INDUSTRIAL A N D ENGINEERING CHEMISTRY
September, 1930
for 14 days. One hundred cubic centimeters of the solution contained 0.966 gram of fibroin. Micro Van Slyke analyses gave an average of 4.6 per cent amino nitrogen. The solution was kept a t 20’ C. and ten consecutive readings were taken. -0 -0 -0 -0 -0
78’
-0 81‘ -0 76’ -0 790 -0 7.50
67C
74= 75c 71‘ Average - 0 75’
-0.740
These results are further confirmation that the fibroin is partially decomposed on dissolving in the neutral salt solutions, since the values obtained on the ground fibroin are considerably higher than those obtained on the dissolved fibroin. Discussion of Results
There is no doubt of the partial breakdown of the fibroin molecule when this protein is dissolved in concentrated aqueous solutions of neutral salts, but as to the exact nature of the decomposition little is known. It is very probable that there preexists in the fibroin structure a labile peptide linkage, and that this is easily broken down on treatment with the salt solutions. The fact that the fibroin was not further broken down on long heating in these solutions is further evidence of this peculiar constitution. Many analyses of fibroin have shown that it is composed of
967
monoamino acids linked together in some unknown manner. It is generally believed that there is a polypeptide arrangement corresponding to Fischer’s original interpretation of protein structure. Lloyd ( 3 ) suggests that, owing to its resistance t o decomposition and digestion by enzymes, there must be a special structural linkage in the molecule. She also describes the fibroin as being immune from bacterial attack. Mann (4),Weyl (Y),and most of the other workers in this field make similar statements. It seems unnecessary to assume that fibroin has a special structural linkage since its physical properties are very similar to those of most proteins. Its immunity is more probably due to its extreme insolubility in the ordinary solvents. Fibroin in colloidal suspension is readily attacked by the proteolytic enzyme, trypsin, regardless of whether the solution is prepared by mechanical or chemical methods. Hubbard ( 2 ) also found that silk could be hydrolyzed by trypsin. Since he used waste silk, it is very probable that the digestion took place here as a result of a fine suspension comparable to solutions prepared by grinding. Literature Cited Abderhalden, 2. p h y s i k . Chem., 178,253 (1928). Hubbard, J . A m . Chem. Soc., 33, 2032 (1911). Lloyd, “Chemistry of Proteins,” p. 53 (1926). Mann, “Chemistry of the Proteids.” Rey and Meunier, C o m p l . rend., 184, 285 (1927). (6) Weimarn, Von, IND. ENG.CHEM , 19, 109 (1927); A n n . Mining Insl., 4, 151 (1913). (7) Weyl, Ber., 21, 1407 (1888).
(1) (2) (3) (4) (5)
st. Pelersbirvg
Relation between Mass Transfer (Absorption) and Fluid Frictionlp2 Allan P. Colburn E. 1.
DU
POWTDE KEMOURS & COMPANY EXPERIMENTAL STATION, WILMINGTOS, DEL.
In processes such as absorption of soluble gases and dehumidification, the rate of mass transfer is dependent on the rate of diffusion of molecules through a viscous gas film adjoining the liquid surface and on eddy currents in the turbulent portion of the gas stream. A n equation is derived for mass transfer from fluids in turbulent motion which is based on resistances in both the viscous film and the turbulent core similar to the method used by Prandtl and by Taylor in obtaining a relation between fluid friction and heat transfer. The resulting equation is:
where K = mass transfer coefficient, f = friction factor, G = mass velocity, pol = logarithmic mean partial pressure of non-condensable gas in the film, M, = molecular weight of solute gas, M,,, = mean molecular weight of gas mixture, and = a function of gas properties and flow conditions which usually has a value of nearly unity. Agreement is shown with experimental data for both dehumidification and absorption.
