&larch, 1932
IKDUSTRIAL AND E N G I X E E R I N G
where S = s,p. gr. of liquid 60’ F./60° F. t = F T = O F. abs. H O = constant H = total heat content, B. t. u. per lb. D, F , C, and E = constants in equation of state when P , T , and V are expressed as lb. per sq. in. a b s , O F. abs., and cu. in. per lb., respectively. The value of Ho is determined by substituting a known value of H a t some pressure and temperature (for example, a t the temperature of the saturated vapor at atmospheric pressure) and by solving the equation for Ho. Substituting the appropriate Tralues of the constants in the above equation’ the following equation is obtained for the lieat content of naphtha from 0” F. (-17.8” (2.) to any temperature ( t o F.) where the distillate exists only in the vapor state: H =
4.0 - 0.7500 6450 -
[ZO 47P
[f t
1 + 6701] + 9331.7
P
+ 0 258P2 - ~3 (28,970 + 203.413) X
CHEMISTRY
301
critical region. This arises partly through the fact that the equations for specific heats of the liquid cannot be guaranteed as exact in this region. I n general, however, the total-heat curves along the dew-point and boiling-point curves should be rounded to meet at the critical point. ACKKOWLEDGMEST The authors are indebted to E. TV. Thiele for valuable suggestions with reference to the calculation of the latent heat, and to E. R. Kirn who aided in the experimental work on gasoline. LITERATURE CITED Bahlke and Kay, IND.ENO.CHEM.,21, 942 (1929). Bureau of Standards, Circ. 279, 77 (1926). Caubet, Z . physik. Chem., 40, 2 5 i (1902). Fortsch and Whitman. IKD. ESG. CHEM.,18, 795 (1926). (-5) Lewis and Randall, “Thermodynaniics,” p. 133, McGraw-Hill,
(1) (2) (3) (4)
1923.
(6) Linde, Xitt. iihcr Porsch-arbeiten w t u T - c
LO7]
7 167
The values of the total heat for various pressures and temperatures have been calculated for naphtha , and the results plotted against temperature in Figure 17. It will be noted t h a t Figure 17 is not complete in the
I-.
dcut.
Ino., 21, fi4
(1905).
( 7 ) Young, Sydney, J. Chem. Soc., 47, 640 (1885). (8) Young, Sydney, Phil. Trans., 177, Part 1, 135 (1886). (9) Young, Sydney, Z. physik. Chem., 29, 193 (1899). RECEIVED November 9, 1931. Presented before the Division of Petroleum Chemistry a t the 82nd Meeting of the American Chemical Society, B u f f d o , N. Y . , August 31 t o September 4, 1931.
Tower-Absorption Coefficients-IV 1,. AI. BENKETCH AND C. W.SIMMONS, Department of Chemical Engineering, Lehigh Uni\rersity, Bethlehem, Pa.
