October, 1934 (8)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Hinchley and Himus, Trans. Inst. Chem. Engrs. (London), 2, 57 (1924).
-
(91 Hine. Phvs. Rev.. 24. 79 (1924). \-,
(10) (11) (12) (13) (14) (15) (16)
I~
Jurges, Beih. Gesundh. In;., 19 (1924). Kirkbride, IND. ENG.CHEM.,25, 1324
(1933).
Ibid.. 26, 425 (1934). Kirschbaum and Kranz, Chem. Fabrik, 7, 176 (1934). Monrad and Badger, IND. ENG.CHEM.,22, 1103 (1930). Prandtl, Physik. Z . , 11, 1072 (1910). Reynolds, Proc. Jlanchestw Lit.Phil. SOC.,14, 7 (1875).
1093
Engineer’s Handbook, p. 1222, McGrawHill Book Co., New York, 1933. (18) Stanton, Brit. Advisory Comm. Xironautics, Rept. & .)fern., (17) Sherwood, Chemical
94 (1913). (19) Stauffer, Roberts, and Whitman, 2, 88 (1930).
IND. ENG.CHEar., Anal. Ed.,
RECEIVED July 30, 1934. Presented as part of the Symposium on Diffusional Processes before the Division of Industrial and Engineering Chemlatry a t the 88th AIeetmg of the Amerlcan Chemical Society, Cleveland, Ohio, September 10 t o 14, 1934.
Diffusion of Vapors through Gas Films T. K. SHERWOOD AND E. R. GILLILAND, Massachusetts Institute of Technology, Cambridge, Mass.
D
IFFUSION plays a tremendously i m p 0 r t a n t
gas flow past the liquid or solid phase, it seems probable that there exists a thin layer of gas moving in stream-line or viscous motion over the surface of Ithe other phase, and that the flow breaks suddenly into turbulent motion beyond the outer boundary of this viscous layer. If this is the nature of the film, it seems probable that the thickness of the laminar layer fluctuates rapidly and to a considerable degree, owing to the turbulent motion of the main gas current. The experimental evidence apparently does not exclude the possibility, however, of a continuous gradation in turbulence from the surface of the solid or liquid phase through the film to the main body of the gas stream. I n the turbulent gas stream the movement of vapor through the gas is mainly by the mechanism of turbulent mixing or eddy diffusion. Eddy diffusion is very rapid, but the distances involved are much greater than in the relatively stagnant film so that it is possible for the eddy diffusional resistance to be as great or greater than the resistance to molecular diffusion offered by the laminar film. The total resistance to diffusion of the gas film may be considered to be the sum of the resistance of the turbulent layer to eddy diffusion and the resistance of the laminar layer to molecular diffusion. The two diffusional mechanisms are quite different in nature. Since the nature of the film is not fully understood, the attack on the problem of diffusion through gas f 3 m s must necessarily be empirical. This empiricism may be veiled by the introduction of the theoretical equations for diffusion in stagnant gases, but appears in the treatment of the “effective film thickness” involved in using this method of approach to the problem. For example, the Stefan equation for the steady-state diffusion of one gas through a layer of a second stagnant gas is :
The nature of a gas f i l m is discussed briefly and the application of the Stefan-Maxwtdl equation to diffusion in gas films is explained. T h e results of experiments on rates of vaporization of various liquids into air are reviewed, and rules are given for estimating the diffusion rate of a vapor through a gas film, o n the basis qf data on the diffusion rate f o r another uapor. Numerical examples illustrate the necessary culculations. Derivations of algebraic equations connecting the absorption coeficient Kca with the height of packing equivalent to one theoretical plate (H. E. T. P . ) are given, and the use of the latter concept in the design of packed towers .for absorption is discouraged.
