Mass Transfer between Phases Role of Eddy Diffusion - Industrial

Eddy Diffusion Mass Transfer in the Central Portion of a Turbulent Air Stream. Industrial & Engineering Chemistry. Towle, Sherwood. 1939 31 (4), pp 45...
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INDUSTRIAL AND ENGINEERING CHEMISTRY

1034

varies from year to year and the amount of water withdrawn, which is the only factor under the control of the users. If the amount of water withdrawn lowers the fresh water head sufficiently, salt water will move u p the formation. I n the early days of Houston, flowing wells could be obtained almost anywhere within the present city limits, and the artesian head in some wells was sufficient to raise the water from 15 to 30 feet above the ground. Kow the artesian head is about 80 feet below the surface in the downtown part of Houston. Between 1920 and 1931 the decline in artesian head averaged about 4 feet a year. Between 1931 and 1936 there was little decline in artesian head in the heavily pumped Houston-Pasadena district. I n the area to the south and southeast of Houston, however, the artesian head declined markedly. In 1937 new wells were put down in Pasadena which are reported to have a combined capacity of 20,000,000 gallons a day, an increase of about 40 per cent over the average pumpage for 1931-36. From March, 1937, when these wells were put into operation, to March, 1938, there was a pronounced decline in water levelsh observation wells in the Houston-Pasadena district, particularly in those within 4 miles of the new wells. I n two wells, 6/8 and 13/4 miles distant, respectively, the decline in water level was 35 feet in the 12-month period. * yu-ia-lie e a r distant down th-jg-ihe-deeper beds, from which _ I _ _ . . -

VOL. 31, NO. 8

a large part of the water in the Houston district is drawn. There is, therefore, a distinct possibility that any large decline in artesian head may result in the encroachment of salty water into the wells of the district. Fortunately such an encroachment is likely to be slow, and can be watched and to a degree anticipated if proper observations are made. The progress report on the ground water resources of the Houston district (S), published in March, 1937, recommends there sh@d be no increase in pumpingjL&-on IS net, ( b ) a d ~ X i K ~ i T t waterZ d & o d d be o m n e d a t a sufficient distance f i m h e - t o avoid undue inJt=&on--Qf ~ a ~ ~ t h ~ ~ s - ; ‘ ~ ~ I ethe n iground B h i n water g reservoir in the heavily pump$ Hguton-Pasadena area, and (c) p r o d i a and w B f i l 5 i e x water should be eIiEiEated, .--_ ”” - . I _

Literature Cited (1) White, W. N., Livingston, Penn, and Turner, S. F., U. S. Dept. Interior, Press Mem. 66,553 (1932). (2) White, W. N., and Livingston, Penn, Ibid., 79,241 (1933). (3) White, W. N., Turner, 9 F , and Livingston, Penn, U. S. Geol. Survey, mimeographed rept., March 1, 1937

PRESBNTED before the Divlsion of Water, Sewage, and Sanitation a t the 95th Meeting of the American Chemical Soaiety, Dallas, Texas. Published by permission of the Director, Geological Survey, United States Department of the Interior.

-------*

Mass Transfer between Phases ROLE OF EDDY DIFFUSION T. K. SHERWOOD AND B. B. WOERTZ Massachusetts Institute of Technology, Cambridge, Mass.

I

KTERPHASE transfer of material is a process of considerable engineering importance, as illustrated by the unit operations of drying, gas absorption, and humidification. I n some cases of mass transfer between a solid or liquid and a fluid moving in turbulent motion, much of the resistance t o diffusion is encountered in a region very near the boundary between phases. According to the simple film concept the entire resistance to interphase transfer of material is represented by a stagnant fluid film a t the interface through which the diffusing substance must pass by the slow process of molecular diffusion. It is generally recognized that the concept of a single stagnant film constituting the entire resistance is an oversimplification of the situation, and that much of the resistance may be in the eddy zone or “core” of the turbulent stream. Any analytical treatment of the whole process may be subject to serious error if it does not allow for the resistance to eddy diffusion, which is a process fundamentally different in character from molecular diffusion. A previous paper (IO) reported a study of eddy diffusion of carbon dioxide and hydrogen in a turbulent air stream. The results show that the rate of eddy diffusion is proportional to the concentration gradient, and that the proportionality constant, or “eddy diffusivity,” is independent of the nature of the diffusing gas. The study was made in the central third of a large round duct and shed no light on the nature of eddy diffusion in the vicinity of the wall. The present study is concerned with the

over-all process of transfer between a liquid surface and a turbulent air stream.

