 for the feed plate or any plate below the feed plate. The method of making these graphical calculations can be best illustrated by working out the following typical example: EXAMPLE-It is desired to design a column which, with a feed containing 20 mol per cent of the more volatile component A , will produce a product containing 95 mol per cent of A and for
Figure 2-Graphical
Calculation of Plate Column
Curve 1-Equilibrium curve Curve 2-Above feed: y n = 0.80 xn + I 4- 0.19 Curve 3-Below feed : ym = 1.79 x,, + I - 0.005 Curve 4-Above feed: yn = 0.60 yn* f 0.32 x n f 0.076 Curve 5-Below feed: y m = 0.60 y I I L * 0.716 x , - 0 003 Curve 6--y = x
+
Solution, basis one mol of product. The mols of feed per mol of prodiict can be found from a material balance. X F F = xc x,w or 0.20 F = 0.95 0.01(4 F 5) F = 4.95 mols The values of the more important quantities are Xe = 0 . 9 5 0 =4 V=4+1=5 X F = 0.20 F = 4.95 s 95 - 5 = 3 95 x, = 0.01 0' = 4 4 95 = 8 . 9 5 Substituting these values in Equations 4, 5, 6, and 7, they become, respectively:
+ + -
+
+
w=
+ yn = 0.60 yn* + 0.32 &, 4- 0.076 Y,,,= 0.60 Y,,~* + 0.716 xn - 0.003 y* = 0.80 %+I 0.19 ytn = 1.79 x r n ~-i 0.008
(9)
(10)
(11) (12)
The curves of these equations are given in the y - x plot of Figure 2 . It will be noticed that Curves 2 and 3 and Curves 4 and 5 intersect a t the value of x corresponding to X F . An examination of their equations will show that this is always a necessity. It may be shown that if the feed is introduced on the plate on which the composition of the liquor is nearest that of the feed, then the number of plates necessary to perform a given separation is a minimum. The value of y above the top plate of the column is 0.95. The value of y below the bottom plate of the column is that in equilibrium with a value of x of 0.01. This value of y is given by the equilibrium curve of Figure 2 as 0.026. The value of x on the top plate corresponding to a value of y of 0.95 above the plate is given by Equation 11 and is the value of x corresponding to a value of y of 0.95 on Curve 4. The value of y below this plate is given by Equation 9 and hence is the value of y on Curve 2, corresponding to the value of x on the top plate and is found by coming vertically down from the point y = 0.95 on Curve 4 to Curve 2. The value of x on the second plate corresponding to this value of y is found by coming horizontally
I N D U S T R I A L A,VD ENGINEERING CHEMISTRY
962
across from Curve 2 to Curve 4. This gives the value of x on the second plate. This process is continued until the feed plate is reached, which is the plate having most nearly the value of x = 0.20. From there on Curves 3 and 5 are used instead of Curves 2 and 4 and the process is carried on as before until a value of y of 0.026 is reached. The number of plates necessary is given by the number of vertical lines giving the change in y from plate to plate. From Figure 2 this is seen to be 24, the feed being introduced on the 10th plate from the bottom.
This graphical method gives a very good picture of the process and is valuable for that purpose. In practice, if the steps are very close together it is probably easier and more accurate to use the equations directly. Although the graphical method can only conveniently be used for binary mixtures, the equations may be used for mixtures of any number of components. It is interesting to note that if the number of mols of overflow per mol of product were infinite Equations 4 and 5 would both reduce to = x and would be represented on the y - x plot by Curve 6. A plot of the minimum overflow above the feed plate as a function of x is given in Figure 3. The foregoing treatment has been for the case in which the vapor entering the rectifying column is produced by vaporizing the liquid coming from the bottom of the column by heat from steam coils or direct firing. With a slight modification it may be used for cases in which steam is blown into the bottom of the rectifying column. B-Packed Columns
The method ordinarily used in solving problems involving rectification in packed columns is to calculate the number of theoretical plates necessary to perform the desired rectification under the given conditions and then, from previous evaluations of active column height in terms of theoretical
Vol. 17, N o . 9
proceeds up the column continuously. In this respect it is essentially different from a plate column. This discussion will be limited to binary mixtures unless otherwise XProducf-lmd stated, and it will be asI sumed as in the precedIm, , y,,Fvl ing section that artifi1 cia1 molecular weights I have been assigned to the I components so that their molal heats of vaporizaLtion are equal. As small a change should be made from the true molecular weights as is possible. The column will not for the present be assumed Fnd-F to be isothermal. The same nomenclature will be used as in Section A. r c yv Also, as in this section, I I the vapor entering the I I bottom of the column will be assumed to be produced by the vaporW' b- V , ization of t h e l i a u o r from the bottom of the L :;,e,-d ; Column column. The treatment may be readily modified in case steam is blown into the bottom of the column. When the driving force may be represented as a partial pressure difference, the equation for the rate of material transfer a t any point in the column can be written as:
