7w
INDUSTRIAL AND 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 pub 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”.