Turbulence and Eddy Diffusion Largely because of its application in aeronautics, the science of fluid mechanics has been developed materially in recent years. The nature of turbulence has received special attention, and many of the concepts and theories (3) proposed bear directly or indirectly on the question of mass transfer in a turbulent fluid. It is impossible to summarize this work briefly, and the reader is referred to the general papers of vcn Karman (6),Rouse (8), Izakson (4), and Bakhmeteff (1). A recent paper by Dryden ( 2 ) gives an excellent summary, with particular reference to diffusion. Turbulent motion is characterized by the random motion of the particles constituting the fluid stream. Individual particles move irregularly in all directions with respect t o mean flow, and it is convenient to think of a fluid in turbulent flow as having a mean velocity, U , in a direction, x, with a superimposed random motion resulting in instantaneous deviations from U a t any point. This instantaneous deviating velocity a t any point has components u, u, and w in the x, ?J, and z directions. Techniques have been developed for measuring u’, which is the root mean square average u(u’ = and the “per cent turbulence” is u’expressed as a percentage of mean velocity U a t the point.

dm),

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

The irregular motion of the turbulent stream results in swirls or eddies, which are small masses of fluid moving temporarily as units. An eddy has a short life and soon breaks up into fragments, which form new eddies. Mixing and diffusion within an eddy may be quite slow, but material may diffuse rapidly by the process of eddy transfer and disintegration. The eddies may be large or small, and various factors have been proposed as criteria of the scale of turbulence. The most common of these is the “mixing length’’ proposed

1035

d U / d y , may be obtained directly from data on velocity distribution or calculated from one of the several theoretical expressions for velocity distribution. Of the latter, that proposed by von Karman seems to be the most acceptable for the velocity of fluids in the eddy zone in round pipes:

u, -

=

--

4

u0 k

[In (1

-

dl.7) + 41 - k] (6)

~

The work described represents a study of the relative importance of molecular diffusion through the laminar surface film and of eddy diffusion in the main body of the turbulent fluid for the case of vaporization of water into a turbulent gas stream. Measurements were made of the watervapor gradient across a turbulent gas stream in a large duct 5.3 X 61 cm. in cross section and 471 cm. tall. The gas was recirculated through the apparatus and a steady state set up, with water vaporizing from a water film on one wall of the duct and being absorbed by a film of strong aqueous solution of calcium chloride on the opposite wall. The apparatus was operated with air, carbon dioxide, and helium. The rate of vaporization was measured, and values of the eddy diffusivity

by Prandtl. This quantity is the average distance the eddy moves before breaking up and losing its identity. It is defined by the Prandtl equation:

~~

were calculated for the central turbulent portion of the gas stream. The concentration traverses show that each film offered 21-36 per cent, and the turbulent core 28-57 per cent of the over-all diffusional resistance. Reynolds number was varied from 3600 to 102,000; the results for air agreed well with those of Towle and Sherwood, obtained by a different experimental technique. The product Ep was found to be the same function of Reynolds number for all three gases and was approximately 1.6 times the eddy viscosity. Turbulence determinations were made by the standard hot-wire technique and employed in a comparison of the results with the modern theories of the mechanism of turbulence. The results were well correlated by means of a semitheoretical equation.

where U, is the maximum velocity (at the center of the pipe), ro is the radius of the pipe, and L is a “universal constant” having a value of 0.384.40. I n streamline flow the shear stress is given by: 7

R is defined by the equation

7

=

7

(at

wall)

ddU/dy)

By analogy, an “eddy viscosity,” E, may be defined to account for the large shear stresses in turbulent flow:

where the bar denotes the average with respect to time, and u’ and v‘ are the root mean square values of u and v. I n writing the last equality of Equation 1, it is assumed that u‘ = v’, and that 1 is the same with reference to both x and y directions. The shear stress r 0a t the wall is given by the familiar expression, 70

=

=

I fpuoz 2

(3)

where f is the usual friction factor for turbulent flow, and Uo is the average velocity of the fluid over the entire cross section of the conduit. The shear stress is zero a t the center of the stream, where the velocity gradient is zero, and

= (P

+ e)(dU/dy)

(7)

I n turbulent flow a t high velocities the shear stress calculated by Poiseuille’s law is negligible as compared with the observed friction; i. e., p is small compared with E . If p is neglected in comparison with E, Equations 4 and 7 may be combined and the result integrated to give:

The integration is carried out on the assumption that B is constant. This type of equation for the “velocity deficiency” ( U , - U)does not have the theoretical basis of Equation 6 but was found by Murphree to fit certain data of Stanton. It was found to fit the present data better than Equation 6. By analogy to molecular diffusion, an “eddy diffusivity” may be defined as

It follows from Equations 1 and 4 that

(5)

Correlation R is not generally known and is frequently assumed to be unity. Friction factor f is obtained from usual friction factor graphs for any Uo. The velocity gradient,

where X A is the rate of transfer by eddy diffusion expressed as gram moles per unit time per unit area, and dC/dy is the gradient of the concentration in the direction of diffusion. Consider the transport of material by the eddies in a direction y a t a point where the average deviating velocity in the As a basis, consider two units of cross-sesy-direction is tional area normal to the direction of diffusion. Fluid is

v.

