268
INDUSTRIAL AND ENGINEERING CHEMISTRY ACKNOWLEDGMEKT
The authors thank F. D. Rossini and the staff of A.P.I. Research Project 6, at the X’aticnal Bureau of Standards, for the careful fractionation and accurate measurement of the properties of the fractions of cut A in Figure 7 . LITERATURE CITED Anderson and Engelder, ISD.EBG. CHEM.,37, 541 (1945): Anderson. Jones. and Engelder, I b i d . , 37, 1062 (1945). Bouzat and Schmitt, C o r n p i r e n d . , 198, 1923 (1934). Bragg, I N D . ENG.C H E X . , AN.4L. E D . , 11, 283 (1939). Brunn, Leslie, and Schicktanz, BUT.S t a n d a i d s J . R e s e w c h , 6 , 363 (1931). Foreiati, Glasgow, Willinghan, and Rossini, Am. Petroleum Inst.. Rept. on Research Project 6 (1945), for hydrocarbons 1, 4, 6-1i; Dairies and Gilbert, J . Am. Chem. SOC.,63, 2731 (1941), for hydrocarbons 2 (corrected), 3 , 4 , 5 , 10-14; Brooks, Howard, and Crafton, J . Research N a t l . B u r . Standards, 24, 33 (1940), for hydrocarbons 3, 4, 6, 14; Brunn and HicksBrunn, Ibid., 10, 465 (1933), for hydrocarbon 15: Glasgow, Ibid., 24, 509 (1940), for hydrocarbons 16 and 17.
Vol. 38, No. 3
ENG.CHEY.,35, 247 (1943). (6) Griswold, IND. (7) Griswold and Ludwig, I b i d . , 35, 117 (1943). ( 8 ) Herington, T r a n s . Faraday Soc., 40,274 (1944). (e) Jackson and Young, J . Chem. SOC.,73, 992 (1898). (10) Lecat, “L’aaeotropisme”, p. V I I I , Brussels, 1918. (11) Ibid.. n. 166. (12) Mair, Glasgow, and Rosiini, J . Research XatE. Bur. Stn,6r/rirdu, 27, 39 (1941). (13) Mair and White, I h i d . , 15, 51 (1935). Marschiier, thesis, P a . State College, 1936. Nagornov, Ann. inst. anal. p h y s . chim. (Leningrad), 3, 5iil’ (1927); Chem. Zentr., 1927, 11, 2G68. E m . CHEM.,36, 805 (1944). (16) Richards and Hargreaves, IXD. (17) Scatchard, Wood, and Mochel, J . Phys. Chem., 43,119 (193TJj. (1% Schultee and Stage, OeE u. KohEe, 40, 66, 68 (1944). (19) Stage and Schultze, 1 h i d , 40, 90 (1944). E N G .CHEW., 25, 733 (1933). (20) Tongberg and Johnson, IND. (21) White and Rose, Bur. Standards J . Research, 9, 711 (1932). (22) Whitmore and Lux, J . Am. Chem. SOC.,54, 3448 (1932). (23) Young, J . Chem. SOC.,83, I, 74 (1903). (24) Zelinsky, J . R u s s . Phys.-Chem. SOC.,43, 1222 (1911). before the fiftieth anniversary Technical Confererice of the Chicago Section, A V E R I C A XC I I E \ I I C ~S L O C I E T Y , November, 1945. PREBEBTED
Empirical Equation for Theoretical Minimum EDWARD G. SCHEIBELI IND CHARLES F. MOKTROSS2 Polytechnic I n s t i t u t e of Brooklyn, .V.2’.
An empirical equation is presented for the calculation of minimum reflux ratio in the fractionation of multicomponent mixtures. The empirical equation is divided into three parts: (1) the determination of the minimum reflux ratio that would be required to separate the key components if all lighter components had infinite volatility and all heavier components had zero volatility; (2) the determination of the incremental reflux required to separate the heavier components from the light key based on the actual volatilities of these heavier components; and (3) the determination of the additional amount of reflux required to separate the lighter components from the heavy key, based on the finite relative volatility of these components. The pseudo minimum reflux ratio can be calculated from the binary equilibrium curve for the key components. The additional reflux quantities are based on empirical functions developed from an analysis of the calculations on a large number of systems. The equation gives a direct calculation of the theoretical minimum reflux ratio and eliminates the trial-and-error calculations of previous methods. The equation has been found to have an accuracy of about 1%in the cases usually encountered in practice.