el
'-------I I [ I
r-
4 --I- --;
I1
iYGr:
)*d
d8
= K,UV(p
- yP)
d L have been substituted for p P and de, rewhere y* and 7 spectively. The partial pressure of the volatile colapcwent in the liquid is represented by p . d In V may be neglected in comparison witla 1 In Y, When which is usually the case, Equation 13 may be writtea
"dL
=
M' (LB- L A ) (14)
LA
Figure 3-Minimum
0--
Overflow above Feed Plate - yn*
095
Yn*
- xn
plates, to calculate the height of column necessary for the particular conditions of the problem. For binary mixtures this method probably gives a fair extrapolation of plant data. It has the objection, however, of not accurately representing the process, because equilibrium is never reached in practice, and also because this method represents the rectification as proceeding u p the column in finite steps, whereas actually it
where M' is the average value of determined from tests on similar columns rectifying the same substances. This is equivalent to assuming the column to be isothermal. d In y In case is not negligible compared with d ~ an , approximate expression for In V in terms of y should be obtained which will allow the integration of Equation 13. In certain instances it might be desirable to express as a function of
ds
L in integrating the right-hand side of (13) or (14)instead of assigning an average value to it. Material balances may be made above and below the point of feed entry in the column as in Section A. Equating input
I.VDrSTRI-4L A N D EAVGI-VEERI,VGCHEMISTRY
September, 1925
and output of the dotted sections in Figure 4, the following equations are obtained: Above feed entry
yv = x o 9- x, Below feed entry
y v = XO'
+
(15) (16)
X w W
963
The value of x a t the top of the column corresponding to a value of y of 0.95 by Equation 22 is 0.95. The value of y af the bottom of the column is that in equilibrium with the waste which from the equilibrium curve of Figure 2 is 0.026. The value of x corresponding to this value of y is given by Equation 23 as 0.019. The values of the integrals of Equations 20 and 21 may now be obtained graphically with the aid of the equilibrium curve of Figure 2 . Table I contains the values of M' values of x ' .
2
for different
On substituting these values of y in Equation 14 the results Table I
are P X
and
*, : [
ax
=
M'L
(18)
- 2'- $CY* - x,) where 0 + 1 and 0' - W hare been substituted for V.
The integrals on the right-hand side of these equations may be evaluated graphically by plotting 1 1 y*
- x - 81 ( x , -
and
y*
y')
W -x -0' (Y* - xw)
against x and obtaining the area under the curvea. The differential form of 17 is = M' [y*
- x - a1 ( x , - y*)J
-x >1
(x,
110 100
-
90
- y*)
I I
This result is the same as was obtained in Section A. Equations 17, 18, and the above inequality were obtained by Lewis4 for the rectification in a plate column where the liquid composition changed continuously from plate to plate. Lewis' equations really apply to the rectification in a packed column and not to a plate column. order t o illustrate the use of the equation developed, the problem given in Section A will be solved for a packed column. The same numerical conditions are t o apply as for the plate column. Since the column is considered t o be isothermal and since the driving force of the material transfer may be represented by a partial pressure difference, Equations 17 and 18 may be used. On substituting the values for 0, 0', W, and xc and xw, Equation 17 becomes:
;.:
6.1 5.9 6.4 9.9 18.5 24.4 33.3 71.5
The values of M'g are plotted against x in Figure 5 . The area under the curves is proportional to the column height. It will be noticed that the minimum area will be obtained if the feed is introduced at the value of x at which the curves intersect. The value of M'L for the whole column may therefore be obtained by integrating under the curve abc. As the point of intersection of the curves is a t a value of x of 0.148, the limits of integration for Equation 21 are x = 0.019 and x = 0.148, and those for Equation 20 are x. = 0.148 and x = 0.950.
In order for there t6 be rectification at any given point above the feed plate, the following inequality must hold : y*
71.5 41.7 14.7
0.962 0.931 0.885 0.836 0.782 0.722 0.647 0.555 0.431 0.354 0.335 0.318 0.298
I
t
I
I
I
I
I
70
t;3 I4 60 .u
-$
$
50 40
ExAMPLE-In
30
20 10
0 0.1 0.2 0.3 0.4 0.5 0.6 07 0.8 0.9 Mol fraction of component A i n liquor-* Figure +Graphical Calculation of Packed Column dL 1 Curve I-Above feed : M' d x 1,26 y * - x o,238 1 Curve 2-Below feed : M' d x = o,559 y * - x + o,oo4
-
and 18
I 0
-
The results of these integrations are given in Table 11, in which the values of M'L are also given for intermediate values of .I!. Equations 15 and 16 with these values of 0, 0', V, x,, and x, are, respectively: y = 0.80 x y = 1.79 x 4
+- 0.008 0.19
THIS J O U R N A L . 14, 492 (1922).