*..... . H E absorption of soluble gases and also the condensation of vapors from mixtures with insoluble or with noncondensing gases are processes involving the transfer of molecules from a region of relatively high concentration to a gas-liquid interface where the equilibrium concentration is low. This phenomenon is similar to that of heat transfer where a temperature gradient, which may be considered as a measure of the gradient in “heat concentration,” causes heat flow. Much success has been met with in the past in evaluating heat-transmission coefficients by considering the transfer of heat and momentum as analogous processes. It is the purpose of this paper to develop a theoretical equation for mass-transmission coefficients in a similar manner by
T
Received June 19, 1930. Contribution No. 37 from the Experimental Station de Nemours & Company. 1 2
E. I. du Pout
considering the transfer of mass and momentum as analogous processes. Relation between Heat Transfer and Fluid Friction
I n considering the mechanism of heat transfer between a pipe and a fluid flowing turbulently through it, Prandtl (9) introduced two concepts (Figure 1): first, a boundary layer in laminar flow next to the solid surface, in which the velocity increases from 0 to U B and the temperature varies from t, to t ~ and ; second, a turbulent core, in which the mean velocity is urnand the mean temperature is t,. He supposed that heat is transferred through the core by convection alone and through the film by conduction alone, and that both processes are analogous to the mechanism of transfer of momentum in the core and film. His resulting equation for the coefficient of heat transfer, based on the difference between the surface
INDUSTRIAL Ailin ENGINEERING CHEMISTRY
968
temperature and the average temperature of the fluid stream, can be expressed as follows: h=
(f/2)C&
1 + UB U,
(y - 1)
(1)
where h = heat transferred per unit time, unit area, and unit temperature difference between wall and mean temperature of fluid; cp = specific heat at constant pressure; G = mass velocity, weight per unit time per unit area of cross seck t i o n ; U B = l i n e a r v e l o c i t y of t h e fluid a t the film b o u n d a r y b e t w e e n film and core; urn= mean linear velocity f of fluid; p = viscosity of fluid; and k = . thermal conductivity; all expressed in consistent units. f is the friction factor in the equation for turbulent flow:
I
F =
,
Figure 1-Velocity and Temperature Distribution near a Surface g Heated ( R a s $ :
(f/2)pmum2
k d (condensed units of k, RT
=
ML-' T-1)
The integrated form of Equation 4 for diffusion across a layer of thickness B due to a partial-pressure difference & p is: ~
where pop and p , are ~ partial pressures of non-condensing gas and and p.' are partial pressures of condensing gas on the two sides of the layer; and p,! represents the logarithmic mean partial pressure of the non-condensing gas in the film. Equation 5 should apply to diffusion across the viscous film next to the liquid surface. The thickness of this film is related to the frictional resistance at the surface as shown by the definition of viscosity (neglecting curvature of the tube surface) : ps1
(2)
where F = frictional resistance of fluid expressed as force per unit surface area of wall, and pm = mean density of fluid. Equation 1 was later derived independently by Taylor (14) and found by Stanton (12) to show good agreement for gases, for which case the denominator is substantially unity, and to show fair agreement for water, with U B / U ~= 0.3.
where F = frictional force per unit surface area as in Equation 2, U B = velocity at boundary between film and core, and p is taken as the average viscosity of the film. Eliminating B from Equations 5 and 6 gives
Mass Transfer and Fluid Friction
The mass transmission coefficient, K , is defined as the weight of material transferred per unit time, unit area, and unit difference in partial pressure between the main fluid stream and the liauid interface. Hence
For mass transfer from a fluid in turbulent motion the partial-pressure gradient of the component being a b s o r b e d or condensed will be as shown by Figure 2, where p , and p~ represent the paztial pressures at the liquid interface and a t the film boundary, and pm represents the average partial pressure in the fluid stream. For convenience, the component transferred will be referred to as the vapor and the other as the gas. The mechanism of transfer will be by convection or eddy motion from the turbulent core to the film boundary, and by diffusion across the viscous film to the liquid interface. While the boundary between the two types of motion is probably not well defined, the present problem is simplified by assuming a definite boundary representing average conditions in the boundary zone. The equation for diffusion of one gas through another was originally derived by Maxwell (8) in 1860 and by Stefan ( I S ) in 1873. The same equation was derived in a somewhat different manner by Colburn and Hougen (3) to interpret their dehumidification data and was given by them as follows: (3) where
where k, =
Vol. 22, No. 9
pi Figure 2-Velocity
I
and Vapor Pressure Distribution near a
