ENGINEERING, DESIGN, AND EQUIPMENT cubic feet of carbon monoxide plus hydrogen. I n calculating the gasifier heat loss for each run, a “thermal balance” temperature was computed and any serious difference between this temperature and observed gasifier temperatures indicated error in the run data. The data from those runs in which t h e temperatures were in closest agreement has been plotted in Figure 9. T h e curves seem t o indicate that, expressed as a function of synthesis gas yield, the heat loss from the gasifier a t constant oxygen t o carbon ratio varies inversely with the carbon throughput and, a t constant carbon throughput, increases directly with increasing oxygen t o carbon ratio. Conclusions a n d recommendations
The operation of a bottom-fired vertical gasifier with two systems of reactant feed has been described. It is believed t h a t many of the difficulties and lack of reproducibility encountered with the first unit are traceable t o the method of feeding and t o t h e changing geometry of the gasifier The indication of i b proved results with steam-conveyed coal leads t o the conclusion t h a t thorough premixing of t h e steam and coal is highly desirable and that perhaps the method of oxygen admission is less important. The mechanical operation of t h e steam-conveyed coal feed system was highly satisfactory in every respect. There seems to be no reason why this system could not be expected t o function smoothly over long periods.
T h e adoption of two opposed burners and the use of a very dense, high-service-temperature refractory apparently reduced refractory damage to the point where sustained operation is possible. It is believed t h a t the impact of the flame on the slag pool does produce undesirable agitation of the slag and promotes refractory loss t o some extent. Further reduction In reactant velocities might help. It might also help t o have t h e burners pitch downward less steeply. Slag temperatures could probably be maintained high enough for satisfactory tapping without such direct flame impingement. The refractory loss and slag agitation would both b e decreased in a unit of larger diameter, so t h a t possibly the present design might be satisfactory for larger scale operations. No matter how i t is achieved, i t is believed that a reasonable increase in refractory life would allow a gasifier of this design t o operate without difficulty, substantially continuously. literature cited
(1) Dressler, R. G.. Batchelder, H. R , Tenney, R. F., Wenzell, L. P., Jr., and Hirst, L. L., U. S. Bur. Mines, Rept. Invest. 5038, 1954. (2) Eastnian, D., Symposium, Gasification and Liquefaction of Coal, at Annual Meeting of the Am. Institute of Min. Met. Engrs., N. Y . ,N. Y., Feb. 20, 21, 1952. Published by The Am. Inst. of Min. Met. Engrs. N. Y., pp. 73-9, 1953. (3) Kastens, M. L., Hirst, L. L., and Dressler, R. G., IND.ENG. CHEM..44, 460-f?6 (1952). RECEIVED for review October
16, 1954.
ACCEPTED March 9, 1955.
Vapor liquid Equilibrium Self-lagging Stills DESIGN AND EVALUATION ARTHUR ROSE
AND
EDWIN T. WILLIAMS
The Pennsylvania S t a t e University, University Park, Pa.
I
T WAS desired to select, from the apparatus described in the literature ( 3 , 6, 7 , 10, 11, 13-16, 18, 19, 21), a superior still
for use a t atmospheric pressure and below. The choice was made most difficult by the disagreement in the published data for several systems which have been run in various stills. Garner ( 5 ) and Kortum, Moegling, and Woerner (8) emphasized this fact when they tested popular equilibrium stills with common mixtures and obtained considerably different results. Recognition of the importance of the boiling point measurement reduced the number of possible choices considerably. Development of self-lagging still
A modification of the Gillespie still ( B ) , designed in this laboratory in 1947 by D. F. Botkin, seemed t o offer the best possibilities. The apparatus is illustrated in Figure 1. The modification was the inclusion of a liquid trap t o permit sampling of the liquid t h a t was discharged from the Cottrell tube. T h e original Gillespie still provided for sampling the boiler liquid, rather than the liquid leaving the Cottrell tube. T h e fallacy of this procedure was pointed out by Fowler ( 4 ) and Othmer (10). This modified Gillespie still was tested in the preliminary stage of the present investigation using the ethyl alcohol-water binary, for t h e purpose of obtaining qualitative information about the possible errors. T o this end the still was run with no lagging or with heating wires on either one, or the other, or both, of two sections of the apparatus-the Cottrell tube-entrainment separa-
1528