INDUSTRIAL AND ENGINEERING CHEMISTRY

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carried a t an average rate U volumes per unit time per unit area in one direction, and a corresponding quantity returns simultaneously at the same average rate. The net amount of solute gas carried is ZAC across the two units of area, where AC is the change in concentration in a length 1 corresponding to the mixing length; hence,

Townend (11) found a constant relation of 0.8/1.0 between u taken without regard to sign, and u’,the root mean square. If such a relation is assumed to apply to v and v‘, then E = 0.40 v‘l. Murphree and Dryden suggest that the corresponding quantity for heat transfer, the eddy conductivity, should be equal to the product of the specific heat and the eddy viscosity. This amounts to taking E as the ratio of eddy viscosity to density, which leads t o

u, the time average of

E = v’lR

(11)

Taylor’s vorticity transport theory (3, 9) leads to E

=

2v’lR

(12)

provided 1 is assumed independent of y. The various theories agree, therefore, in that E should be proportional to the product of v‘ and 1. This proportionality may be tested by the data obtained in the work to be described.

VOL. 31, NO. 8

series of 0.127-cm. 0. d., 0.076-cm. i. d. nickel sampling tubes pointing into the air stream and placed a t intervals across the duct. A total of eighteen tubes was provided, although only tubes 2, 3, 5, 7, 9, 10, 12, 14, 16, and 17 were used in most tests. These traverses were usually made at horizontal plane A of Figure 1, although five runs were made with sampling traverses a t B . I n both positions the line of the traverse was 22.2 cm. from one narrow-end vertical wall of the duct. This position was taken as representative of the whole cross section. The air samples were aspirated s l o ly ~ through IO-cm. weighed drying tubes; large aspirator bottles were used to obtain samples of 7400 cc. through each tube in 16 t o 20 minutes. In runs with air the desiccant was anhydrous calcium sulfate; anhydrous ma nesium perchlorate was used in the runs with helium and carton dioxide. Care was taken that the first weighings of the drying tubes should be made with the tubes containing the same gas (air, helium, or carbon dioxide) as was to be drawn through them. The partial pressure of water vapor in the gas samples was calculated from the weight of water absorbed in relation to the amount of dry sample aspirated. Since rubber was found to absorb water vapor, the sampling tubes leading t o the drying tubes were constructed almost entirely of glass or metal. In most runs gas velocity was measured by obtaining Pitot tube traverses at three positions in the same horizontal plane as the Sam ling traverse (planes A or B of Figure 1). Movable Pitot tufSes equip ed with Vernier scales t o indicate exact position relative to t i e wetted walls were located at three points, 30.5, 14.0, and 2.9 cm. from the dry narrow-end wall. The flow was assumed to be symmetrical about a center line normal t o the wetted walls, and the total gas flow was obtained from the average a t each traverse by the Gauss method of integration (the traverses were spaced so that this might be done). At very low velocities a sharp-edged orifice was installed in the round return duct in order t o obtain larger manometer readings and so improve the accuracy. A micromanometer was used, with ethyl alcohol as a manometer fluid. The density of the aloohol was checked occasionally by means of a Westphal balance.

Experimental Procedure

DAMPER GLASS

The procedure followed involved the measurement of the moisture concentration gradient in a turbulent air stream across which water was being transferred a t a steady rate. Figure 1 shows the apparatus:

It consisted primarily of a vertical rectangular duct, 5 . 3 X 61 cm., through which air was passed vertically from bottom to top, and a return duct and blower. The large left-hand wall of the rectangular duct was covered with a moving film of water supplied through a weir of 40-mesh (per inch) brass wire screen a t the top. The right-hand wall was covered with a film of strong aqueous solution of calcium chloride, supplied with another similar weir at the top. The right-hand wall covered with calcium chloride solution was 61 cm. wide and 471 cm. from top to bottom, The solution was withdrawn from a narrow horizontal opening a t the bottom. The water sup lied at the top was withdrawn through a similar slot about haEway down and returned to the top by a small pump through a measuring reservoir. Evaporation from the upper portion of the left wall was measured by noting the depletion of water from this small reservoir. Water was supplied t o the lower part of the left wall through a third wire weir and was removed a t the bottom through a third removal slot. The narrow vertical ends (5.32 cm. wide) of the duct were ke t dry by small vanes extending out from each corner, as indicatetin Figure 1. The apparatus was filled with air, carbon dioxide, or helium, circulated by the blower at velocities corresponding to Reynolds numbers of 3600 to 102,000 in the rectangular test section. Water evaporating from the left wall traversed the turbulent air stream and was absorbed by the film of calcium chloride. The lower part served as a “calming” or “conditioning” section, and the rate of vaporization of water was measured only in the upper part. I t will be noted that the length of this calming section was a proximately fifty times the width of the duct. The galvanize$ iron walls of the duct were reinforced by soldering horizontal angle irons on the outside a t intervals of 30 cm., so reducing the extreme variation in internal dimension to the limits 4.92-5.56 cm. Except for a very few points, the variation was from 5 . 1 to 5.4 cm. Better uniformity of duct width seemed impractical with galvanized-iron construction. Windows were provided a t top and bottom of the rectangular duct to make sure that the liquid films were completely wetting the side walls. A traverse of the concentration of water vapor in the air stream was obtained by withdrawing air samples through a

