INDUSTRIAL AND ENGINEERING CHEMISTRY
August, 1945
the morphology of Butyl rubber and soap. Considering that these photomicrographs were taken with dark-field illumination and remembering the difference in resolving power of the instrument, the eimilarity of the illustrations to those obtained with the electron microecope (I,6,14)is evident. In the case of a puttylike dilatant material such as some of the recently developed silicon resins, a small piece of the substance can be placed on the wire gauze and be expanded until fine fibers form (Figure 10). I n the study of clay gels the thixotropic gel is merely spread on a glass slide (9) and allowed to dry. One can therek" observe the formation of a coherent film consisting of a network (Figure 11) caused by filamentous aggregation of the clay particles. This type of ultra-illumination by incident light makes possible various manipulations under the microscope without enhancing the clearness of the observation by blurring the picture with undesired reflections or the like. For example, it is possible to study elastic deformations by forming a blob of elastomer on the points of two micromanipulator needles and then carefully moving them apart; or a coagulum of rubber latex can be formed on the ends of a micropincette, and the pincette is then allowed to open, as Figure 12 shows. The resulta so far obtained with this technique substantiate the assumption that the morphology of lyogels of quite different chemical composition is very similar (1'7, 18); this finding offers an explanation for the analogy of some of their properties. In conclusion, the authors would like to stress that this contribution offers only a condensed discussion of their finding8 and
7189
a limited number of illustrations. They hope to add more detailed results as soon as possible, but believe that the information presented here should not be withheld any longer. LITERATURE CITED (1) Anderson, T. F., "Advances in Colloid Science", Vol. 1, p. 363 ff., New York,Interscience Publiahers, 1942. (2 Baohmann, W., KoU&Z., 9,312 (1911);23, 86 (1918). (3{ Baahmann, W., I. unorg. Chm., 73,No. 2 (1911) (4) Darke, W. F., MoBain, J. W., and Salmon, C. S., Proc. Roy. Soc., A98,396 (1921). (6) Hall, C. E., Hauaer, E. A.,le Beau, D. S., Schmitt, F. O., and Talalay, P., IND. ENO.CHIOM., 36,834 (1944). (6) Hauser, E. A,, Chm. Fabrik, 4,277 (1931). (7) Hauser, E. A., "Colloidal Phenomena", New York, McGrawHill Book Co., 1939. (8) Hauser. E. A., Ku&chuk, 7 , 188 (1931). (9) Hauser, E. A., and le Beau, D. S., J . Phye. C h m . , 42, 981 (1938);43,1037 (1939). (10) Heine, H., U. S. Patents 1,840,448(1932)and 1,936,444(1933). (11) Heine, H.,I . Wias. Mihmkop., 48,460(1931). (12) Lawrence, A. C. S., "Soap Films", London, C.Bell and Sons, 1929. (13) Maclennan, K.,J . Soe. C h m . Id.,42,393T (1823). (14) Marton, L., MoBain, J. W., and Vold, R. D.. J . Am. C h m . SOC.,63,1990 (1941). (15) Seifria, W., Colloid Symposium Monograph, 3, 286 (1926). (18) Vold, R. D., and Ferpuson, R. H.,KoUoid-I., 11, 146 (1912). (17)Weimam, P. P. von, in J. Alexander's "Colloid Chemistry", Vol. 111, p. 89,New York,Chemical Catalog Co., 1931. (18) Weimam, P. P.von, Rubber Chem. Tech.,2,108 (1929). (19) Zsigmondy, R.,and Bachmann, W., KoU&Z., 11, 146 (1912).
MINIMUM WORK IN MULTISTAGE COMPRESSION HAROLD G. ELROD, JR. U. S. Nom1 Academy, Annapolis, Md.