tor section and the liquid trap section. The following results were obtained. When the Cottrell tube was unlagged, partial condensation of the binary mixture occurred. When the Cottrell tube was heated, incorrect boiling points were obtained. When the liquid trap was unlagged, the vapor above the liquid condensed and enriched the liquid sample. When the liquid trap was heated, total vaporization of some of the liquid droplets on the walls of the trap resulted in a lowering of the vapor concentration. These results were obtained by exaggeration of t h e underheating and overheating problems. It was felt t h a t good results were possible if the Cottrell tube, entrainment separator, and liquid trap could be kept in an adiabatic condition, but would be hard t o achieve with any of the usual types of automatically controlled heaters. Fowler ( 4 ) subsequently described this modification of t h e Gillespie still, and Brown and Ewald ( 2 ) employed a still modified in this way in several experimental investigations. It was decided to design a new equilibrium still, and a list of essential design features was compiled. Requirements of Equilibrium Still. The apparatus should be based on the Cottrell tube, because of its known value in determination of boiling point (18). T h e Cottrell tube, entrainment separator, and liquid trap should be automatically self-lagging, without the aid of complicated control mechanisms.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47,No. 8
ENGINEERING, DESIGN, AND EQUIPMENT T h e sampling system should have the following features.
~
No stopcock grease should come in contact with the test mixture. T h e samples should be taken while the still is in normal operation, and at the pressure prevailing in t h e still. T h e lines in the sampling systems should be flushed b y part of t h e liquid to be sampled. It should be possible t o chill the sample bottles. (They therefore may not contain moist air.) Construction of Self-Lagging Still. An all-glass, self-lagging equilibrium still, with a convenient sampling system, and a capacity of about 230 ml. was designed on the basis of the requirements listed (Figure 2). T h e central tube leaving t h e top of the boiler is the Cottrell tube. The liquid and vapor rising through it are separated after striking the baffle above t h e Cottrell tube. The liquid flows through t h e passes around the Cottrell tube, as indicated by the arrows, and returns t o t h e boiler. T h e vapor passes to the condenser through a 54-mm. tube. The condensate returns to the boiler through t h e condensate trap, from which samples can be withdrawn. Samplers are located a t each side of t h e apparatus. T h e tubes which extend downward from t h e liquid and condensate traps are bent behind the apparatus and terminate in wider (16-mm.) tubes (not shown). When t h e apparatus is being set up, each 16-mm. tube is extended vertically, and, after mercury and a n iron core have been placed in the 16-mm. tube, it is joined t o the top of the condenser. The mercury fills each 4-mm. tube to a point, above the Y , and prevents liquid egress during normal operation. When t h e iron core is raised b y means of a solenoid, the mercury level drops below the Y and liquid flows into the samplers.
A NEW SELF-LAGGING ATMOSPHERIC STILL
. . . permits accurate boiling point determination . . . allows sampling at any pressure while operating
n-Heptane-toluene proved best binary test mixture
A standard binary test mixture was wanted with properties that would maximize the possibility of detecting deficiencies in equilibrium stills when operated near atmospheric pressure. Requirements of Test Mixture. The components should be easy to obtain, to purify, and to keep in a final purified condition. The mixture should be amenable t o accurate analysis by a common, convenient, analytical method. The relative volatility should be small enough to permit several uniformly spaced determinations t o be made in the range between 20 and 80 mole % (liquid composition). T h e system should be capable of correlation by the Carlson and Colburn forms of the van Laar or Margules equations, because these are relatively simple to use. As these forms are for isothermal conditions, t h e boiling point range should be less than 15' C., so t h a t there will be no appreciable temperature effect on the activity coefficients. The determination of refractive index with a four-place refractometer was chosen as the common, convenient, analytical method. This choice in turn established a requirement of a refractive index spread of 0.1 unit between the pure components, as it was desired t o analyze the samples with an accuracy within about 0.1 mole %. A review of the systems reported in the literature resulted in the selection of the n-heptane-toluene binary as a standard test mixture for use in testing vapor liquid equilibrium stills a t atmospheric pressure. This mixture meets all the stated requirements. Preliminary still tests determined boiling point accuracy and equilibrium time
Figure 1. E.