role in industrial processes because the capacity and consequently the fixed charges on many types of i n d u s t r i a l equipment are determined by the rate of diffusion of heat or of material f r o m o n e p h a s e t o a n o t h e r . Diffusion in gas films is one of the most important technical problems which confronts the chemical engineer. The rate of interaction of a gas with a l i q u i d o r solid m a y be governed by the rate of a slow chemical reaction, or by the slowness of diffusion t h r o ugh the liquid phase. In many -of the most i m p o r t a n t of s u c h operations, however, the rate a t which the operation proceeds is determined by the slowness of diffusion in the gas phase. This condition may be summarized by the statement that “diffusion through the gas film is controlling.” Such is the case in the absorption by liquids of soluble gases from mixtures with an inert gas; in adsorption of gases by solid adsorbents; in the air-drying of wet solids; in vaporization of water or other liquids into air, as in humidification; in condensation of vapors from mixtures with noncondensable gases, as in dehumidification; in the evaporation of solvents, as in drying of lacquers; and in the combustion of solids a t high temperatures. In interphase reaction, the distribution of diffusional resistance in the gas phase depends mainly on the velocity characteristics of this phase. For the case of a quiescent gas phase, as in a bubble rising through a deep liquid layer, the diffusional resistance is constant throughout. If the gas phase is in laminar flow, the diffusional resistance is still constant at all points, but the progressively lower mass velocity of the gas as the interface is approached causes a very different distribution of the diffusing substance from that which would be encountered with a stagnant gas. In the case of turbulent flow a large resistance to diffusion in the gas phase is encountered in a narrow region adjacent to the iquid or solid phase. This fact has been well understood for a number of years, as evidenced by the widespread use of the phrase “gas film.” The physical nature of this film is, however, not well known, although it appears certain that it is not the simple layer of perfectly stagnant gas which affords such a convenient, although erroneous, mental picture of the reason for the concentration of the diffusional resistance near the phase boundary. Under conditions of appreciable
where N A
diffusion rate per unit interfacial area, gram moles/ (sec.)(sq. cm.). D = diffusivity of system, sq. cm./sec. P = total pressure, atm. R = gas constant: (cc.)(atm.)/(gram mole)(’ K.). T = abs. temp., K. z = thickness of stagnant gas layer, cm. k~ = film coefficient, grams/(sec.)(sq. cm.) (atm.). =
INDUSTRIAL AND E N G I N E E R I N G CHEiMISTRY
1094
M A = mol. weight of diffusing gas = partial pressures Of the diffusing gas or vapor at the boundaries of the layer, atm. pel, p B 2 = partial pressures of the stagnant or non-diffusing gas at the two boundaries of the layer, atm. pn.u = logarithmic mean of p e n and pel AI, P A 2
When this equation is applied to diffusion through a gas film, the thickness, 2, may be interpreted as the effective thickness of the film. This thickness has no great theoretical significance, but it may be used empirically as are the various forms of gas film coefficients. Values of D for various gas systems may be obtained from the standard tables of physical constants or be estimated by means of an empirical correlation published recently (1). An experimental study of the application of Equation 1 to diffusion in gas films has been described by the authors in another paper ( 2 ) . A wetted-wall column was used in these experiments; the liquid to be vaporized flowed in a thin film down the inner surface of a vertical tube through which air was forced, either parallel or countercurrent to the movement of the liquid film. The rates of vaporization of nine liquids were measured a t various temperatures, using various air speeds. The driving force (pal - P A * ) was varied by changing the temperature of the liquid circulated. The mean pressure of inert gas, P B M , was varied by operating the apparatus a t pressures ranging from 110 to 2330 mm. of mercury. The diffusivity was varied by using several liquids. The effect of air turbulence was studied by varying the air velocity over a wide range. The data showed the rate of vaporization (1) to be directly proportional to the driving force, as called for by Equation 1; ( 2 ) to be inversely proportional to the mean pressure, P E M , of air in the gas film, as called for by Equation 1 ; (3) to vary as the 0.83 power of the Reynolds number for air flow through the pipe; and (4) to vary directly as the 0.56 power of the diffusivity, D. The data were expressed empirically by the equation : =
0.023
(d ~ ) " * 8(3
X
where d = inside diam. of column u, p , = velocity, density, viscosity of turbulent gas stream, respectively Each of the three groups in Equation 2 is dimensionless. The relation between the rate of vaporization, or of the effective film thickness, 2, and the group p / p D is particularly significant in that it suggests a method of prediction of the performance of apparatus involving diffusion in gas films, on the basis of data on the same apparatus with a different vapor. The familiar gas film coefficient, k G , represents the rate of diffusion as weight per unit time per unit area, per unit driving force. It is apparent from Equations 1 and 2 that k~ varies : (1) Directly as the 0.56 power of the product diffusivity times total pressure. (2) Inversely as the mean inert pressure ~ B M(the product D P is independent of pressure). (3) Directly as the Reynolds number to the 0.83 power. (4) Directly as the molecular weight of the diffusing vapor (in addition t o the effect of molecular weight on D).