T
HE theoretical minimum reflux ratio in distillation is the lowest reflux ratio at which a given separation can be obtained through the use of a n unlimited number of trays. At reflux ratios greater than the minimum, a finite number of trays is required. At total reflux the smallest number of trays is required. From correlations based on the minimum reflux ratio and the minimum number of trays, it is possible t o estimate the number of trays for any reflux ratio above the minimum (1,4). I n binary systems the minimum reflux ratio occurs when the 1 2
Present address, Hoffmann La Roche, Inc., Nutley, N. J. Present address, The Lummua Company, New York, N. Y.
operating lines for both the stripping section and the fractionating or enriching section of the toTver intersect a t the equilibrium curve. Thus, in these cases the minimum reflux ratio is readily calculable. However, in multicomponent distillation the problem is more complex since the equilibrium curve is a function of t h c reflux rat,io. As tray calculations are carried out up the column from the bottoms product, a point of maximum concentration of thc light key will be reached, and additional trays will not change the liquid and vapor compositions. This is called the “stripping section pinch”. Similarly, a point’ is finally reachcd in the fractionating section where the heavy key component reaches its maximum concentration, and additional trays will not change the compositions of the vapor and liquid. This is called the “fractionating section pinch”. Fenske (3) and Uiidcrrvood (8)derived equations for calculat’ing these pinch compositions directly. Jenny (6) gave the first exact definition of minimum reflux in multicomponent distillation. At minimum reflux in tt multicomponent system the two pinches do not occur a t the intersection of the operating lines, as in a binary system. The pinch region in t,he fractionating section occurs above the pinch region in the stripping section. The fractionating pinch composition contains only the components in the overhead, such as the light, and heavy key components which are being separated and tho components lighter than these keys. The styipping pinch similarly contains only the components present in the bottoms product, which consists of the key components and those heavier than the keys. The correct minimum reflux ratio is that which gives a match of the components when tray calculations are carried out between the pinches. The calculations are made by adding a trace of the lighter components to the stripping pinch and a trace of the heavier components to the fractionating pinch. The rigorous method of determining the minimum reflux ratio is an extremely tedious trial-and-error process. Several shorter methods have been developed (1, 5 ) . They are based on the
March, 1946
.
269
INDUSTRIAL AND ENGINEERING ]CHEMISTRY
assumption that the ratio of the keys in the pinches a t minimum reflux are equal. These methods depend on a series of trials t o obtain the desired condition. Gilliland (4) developed a formula for calculating minimum reflux based on this assumption. The equation is directly applicable to separations where the feed is completely liquid; for other conditions of the feed a trial-anderror method is required. The equation was observed t o give a low value, and Robinson and Gilliland (7) presented a second equation which gave a limiting high value. They recommended the average of the two values as the desired minimum reflux. Colburn (2)showed that the ratios of the keys in the pinches are not equal at minimum reflux. The ratio of the light key to heavy key in the stripping pinch is always greater than the corresponding ratio in the fractionating pinch, and Colburn developed an empirical expression for this relation. The method thus consists of calculating the pinch compositions a t different reflux ratios until the empirical relation is satisfied. The reflux ratios determined by Colburn’s method have been found t o agree t o about 1% with the ratios obtained by the rigorous tray calculations. Obviously, a direct straightforward calculation of minimum reflux ratio would be most desirable. The rigorous method is too complex to yield an exact algebraic equation; therefore, an empirical approach must be followed. This can be developed by considering the binary equilibrium curve of t h e keys, and by assuming that all components ’heavier than the heavy key have zero volatility and, hence, do not have a partial pressure in the vapor phase, and that all components lighter than the light key have infinite volatility and thus have no concentration in the liquid. Under these conditions the only effect of the presence of the other components will be on the slope of the operating lines on the two-component diagram. The equilibrium curve will correspond to the binary curve for the key components. However, the operating lines for the fractionating and the stripping section are altered to give a different locus of their intersections a t different reflux ratios. This locus is called a “feed line”, and the minimum reflux ratio is that which corresponds t o the intersection of the feed line with the equilibrium curve. PSEUDO MINIMUM REFLUX RATIO
Figure 1 shows the location of the different feed lines for the determination of the pseudo minimum reflux ratio. Line a shows the case of liquid feed at the temperature of the feed tray, containing no components lighter than the keys. Line b is for the case of liquid feed at the feed tray temperature, containing some components lighter than the keys. Line c represents the case of total vapor feed at the feed tray temperature, containing no components heavier than the keys. Line d shows the casd of total vapor feed at the feed tray temperature, with some components present heavier than the keys. For the case of partial vapor feed with components both heavier and lighter than the keys, the feed line will lie somewhere between d and b, depending on the relative amount of vapor and liquid. When components lighter than the keys have infinite volatility and the components heavier than the keys have zero volatility, the equilibrium curve does not vary with reflux ratio. Thus the minimum reflux ratio can be readily calculated as in any binary system:
where Rd = pseudo minimum reflux ratio, based on key components x p = overhead product composition based on key componentgody ‘ys = vapor composition at intersection of operating lines a t minimum reflux = intersection of feed line with equilibrium curve x i = liquid composition at intersection of feed line with equilibrium curve
a 0
J z > W Y I-
r
c1 LL
0
z
0 I-
o 4 a LL _I Y
0
z
0 MOLE FRACTION OF LIGHT KEY IN LIQUID
Figure 1.