Boiler
Modified Gillespie still
C. Cottrell tube
E.
Entrainment separator
The sample bottles are attached b y spherical-joint clamps, and the 3-mm. holes through the fixed cap and bottle are in alignment. Therefore a sample flows into t h e bottle and overflows through t h e 3-mm. tube, thus flushing the bottle. The caps should have as little open space as possible, t o prevent change of sample concentration b y vaporization. The internal heater is t h e same as t h a t of Gillespie (6). The external heater is made b y wrapping the boiler with asbestos tape, winding on 6.5 feet of S o . 29 B. and S. gage Nichrome wire (5.12 ohms per foot), and covering with more asbestos tape.
August 1955
Accuracy of Boiling Point Determination. Several sources of error should be considered in boiling point measurements in the self-lagging still. If the superheat of the liquid boiling in the still pot is not dissipated during passage up t h e Cottrell tube, the temperature will be higher than the correct boiling point. Another source of error is the pressure drop necessary t o force the vapor into the condenser. The pressure on the system is measured a t the top of the condenser, whereas the boiling point is measured at the top of the Cottrell tube. If the pressure differential between these points is sufficiently high, t h e temperature measured is not the boiling point that corresponds t o the measured pressure. In order t o evaluate these t\To sources of error in the selflagging still, a series of'temperature measurements was made on boiling water, with the heat input varied from 50 t o 300 watts. The higher heating rates should accentuate pressure drop and superheat errors. KO temperature variations outside the limits of accuracy (0.2"C. for preliminary experimental runs, 0.05" C. for final experimental runs) n ere observed. Three hundred watts was the upper limit, because a t this heat input the vaporization is so rapid that the Cottrell tube is virtually filled with vapor. No system would be run a t this heat input, because the poor liquid circulation would cause a long equilibration time. The measured temperature may also be in error because the heat losses by radiation or conduction from the thermocouple
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1529
ENGINEERING, DESIGN, AND EQUIPMENT junction are not negligible. While the boiling point tests and experimental runs were being made, two readings of each temperature were taken. For the first reading the Cottrell tubeentrainment separator section of the apparatus was covered with a n aluminum radiation shield, and for the second reading it was uncovered. This procedure did not cause a temperature change detectable by the temperature-measuring system, t h e limit of precision of which was 0.02" C. This was accepted as evidence
heat input to the still at 180 watts. A4nalyseswere made with a Valentine five-place refractometer which was reproducible t o within 0.00005 R.I. unit. T h e refractive index spread of t h e system was 0.11 unit. Temperature measurements were made with a Leeds & Northrup self-compensating potentiometer, using a calibrated copper-constantan thermocouple. The precision of temperature measurement was 1 0 . 2 O C. The y points all fell within 0.05 mole % of a smooth curve and all the t - z data within 0.2" C. of a smooth curve. T h e Ayrmeof the z - y values was 0.0035 as calculated by the method of Rose and others (12). It was concludcd t h a t equilibrium was reached within 0.5 hour.
2-
Still operated smoothly at 760 mm. of mercury
Making a Run. The procedure outlined here is for operation of the equilibrium still at 760 mm. of mercury. The prevailing atmospheric pressure was about 730 mm. of mercury. T h e procedure would have t o be modified if determinations were made under vacuum, but all the runs made for the present paper were a t 760 mm. of mercury. The procedure gave smooth operation without bumping in the initial start-up or surging during the run.
Figure 2. 6. C.
Self-lagging equilibrium still
Boiler Cottrell tube
CT. IT.