LIfole-Jow ratio and free-volume concepts are introduced into the derimtion of a general absorption equation which is applicable equally to systems obeying either Henry’s or Raoult’s law
M
UCH has been written concerning the mechanism of gas absorption by countercurrent tower processes, and much emphasis has been placed on the graphical solution of tower-design problems. It is the purpose of this paper to give a mathematical analysis of countercurrent tower processes, to derive a n absorption equation, and to apply this equation to some experimental data. DOiiNAN-MASOB COXCEPT
The derivation of a satisfactory equation representing the continuous countercurrent absorption process, such as is used in gas scrubbing, depends chiefly upon the function representing the rate of solution of the solute gas in the extractor. This rate of solution Wac given by Donnan and Masson (3) to be
where m n
= =
gram concentration of solute in carrier gas gram concentration of solute in extractor
At equilibrium, when no absorption occurs, f ( m ) , sunif function of m, and n are equal. In addition, in a system obeying Henry’s law, f ( m ) equals k ( m ) where h: is Henry’s coefficient. Cantelo ( 1 ) utilized this fact t o transform the original equation of the rate of solution to
where K
=
Cantelo dissolution coefficient, which, for strictly specified conditions of gas and extractor flow, etc., will remain a constant
With the aid of a second equation based on the fact that a steady state prevails mithin a countercurrent absorption unit, Cantelo integrated Equation B obtaining a formula for investigating tower-absorption coefficients. LEKIS COSCEPT It was proposed by Lewis (/t) and Whitillan and Neats (6) that the rate of solution of soluble gas in extractor can be more accurately represented by
where P , - PJ = difference between partial pressure of solute in gas and vapor pressure of solute in extractor Here equilibrium conditions prevail when P , = PL. The Lewis form may be equated to the Donnan and Masson form by applying Henry’s or Raoult’s law and converting the partial pressures to gram concentrations. Thus in Pr X H
= = =
PI = H X vapor pressure of solute in extractor its mole fraction either Henry’s or Raoult’s factor
However, the fundamental difference between these equations for the rate of solution is in the units expressing solute concentrations. The first form which uses gram concentrations introduces a n inaccuracy, owing to the fact that the concentrations in Henry’s and Raoult’s laws must be given in mole fractions. The Lewis rate-of-solution equation readily lends itself to evaluation in mole-fractional quantities, and i t is therefore used as the basis of the derivation in this article.
I N D U S T R I A L A N D E N G I N E E R I N G C H E &I I S T R Y
302
FUNDAMENTAL ASSUMPTIONS A countercurrent tower-absorption process is assumed in which there exists a steady state. Thus, at any chosen point in the tower, all factors, such as temperature, pressures, or rates, are constant. Mathematically this means that a t any fixed point in the tower all derivatives with respect to time are zero. Moreover, the extractor is taken to exert a negligible vapor pressure, and the carrier gas to be inert. For example,
VoI. 24, No. 3
This equation states t h a t there is no change in the quantity of carrier in its passage through the tower. It can be transformed into terms of partial press:
It can also be shown t h a t I
19
I
I
I
I
EFFECT OF MOLE-FLOR RATIO ON ABSORPTIOH COEFPICIENT Rg W N W OIL SYS. WATER SY8.
-
E. Ja A. C
Equation 3 is the equation adopted as representing the rate of solution, expressed as rate of change of partial pressure of solute in the gas as a function of ( P p - PJ). Po P, or
moxa
MW
WIO
y
-
2/
=
p,,
-I
area of cross section X free volume/unit gross volume, sq. dm. h = height above gas inlet, dm. A h = free volume of tower, liters = gas rate up tower, dm./min. p = density of carrier gas, grams/liter m = molecular weight of carrier, grams J4 = molecular weight of extractor, grams w = carrier rate, grams/min. V’ = extractor rate, grams/min. j = mole-flow ratio P , = partial pressure of solute in gas, atm P , = total average pressure of gas in tower, atm PI = vapor pressure of solute in extractor, atm. Z = Henry’s or Raoult’s coefficient, atm. K = absorption coefficient e = unit of time, min. y = moles solute/mole carrier at time e x = moles solute/mole extractor at time e Subscripts (0) refer to conditions at h = zero No subscripts refer to conditions a t h = height of tower .4
=
mole fraction of solute in gas (4)
P
,
(1
+
1/12
- p,
=
P,,
I
P , - P, P,, d2/ dP, = ___
FIGURE 1 these conditions obtain in a benzene scrubbing unit where coke-oven gas is the inert carrier and the petroleum-oil extractor is nonvolatile at the temperature of the system. The recovery of hydrocarbons from natural gas is a similar process However, in the case of water, aqueous solutions, and organic solvents being used as the extracting medium, there may he error introduced into the calculations if these extractors exert an appreciable vapor pressure. The magnitude of this error will depend upon the ratio of the vapor pressure of the extractor to the total gas pressure within the tower, and to the partial pressure of extractor in the entering gas, i. e., the degree of saturation of entering gas. Finally, i t is assumed that the systems obey Henry’s or Raoult’s law and that these coefficients are known at the temperatures and pressures involved. The nonienclature which follows applies to the derivation of tower-absorption coefficients using mole-flow ratios and free volumes, and expressing concentrations or partial pressures in terms of mole fractions.