Location of Feed Lines for Determination of Pseudo Minimum Reflux Ratio
The slope of the feed line depends on the ratio of liquid to vapor in the feed. To calculate the pseudo minimum reflux ratio, a pseudo-ratio of liquid to vapor must be used: m =
X vL X
XD - ZZXA -M ML v - ZMD BMA
pseudo ratio of liquid to vapor in feed mole fraction of feed as liquid mole fraction of feed as vapor total mole fractions of components heavier than heavy key in feed total mole fractions of camponents lighter than light key in feed moles of liquid feed moles of vapor feed total moles of components heavier than heavy key in feed total moles of components lighter than light key in feed I
Figure 2. Relation between Feed Line and Pseudo Minimum Reflux Ratio in Gasoline Stabilizer
INDUSTRIAL AND ENGINEERING CHEMISTRY
270
If zj is defined as the equivalent feed composition based on the key components as follows, zj = X B / ( X B
+ Xc)
it is possible to draw a feed line (Figure 1) through point zj with a negative slope equal to the pseudo ratio of liquid to vapor. The intersection of this line with the equilibrium curve locates (zi, ut)which is used in the calculation of the pseudo reflux ratio. If the equilibrium curve is not readily available but the relative volatility is known, it is possible to calculate the value of 2% algebraically. The equation of the feed line is: y =
zj
+ m(z, -
5)
The equation of the equilibrium curve is: = O I ~=
1
+
OIBz (OIB
-1) z
relative volatility of key components
Solving these two equations simultaneously, the following equation can be developed to give the intersection of the two lines:
((YE
2,
-1) (1
=
where
+ m) x j F c X B
o1B
- m ==!
yl[(aB
lighter components to have infinite volatility. The second term represents the additional amount of reflux which must be added to separate the heavier components from the overhead, and the third term represents the amount of reflux required to separate the lighter components from the bottoms because of their actual relative volatilities. Equation 5 has been found to check all the cases given by Colburn ( 8 ) to an average error of about 1%. The equation makes no provision for a split key, but if the amount of this component going overhead can be estimated, it can be combined with the light key and the balance can be combined with the heavy key in Equation 5. The reflux ratio so calculated agrees fairly well with the value obtained by the tray calculations. In addition to the cases previously mentioned, twenty-three other cases mere studied in which the feed was total liquid and contained heavier than the heavy key up to 90% of the total feed. The relative volatility of the keys varied from 1.1 to 6, the relative volatility of the heavy components ,varied from 0.9 to 0.2, and the agreement with the tray calculations was very good. When (YD is zero, the correction term is zero, and it is obvious that the equation holds because of the basis for the empirical terms. The equation was applied to twenty-four cases in which the
- 1 ) ( 1 + m)
2m
((YE
Vol. 38, No. 3
-
XB
XB f
XC
- 1)
liquid composition a t intersection of operating 1ines.at minimum reflux = mole fraction of light key in feed = relative volatility of keys a t feed tray = mole fraction of heavy key in feed
- mIa
+ 4m
(ag
- 1) (1
+ m) A + xc XB
(3)
feed was liquid and contained lighter than the light key up to 90% of total feed. The relative volatility of the keys varied from Xg 1.2 to 3, the relative volatility of the lighter components varied OIB from 2.5 to 8, and the calculated minimum reflux ratios agreed Xc closely with the values determined by tray calculations. When The proper sign in front of the radical is chosen to give a value of O I A is infinite, the correction term is zero; in this case it is also z between zero and unity. Obviously, negative values or values obvious that the equation holds because of the basis for the greater than unity correspond to the wrong intersection of the empirical terms. feed line with the equilibrium curve. For the thirteen cases