Condensate trap Liquid trap
that losses were practically negligible up t o the highest operating temperature, which was 110" C. For higher temperatures a radiation shield and/or insulation should be provided, if tests such as the above indicate the need. Absence of appreciable heat loss by conduction from this type of still has been demonstrated by showing that temperatures and compositions are unchanged when an electrically heated jacket is added. This confirms the prediction based on approximate calculations of the heat transferred by conductivity through the series of resistances between the thermocouple and the surrounding atmosphere. The fact t h a t the vapor compositions calculated from t h e measured boiling points deviated only 0.19% from the measured compositions also indicates that errors in temperature measurement are small. OF HEAT INPUT TO Tests of Equilibration Time. EFFECT STILLPOT. With n-heptane-toluene as the test mixture, a series of 1-hour runs was made with the heat input ranging from 50 t o 300 watts. An approximately 50 mole % solution was used each time and the results were compared with the data of Steinhauser and White, as these had been proved to be the best of the published ones. For heat inputs between 100 and 200 watts the vapor compositions agreed within 0.3 mole % with those of Steinhauser and White (I?'). Below 100- and above 200-watt input the vapor compositions were up to 3 mole 7 0 lower than those of Steinhauser and White. This indicated that the equilibration time was longer than 1 hour outside of the 100- to 200watt range. Below 100 watts, the vapor circulation is slow; above 200 watts, the liquid circulation is slow. DETERXINATION OF EQUILIBRATION TIME. T h e determination of equilibration time was combined with a preliminary determination of the equilibrium curve of n-heptane-toluene.
A series of runs was made with the self-lagging equilibrium still; the time of run was varied from 0.5 t o 3.5 hours, with the 1530
The air was removed from the apparatus b y successive evacuation and filling with nitrogen. With the nitrogen turned on, t h e cap was removed from a filling tube and connected t o t h e top of the condenser, and a buret was suspended above it. T h e flow of nitrogen through the condenser prevented entrance of air. T h e external heater was turned on and set a t 50 watts. After the still pot had become warm, the mixture t o be tested was poured into the buret, and then allowed to enter the apparatus. When 100 ml. of the mixture had entered the pot, the condenser water was turned on. The internal heater was then set at 30 watts and the external heater a t 100 watts. Addition of the mixture through the buret was continued. The material in the still pot had, of course, already begun to vaporize and t o condense in the liquid trap. This vaporization continued until sufficient liquid had entered the pot t o cause vapor-lift action to begin. T h e object of this preheating and slow filling from the buret is t o heat the liquid as i t enters the apparatus, thus avoiding bumping. Filling was continued until t h e li uid level in the condensate chamber was above the circular %elf. The external heater was then set a t 150 watts. T h e apparatus was allowed to run for a few minutes, so as t o heat the entire charge, and then t h e sampling valves were activated, and both sampling bottles flushed with the hot test mixture. T h e bottles were then removed and dried by blowing out with air. The tubes which extend into the sampling bottles were dried by insertion of a wisp of paper toweling, and t h e sample bottles replaced. Sufficient material had been withdrawn in the flushing operation t o lower the liquid level in the condensate return line t o a point about 1 inch below the top of the overflow tube in the condensate trap. T h e cap was replaced on the filling tube and the pressure rose t o 760 mm. of mercury, where it was held by t h e automatic control system. The entire operation took about 10 minutes. When a record was kept of the time of a particular run, it was timed from the start of Cottrell tube action. h second flushing, with the actual sample, occurs a t the time of sampling.
.. .
and consistent experimental results were obtained for n-heptane-toluene
.
Data of Other Investigators. Three previous determinations of the z - y - 1 relationships of n-heptane-toluene have been reported (1, 16, 1 7 ) . I n order to compare these objectively, an equilibrium curve was predicted by the trial-and-error method (12, 19) from the data of each of these investigations, and compared with the corresponding visually smoothed experimental data by calculation of t h e root-mean-square deviation between the predicted and experimental values of vapor composition, a t nine evenly spaced values of liquid composition. The results of the correlations are shown in Tables I, 11, and 111. The data of Steinhauser and White are clearly more consistent. For the prediction incident t o each correlation, t h e value of ya at the base point was chosen to the nearest 0.2 mole 7 0 ,
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Vol. 47, No. 8
ENGINEERING, DESIGN, AND EQUIPMENT Table 1.
X
Correlation of Bromiley and Quiggle Data on n-Heptane-Toluene at 760 Mm. of Mercury
Visually Smootlied Experimental Y t
0 0.1 0.2 0.3 0.4
760 760 760 760 760 760 760 760 760 760
0.5O
a
0.6 0.7 0.8 0.9 I .o Base point.
Table
II.
-
0.5a
...
Aljrrns = 0 00830
Correlation of Sieg Data for n-HeptaneToluene a t 760 Mm. of Mercury
Visually Smoothed Experimental
0 0.1 0.2 0.3 0.4
7fi3:5 766.8 766.6 764.0 760 0 758.5 759.3 761 .O 761.3
...
