Punched Card Calculation of SixComponent Distillation Columns with Heat Balancing ASCHER OPLER AND ROBERT C.HEITZ Western Diuision, The Dow Chemical Co., Pittsburg, Calif.
4
Although de hiled plate-by-plate calculations are considered the most reliable method for designing multicomponent continuous distillation columns, they are extremely laborious. To facilitate this operation, a method for performing these calculations on IBM punched card machines has been developed which represents a practical compromise among speed, accuracy, and flexibility. The calculation employs a modified Thiele and Ceddes method for preliminary orientation and the Lewis and Matheson method for final balance. Thermal effects and departure from ideal behavior are handled by successive
repetitions of the column calculation. The machine method is especially suitable for this type of procedure, as an entire section of a column may be calculated at speeds as high as five plates per minute. Flash equilibrium calculations are also made using much the same machine setup and the same master cards. The machine method greatly facilitates calculation of complete temperature and composition profiles for a wide range of feeds and operating conditions. This should make available to both the designer and column operator much more complete information than it is now practicable to obtain.
NTEREST in the possibilities of applying computing machines to chemical engineering calculations has been steadily mounting. Within the past few years, the distillation problem haa been approached by several groups using analog computers (I, 6) and digital punched card machines (8, 10-18). Goelz and Calvert (6)have devised an analog computer for calculating plate composition in a six-component continuous distillation column in thermal balance. Bubb, Nisle, and Carpenter ( I ) have described a computer of the analog type for vapor-liquid equilibrium calculations of seven-component systems. Donne11 and T u r b i ( 8 ) have published a punched card machine procedure for determining boiling point, dew point, and vaporliquid equilibrium. Rose and Williams (If) and Rose, Williams, and Dye (18) have performed various calculations on electronic punched card operated equipment including continuous binary rtnd ternary distillation a t constant relative volatility and batch distillations of binary mixtures with appreciable holdup. They have discussed and proposed solutions to perplexing problems of nonideality. Opler and Heitz (IO) have described a machine method of calculating four-component distillation columns making certain simplifying assumptions.
I
of eight. The additional capacity simplfics wiring and enables extension to problems with more factors. During the plate-byplate distillation calculation the machine stores 23 four-digit factors and still maintains capacity t o devdop two eight-digit products. The digital inaccuracy previously reported (IO), when using the 602 calculating punch, has been largely eliminated in using the 602-A machine. Use of a standard machine would probably restrict the distillation calculations to four components using the method developed here. However, the vapor-liquid equilibrium method described below can be used with possibly more than six componente with a standard machine. The distillation calculation technique can be modified to enable more than six components to be calculated with a standard machine (I@, although with some sacrifice in speed. To perform the machine calculations described below, one needs a 602-A calculating punch, other usual punched card auxiliary machines, data cards (discussed below), and three moderately complex control panels for the 602-A machine. On written request, the writers will supply copies of the planning charts for the three panels.
PUNCHED CARD EQUIPMENT USED I N DISTILLATION CALCULATIONS The common IBM punched card machines and their operation have been adequately described by Eckert (8, 4), King (‘Y), and others. The machines may be classified in regard to their scientific calculating functions as (1) originating (key punch reproducer, summary punch), (2) manipulating (sorter and collator), (3) calculating (tabulator and calculating punch), and (4) result-listing (tabulator, interpreter, or end-printing reproducer). Many of the calculations described here were performed originally with the 602 calculating punch, but the present work waa largely performed with a 602-A calculating punch. The particular machine employed is equipped with eight storage units instead of the conventional six, eight counters instead of six, 12 pilot selectors instead of seven, and 12 coselectors instead
INTRODUCTION TO DISTILLATION CALCULATION METHODS
The basic distillation calculation method of Lewis and Mathe8on (9) is used here. The principle of machine solution of the equations has been discussed previously (IO). Since developing the basic technique, principal attention has been given to modifications that will extend the method to include more rigorous calculations. A method of calculating which considers variations in reflus ratio is described here. To enable calculations to be made when nondistributed components are present in the fccd, and when the key components are not adjacent, the Thiele and Geddes (16)method has been adapted to machine calculation. A case is illustrated which would probably be considered tedious by most chemical engineers, and the complete calculation is shown. 2465
INDUSTRIAL AND ENGINEERING CHEMISTRY
2466
Vol. 43, No. 11
LEWIS AND MATHESON-TYPE CALCULATIONS Tlir inatrial balance and equilibrium equations have been coiilbiiied to form four equations:
Table I.