studied in which the feed contained 10% The pseudo minimum reflux ratio based on the keys can be of the key components, the maximum error was 18% and the calculated by the following equation (8): average deviation from the reflux ratios determined by tray calculations mas 7.7% in these extreme cases. It is possible that the error might be greater if the feed contained less than 10% of the key components. I n some of these cases the previous empirical method of Colburn gave considerably greater deviations where zp = concentration of light key in overhead expressed as up to 5oY0 as compared with the minimum reflux determined fraction of total amount of keys in overhead by actual tray calculations. However, these cases are rarely cncountered in practice. If the overhead contains a very small amount of heavy key so Five extreme four-component systems were studied where the that the second term is negligible, feed mas liquid and contained only 20 to 3oy0 of the keys, and in these cases also the agreement was excellent. Ten cases of vapor feed were studied where the feed contained 72% of a third component either heavier than the keys or lighter t,han the keys, and the agreement in these cases was also excellent. and thus the second term may be considered as a correction term Table I compares some of the calculated minimum refor an impure overhead. flux ratios and the values determined by actual tray calculaWhen the minimum reflux ratio base4 on the keys has been tions. Cases I, IV, and VI1 were chosen from the work of determined, additional terms are introduced to correct for the Colburn ( 2 ) ; the others illustrate some of the extreme cases actual volatilities of the additional components. These terms studied to establish the empirical terms in Equation 5 . Cases were developed empirically. The minimum reflux ratio for the I to VI11 are for total liquid feed at the temperature of the feed total mixture is given by: tray, and cases I X and X are based on total vapor feed a t the feed tray tempera+ (xc+ zxD) + ture. I n some of the extreme cases such x_. LYA (l + (5) ffB - 1 OID as VI. the empirical term comprised over 90% of the- total calculated ;due, The denominator of the first term represents the fractional and the agreement was generally within 10%. This constitutes a strong test for thc empirical relation developed in amount of the feed passing overhead. The terms inside the Equation 5 , However, in cases usually encountered in pracbrackets represent the reflux expressed as a fraction of the total tice, the equation gives minimum reflux ratios with an accuracy feed. The first term is the reflux required to separate the keys, assuming the heavier components to have zero volatility and the of approximately 1%. 2% =
=)I
March, 1946
INDUSTRIAL AND ENGINEERING CHEMISTRY
271
OF CALCULATED AND DETERMINED MINIMUMREFLUX RATIQS TABLEI. COMPARISON
Case NO.
XA
I
0.30 0.16 0.05 0.30 0.16 0.08 0.25 0.20 0.16 0.16
I1
I11 IV V VI VI1 VI11 IX
0:io
0.72 0.90 0.25 0.30
X
Relative Volatility Based On = C
Feed Composition XB XC
o:i2
XD
.. ..
0.40 0.72 0.90
.
aB
aD
2.00 .2.00 1.10 2.00 2.00 2.00 2.00 2.00 1.20 1.20
0.50 0.50 0.90
ad
..
4:OO 4.00 4.00 4.00 4.00
..
0:25
0.30 0.72
..
4:OO
.. ..
0:50 0.50 0.50
..
APPLICATION OF EQUATION TO VARIOUS SYSTEMS
coNsTANT
coMPoNENT~ OF R~~~~~~~vOLATILITY. A sample calculation for case VI11 (Table I) illustrates the application of the method. The material balance on the tower, based on 100 moles of feed per hour, is: Feed Mol&/Hr, 30 20 20 30 100
A
B
C
D
Overhead, Molas/Hr. 30.0 19.5 0.5
(2
- 1) (1 - 87)
xi =
0:5 19.5 30.0 __. 50.0
c6
Ca
-
-
a
-
Thus the minimum reflux ratio for the complete separation can be calculated: 1 (0.20) (1.472) (0.20 0.30) R M = 0.20 0.30
+
+
+
+y =
1 (0.294 0.5
-
-2.50 -1.39 -1.11 -3.00 -2.33 -0.72 0
2.00 2.29 19.0 0.714 0,179 0.010 0.865
0.589 17.35 1.85
2.311 4.05 96.1 0.928 0,500 0,376 1.136 0,912 21.12 2.13
2.295 3.95 96.0 0.925 0.49 0,361 1.12 0.89 20.95 2.01
Feed, Moles/Hr. 26 9 25 17 11 12 100
Overhead, Moles/Hr. 26.0 9.0 24.6 0.3
..