L~
! I
X
Predicted A = R = 0.1023, 0.1252, Y T
0 0.163 0,292 0.398 0.492 0.577 0,659 0.740 0.823 0.915 1.0
0.6 0.7 0.8 0.9 1.0 a Base point.
t
T
110.62 107.98 105.86 104.19 102.84 101.65 100.65 99.86
760 760 760 760 760 760 760 760 760 760 760
99.28
98.78 98.45
Predicted __ A = B = 0.1115, 0,1278, Y 7r 0 0.1609 768.' 1 0.2944 758.6 0,4057 759.4 0.5026 760.6 0.590 760.0 0.6718 758.7 0.7607 759.2 0,8293 760.6 0.9113 760.9 1,0000 ... Ayrms 0,00872
and the final calculations were made by mesns of nine-point checks (fa or 19). Discussion of Results. The preliminary runs, in which the boiling points were determined within 0.2" C., gave Ayrms = 0.0035. Kine additional runs were made, covering a liquid composition rang61 of from 10 t o 90 mole yo. The time of each run was 2
hours or more. The temperature-measuring apparatus permitted determination of the boiling point with a precision within 0.02" C. and an accuracy within 0.05" C. T h e temperatures deviated less than 0.02" C. from a smooth curve. These final experimental z - y - t data are given in Table IV. The z - y points of the preliminary runs are included, because the compositions were measured with t h e same degree of accuracy as those of the final runs. Table V gives the results of the correlation. T h e equilibrium curve predicted from the boiling point data showed an average deviation from the experimental equilibrium curve of Ayrms = 0.0019. This represents a greater consistency than had been exhibited by any data tested, including the many systems examined in connection with the development of the trial-and-error method of prediction and correlation ( 12 ) . The experimental equilibrium data are plotted in Figure 3. The boiling point curve is plotted in Figure 4. Still modifications permit its use as boiling point apparatus
Alternative Sampler. If it is possible to use a stopcock lubricant with a given system, it may be run in a self-lagging still on which the sampling devices have been modified, as shown in Figure 5 . Use a s Boiling Point Apparatus. The modification illustrated in Figure 6 is proposed to permit the use of the self-lagging still as
Table 111. Correlation of Steinhauser and White Data for n-Heptane-Toluene at 760 Mm. of Mercury
X
Visually Smoothed Experimental Y t
0 0.1 0.2 0.3 0.4
0.5a 0.6
0 7 0.8 0.9 1.0 a Base point.
Figure 3.
Experimental vapor liquid equilibrium data for n-heptane-toluene a t 760 mm. X. Y.
August 1955
M o l e fraction n-heptane in liquid Mole fraction n-heptane in vapor
Figure 4.
T
760 760 760 760 760 760 760 760 760 760 760
- PredictedA = B = 0.1467, 0 1386, Y r 0 0 1704 0 3017 0 4091 0 5026 0 687 0 6668 0 7460 0 8248 0 9088 1 0000
76i :o
759.0 761.0 760.1 7 6 0 .a 761.4 760.7 759.5 758.9 ,
.
Ayrrns = 0,00472
Boiling point composition data for n-heptanetoluene a t 760 mm.
T. X.
Boiling point C. Mole fraction n-heptane in liquid
INDUSTRIAL AND ENGINEERING CHEMISTRY
1531
ENGINEERING, DESIGN, AND EQUIPMENT
Table IV.
(Temperature measurements accurate to
Y
2
0
0
0 040 0 086 0 094 0 106
0 073 0 147 0 156 0 174
0,142 0.176 0.181 0.264 0.265
0 223
0.265 0.273 0.352 0.365
0.276 0.347 0.417 0.489 0.503
0,375 0.450 0.510 0.573 0.585
t,
c.
r;;
110 59 107 88
1 268
106.09
1,209
104.47
1.158
102.43
1.090
101.52
1.064
=to.05'
Y
X
0 614 0 657 0 766 0 800 0 800
0 0 0 0 0
n 8x7 -._I.
n- . mi
678 710 799 827 8275
0.890 0.893 0.903 0.924
0.904 0.906 0.914 0,934
0.964 0.971 0.976 0.984 0.994
0.969 0.974 0.979 0.986
...