:I'?.,
1 L:+l __ K , V:
2, = x n + 1 -
- xw
I W 93 93 94 94 95 95
v',
97
Equations 1 and 2 are used in calculating the stripping section; Equations 3 and 4 for the rectifying section. Equations 1 and 3 are used in calculating up the column; Equations 2 and 4 in going down the column. In the machine operation, wiring panel 1 is first used. The initial cornposition is read into internal storage. The necessary parameters describing the system a t T o C. are next fed in on a single card. The machine solves j seta (one for each component) of the proper one of Equations 1 to 4,obtaining the xn for each component. Then it finally solves Equation 5:
L' =
2
(Xn)i
- 101
i=l
Depending on the sign of U,the machine selecta one of two courses of action: ( a ) rejects all trial values of znand feeds in a new set of ammeters at temperature T f 1 (depending on the direction o f t h e calculation) or ( b ) punches the computed composition and clears out the initial set of values, replacing it in the internal storage with the newly calculated composition. This process continues right up (or down) the section of the column. The cards are now sorted on the result field into the punched and the unpunched. The punched cards are arranged In proper order and listed by the tabulator. One pitfall is likely to occur in this calculation. If the change in temperature from one plate to the next is less than 1 C., U in Equation 5 will ap ear negative for each succeeding trial temperature, The calcukted total will rapidly decrease in widening &ps-99,91, 76, and 34%, to cite an example. One solution to this problem IS to have J f sets of each temperature card, where M ia the estimated maximum number of plates a t the same T. A better procedure, which is followed here, is to use an auxiliary counter to teat Equation 6. j
s
=
(G)* - 97
Comparison of Results Obtained by DiAerent Method of Adjusting to 100 Mole Balancin after Each A a t e MethylCarbon ene Chlorotetrachloride form chloride 9.88 66.27 23.83 9.47 23.88 66.64 8.74 67.32 23,93 7.86 68.18 23.95 6.70 68.53 24.76 5.30 67.79 26.89 64.24 32.1? 3.63
Balancing after Column Calculation MethylCarbon ene Chlorotetrachloride form chloride 9.90 9.48 8.77
7.90 6.73 5.30 3.63
66.26 66.61 67.20 67.94 67.88 87.79 64.24
23.83 23.89 24.02 24.15 25.38 26.89 32.12
plates immediately above, but this is gradually overcome in progressing up the column.
HEAT BALANCING One of the simplifying assumptions frequently made is that of constant molal overflow. As this assumption can often produce results far from the actual case, it has been found desirable to make correction for the variation in reflux ratio due to the evaporation and condensation that take place on each tray. The method used consists of six steps: 1. Calculation of the liquor composition wuming constant molal overflow 2. Calculation of the vapor composition using Equation 7
vi = K15, (7) 3. Calculation of the liquid and vapor enthalpies using Equation 8
4. Calculation of the corrected liquid and vapor flows a t several points in the column using the heat balancc relation (9)
+
+
L A VnHn Ln+lhn+l V n - I Ha-I (9) A convenient way to use this equation, taking the moles of va-
por leaving the feed tray as a starting oint, A i t o make the subW ,letbding to the stitutions L, = Vn-, - D or Ln = < + I equations
+
L,
~
(V/Ht
-
-
I-/ +IX h f + ~ ) DH,,-i Hn-I - hn
-
[(n - 1) 4-f_11p (10)
i=l
When S in Equation 6 is negative, the machine stops instantly and signals an error. .4dditional cards of the same temperature may be fed singly into the machine and the calculation handled in this manner. By using this technique, maximum s eed is obtained, yet the machine is alert to possible trouble. #he punched values must total between 101 and 97%. As the mean difference in U between successive trials generally approximates 2%, the final results will average 100 f 1%. The ratios of the com onents will be close to the correct values, but they will not totayexactly 100 mole %. After the column calculation is completed, wiring panel 2 is placed in the machine and the cards containing the punched results are passed through. This divides each composition by the total mole per cent to produce a distribution that totals 100%. In this connection, the question arises as to the comparative accuracy of dividing after each plate is calculated (not convenient in the machine method) or after the column is calculated (the method employed here). If the trial results are sufficiently close to 100% 80 that the succeeding plate calculation yields the same equilibrium temperature, there is little difference in the final results obtained by the two methods. In Table I a comparison is shown between compositions obtained with balancing to 100 mole % after each plate calculation and after the entire column calculation. The difference in equilibrium temperature of the second plate produced a difference in the compositions of the
(11)
Here the term in brackets is constant. The suhscri t, f, refers to the number of the feed tray. The heat Iw througi the walls of the column per tray is q. The F X Q, must be the total heat content of the feed vapor plus liquid, 5. Calculation of new reflux ratios using the results of step 4. 6. Treating the reflux ratios as functions of temperature, the reflux ratios are interpolated at 1 O C. intervals. These new values are used t o re lace the reflux factors in step 1 above. Steps 4, 5, a n 8 6 may be performed by the 602-A machine or as a hand calculation. The assumption is here implied that the heat content of a given stream in the column is a function of temperature alone. Actually this will be very nearly true, as the compositions a t a given temperature level in the column will not change greatly on recalculating with the new reflux ratios. If desired, the heat balancing can be repeated after getting the compositions with variable L / V , This will result in even closer balancing.