These rules may be used, for example, to predict the performance of a packed tower for the absorption on a soluble gas such as acetone, provided data are available on the same packing used for the absorption of another soluble gas, such as ammonia. They also suggest methods of correlation of gas film coefficients for packed towers and other similar apparatus. For example, i t follows from Equation 1 that l / r is proportional to ~ G ~ B M T / D P and M A ,it seems probable that values of this latter group might be correlated for -c.arious gases. Unfortunately, the effect of diffusivity has been determined
Vol. 26, No. 10
only for the wetted-wall apparatus, and it may be that the effect of D on ko for a packed tower Tvill be somewhat different. The exponent 0.56 for 1 would be unity if the mechanism were wholly molecular diffusion and would perhaps be zero for pure eddy diffusion. The observed fractional exponent is dependent on the relative resistance of laminar and eddy layers. That it is not sensitive to turbulence conditions, however, is indicated by the fact that it remained constant as the Reynolds number was varied over the range from 1800 to 26,000 in the wetted-wall column experiments. For convenience, values of D for a number of typical systems are given in Table I. TABLE I. DIFFUSIVITIES IN AIR AT 30" C. SUBSTANCE c12
so2 coz
0.109" 0.119~ O.12sa
H C1
0. 166a 0.149" 0.201"
Hz0
0.2636
0 , 138a
NO2 NO
NH3
Acetone a
b
AND
SUBSTANCE Methyl alcohol Ethyl alcohol Isopropyl alcohol n-Butyl alcohol see-Amyl alcohol Ethyl acetate Chlorobenzene Toluene Aniline
0 3 0
0.097n
1 ATMOSPHERE n,, 0,140a 0.110a
0.101b O.088b 0.072b
0.089s 0.075b 0.088b
O.075b
Calculated by empirical equation of Gilliland ( 1 ) . From experimental d a t a of Gilliland ( 1 ) . ILLUSTRATIVE PROBLEM
1
A wetted-wall column operating a t a total pressure of 518 mm. of mercury is supplied with n-ater and air, the latter a t the rate of 120 grams per minute. At a mean driving force of 62.5 mm. and a calculated mean inert pressure of 449 mm., the rate of vaporization of water in the apparatus is 13.1 cc. per minute. The same apparatus, now a t a total pressure of 820 mm. of mercury, is supplied with air a t approximately the same temperature as before a t a rate of 100 grams per minute. The liquid vaporized is n-butyl alcohol, which is supplied a t such a temperature that the mean driving force is 24.4 mm. and the mean inert pressure is 799 mm. The specific gravity of the liquid is 0.807. The problem is to estimate the rate of vaporization of n-butyl alcohol, as cubic centimeters per minute. SOLUTION. From Equation 1 it is apparent that the weight rate of vaporization is proportional to k~ and to the driving force. Hence: n-butyl alcoholvaporization = 13.1 X 1 0 24.4 &(ah.) 0.807 62.5 )-k The ratio of ko(alc.)/kG(water) follows from the four rules' given above: k c (alc .) 0.83 74
(g)
- =
X - = 1.08 18
The ratio of Reynolds numbers is assumed to be the same as the ratio of air rates, since the apparatus was the same, and the air temperature approximately the same. Hence, the rate of vaporization of n-butyl alcohol will be: 1.0
24.4
13.1 X __ X - X 1.08 = 6.8 cc./min. 0.807 62.5 The conditions given are those of actual experimental tests. For the run on n-butyl alcohol the observed rate of vaporization was 6.9 cc. per minute.
ILLUSTRATIVE PROBLEM 2 Assume, for example, that a test on a certain packed tower absorbing ammonia in water a t 2 atmospheres and a t 25' C. gave a value of koa of 0.016 kg. per minute per cubic meter per mm. when using a water rate of 78 kg. per minute per square meter of tower cross section and an average gas rate 1 Since t h e product D P is independent of pressure, rule 1 is equivalent to a statement t h a t lzo varies a s the 0 56 power of t h e diffusivity taken a t one atmosphere.
October, 1934
INDUSTRIAL AND ENGINEERING CHEMISTRY
of 20 kg. per minute per square meter of tower cross section. The tower reduces the ammonia concentration of the air mixture from 20 to 1 per cent ammonia by volume. Using these data, the problem is to estimate the value of ~ G Ufor the absorption of acetone from air a t 0.5 atmosphere, using the same type of tower packing, temperature, gas, and water rates as in L the test on ammonia. The concentration is to be reduced from 5 to 1 A per cent acetone by volume. r--0 -&--1 SOLUTION.In calculating values I ’--