A
PERUSAL of the literature indicates that the general con-
dition for minimum work in multistage comprmion is not widely known. Since very high pressures are now being used in chemical synthesis processes, it is believed that the following derivation has practical value. Consider first the simple two-stage process shown in Figure 1. Isentropic compression with intercooling to a fixed temperature, T,,is assumed. The work of compression is given by
W
hi
- ha + h2 - hi
5
dhr
- dha + d h
(2)
= T
9
Equation 7 then results: (7)
(1)
Now consider variations of W produced by changes of the intermediate pressure pa: dW
(5)
(2)
where h4 varies at constant pressure, p4; ha varies at constant temperature, Ta; h varies a t constant entropy, sz = Si; h~ is k e d . When W is a minimum, dW = 0. Or
The subscripts in Equation 3 refer to the locations of the derivatives. We can simplify Equation 3 with the following general thermodynamic relations, (4)
The validity of Equation 7 does not depend upon the assumption of isentropic compression; a constant compression efficiency, the same for both stages, might have Seen assumed instead. Even if there are more than two stages of compression, Equation 7 must, nevertheless, apply to every intermediate pressure with respect to adjacent pressures. Otherwise, variations of the total work would not be zero for all possible variations of the intermediate pressures. CONVENIENT RULE FOR DESIGN
If in Figure 1 the isobar joining points 3 and 2 is a straight line,
7w
INDUSTRIAL A N D ENGINEERING CHEMISTRY
T4
and consequently
=
Tz
But Hence the necessary and sufficient condition for the validity of Equation 9 is that the specific heat at constant pressure of the working fluid be a function of temperature only. This condition is sufficiently liberal to permit the use of Equation 9 in almost all design problems. Furthermore, this equation is applicable when the substance is not intercooled to initial temperature. PERFECTGASES. For a perfect gas, defined by p v = RT
(11)
two columns shows that the principle of equal work would be slightly worse than any of the foregoing methods. A significant feature of the comparison is the flatness of the minimum for total work. The difference in the values of total work is only 1%.
BY EQUATIONS 7 AN^ 14 TABLE I. COMPARISON OF SOLUTIONS
Suction, 1st stage
Tz
(12)
Diecharge, let stage
TI
0
Suction, 2nd stage
P t
A
V
do
(13)
Moreover, 54 = 5s and se = 81. It has been shown (3)that under these conditions PZlPl = P4lPs
P t h
If the gas is intercooled to initial temperature,
Ta
P h
Cpis a function of temperature only (8). Hence, by Equation 9:
Tc
(14)
Vol. 37, No. 8
a
Discharge, 2nd stage
Work (isentropic) 1st stage 2nd stage Total
P
fi
p
Eq.7 352.8 670.0 1349.2 2200.0 1246.0 1618.6 0.4360 2200.0 670., 0 1169.1 0.1879 0.0010125
Eg. 14
352.8 670.0 1349,2 1400.0 1082.5
.... ....
1400.0 670.0 1269.6
.... ....
5500.0 915.0 1248.4
5500.0 1084.7 1427.6
269.4 77.3 546.7
192.1 168.0 350.1
and (1) that h*
- ha
= h2
- h,
Thus the rule of equal pressure ratio and equal work has been derived for perfect gases without recourse to the usual assumption of pv’ a constant.
CONCLUSIONS
1. A general method has been established for selecting optimum interstage pressures. 2. The following generalizations m y be induced from the illustrative example presented: (a) The thermodynamic optimum is not sharply defined. (a) &lection of interstage pressures by the standard methods of equalizing the stage pressure ratios or the stage work will usually be satisfactory when the working substance is intercooled to initial temperature. 3. The best approximate method for determining interstage pressures is to equalize the isentropic discharge temperatures of all stages. This method is valid whatever the intercooler terminal temperatures may be. The conclusions of this article are based solely upon a consideration of fluid properties. As mentioned earlier, they are valid if the compression efficiency may be assumed constant and the same for a11 stages. However, other factors such as volumetric efficiency, pressure drop in the heat exchangers, heat transfer surface required, etc., have been neglected; it is left for the practical design engineer to estimate by experience the effects of these items. LITERATURE CITED (1) Keenan, J. H., “Thermodynamics”, p. 99, New York,John Wiley & Sons, 1941. (2) lbid., p. 100. (3) Keenan and Kaye, J. Applied Mechanics, Trans. Am. SOC.Me&. EWr8.. 65, A-123 (1943). (4) Keenan and Keyes, “Thermodynamic Properties of Steam”, New York, John Wiley & Sons, 1936.
Figure 1
NUMERICAL EXAMPLB. Although the following problem is purely academic, it waa chosen because the properties of steam are readily accessible to the reader (4): “It is desired to compress steam in two stagea from 352.8 pounds per square inch absolute and 670” F. to a pressure of 5500 pounds. Intercooling to 670”F. is assumed.” Table I compares a solution obtained by Equation 7 with one obtained by Equation 14. The comparison indicates that for this problem Equations 9 and 14 are practically equivalent, and that Equation 7 is better than either. The trend between the
Microbial Amylase Preparations (Correction) An error has been pointed out in t8hecaption of one of the photographs used to illustrate the article with the above title AND ENQINEERING CHEMISTRY, June, 1945). On (INDUSTRIAL page 523 the caption should read as follows ’ Centrifuge and Sperry Pressure Filters Through an inadvertence this equipment w m labeled “Shriver Pressure Filters”.