“‘M

’$1 r z ? l

WATER

I

’

- TEST SECTION

2 - CALMING SECTION

‘I, i

CaCI, SOLUTION

FIQURE 1. DIAGRAM OF APPARATUS

All tests were carried out a t ap roximately room temperature, and small coolers placed on bot% water and calcium chloride solution feed lines served to maintain constant temperatures of the two li uid feeds. Temperature measurements included inlet and ouaet wder to the test section, inlet and outlet solution, and inlet water t o the calming section. Samples of solution were taken three times during the run, and vapor pressure determinations were made on each. They were obtained by bubbling air through the solution sample and determining the water vapor content of the equilibrium mixture by the same gravimetric technique described above.

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

In starting a run the drying tubes were weighed, water was introduced to the main apparatus, and the water pumps were started. The right-hand wall was first wet with water to aid in obtaining a complete film. In order t o spread the water or solution completely, it was usually necessary t o rub the surface with a small rag on the end of a long rod. Sampling-tube lines were flushed out by maintaining a slight vacuum for about 30 minutes. During this time solution and water rates were adjusted so that the calculated velocities of the films on both walls would be equal. The gas samples were then taken over a period of 16-20 minutes, before and after which the various temperatures were read and solution samples taken. Vaporization from the test section was observed at 15-minute intervals for a period of 3 hours. Between readings the Pitot tube traverses were made and pressure drop measurements recorded. A final sample of solution was taken, the pumps were stopped, and the apparatus was drained, washed, and dried. Turbulence conditions in the duct were tested by measurements of the deviating velocity, u', obtained by means of the standard hot-wire anemometer technique. The instrument used, together with the oscillograph and other auxiliary equipment, was loaned by the Department of Aeronautical Engineering, M. I. T., and the actual measurements were made by Carlyle Jacob of that department. The hot wire sup ort was installed with a Vernier scale similar t o that used witf the Pitot tube traverses, and turbulence measurements were recorded at various gas velocities with duct walls wet and dry, the traverses being made at the same lace as the sampling traverses. The results are reported as ' per cent turbulence," which is the value of u' as a percentage of mean velocity U at the same point.

P

Calculation of Results The gravimetric humidity determinations on the gas samples were converted to partial pressures of water vapor and plotted against position in order to show the water vapor concentrations. The resulting curves were found to be essentially straight over the main central portion of the duct width, and the eddy diffusivity mas calculated from the slope of the partial pressure curve in this region by means of the relation

where p , is the partial pressure of water vapor in the gas, y is the distance from the surface of the water film, and N A is the rate of vapor transfer across the duct. The gradient d p J d y obtained from the graph of p w us. y was adjusted as described below to correspond to the evaporation rate determinations. The over-all driving force, pw* - pc*, for the upper test section was slightly different than the difference, pw* - p,*, at the point where the sampling traverse was made. For the test section as a whole, p,* was taken at the temperature (tl t2)/2, where t1 and tz are the inlet and outlet water temperatures. The temperature drop of the water was usually about 2" C. The corresponding value of pe* was taken at the temperature '/2[t4 (t, t ~ ) / 2 ]where , t 4 and t h are the temperatures of the solution entering and leaving the apparatus as a whole. An allowance for the slight dilution due to absorption of water vapor by the solution was made on the assumption that the dilution was linear from top to bottom of the duct. For the conditions a t the sampling traverse, p,* was taken at the temperature tl - (tl - t2)/7, and p,* was taken a t the temperature t4 ( t s - t4)/16 with a correction for dilution. Since the rate of vaporization was always less than one per cent of the rate of flow of solution, the dilution correction was usually negligible. The water vaporized was plotted against time, and the points were fitted by a straight line, using the method of least squares. The slope of this straight line (as cc./second) divided by 18 and by the area of the water film (12,630 sq. cm.) gave N A . The gradient d p w / d y , corresponding to the average conditions in the test section, was obtained by multi-

+

+ +

+

1037

plying dp,/dy obtained from the concentration traverse by the ratio of (p,* - pc*) for the whole test section to (pw* po*) a t the sampling traverse and the corrected d p w / d y employed in Equation 13 with the observed N A .