The ratio of b p ~ / A p ,to U B / U m can be evaluated by use of Equation 8 and a somewhat similar equation relating mass transfer and friction resistance in the core. To obtain this relation for the core the same reasoning is used as that of Reynolds (11) in deriving a relationship between fluid friction and heat transmission. The ratio of momentum lost by skin friction between two sections dy apart to the total momentum of the fluid will be the same as the ratio of the material actually supplied to the film boundary to that which would have been supplied if the whole of the fluid had been carried to the boundary. Placing the above assumption in mathematical form gives:
Ku = mass transferred in unit time to unit surface area de k d = diffusion coefficient (condensed units, L2T-') M , = molecular weight of diffusing material
R
= gasconstant T = absolute temperature P = total pressure p , = partial pressure of inert gas, at distance x pu = partial pressure of diffusing gas, at distance x x = distance in direction of diffusion
Hanks and McAdams (6)introduced a new diffusion coefficient, k,, which permits the above equation to be written in the more simple form:
where
e = volume of fluid flowing in unit time
pm =
r
dp,
mean density of fluid
= radius of tube
= decrease in partial pressure of vapor over distance dy
The amount of vapor transferred from the fluid core to unit surface area in unit time is:
INDUSTRIAL A S D ENGINEERING CHEMISTRY
September, 1930
Eliminating Q from Equations 9 and 10 and noting that pm = PM,/RT one obtains
969
2 = 5.3 dj
Urn
According to kinetic theory, the dimensionless group, p / p m k d , where k d is the coefficient of self-diffusion, is a con-
I n deriving Equation 3 for diffusion of one gas through another, it was shown that the term P/p,f corrected for the actual flow of vapor and gas to the condensing surface. A similar correction should be made for the transfer of vapor in the core, since besides the usual turbulence and eddy motion accounting for Equations 9 to 11there may be assumed to be a component of flow toward the film boundary which will again be proportional t o P/pe,. The corrected statement of Equation 11 will he:
Combining Equations 7 and 12 gives:
APB Pm -A=
w lM, kw M m
--P i a
UB -
-
(13)
stant for any one gas. For a mixture of two gases the value of kd is practically independent of csmposition, so that the group
will vary only as p / p m varies with the composition. For some pairs of gases this ratio does not vary much. The value of the group is practically independent of temperature and pressure; this can be roughly checked from the way the individual terms are affected. Thus the function can be used quite simply in most cases. Values for the three factors a t standard conditions can be substituted for the two extreme cases of 100 per cent vapor and 100 per cent inert gas, and a straight-line relation assumed to hold for mixtures. For some pairs of gases an average value of p/pmkd can be used over a wide range of concentrations as indicated by the examples of Table I. Where no data were available for the diffusion coefficients, they were estimated from the molecular weights of the gases. The method of making this estimation is given by Wintergerst (16) and also by Arnold (1).
Urn
Tahle I
Substituting Equation 13 in 8 gives the general equation: A
.P BASEDON
B 100% ,- B
where Q = 1
+ 2 (2-- 1) Urn
HzO
(15)
Pmkd
H20 NHa NHs NHa NHa
For flow in tubes, where F is given by Equation 2, Equation 1 4 can be rewritten:
so2
NO2 CEHE C6H6 CsH6
"B
p
0.53
U Y
I
~
0.602 0.44 1.195 0.67 0.565 0.265 1.11 1.12 2.84 1.72 1.33
100% ," A
100% B
0.763 0.703 0.565 0.806 1.000 1.016 0,661 0.875 5.7i 2.47 3.41
0.789 0.703 1.103 0.825 0.770 0.605 1,058 1.064 1.976 1.382 1.175
where G = mass velocity = pmum(consistent units)
Equation 16 for mass-transmission coefficients is quite similar to the Prandtl equation for heat-transmission coefficients as seen by comparing with Equation 1. The new features for correlating absorption and dehumidification data M , and -.P are the terms Values of -!!for M,, pmkd Pmkd many systems are of the order of magnitude of unity, as shown in Table I. Taking U B / U ~= 0.53, the term @ is usually nearly one and not very important, as shown in Table I. Note that Equation 16 is for conditions a t a gisen position along a tube and does not represent the average conditions over any given surface, and that consistent units have been employed throughout the above derivations. EVALUATION OF @--The ratio of U B to urnwas found by Stanton (12) from tests on heat transmission to be about 0.3 for water. I n a recent paper Prandtl (10) utilized the Karman ('7) relation that the velocity in the turbulent core varies with the seventh root of the distance from the surface and in the viscous film as the first power of the distance t o show that
Agreement with Data
DEHuMmFrcmIoiv-An investigation of the condensation of water vapor from a wide range of mixtures with air in an annular space between vertical pipes was reported by Colburn and Hougen (3). They found that a plot K/G0.8 against 1/pje gave a slope of 1.5 on logarithmic paper, instead of a slope of unity which had been predicted. When the data are replotted as shown by Figure 3 with K/GOs against M , / M , p , , , a straight line through the origin results on arithmetic paper, which shows that the M M , / Mfactor m is helpful. The empirical equation resulting from Figure 3 is:
(=gm)
(17)
The friction factor for isothermal turbulent flow in long tubes, over the range of G d / p (in consistent units) from 2000 to 2,000,000, can be approximated by either of the relations f = 0.09
(&)
where K' = pounds per hour per square foot per inch of mercury diff ererice G = mass velocity in pounds per second per square foot of cross section p o t = logarithmic mean partial pressure of air in film, inches of mercury