THIELE AND GEDDES METHOD The discussion above has dealt with the calculation of individual sections of a column. In that case, the initial composition and an estimated or desired terminal composition must be
November 1951
INDUSTRIAL
A N D ENGINEERING CHEMISTRY
known. In the case of a complete two-section column, the problem is more complex, The top and bottom compositions cannot be completely fixed, Usually the conditions for which the column is to be built can be interpreted in terms of a specified separation of two of the components, called the ''keys." Then the resulting separations of the other component3 at a given heat input and optimum (minimum) number of trays are fixed. The determination of these separations and the remlting cornpositions at the feed tray which must match usually requires some trial and error plus experience. Here will be found one of the advantages
of machine calculation; many trials may be made with little more effort than is involved in a single trial. Trial and error calculations may be reduced by a modification of the method of Thiele and Geddes (16). Not only may the same wiring panel 1 be used, but also the identical cards as p r e pared for the Lewis and Matheson calculations. The fundamental equations are obtained by dividing Equation 1 through by xw to produce Equation 12, and dividing Equation 4 by xd to produce Equation 13,where T, is the ratio of xnto Ed or to xw.
L. + z 1 L+I 1 D rn+l - --
rn = rn-i rn
5
VL1 - 1 7
K n
Kn
Vn
ZK
(12)
(13)
The steps in the modified Thiele and Geddes method are as follows: 1. Assume a number of plates above and below the feed believed t o be somewhat in excess of the number which will be required. 2. Assign temperatures t o these various plates in accordance with some study of hhe volatilities and boiling points of the components. Choose trial values in like manner for the vapor flows in the column. The liquid flows are then calculated from the total overhead and b o t t o m flows. If these are not set by the conditions of the column design, values must be assumed, ordinarily in such a manner as t o require the lowest total number of trays. Use wirin panel 1, but change the trial and L1meChBnism,, so that c%oice of Lewis and Matheson calculations is always made. For the initial conditions r 1 for each constituent. The machine will calculate m - 1 or rn+i for each plate. This calculation is very ra id (five plates per minute for six componente). Therefore, it may &eadvisable t o carry this out for eeveral reflux ratios initially. will show 4. The listing of the array of for each the ratio of the concentration of each component in each plate t o ita concentration in the overhead or bottom. 5. With the arrays produced in step 3 in the hands of the chemical engineer, the over-all separations for the keys can be compared m t h those desired. Choosing the reflux and number of trays above and below the feed which match- best for the keys, the over-all "splits" for the other components are comguted t o yield an estimated overhead and bottoms com osition. 6. T o check the assumed temperatures, tiese estimated bottoms and overhead cornpositions are used in a Lewis and Matheson-type calculation. From this point on, the solution is succee sively refined by alternate use of the two methods. The original assumed overhead and bottoms quantities may be revised if Indicated. After balance is obtained at constant molal overflow, a cycle of heat balancing as described above may be introduced.