C.)
Correlation of Final Runs on n-HeptaneToluene a t 760 Mrn. of Mercury
(Temperature measurements accurate to f 0 . 0 5 ' C.)
t 100 17
Predicted 1 028
99 29
1 009
98.81
1.003
98.48
1.002
98,46 98.43
1.002
0.513 0.591 0.996 , , _ 98.43 0.584 0.651 100.77 1,042 1.000 , , . 98.43 a Activity coefficient of n-heptane calculated from experimentd data.
a boiling point apparatus. The parts of the still not shown are exactly as indicated in Figure 2, except t h a t the samplers may be of the form shown in Figure 5. A mixture of known composition is made b y use of the analytical balance, and the still is charged and operated as previously described. T h e boiling temperature is read, and then t h e pressure is increased and the heat turned off to stop t h e boiling. T h e contents of the liquid trap are drained completely, and fresh original solution is introduced through the opening in the top of the still. A filling tube must be used t o direct the liquid into the chamber t h a t the thermocouple well enters, so that none can enter the condensate trap. T h e apparatus is restarted and the boiling temaerature measured. Successive redacements of t h e liquid-in t h i liquid trap are made until no change in boiling point occurs. When this happens, i t is because the liquid removed from the trap had t h e same composition as t h e liquid introduced, T h e boiling_ point measured is t h e boiling point of _ t h a t liquid. T h e ordinary Cot,trell apparatus, T J and the modification designed by Swietoslawski (I@, d o not measure the boiling point of the liquid introduced, aB the liquid is introduced into the still pot and the boiling point, corresponds t o the liquid leaving the top of the Cottrell tube. They depend on relatively little vapor formation and postulate that the W liquid composition cannot change appreciably. - T h e Figure 5. Modified sampler modification introduced does not inT. Liquid or condensate trap terfere with its operation as an equilibrium still, BO one apparatus may serve in place of two. Design for Vacuum Work. If the self-lagging still is t o be used for the determination of vapor liquid equilibria under vacuum, the Cottrell tube should be of wider diameter, and the
1532
Table V.
Final Experimental Data for n-HeptaneToluene at 760 Mm. of Mercury
A =
Visually Smoothed Experimental xa y= t, 7r rH"
O.lt94, Y
0 0.1 0.2 0.3 0.4 0.56 0.6 0.7 0.8 0.9 1.0
0 0.1692 0,2980 0.4002 0,4972 0.683 0.6651 0.7461 0.8276 0.9119 1.0000
c.
0 0.166 0.294 0.4005 0.497 0,6825 0,664 0.744 0.8275 0.912
110.69 107.73 105,62 103.88 102.59 101,52 100.60 99.82 99,26 9%78 98.43
760 760 760 760 760 760 760 760 760 760 760
1.273 1.196 1,160 1,101 1,066 1,039 1,020 1.010 1.003 1,000
7r
= 0.1132 7;
760.8 761.7 754.8 760.0 760.0 769.0 758.3 759.4 759.4
1,299 1.215 1.151 1,102 1,066 1.039 1.021 1 ,009 1 ,002 1 ,000
-
a Visua! smo0thin.g of 2 21 data from preliminary and final runs. Actlvlty coefficient of n-heptane calculated from visually smoothed experimental data. Predicted from final t - x and 7r - x curves. d Activity coefficient calculated from van Laar constants. e Base point. A p m s = 0,0019.
*
condensate trap should be as small as sampling demands will allow. T h e diameter of the Cottrell tube needs to be increased, because the extra volume occupied by the vapor under vacuum would cause poor liquid circulation in the same manner as an excessive heat input did in t h e atmospheric pressure runs described in the discussion of equilibration time. The Condensate trap needs t o be raduced in size, because t h e condensate for a given volume of vapor is much smaller under vacuum, and the '%urnover" in the condensate trap would be too slow.
