minutes a t the 33 ml. per minute slow flow rate whereas it had been 6.2 minutes a t the 130 ml. per minute rate. The amount of trimethylolpropane was then determined by relating the area for a sample to that obtained for a standard. A straight line relationship between the trimetliylolpropane content and the area was (demonstrated.
Since a urethane functional group is a “half-ester”, it was theorized that the aminolysis technique might be used on polyurethane foams as well as polyesters. The scheme was applied to urethane foam whose trimethylolpropane content was known to be 1.16%. Six determinations of trimethylolpropane in this urethane foam gave results ranging from 1.10 to 1.24%. The average mean deviation was -0.03%. I n the above work it was noted that the diacetate of diethylene glycol was eluted in both the polyester and the polyurethane foam work. The diethylene glycol might have been determined in a manner similar to that described above for the trimethylol-
RESULTS AND I)ISCUSSION
Following the itbove analytical scheme, trimethylolp.ropane was determined on lab-prepared adipate polyester of known trimethylolpropane content as well as several commercially available polyesters known to be used in flexible polyurethane foam production. The results are summarized in Table I.
propane. I n fact, it is probable that any poly01 whose acetate can be characterized by gas chromatography might be determined similarly. LITERATURE CITED
(1) Dimbat, M. Porter, P. E., Stross, F. H., ANAL.&EM. 28, 290 (1956). (2) Esposito, G. G., Swann, M. H., Ibid., 33, 1854 (1961). (3) Kappelmaier, C. P. A., Mostert, J., Boon, J. F., Verfroniek 27, 291 (1954). (4) Percival, D. F.,’ANAL.CHEM.35, 236 (1963). ROBERT E. WITTENDORFER
Plastics Division NOPCO Chemical Co. North Arlington, N. J.
Some Zone-Refining Calculations on Phena nthreiie-Anthracene System SIR: I n recent yews, a number of investigators (3, 6, 6) have pointed out that controlled freezing methods may be used as an aid in elucidating uncertain regions of many phase diagrams. A project utilizing these techniques is currently in progress a t these laboratories and the work has involved the development of a computer program for the computation of zone refining concentration profiles. The calculations to follow are an example: They are based upon the zone refining experiment and phase diagram reported by Joncich and Bailey (2) for the phenanthrene-anthracene system, . and although performed primarily because this system with its discontinuity ap-
C*(x
peared to provide ideal test input data, the results are presented ?%cause of their possible general interest.. A mathematical model is provided by the following relationships for the solute concentration at a distance x along the nth pass: dC*(x
+ I ) - Cn(X)l/Z < x < ( L - 2)
+ 6)/dZ
= [C*(Z
+ 6) - C(Z)l
(3)
. C*(x)
(4)
where
C
=
C* =
[Cn-dx dC*(x
[C*(Z C(Z) = ko(C)
+ & ) / d x= for 0
+ 6 ) - C * ( x ) = [I - exp(f6/D)]
(1)
D L
+ 6) -
f 1 6
Cn(Z)I/(L - z) (2) for (t- 2) < x < L
= = =
= =
solute concentration in the solid solute concentration in the liquid diffusivity i n the liquid length of material involved growth rate zone length effective thickness of the diffusion layer
i
B
70
-
-
EXPERIMENTAL RESULTS OF
a a
a
407
\‘
40-
sp
30 20
-
10-
0
-i-L_L
0
05
IO
15
I
20
25
ZONE
Figure 1. passes
30 35 LENGTHS
40
45
50
55
Concentration profiles for 1, 2, 5, and 18
05
IO
1.5
20
25 3 0 3 5 40 ZONE LENGTHS
45
50
55
Figure 2. Concentration profiles for different growth rate parameters Experimental results of Joncich and Bailey superimposed on calculated profiles VOL. 36, NO. 4, APRIL 1964
931
Equations 1 and 2 are the well known differential mass balance equations (4) for zone refining, Equation 3 is based on the Burton-Prim-Slichter model (1) to allow for mass transport and Equation 4, the segregation coefficient by definition. The problem was programmed in machine code for the Air Force Cambridge Research Laboratories, Research Data Evaluator (a Univac AN/USQ-17 computer). using a package of routines which have been developed primarily for the general problem of fitting experimental data to mathematical models. The program permits concentration profiles to be calculated for an arbitrary initial solute distribution and for an arbitrary dependency of ICo upon solute concentration. The concentration profiles in Figures 1 and 2 were obtained by integrating Equations 1 and 2 using a fourth-order Runge Kutta formula and 110 intervals. The ko dependencies were taken from the phase diagram presented by Joncich and Bailey and supplied to the computer in the form of two fourth-order
polynomials. Figure 1 shows the resulting concentration profiles for 1, 2, 5 and 18 passes; Figure 2 shows the experimental results of Joncich and Bailey superimposed upon calculated profiles for different growth rate parameters. The following equations, derived from the range 0 < z