2 1 DISTANCE FROM CENTER-LINE O f DUCT CM

-

0

1

2

CONCENTRATION TRAVERSES ACROSS FIGURE2. TYPICAL TURBULENT AIR STREAM I n the series of runs the water rate varied from 10 to 42 cc./ second and the solution rate from 13 to 55 cc./second. At these rates both films would flow in streamline motion, and the liquid velocities a t the gas-liquid interfaces were calculated by means of the equation:

These calculated interfacial velocities were of the order of 20 cm./second. The Reynolds number for gas flow through the duct was necessarily defined quite arbitrarily. The average gas velocity was taken as the mean of the two Pitot traverses nearest the center line of the duct, a t the same horizontal plane. The average velocity of the third Pitot traverse (nearest the narrow-end wall) was considerably smaller than a t the first two and the average of the first two seemed more representative of the conditions in the vertical plane of the sampling traverse. The Reynolds number was based on gas velocity relative to the films, obtained by adding the surface velocity of the films to the average gas velocity as defined above. The equivalent diameter was taken as four times the hydraulic mean radius, the latter being taken as half the clearance between films, or 2.63 cm. It will be noted that the calculation of the eddy diffusivity does not depend on the definition of method of calculation of the Reynolds number.

Results Figure 2 shows three typical concentration traverses, plotted as partial pressure of water vapor against position in the duct. I n each case the left vertical boundary of the graph represents the water wall, and the right boundary represents the wall covered with calcium chloride solution. The extreme lefthand point represents the vapor pressure of the water (pw*) and the extreme point at the right in each graph represents the equilibrium partial pressure of water over the calcium chloride solution (p,*). The examples were selected to show the change in nature of the concentration traverse with change in Reynolds number. At the lowest Re (run A-19) a smooth S-shaped curve is obtained with no evidence of a film at the wall. At the higher Reynolds numbers the curves are essentially straight over the large central portion of the duct, and an abrupt drop in partial pressure occurs in a narrow region near the wall.

INDUSTRIAL AND ENGINEERING CHEMISTRY

1038

At these higher flow rates it is evident that the resistance of each "film" is roughly the same as that of the main central turbulent portion. I n the case of the third curve (run A-13) the drop at the calcium chloride film is slightly larger than that a t the opposite water wall; therefore the partial pressure a t the center is not the mean of pw* and p,*. This was noted in the majority of cases and may be due to a concentration gradient in the viscous liquid solution film which causes the surface to exert a higher vapor pressure of water than in equilibrium with the main body of the solution. However, the calculation of the eddy-diffusivity for the main central portion of the duct is not dependent on this questionable quantity, PO*.

Values of the eddy diffusivity, E , were calculated from the data by means of Equation 13. The quantity dp,/dy was taken as the slope of the concentration traverse (such as those illustrated in Figure 2) over the main central portion of the duct. At the higher values of Re a straight line was drawn through all the points (except those a t the wall) and its slope was evaluated. At low values of Re, as in run A-19 of Figure 2, a straight line was drawn over a range of about 1.5 em. on either side of the center. The values of E calculated in this wag apply strictly only for the center of the duct; but it is apparent from the straight lines obtained that, except for the lowest Reynolds numbers, E is almost constant over 80 to 90 per cent of the cross section. The values of E obtained are given in Table I and are shown in Figure 3 plotted

VOL. 31. NO. 8

TABLE I. SUMMARY OF RESULTS Run No

L'o

Evapn. Rate Cc. water/

Re

W d u HQ/

Mm.

E Sq. om./ see.