-
ILLUSTRATION OF USE OF COMBINED TECHNI QUEs
4
In order to demonstrate the use of the combined Lewis and Matheson method, Thiele and Geddes method, and heat balancing techniques, a suitable is presented in some detail. Each stage Of the entire process is illustrated in one Of the accompanying tables and discussed below. This is a case in which two components were present between the keys and in which two other components of the feed were nondistributed in the products removed. The example has been carried through to a condition of balance that would be satisfactory in practical work. It is readily apparent that, by continued repetition, the calculation may be brought into as fine a balance as desired. (BY balance is meant the degree to which the concentrations a t the feed tray calculated
2467
Table 11. Results of First Thiele a n d Geddes-Type Calculation (1) 1.00 0.10
80.03t:
0 03 0 03
(2)
(3)
1.00
loo
1.03
:I
0.35 0 28 0 25
::$! 0.03
;.:0.20
....
0'19 .... ....
0.03
....
....
.. .. ,. .. .. ..
9 283.4 80 3 22.0 5.4 1.0
*.
2 07
(4) 100 3.70
(5) 1 15
t.3": 4"3 i60: 273 l! 0 98 0.71 0.60
t.:Q 0.44
o'38 659.4 470.8 313.7 184.4 102.3 56.7 29.7 15.5 7.6 3 2 1.0
2 65 1.87
275 277
1.06
282
O"' 6.3 7.5 8.1 7.8 6.9 6.0 5.2 4 1 3 2 2.1 1.0
283 0 32 0 33 0 37 0.37 0 37 0.38 0 41 0.50
: $:
iLy
0.60
0 86 1.00
T
(6) 1 50 412 2068 8278
io 50 60 65 70 72 74 76 78 80
..... *....
..... .....
..80
0 28
0.28 0.29 0.32 0.32 0.32 0.32 0 33 0 39 0 66 1 00
85 90 91 92 93 94 95 100 110
Table I l l . Results of First Lewis and Matheson-Type Calculation (1)
(2)
(3)
(4)
(5)
2 75 0 15
40 24 20.96 159'J 13 96 12 16 10 45 8.48 34.40 18.49 7.94 2.93 0 92 0.24 0.05
32 14 32.25 2938 25.58 22 14 18 42 15 18 9 21 9.11 7.14 4.72 2 62 1.17 0.40
24.50 44.04 47.51 46.11 40 90 34 19 28 39 30 93 44 73 53 91 55 51 48.39 33.70 17.43
0 32 2 45 698 14.24 24 73 36 92 47.92 20.48 22 46 25.61 30.82 40.86 54 21 64.82
....
.. .. ..
,. .. .. .. ,.
.. ..
T
(6)
... ... ... ... ... ... ..
61
65 68 70 73 77
4 97 5 21 5.47 5.96 7 20 10.63 17 30
..
70 78 84 89 97 106
from the overhead composition down the column match the concentrations calculated up from the bottoms stream.) In this example, 100 moles per hour of a six-component liquid mixture enter a distillation column. Sixty-three moles per hour of roduct containing 1 / 1 ~of component 5 are t o be taken off as distifak product. Thirty-seven moles per hour are to be removed from the bottom and are tocontain l/low of component 2. The column is t o o erate a t atmospheric pressure (760 mm. of mercury[ The resuis of the calculation should show the number of p ates above and below the feed at some determined reflux ratio, the temperature of each plate, and the total composition of the distillate and bottoms product. The compos~t~on of per cent is "follows: the feed in 1. Methyl ohloride 2. Methylene chloride 8. Chloroform 4. Carbon tetrachloride 5 1 12-Trichloroethane 6: l:l:1,2-Tetraohloi~than~
1 7 25.4 20.4 22.1 24.0 6.4
Ste 1 (Table II). In order to select a suitable reflux ratio and to m a t e a crude estimate of the composition of the produrts! a Thiele and Geddes-type calculation was run at reflux ratios ( L / V ) of 0.20, 0.25, 0.30, and 0.40. A set of 10 postulated temperatures waa inserted above and below the feed trav. The results of this calculation for reflux 0.40 are shown in Table 11. These are the ratios of the concentration of the components on each tray to the concentration of these components in the over-
he^$^)$??^^ ::izx;n
factors in the two sections, trial overall separation factors for a column of any number of trays may be obtained. I n this case, the separation factors omitting the eight center plates were selected, because they matched reasonably well the re uired separation of the keys. On the basis of this, the moles components 3 and 4 could be tentatively p a p titioned between the overhead and bo!tom products. Thus tentative bottoms and distillate compositions were developed for use next step* These &??pear at the top and bottom Of
og
Step 2 (Table III). T o check on the assumed tempwatures and assumed product compositions from step 1, a Lewis and M a t h 4 SOn-tYpe calculation was next performed. Using Equations 1 and 4 with a reflux ratio of 0.40, calculation was carried toward the feed, resulting in Table 111, On the baais of this calculation, a better set of temperatures was selected for the next Thiele and Geddes calculation.