.. 59.9
-
Bottoms, Moles/Hr.
.. 0:4 16.7 11.0 12.0 40.1
3 7-12
+4
(- ;)
(2
-
(1
-
;)og
- - (2-1)
0.975 (1.00 - 0.975) (2) (2 1) (0.610) (1 0.610) (2 -1) = 1.600 0.128 = 1.472
-
0.429 0.500 0.600 0.726 0.845 0.577 0.610 0.288 0.385
P?,
Obviously the proper value of xi is 0.610. The value of xp based on the keys is 0.975.
-
o.go0
m m
m
-
3
2
=
Minimum Reflux XB Ratio X a f z X a Rn; Calcd. Actual
Xi
The feed consisted of 66 moles of vapor and 34 moles of liquid per hour. For this feed condition the pseudo ratio of liquid t o ___ 23 - 0.355 vapor is calculated as follows: m = 34 66 - 35
I
RL
m
As a n illustration, the equation was applied t o the example studied by Jenny ( 6 ) . The material balance on the tower follows:
CI
0 2 0o e - + ;* 4 [ ( 2 - 1 ) ( 1 - 3 ) m7 - 20.20 +2
Trace 7.5 20.0 Trace 10.0 5.0 Trace 2.5 Traoe Trace
CJ C4
-7
-0.30
0.6
2.0 Trace 1.0 1.3 Trace 2.5 Trace Trace
CP
The pseudo ratio of liquid is calculated as follows: m = - -1- -- -0.30 =-
Trace
Qottoms, Moles/Hr.
.. 50.0
Calcn. of Pseudo Minimum Reflux
Rejection Retention of c, % of B , yo
( 1
+
;)I
-I
+ 0.050 + 0.112) = 0.912
The calculated minimum reflux ratio, 0.912, agrees closely with the value of 0.89 determined by actual tray calculation. COMPONENTS OF VARIABLE RELATIVE VOLATILITY.I n general, the relative volatility of two components varies with temperature. Since all towers necessarily have a lower temperature at the top than at the bottom, the relative volatilities of the different components vary somewhat from tray to tray. However, the equation developed in this paper can be reliably applied t o such systems by using the relative volatility 1 R M = __-_ M B MA of the components at the proper feed tray temperature.
+
The binary equilibrium curve for propane and butane a t 300 pounds per square inch absolute was calculated from the same K value chart that Jenny used (6). Then, through the value of XF = 0.595 on the 45" line, the feed line was drawn with a slope of -0.355 (Figure 2). This feed line intersected the equilibrium curve at x; = 0.470 and uc = 0.639. Thus the pseudo minimum reflux ratio can be calculated. The overhead product composition, zp, is equal to 0.988 based on the key components:
RL
=
0.988 0.639
- 0.639 = 2.065 - 0.470
Also based on the key components, the feed tray temperature can be calculated by either the aew point of the vapor or the bubble point of the liquid a t the intersection of the feed line and the equilibrium curve. This temperature was calculated as 184" F., and the relative volatilities at this temperature can be used in Equation 5 for the minimum reflux ratio of the complete mixture with an accuracy sufficient for all engineering requirements; however, the K values at this temperature may also be used directly in the equation without first calculating the relative volatilities. The general equation may be readily expressed in terms of moles of feed and of K a t the feed tray temperature:
KB
I N D U S T R I A L A N D E W G I N E E R I N G CHEMISTRY
272
A t 184" F. and 300 pounds per square inch absolute the components have the following K values: K
h 7
Thus the iiiininiiiiii rcfliix ratio is calculated as follo\~-s:
1
+ 26 + 9
(25) (2.065)
11 11 + 12 + 17 + 100 (136
- ~- -
0 3 1.36
'1.36 0.141 = 1 (51.6
The value of 0.955 agrees with the calculations of ,Jenny, who found that a reflux ratio of 1.0 was too high and a reflux ratio of 0.9 was too low, and concluded that the true minimum reflux ratio was about 0.95. The method of Colburn (2) gave a value of 0.96 for the minimum reflux ratio. Thus the empirical equation developed in this paper applies to this problem with extreme accuracy. LITERATURE CITED
r R M = 25
Vol. 38, No. 3
~
-
0.68
+ 1.8 + 3.9) = 0.933
(1) Brown, G. G., and Martin, H . Z., T r a n s . Am. Inst. Chem. Enur..,
35,679-708 (1939). (2) Colburn, A. P., Ibid., 37, 805-25 (1941). (3) Fenske, -M.P., IXD.ENG.CHEM.,24, 482-5 (1932). (4) Gilliland. E. R., I h i d . , 32, 1220-3 (1 940). (6) Hogan, J . J.,Ihid., 33, 1132-8 (1911). (6) Jenny, F. J., T r a n s . -4m. Inat. Chem. Engrs., 35, 635-77 (1939,. (7) Robinson, C. S., and Gilliland, E. R., "Elements of Fractional Distillation", New York, McGraw-Hill Book Go., 1939. (8) Underwood, A. J. V., Trans. Inst. Chem. Engrs. (London), 10, 112-58 (1932).