=
l + Y
1+ Y
Equation 4 is supported by Avogadro’s and Dalton’s gas laws.
rz ; l=+ rz z=-
z
(5)
PI - P,
Equation 5 is a statement of Henry’s or Raoult’s vaporpressure law, with concentrations expressed in moles per mole.
Equation 6 is a solute-balance relationship between carrier and extractor.
zw
Wm Let = f,the mole-flow ratio, and i = y1 - jx,,a constant depending upon terminal conditions, or Equation 6 becomes y=i+fx
and
P, =
(b
(Y - 4 Y -4
+
Equations 3 and 2 are combined, and the necessary substitutions performed to eliminate Pg, PI, and x, yielding
f
A countercurrent absorption process may be represented by the equation of continuity: dh A ~ - = w d8
and
cf-i)
= D
Equation 7 is integrated according to the form
dl4
s a
+ by + cy2
where a, b, and c = coefficients in quadratic denominator of Equation 7. Integrahng and evaluating the constant,
INDUSTRIAL AND ENGINEERING CHEMISTRY
March, 1932
d b 2 - 4ac and b* > 4ac for real results. a fact that is borne out by substituting experimental values. Also a = iz b P,, D Z ( i - 1) c = P., -z
TABLEI. ABSORPTION OF BENZENE
v =
+
. 4 B s~ O.-. R P~ . .
I
+ V) In (2cy + b +
( 2 ~ n h -V)
- (2cD
-b -
BENZENE GAS RATES Gas .TEST Inlet Outlet Air Oil (in) % L./min. L./mzn. C. % 6.00 6.01 4.01 4.38 3.64 5.01 3.88 3.50 3.99
V)
S o w dividing through by 2c and multiplying through by u ,
APPLICATIOKS The above equation was used in calculating the towerabsorption coefficients for several systems, including volatile and nonvolatile extractors and systems controlled by Raoult’s and Henry’s laws. dininions and Long ( 5 ) used a countercurrent absorption tower, packed with Raschig ring filler for the absorption of benzene by mineral oil, and operated it in accord with the fundamental assumptions of the absorption equation. This is the case in which the extractor is nonvolatile and in which the system conforms with Raoult’s law. The absorption coefficient was calculated over a fivefold variation in mole-flow ratio, yielding a substantially constant, value as shown in Table I, and in Figure 1, curve d .ipplying the earlier equation of Cantelo (1) to these results, an absorption coefficient is obtained which decreases exponentially as flow ratio or extractor rate increases. The second case, in which the absorption process followed Iienry’s lam-, is given in the data by Cantelo ( 2 ) on the extraction of carbon dioxide from air by water in a countercurrent system. All the assumptions specified in the equation developed in this article were realized in this system, except that water (the extractor) exerted a vapor pressure. However, the error introduced in calculating the absorption coefficient was slight because the entering gas contained nater vapor corresponding to the humidity of the air used. Although no data are given, a minimum humidity of 50 per cent generally prevailed. Then, neglecting any changes in gas volume entailed by vaporization of extractor, the absorption coefficients were calculated with the above formula, using the data in Table 11. The resulting absorption coefficient was plotted against mole-flow ratio as shown in Figure 1. cun-e B.