2468
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 43, No. 11
Figure 1. Punched Card Used in Six-Component Calculations of Lewis and Matheson Type Step 3 (Table IV). Another Thiele and Geddes-type calculation was performed using the tem eratures selected after Table 111 was calculated. After the cal%lation shown in Table IV had been completed, over-all separation factors were again calculated. On a basis of this information, still better apportionment of components 3 and 4 between the top and the bottom could be made. The new calculated products are shown a t the top and bottom of Table V. Step 4 (Table V). Starting with the im roved trial product compositions, step 2 was repeated. In or&r to study the disappearance of component 6 in the u per section, an additional calculation was made away from the Led tray using Equation 3. Equation 2 was used in a similar fashion below the feed, but showed that the concentration of volatile component 1 was less than 0.01% on the first tray below the feed. For the first four steps, the same deck of working cards was used. This reduces the calculating time considerably. The average machine time for the Lewis and Matheson calculation is 4 minutes for a 20' boiling range (six components). As there is no trial and error in the Thiele and Geddes calculation, the calculation time is five plates per minute. Step 5 (Table VI). A t this oint, heat balancing \vas introduced. The results on Table were adjusted to 100 mole % and the corresponding vapor compositions were calculated and adjusted. By combining the plate composition cards with the corresponding molar enthalpy cards, the enthalpy of each phase was calculated. Equations 10 and 11 were used to solve for L and 4'. A heat loss q = 5000 calories per plate was assumed. By adding D or subtracting W from L or L' new values for V and V' were obtained. Finally, after listing the new reflux ratios and their corresponding tem erature, interpolation was made for 1' intervals (Table VI). dnceforth the new reflux ratios were used in calculation. Step 6 (Table VII). Starting with the new variable reflux ratios, new working cards were prepared by multiplying the K values or their reciprocals by the a r p r i a t e factor. With theso new working cards, the Lewis and atheson method was used for the final calculation. The results are listed in Table VII. The steps in making the balance at the feed tray were: 1. Calculation by Lewis and Matheson method using new working cards and same top and bottom compositions. The sixth component was entered a t tray 7 a t 0.04%. This calculation is not shown. 2 Manual calculation of liquor leaving feed tray, assuming feed entered on tray 4, gave a poor match, component 4 matching only above tray 5, component 5 matching above tray 2. 3. The manual calculation of liquor leaving feed tray with feed on tray 5 showed fair agreement. To improve the matching, component 6 was introduced in liquox leaving tray 8 at the value 0.13% and component 3 was taken as 1.27% in W instead of 1.22%. The final results are listed above. Plate 5'(f) shows the composition of the liquor leaving the feed tray machine calculated from the bottom. Plate 5 (f), between plates 4 and 5'(f), shows the composition of t,he liquor leaving the feed tray
manually computed from the marhine value for plate 6 (calculated from top) plus the feed.
VAPOR-LIQUID EQUILIBRIUM CALCULATIONS The equilibrium composition of each phase in a liquid-vapor system may be readily calculated using similar machine techniques. The same data cards are used and only one additional wiring panel is required. Donne11 and Turbin ( 2 ) have described how this calculation may be carried out by a stepwise procedure. Starting with the equilibrium constants a t the desired temperature, they calculate the liquid compositions a t various ratios of vapor to liquid until they obtain a set of values whose sum is 100%. This procedure yields the value of v to the nearest 0.5% after 30 minutes' calculating time. The following procedure has been devised with a view of computing the entire curve of v us. temperature as well as the equilibrium compositions of bot,h liquid arid vapor at, each temperature. Table IV. Results of Second Thiele and Ceddes-Type Calculation (1) 1.00 0.06 0.03 0.03 0.03 0.03 0.03
.... .... .... .
.
I
.
....
443.7 22.8 1.0 ~~~
(2) 1.00 0.50 0.34 0.30 0.27 0.24 0.20 931.9 514.4 284.2 139.7 55.1 18.2 4.8 1.0 ~~~~~
(3) 1.00 0.97 0.80 0.70 0.63 0.55 0.47 16.3 16.5 16.8 15.0 10.6 6.3 2.9 1.0
(4) 1.00 1 70 1.60 1.62 1.50 1.30 1.10 1.07 1.41 2.00 2.69 2.91 2.68 1.88 1.00
(5) 1.00 6.65 17.11 35.35 63.75 99.83 134.76 0.30 0.31 0.33 0.37 0.44 0.60 0.82 1.00
T
(6)
.61.