These data were for a point 3 feet from the entrance to the condenser and were slightly high owing to entrance turbulence. For a point far removed from the entrance the corresponding equation is, according to Colburn and Hougen's entrance factor,
0.21
To make the units consistent, K should be expressed in terms of seconds instead of hours. Then from a curve given by Walker, Lewis, and Mc.4dams (16). Combining Equations 17 and 18 gives:
(22)
970
IATDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 22, No. 9
Equation 16 can be changed into the form of 22 by substituting the empirical relation for the friction factor given by Equation 19. Atherton (2) found that the friction factor for annular spaces was higher by about 25 per cent than that in tubes having the same hydraulic radius. Equation 19 will then become for annular spaces:
of the absorbing liquid and the tower is long enough so that entrance turbulence is not an important factor, the pressure drop should be due mainly to skin friction and the absorption process will occur according to the mechanism pictured in the preceding derivation; Equation 16 should then apply. If the velocity of the absorbing liquid is high in comparison with that of the gas, the process is complicated, since it seems 0.1 f’ = 0.075 likely that the skin friction will be dependent on the velocity of the gas relative to that of the liquid and that this relative where d‘ = equivalent diameter or clearance. The term @ velocity should be used in Equation 16. A further complicacan then be calculated as follows: The experimental average tion results since the liquid will carry a film of gas rapidly value of G was 0.25 pound per second per square foot, p = downward with it from a zone of low vapor concentration (in the case of countercurrent flow of gas and liquid) to a zone of high concentration. This gas film must be supplied with vapor so that a portion of the vapor calculated as diffusing across the film remains in the film itself, and thereby decreases the actual absorption. Many data on flm towers are available in the literature (6,6)but these are mainly for high liquid velocities in comparison with the gas. An attempt is being made to correlate these results with amathematical expression of the film movement described above. One i n v e s t i g a t i o n in Figure 3-Agreement of Dehumidification Data of Colburn a n d Hougen w i t h Theoretical which the gas velocity was high in comEquation p a r i s o n w i t h the liquid was that of 0.000012 pound per second per foot, and d’ = 0.292 foot, Greenewalt (4) for the absorption of water iapor from air by By Equations 19a and 20, f = 0.013, and U B / U ~ = 0.605, sulfuric acid. His data for the case of a Venturi entrance to From Table I, p/pmkd = 0.58, and @ = 1 0.605 (0.58 - 1) the absorption tower are shown by Figure 4, in which values = 0.746; this illustrates the point that @ is near unity, of absorption coefficients, K , expressed as grams per hour per Equation 16 then becomes square centimeter per atmosphere difference, are plotted against the effective relative mass velocity, GR,pounds of gas 0.0066 Go.*M , per second per square foot of cross section. This velocity was K = (23) Pat M m calculated by multiplying the density of the gas by the relative linear velocity of the gas compared to that of the liquid. Comparison of Equations 23 and 22 shows excellent agree- The line shown is predicted by Equation 16, using Equation 19 ment between theory and practice. for the friction factor, and shows fairly good agreement with the experimental points except at the lowest gas velocities.
(&)
+
Acknowledgment
The writer wishes to acknowledge the valuable suggestions and criticisms of T. H. Chilton and W. H. McAdams. Literature Cited
Figure 4-Agreement
of Absorption Data of Greenewalt w i t h Theoretical Equation
ABSORPTION IN FILMTowERs-The conditions for which Equation 16 was derived are sometimes found in film towers for absorption of soluble gases where the main resistance is the gas Ha. Where the velocity of the gas passing through the tower is large in comparison with the downward velocity
Arnold, forthcoming publication from Massachusetts Institute of Technology. Atherton, Trans. A m . SOL.Mech. Eng., 48, 145 (1926). Colburn and Hougen, IND.ENG.CHRX., 22,522 (1930);also forthcoming bulletin of University of Wisconsin. Greenewalt, IND. ENG.CHEM.,18, 1291 (1926). Hanks and McAdams, Ibid., 21, 1034 (1929). Haslam, Hershey, and Kean, Ibid., 16, 1224 (1924). Karrnan, 2. angew. Math. Mech., 1,233 (1921). Maxwell, Phil. M a g . , 20, 21 (1860); 36, 199 (1868). Prandtl, Physik. Z., 11, 1072 (1910). Prandtl, Ibid., 29,487 (1928). Reynolds, M e m . Proc. Manchester Lit. Phil. Sac., 14, 7 (1874-5); “Scientific Papers of Osborne Reynolds,” Vol. I, p. 81, University Press, Cambridge, 1900; also in Gibson’s “Mechanical Properties of Fluids,” p. 173, Van Nostrand, 1924. Stanton, in Glazebrook’s Dictionary of Applied Physics, Vol. I, p. 401, Macmillan, London, 1922. Stefan, Silzb. A k a d . Wiss. W i e n , 68, 403 (18731; Ann. P h r s i k , 41, 725 (1890); see also Lewis and Chang, Trans. A m . Xnst. Chem. Eng., a i , 127 (1928). Taylor, Brit. Advisory Committee Aeronautics, Refits. Memo. 272 (1916). Walker, Lewis, and McAdams, “Principles of Chemical Engineering,” p. 87, hlcGraw-Hill, 1927. Wintergerst, A n n . P h y s i k , 4, 323 (1930).