Thermal Evidence of rvstallinitv olymers b
d
vc'. 0. BAKER
AND C. S . FULLER Bell Telephone Laboratories, Inc., Murray H i l l , S. J .
Unusually sharp differences between physical states at molding or fabricating temperatures and those at use temperatures are striking properties of newer thermoplastics such as polyethylene and polyamides. This report shows that such polymers, unlike polystyrene, methacrylates, and many cellulose derivatives, show relatively sharp phase transitions on cooling from a fluid state. Timetemperature studies indicate that these transitions resemble the setting of an ordinary crystalline organic material. Correspondingly, the large shrinkage after molding of these compounds is explained. However, the cooling curves also exhibit some striking differences from the freezing of usual small molecules. Supercooling can be extensively induced and controlled and utilized technically for appreciable times, w-hereas it is very unstable with small molecules. Further, it depends on the molecular weight and polar structure of the long chains. Con\ersely, the ordering denoted by the phase transitions is never complete; i.e., some 'Gsupercooling"is always present in the solid and affects physical properties.
heat capacity (sa), heat of fusion (6), density ( S T ) , and stressstrain dependence on temperature; ( d ) x-ray estimation of crystallinity in unorient'ed, unstressed polymers (16, $3). While these other researches have usually treated final crystallinity as special cases for different' polymers (mostly natural), the present report investigates the existence of spontaneous local ordering over a narrow temperature range, which xould account for the final states reported in the preceding references. That is, a phase change whose thermal effects are closely analogous to those observed in the crystallization of ordinary molecules has been sought in the solidification of polymers containing a wide range of macromolecular weights. From such phase changes may be inferred the presence of ext,ensivc, cooperative, local ordering in a fashion supplementing the usual x-ray or electron diffraction patterns. Further, i t was desired t o see if the sharp shrinkage and change in physical properties known to occur on cooling these polymers from molding or fabricating temperatures were associated x i t l i evidence of latent heat evolution, marking a phase chango.
T
POLTCTHTLESE. Material with electrical properties indicating freedom from polar impurities, and with a high-temperature solubility (toluene a t 100" C.), indicating freedom from cross linkage, was obtained from Imperial Chemical Industries (111 qr/c = 0.775, for c = 0.7000 gram/100 cc. in Tetralin a t 80" C.). POLYESTEI~S were prepared from purified ingredients, as previously described (2). The average molecular n-eight,s of theso compounds were accurately determined, and unless othorwise noted lay in the range -Ifw(n-eight average) 18,000to 22,000. P o L Y A h i I D E s were ohta.ined from the Du Pont Company or prepared experimentally; they vere pure, linear polymers having negligible ash content and average molecular weights of M , 20,000 to 25,000 and In ~ ~ in/ cresol c of 0.9 to 1.2, except for the low-viscosity sample.
MATERIALS
HE concept of crystallinity or local order in high polymers has been studied in this investigation by means of timetemperature curves (35). The object was t o seek phase changes nhich would reveal further the significance of the following earlier investigations of molecular order in high polymers: (a)x-ray detection of crystallinity in natural, highly oriented fibers such as cotton (19, $6) and silk (18); ( b ) x-ray detection of crystallization from externally applied stress, as in stretched rubber (ai?) and relation of this crystallinity to the physical properties of the stressed polymer ($0) ; (c) thermodynamic or mechanical determination of crystallinity, TTith or without orientation, from