Cm./sec. mzn. cm. Air Flow, Traverse at Section B A-1-L 725 6.89 50,600 0.645 14.5 A-2-L 352 4.79 24,600 0.816 8.1 A-3-L 990 6.69 68,500 0.460 19.7 A-4-L 550 6.21 37,700 0.705 12.2 Air Flow, Traverse at Section A A-1 726 5.96 51,700 12.1 0.665 A-2 726 6.88 51,400 0.898 13.4 A-3 716 7.10 51,000 0.694 13.9 A-4 512 4.98 37,000 0.621 10.9 A-5 355 3.97 25,800 0.591 9.2 A-6 171 3.13 13,000 1.04 4.1 A-7 108 4.32 9,200 1.42 4.2 A-8 342 5.27 25,100 0.936 7.7 A-9 613 4.58 43,000 0,747 8.4 A-10 444 4.26 32,300 0.808 7.1 A-11 261 3.20 19,200 0.847 5.2 A-12 873 5.97 60,700 0.580 14.0 A-13 1019 6.33 70,000 0.540 16.0 A-14 955 6.91 65,700 0.539 17.5 A-15 845 6.23 59,100 14.0 0.603 A-I6 991 7.95 67,800 16.5 0.657 A-17 7.65 53,500 799 0.797 13.3 A-18 29.66 1.58 3,600 1.4 1.527 A-19 63.5b 2.26 5,700 1,426 2.2 A-20 834 6.70 57,200 ' 14.2 0.647 Carbon Dioxide, Traverse at Section A c-1 186 2.20 26,600 1.00 3.0 297 3.39 41,200 0.691 C-2 6.6 3.80 49,800 0.598 c-3 37 1 8.6 634 5.94 83,100 0.686 11.8 c-4 792 7.14 102,000 0.587 16.5 c-5 C-6 9,600 1.36 56 b 2.29 2.3 c-7 65,000 0.898 10.6 498 6.95 Helium, Traverse at Section A H-1 20.2 7.46 12,000 1067 0.495 H-2 8,600 0.685 13.0 709 6.62 660 6.54 7,400 0.640 H-3 13.8 5,400 8.3 H-4 491 4.43 0.728 8.81 11,600 0.531 22.6 H-5 1082 a The values tabulated represent, the slope of the straight central portion of the concentration traverse (as in Figure 2) multiplied by the ratio of (pu* p c * ) for the whole test section to ( p w * - PO*) at the samplin traverse (see "Calculation of Results"). This correction was less than 6 2 in most cases, and was greater than 10% i n only four runs. b Values of Uo tabulated are the mean of the two average velocities obtained by Pitot traverses at the center and at a point 1 4 . 0 om. !ram one narrow end wall. Values of R e are based on this Uo. The ratio of thls average to the avera e for the whole duct, based on all three Pltot traverses, was noted, an8 this ratio was used t o estimate Lio in those runs at low velocities where it was necessary to use an orifice t o measure the total flow. The values marked b were obtained in this way.

-

REYNOLDS NUMBER

-

4

(U t U f ) P a,

FIGURE 3. EDDYDIFFTJSIVITY AS A FUNCTION OF REYNOLDS NUMBER

against Re. Three lines are obtained for the three gases, the ordinates being roughly in inverse ratio to the densities of the gases. The points scatter somewhat, but i t is evident that the data for the diffusion of water vapor through air in a rectangular duct agree remarkably with the results of Towle for eddy diffusion of carbon dioxide and hydrogen in the central portion of a turbulent air stream in a round pipe. The slopes of experimental curves are always much more uncertain than the actual data plotted, and it is apparent that one of the major sources of error causing the spread of points on Figure 3 is the uncertainty as to the exact slope of the concentration traverse. Although special pains were taken in making the humidity determinations, an extreme accuracy is necessary if really good values of the slopes are to be obtained, The rate of evaporation was quite constant and was measured fairly accurately, so that no serious error was believed to have been introduced in measuring N A . The

location of the concentration traverse (35 em. from the top, 22.2 em. from one end wall of the duct) was chosen rather arbitrarily as representative of the entire area of the upper or test section, so that even though the correction, as described previously, was made to allow for the difference between the mean p,* - p,*, for the whole test section and the value of p,* - p,* a t the traverse, a more or less constant small error may have been introduced. The greatest potential error was probably in the assumption of steady-state conditions. At the bottom of the calming section, where the air entered, the water vapor distribution was uniform across the duct, and it was assumed that the measured gradient a t section A (Figure 1) represented the final equilibrium condition. This seems reasonable, since the distance up to this point was 436 em., or over eighty times the width of the duct. Traverses at section B , near the bottom of the test section, gave slopes about 15 per cent flatter (and values of E ahout 15 per cent greater) than a t section A . It is believed that this was due to the disturbance of the stream by the exit slot and feed weir just below this point on the water wall, rather than to lack of attainment of a steady state, and that the traverses at section A very nearly represent equilibrium conditions. The turbulence determinations were made at the same positions as the concentration traverses a t section A . Corresponding data on velocity distribution (series DA) were obtained a t the same point, although evaporation rate data and concentration traverses were not obtained in these tests. Figure 4 represents a typical set of data; one curve in each

AUGUST, 1939

INDUSTRIAL AND ENGINEERING CHEMISTRY

case is for the duct operated with dry walls, the other is obtained under the usual test conditions with water and solution films covering the walls. The effect of the liquid films was negligible. Because these velocity traverses were not only made a t the same position in the duct as the usuaI concentration traverse, but were made with somewhat greater care than those velocity traverses obtained in the course of the usual tests, they may be taken 700 as best representing the flow condi” :: tions corresponding to concentra600 > tion traverses at 12 k the same average > gas velocities. IQ 3 The per cent tur5W bulence varied . n little with Reynolds number; i t 2 s was 3.1 per cent Ia t the center for $ 4 air a t Re = 79,900, B 2 and 4.4 per cent at Re = 11,200. Similar values 0 2 1 0 1 2 were obtninedwith USTU(CE FROM CENTER- LINE O f DUCT carbon dioxide and FIGURE 4. TYPICAL VELOCITY AND helium. TURBULENCE TRAVERSES Static pressure determinations were made a t section A and a t a point 173 cm. upstream. Friction factor f was calculated from the results, and the points for air are shown plotted against Re in Figure 5 . The best line through the points falls in approximately the normal position for turbulent flow.