J
Self-lagstill modified for
use
boiling Point apparatus
0. Opening for introducing shution
Purification and Properties of n-Heptane. The n-heptane used in this investigation was Phillips ASTM grade. It was purified b y distillation through a column 4 inches in diameter packed with 0.24 X 0.24 inch stainless steel protruded Cannon packing. T h e column was e uivalent t o between 100 a n 2 120 theoretical plates. T h e reflux ratio used was 100 t o 1. The heart cut only was used in this work. It had a refractive index a t 25' C. of 1.38518. T h e National Bureau of Standards ( 9 ) value is 1.38517. This n-heptane was used as a primary standard in the calibration of the thermocouple. T h e vapor pressure of n-heptane, for use in t h e calculations, was calculated from the following Antoine equation: loglo P = 6.90319 1268.586 (216.954 t':C.)
+
('I
Purification and Properties of Toluene. The toluene used in this investigation was from t h e J. T. Baker Chemical Co. It was distilled in t h e same manner as t h e n-heptane and t h e heart cut was passed through silica gel. It was stored over calcium chloride. I t s boiling point was 110.59' C., and its refractive index at 25' C. was 1.49405. The National Bureau of Standards values (9) were 110.623 and 1.49413, respectively. T h e vapor pressure of toluene was calculated from t h e following Antoine equat,ion: log,, P
6.95334 -
INDUSTRIAL AND ENGINEERING CHEMISTRY
1343.943 -(219.377 $- t o C.)
('I Vol. 47, No. 8
ENGINEERING, DESIGN, AND EQUIPMENT Refractive Index Curve. Small weighing bottles of about 30ml. capacity, with female-joint tops, were found convenient for weighing out the liquids in the determination of t h e refractive index curve. T h e fact t h a t the grinding on the bottle proper was on t h e outside prevented accidental wetting of t h e ground portion as t h e liquids were poured in from small graduated cylinders. The smallest quantity of a n y one liquid weighed in these determinations was 1 gram, and because the balance was accurate to 0.0002 gram, t h e combined error for any sample could not be over 0.04%. Pressure-Regulating System. The barostat used was that designed by Willingham and Rossini ( 2 0 ) a t t h e National Bureau of Standards. It was thoroughly cleaned, then heated under high vacuum and filled with clean, redistilled mercury, a vacuum being obtained b y carefully boiling out dissolved air under vacuum M hile the filling was in progress. The barostat was installed in an air thermostat, the temperature of which was controlled t o u-ithin 3" C. b y a Fenwall thermoswitch, which was surrounded by the circulating air stream. T h e temperature of the mercury lagged behind t h a t of the air, and consequently did not change over as a i d e a range. 4 dibutyl phthalate manometer registered the difference between the system pressure and t h a t of t h e atmosphere. B y reading this manometer and the barometer, it was possible t o calculate the pressure within the apparatus. Kitrogen was supplied t o the barostat through a diaphragm valve, from a standard nitrogen cylinder. When the mercury column reached the upper contact of the barostat, a circuit was completed which activated a relay t h a t opened a solenoid valve and allowed escape of nitrogen through a needle valve. All runs were made a t 760 mm. of mercury T h e prevailing pressure in State College is about 730 mm. The pressure variation due t o the variation of the temperature of the mercury was about 0.1 mm. This far exceeds pressure variation due t o the opening and closing of the solenoid valve. It was easy to adjust t h e diaphragm valve on the nitrogen cylinder and the final outlet needle valve, so that the pressure variation could not be detected on the dibutyl phthalate manometer. Temperature Measurement. The temperature-measuring system consisted of a calibrated single-junction copper-constantan thermocouple, a Rubicon galvanometer with a sensitivity of 1.1 pv. per mm., and a Leeds & Northrup Type K-2 potentiometer, which could be read to 0.1 mv. The calibration was accomplished b y reading the electromotive force with carefully purified n-heptane, and with pure water boiling in the apparatus. The difference between the observed voltage and the voltage of a standard sample was taken a t each point and plotted against the observed voltage. A straight line was drawn from the zero point ( A E = 0, EOb4 = 0) which passed between the plotted points. The distance of t h e
calibration points from the line was within the limits of accuracy (0.05 O C. ) of t h e temperature-measuring equipment. Subsequent voltage readings were corrected b y adding t h e AE for each When the boiling point of the carefully puSified toluene was measured, it was 110.59",as compared t o a National Bureau of Standards value of 110.62'. This serves as a further indication of the accuracy of the temperature measurements. Literature cited
Bromiley, E. C., and Quiggle, D., IND.ENG.CHEX.,25, 1136 (1933).