-
From n = 0.019 to x
0.050 0.100 0,200 0.300 0.400 0.500
Table I1
MI.' 1.33 2.63 4.26 5.05 2,6S
6.21
From r = 0.019 to x = 0.600
0.700 0,800 0,850
0,900 0.950
.u '1. $.hl i , f'l 3
8 .4 f i 9.7; 11 :,2 1:i.9:
INDUSTRIAL AhrD ENGINEERING CHEMISTRY
964
In Figure 6 x has been plotted against fraction of column height for the plate and for the packed columns. It will be noticed that for the plate column the value of x at the point a t which the feed should be introduced is 0.20, whereas for the packed column it is 0.148. For the plate column the feed should be introduced a t a point which is 10/24 of the active height; for the packed column it should be introduced a t a point which is 6.15/24 of the active height.
Vol. 17, No. 9
(27) and
where the primes are used to distinguish the two volatile components and 4V' is a constant similar to M'. These equations cannot in general be exactly integrated as they stand, because both y*' and y*" are functions of both y' and y" and hence the variables are not separated. The relation between y' and y" is obtained by eliminating LE - L A from these equations and is
Unfortunately this relationship cannot in general be expressed in algebraic form, and hence Equations 27 and 28 must be integrated by approximations. The quantities such as M' and R' are measures of the efficiencies from a distillation standpoint of different types of packing for a given height. Figure 6-Comparison
of Plate and Packed Column
In cases in which the driving force of the material transfer is a concentration difference (liquid film controlling) the equation for the rate of material transfer a t any point in the column may be written:
or following the same steps as for the preceding development,
where R' is the average value of tests. dIn V
KGdetermined from column
I n cases where d L is negligible compared with
d In y
Equation 25 becomes
Equations 24 or 25 may be used with Equations 15 and 16 in a manner analogous to the case in which the driving force of the material transfer is represented by a partial pressure difference. When more than two components are present, certain calculation difficulties are met when the methods of this section are applied. This will be shown by briefly considering a three-component mixture in which the driving force of the material transfer for both more volatile components may be represented as a partial pressure difference. For this case equations analogous to (14) would be obtained for both volatile components. These may be written as:
Calculation of Points in the Ethyl Alcohol-Water Distillation Curve By E. Oman and A. Gunnelius [Abstracted by A . R . ROSEfrom Teknisk T i d s k r i f t . 66, 33 (1925)l
OILING points for alcohol in water solution determined within an err01r of * 0.02O C., are given in the following table: Wt.
B. P.
3%
oc.
7c
6.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 24 23
99.44 98.88 97.88 96.91 96.05 95.18 94.34 93.55 92.83 92.13 91.45 90.82 90.18 89.57 88.97 88.40 87.85 87.34 86.86 86.42 86.03 85.70 85.40 84.90 85.14
25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
Wt.
B. P. O C .
84.68 84.47 84.27 84.09 83.91 83.73 83.60 83.48 83.28 83.12 82.98 82.84 82.70 82.56 82.43 82.30 82.19 82.08 81.97 81.86 81.76 81.67 81.58 81.49 81.40
Wt.
B. P.
%
c.
50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74
81.31 81.23 81.15 81.07 80.99 80.91 80.83 80.75 80.67 80.58 80.50 80.42 80.34 80.26 80.18 80.10 80.02 79.94 79.87 79.80 79.72 79.64 79.56 79.48 79.41
Wt.
% 75 76 77 78 79 80 81 82 83 34 85
86 87 88 89 90 91 92 93 94 95 96 97 98 49 100
2. P. C.
79.34 79.26 79.18 79.10 70.02 78.95 78.87 78.80 78.72 78.65 78.57 78.50 78.43 78.37 78.30 78.24 78.19 78.14 78.10 78.06 78.04 78.04 78.06 78.08 78.10 78.13
To calculate the alcohol content of any distillation vapor from an alcohol-water mixture: from the composition of the mixture find the boiling point in the table; the temperature being known, get the vapor pressure of water; calculate the molar percentage of the water in the mixture and from this the partial pressure of the water, and by subtraction the partial pressure of the alcohol; the molecular weight of the water and of the alcohol multiplied by their respective partial pressures will give the weights of each in the vapor, from which per cent composition can be calculated. The facts required for the calculation are readily available except the molecular weight of alcohd vapor, and this may be determined from the following: The molecular weight of alcohol in vapor from an aqueous solution having 10 per cent alcohol has been found to be 43.15, and from a 70 per cent solution, 26.05. The variation in molecular weight for 1 per cent alcohol is 0.285.