.p

E

0.010 Q.006 0.006

f 0.004

I

I

I

I

I 1 1 1 1

11. ANALYSIS OF TYPICAL DATAON VELOCITY TRAVERSZ

TaBLE

aa - u, Cm. 2.25 2.00 1.50

133 86 49 30

1 .oo

0.50 0.25 0

d U/dy,Cm./(Sec.) (Cm.) Calod. from Equation 8

----Obsvd.--210 185

205 179

130 89

44 22

0

200 178 134 89 45 22 0

Calcd. from Equation 6 315 190 101 65 45 36 25

From the derivation of Equation 8 it follows that the eddy viscosity, E, must be a constant over the range in which the parabolic velocity deficiency relation holds. Values of c were obtained from logarithmic plots of (Urn - U ) against (a0 - y ) and the values of e calculated, using friction factors from the solid line of Figure 5. The results are shown in Figure 6 as E us. Re, where one line serves to represent the values for all three gases. 0.025

~

-Y

0.020

I

B

a ,0.015

2 + *

g

0.010

2 > z-

8

0.005

V

By plotting (Urn - U ) against (a0 - y) on logarithmic graph paper it was found that the velocity traverses could be well represented by Equation 8. This is illustrated by the comparison of slopes given in Table I1 where the values calculated from Equation 8 check the observed velocity gradients much better than those calculated by differentiating Equation 6. Since a comparison of derivatives is a severe

20 000 40 000 60000 100000 REYNOLDS NUMBER

slopes of the velocity traverse a t two points equidistant from the center line.

I

Comparison of Results with Theory

0.002 I I I 1 I 6000 IO 000

1039

I 200 000

FIGURE5. FRICTIONFACTOR^ CALCULATED FROM DATA ON PRESSURE DROP

test, it is evident that Equation 8 fits the data remarkably well. The velocity traverse given in Table I1 was one of series DA, obtained a t the same position in the duct as the concentration traverses. I n this test the duct walls were dry, but it is evident from Figure 4 that the presence of the water and solution films did not greatly affect the velocity traverse. The values of d U / d y in the “observed” column of Table I1 were obtained by careful measurement of the

0 REYNOLDS NUMBER

FIGURE6 . EDDYVISCOSITY us. REYNOLDS NUMBER (DATA OF SERIESDA ON VELOCITY TRAVERSES)

As mentioned previously, one suggestion is that e should be equal to the ratio E / p , or that Ep = e. Figure 7 shows the data of Figure 3 replotted as Ep us. Re, with a dotted line representing the data of Figure 6 on eddy viscosity. Although i t is apparent that Ep and e are not equal, they seem to bear a constant relation to each other, expressed approximately as E p = 1 . 6 ~ . Since u’ = l(dU/dy), then from Equations 1 and 7, neglecting ,u as compared with E,

whence E = 1 . 6 5 = 1.6Rv’l P

(16)

Comparison of Equation 16 with 11 and 12 shows that the data indicate a proportionality constant about midway between those predicted by the Prandtl momentum-transport theory and the Taylor vorticity-transport theory. The turbulence determinations provide a means of calculating R. If u’ and v’ are assumed equal [Wattendorf ( l a ) found them to be approximately equal for air flow in a similar channel], then

VOL. 31, NO. 8

INDUSTRIAL AND ENGINEERING CHEMISTRY

1040

The turbulence measurements give (u’)z as a function of y, and f is obtained from Figure 5. Figure 8 shows typical values of R, compared with certain data of Wattendorf and of Reichardt (7, l a ) . Similar curves were obtained for carbon dioxide and helium (series DC and DH). The values of R for carbon dioxide were somewhat 0.030 less than for air, but the trend with Re was similar. At 0.025 the highest value of Re (102,000), 0.020 however, R for carbon dioxide was EP low and constant 0.015 at 0.18 almost all the way across the duct. 0.010 As the introduction brought out, eddy diffusivity E 0.005 should be proportional to the prod0 uct v’l. It is of 0 20000 4( interest, therefore, REYNOLDS NUMBER to compare the exFIGURE7. COMPARISON OF DATAON perimental value EDDYDIFFUSIVITY AND EDDYVISCOSITY of E with v’l as calculated on the basis of the various theories. R will be taken as unity or grouped with the proportionality constant. The deviating velocity, v‘, may be obtained in various ways: It may be assumed equal to u’and obtained from the turbulence measurements, or it may be taken equal to u’and obtained from