Brown, J., and Ewald, A . H., Australian J . Sci. Research, 3A, 306-23 (1950); 4A, 198-212 (1951).
Cornell, L. W., and Montonna, R. E., IKD.ENQ.CHEM.,25, 1331-35 (1933).
Fowler, R. T., J. SOC.Chem. Ind. (London), 68, 131-2 (1949); Ind. Chemist, 24, 717, 824 (1948).
Garner, F. H., Ibid., 25, 238 (1948). Gillespie, D. T. C., IND.ENG.CHEM.,ANAL.E D . ,18, 575 (1940). Jones, C. A., Schoenborn, E. >I., and Colburn, A. P., IND. ENG.CHEM.,35, 666-72 (1943). Kortiim, G., Moegling, D., and Woerner, F., Chem. Eng. Tech., 22, 453-7 (1950).
Natl. Bur. Standards, Circ. C461, 39, 43, 123, 126 (1947). Othmer, D. F., IND.ENQ.CHEM.,20,743-6 (1928); Anat. Chem., 20, 763 (1948).
Regnault, Mem. mad. inst. France, 26, 727 (1862). Rose, Arthur, Williams, E. T., Sanders, W. W., Heiny, R. L., and Ryan, J. F., IND.ENG.CHEW,45, 1568-72 (1963). Rose-Innes and Young, Phil. Mag., (V) 47, 353 (1899). Saddington, A. W., and Krase, N. W., J . Am. Chem. Soc., 56, 353-61 (1934).
Sage, B. H., and Lacey, W. N., Trans. Am. Inst. Mining Met. Engrs., 136, 13G-57 (1940).
Sieg, L., Chem. Eng. Tech., 22, 322-6 (1950). Steinhauser, H. H., and White, R. R., IND.ENG.CHEY.,41, 2912-20 (1949).
Swietoslawski. W., J . Chem. Educ., 5, 469 (1928). Williams, E. T., Ph.D. thesis, Pennsylvania State College, University Park, Pa., 1952. Willingham, C. B., and Rossini, F. D., API Research Project 6, December 1945. Zawideki, J., 2. physik. Chem., 35, 129 (1900). ACCEPTED February 2 , 1955. RECEIVED for review April 21, 1953. Presented before the Division of Industrial end Engineering Chemistry a t Los .4ngeles, the 123rd Meeting of the AMERICAN CHEMICAL SOCIETY, Calif,
Applications of Matrix Mathematics to Chemical Engineering Problems ANDREAS ACRlVOSl
AND
NEAL R. AMUNDSON2
Departmenf of Chemical Engineering, University of Minnesofa, Minneapolis 14, Minn.
R
ECENTLY, chemic81 engineers have become increasingly aware of the usefulness of the calculus of finite differences as a mathematical tool for solving problems in stagewise operations. This particular section of mathematics has remained dormant for some years, because of the restricted use it has found in most branches of applied mathematics, where mostly continuous, rather than discrete, phenomena are studied. There are, however, many textbooks on the calculus of finite differences, including those of Boole ( 3 ) , Jordan ( I S ) , Milne-Thompson ( I $ ) , Wallenberg and Guldberg (271, and Norlund (21). Applications of the calculus of finite differences t o a variety of relatively simple chemical engineering problems have been made 1 Present address, Department of Chemistry and Chemical Engineering, Univeisity of California, Berkeley, Calif. * Present address (on leave), Cambridge University, Cambridge, England.
August 1955
by Tiller and Tour ( 2 5 ) , Tiller (ZC), Mason and Piret ( l 7 ) , Lapidus and Amundson (16), and Amundson ( I ). This article presents a method t h a t will enable the chemical engineer t o deal with more involved problems t h a t arise in connection with the unsteady-state behavior of stagewise operations; there has been, of late, considerable interest in these chemical engineering problems. This new approach will make extensive use of matrix algebra, the salient features of which are described. It is surprising t o learn t h a t most results and theorems of practical interest concerning matrices were discovered almost a century ago. However, their usefulness in many branches of applied mathematics was not appreciated until very recently. The present interest in matrices was first stimulated by the work of Born, Heisenberg and P. Jordan in quantum mechanics, a n d
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