1.00 11.10 50.40 89.94 179.86
86
66
70 73 77
.... ....
0.29 0.29 0.29 0.31 0.33 0.41 0.60 1.00
69 69 73 81 87 96 105
~~
Table V. Results of Second Lewis and Matheson-Type Calculation (1) 2.75 0.15 0.13 0 .-11 0.09 0.08
... .
., .. .,..
(2)
40.24 21.28 15.09 13.35 11.15 8.76 19.28 '2.20 3.73 1.24 0.33 0.07
(3) 31.66 32.01 28.13 25.09 20.66 15.73 21.22 18.44 12.70 7.52 3.52 1.22
(4) 25.03 44.00 49.67 45.63 39.10 32.34 32 '27 43.00 48.56 44.75 32.17 17.10
(5)
(6)
T
0.38 2.53 7.00 15.88 27.45 38.45 21 35 23.96 29.16 39.40 53.60 64.32
...
..
. I .
0.04 0.32 1.82 4.62 5.02 5.37 5.82 7.06 10.35 17.30
60 64 66 70 ,.
..
74 80 88
'25 104
INDUSTRIAL AND ENGINEERING CHEMISTRY
November 1951
Table VI. Reflux Ratios (at Each Trial Temperature) Calculated to Maintain Approximate Heat Balance in Column
Ln vu-
T
& T
I
0.3803 0.3856 0.3903 0.3944 0.3979 0.4008 0.4030 0.4042 0.4048 0.4049 0.4049 0.4050 0.4050 0,4050 0.4051 0.4051 0.4051 0.4051 0.4052 0.4052 0.4052 0.4052
57 58 59
60 61 62 63 64 65 66
67
.68
69 70 71 72 73 74 75 76 77 78
70
V'n 1.371
83 84
1:3i1 1.372
90 91
1:aiz
96 97 98 99 100 101 102 103 104 105 106 107 108
1 3i3 1.374 1.375 1.377 1.380 1.383 1.388 1.396 1.404 1.412 1.422 1.434 1.440 1.469
*.
..
5. Using wiring panel 2, pass all cards through the machine t o adjust figures to total 100 mole %. When low boiling gases are present, they are generally treated as inert diluents. In this case, Equations 14 and 15 become
(16-17) where j is the number of the inert component. A minor wiring change is controlled by j t o change the original equations to 10 and 17 for the inert component in the machine solution. To illustrate the method, the equilibrium a t 328-mm. pressure for a mixture of nitrogen and five chlorinated hydrocarbons is shown in Table VIII. Only 36 minutes of machine time were required for the calculation, including adjusting each composition to 100 mole %.
1.373
..95
At selected values of u, the trial and error calculation is carried out at successive temperatures separated by intervals of 1' C. When the first t o h l composition less than 101% ie encountered, the machine stops and signals immediately. The remainin unteeted temperature cards are removed from.the machine mf the "trailer" card is passed through the machine t o receive the results punched from the storage units. A new u is selected and the calculation is repeated. From the smooth slope of the u T curve, the group of temperatures tested can be made small after one or two calculations), while still maintaining ood proba ility that the equilibrium temperature will be incluted in the group tested. The equations used for solving for the compositions are:
CALCULATION OF ENTHALPY CHANGE ON PARTIAL CONDENSATION OR EVAPORATION Once the composition of the liquid and gas phase is calculated, it is a simde matter to calculate the enthalw of each mixture. Using wiring panel 1, the mole per cent of-each constituent is multiplied by the enthalpy of the corresponding phase at the temperature indicated. Instructed by panel 1, the machine calculates the partial molal enthalpy of each constituent and the total molal enthalpy of the phase. After these are tabulated, the total may be found by applying Equation
-
b
xi =
vi
1
(1
- v)
+
(14)
VKiZi
1 (1 - v ) + uKi ziKi
2469
(15)
These are calculated with the 602-A machine using wiring panel 3. I n this procedure, a lead card inserts u and (1 v ) into a storage unit. Succeeding cards are numbered 1t o n (n = 6) and carry K ~ zi,, and ( K ~ ~