1.10 1.20 0.92 0.78 0.67 0.98 0.52 0.48 0.42
5.38 5.66 5.66 5.95 5.95 5.66 5.38 5.66 5.66
0.04 0.06 0.11 0.09 0.12 0.12 0.14 0.15 0.21
24.0 29.0 2 4 . 0 30.5 2 7 . 0 29.5 2 3 . 5 28.7 26.4 30.0 28.0 33.0 2 8 . 3 34.2 29.2 3 4 . 5 25.8 2 8 . 5
760.7 757.3 757.3 753.1 758.4 755.4 757.0 748.0 754.5
0.608 0.875 1.618 1.249 1.669 1.775 2.171 2.245 3.086
5.67 5.44 4.65 5.78 5.61 5.23 5.89 5.99 6.56
TABLE 11. ABSORPTION OF CARBON DIOXIDE ABSORPPRES- MOLE- T I O N TEMPERATURES SURE FLOW COEFCOI GAS RATES Gas GAS RATIO FICIENT TEBT Inlet Outlet Gas Water (inlet) Water Av. f K % % L./min. L./min. C. a C. M m . H g 1 2 3 4 5
Thus Equation 12 is the final form of the absorption equation which is applied to the following absorption processes.
PRES- MOLE- TION SURE FLOW COEVOil GAS RATIO FICIENT (out) (IN) f K a C. MA..H*
TEMPERATURE0
stituted into (8),and the result rearranged giving (2cD - b
303
17.8 1 2 . 4 11.8 7 . 2 25.4 18.0 21.6 12.0 25.6 1 3 . 4
3.38 3.46 4.02 3.88 4.05
2.24 2.94 2.72 5.21 6.06
15.6 18.2 11.6 21.2 12.0
9.2 8.2 7.0 7.0 7.2
758.3 749.0 756.0 755.6 763.0
1060 1297 1179 2301 2600
0.523 0.552 0.600 0.658 0.775
6 7 8 9 10
15.2 18.4 24.6 13.8 26.4
8.2 12.3 18.2 10.6 20.6
3.40 3.28 3.54 3.85 3.48
4.39 2.57 2.04 1.72 1.56
18.8 15.4 17.4 17.6 16.6
6.8 9.0 9.2 8.4 9.2
755.3 758.3 759.0 748.0 759.0
2030 1265 1010 696 804
0.645 0.434 0.359 0.345 0.297
11 12 13 14 15
13.8 25.8 13.8 13.6 20.8
9.8 16.0 8.6 7.8 10.6
3.57 3.63 3.57 3.57 3.91
1.73 3.26 3.20 5.05 6.13
18.8 17.4 18.4 14.0 10.4
8.6 9.2 8.4 6.4 7.4
749.0 759.0 749.6 755.0 755.2
757 1604 1398 2152 2566
0.443 0.537 0.535 0.592 0.790
Some increase of the absorption coefficient with increase in mole-flow ratio may be noted from the curves. The reason for this is the experimental difficulty in maintaining a constant free volume under operating conditions. From the equation derived, it may be seen that the coefficient is inversely proportional to the free volume, Ah. An increase in the velocity of the absorbing liquid results in a decrease in the free volume of the tower, and hence an increase in the absorption coefficient. This investigation is being continued, using semi-commercia1 countercurrent scrubber units and bubble-cap towers. The resulting practical applications of the equation to industrial operation and design will appear in subsequent articles. CONCLUSIONS 1. The experimental results indicate the validity of the above-derived equations when the solute obeys Henry’s or Raoult’s law, and the extractor is nonvolatile. 2. For a given system, the absorption coefficient is substantially constant, and the magnitude of the coefficient is a characteristic of the system. 3. At elevated mole-flow ratios, an upward trend in the absorption coefficient is observed, an effect explained by the diminution in free tower volume associated with increasing extractor rate. ACKNOWLEDGMENT
This investigation was carried out under the Henry Marison Byllesby Memorial Research Fellowship in Engineering. LITERATURE CITED (1) Cantelo, Chem. M e t . Eng., 33, 680 (1926). (2) Cantelo, Simmons, Giles, and Brill, IND.ENG.CHEM, 19, 989 (1927). (3) Donnan and Masson, J. SOC.Chein Ind.. 39, 236T (1920). (4) Lewis, J. IND.ENG.CHEM, 8, 825 (1916). ( 5 ) Simmons and Long, Ibid., 22, 718 (1930). (6) SThitman and Keats, Ibid., 14, 186 (1922). RECEIVED December 1, 1931.