values of u‘,1 is obtained from Equation 21, with d U / d y from a differentiation of the form of Equation 8 best fitting the velocity traverse. I n method B, v’ is taken equal to the exand 1 is obtained from Nikuradse’s perimental values of u’, curves. I n method C, v’ is obtained from Equation 18 and I from 21, with d l J / d y calculated from a differential form of Equation 6, with k = 0.38. The results of these calculations for the illustrative case are given in Table 111, where the values of v’l may be compared with the constant value of 16.8 for E found experimentally. By the first method 1 is infinite a t the center, where d U / d y is zero. ~~

TABLE111. THEORETICAL VALUESOF v’l

-

Method A 15.9 16.7 17.9 19.9 23.7

ao 21 2.00 1.50 1.00 0.50 0.25

0 0.25 0.50 1.00 1.50 2.00

Method B 13.0 16.8 17.0 14.6 12.3 12.2 13.9 15.3 17.3 17.6

2;. 8 20.9 18.2 17.5

..

..

Method C 10.3 14.5 14.9 11.0 6.7 0 6.7 11.0 14.9 14.5 10.3

By the third method v’l goes through a maximum on both sides of the center, where it is zero. The values of v‘l by this method are calculated by the relation:

The maximum value is reached when (1 - y/ao) has the value 4/9. If proper allowance were made for the drop in R in the central region, v’l over the main central portion of the duct might be expected to remain approximately constant a t its maximum value. This maximum value is found by substituting (1 - y/ao) = 4/9 and is: (25)

A third source of values of v’ is the relation,

in which u‘ is obtained from turbulence measurements. Similarly, mixing length 2 may be obtained in various ways:

I-

If the value 0.38 is substituted for k , this semitheoretical equation for E becomes: E = 0.08 Uoaodj (26) Values of E calculated from this equation are compared in Table I V with values of E read from the curves of Figure 3 for several values of Re chosen arbitrarily. The agreement is a h o s t within the experimental precision; this indicates that Equation 26 may be of general value in predicting values of E for the main “core” or central turbulent portion of a gas stream. It is noteworthy that this calculation requires only the average velocity, the friction factor, and the dimensions of the apparatus. TABLEIV.

It may also be obtained from Nikuradse’s data, which represent accepted values of 1 obtained from experimental data by Equation 21. I n Equations 20 to 23 the gradient d U / d y may be obtained in several ways-for example, by differentiating Equations 6 or 8, or from the slopes of the velocity traverses. A large number of possible methods of calculating v’l present themselves, and a number were tried out. One of these, the combination of Equations 18 and 21, with d U / d y obtained from 8, results in E = e / p and has already been discussed. Three other methods will be discussed briefly and the values of u‘l calculated from the velocity traverse obtained in run DA-5. I n this run Re = 69,000, for which E from Figure 3 is 16.8. I n method A, v’ is taken equal to the experimental

E,

Gas

Air Air Air Air Air Air

RE 11,100 25,800 40,200 56,200 69,000 79,900

VALUESOF E FROM EQUATION 26

Calcd. 3.1 6.3 9.4 12.6 15.3 17.5

E

Ex&.

(Fig.3) 4.2 7.7 10.7 14.1 16.8 19.0

Gas

Re

COz

38,300 79,000 108,000 5,300 11,100

COz

Con H~ ~e

E Calch. 5.1 9.5 12.5 10.1 20.6

E

E&l. (Fig. 3) 6.7 12.4 16.6 9.2 20.0

I n the chemical engineering literature the most significant article relating to the mechanism of heat and material transfer in turbulent flow is perhaps that of Murphree (6). He assumes that the eddy viscosity E is constant over the main central portion of the turbulent stream, and that the velocity distribution is accordingly given by Equation 8. He assumes that, in a “film” or region near the wall, E is propor-

INDUSTRIAL AND ENGINEERING CHEMISTRY

AUGUST, 1939

tional to the cube of the distance from the wall; it thus defines a velocity distribution in this region (by suitable integration of Equation 7). The “film” thickness, yf, is determined by locating the intersection of these two branches of the velocity distribution curve: For heat transfer Murphree assumes the eddy conductivity to be equal to the product of E and the specific heat of the fluid, which corresponds to E p = E for mass transfer. Thus, in the “film” Murphree would employ a diffusivity (D e / p ) where D is the molecular diffusivity and E is assumed proportional to y3.

+

-.-

05

flow. The product of eddy diffusivity and gas density is approximately 1.6 times the eddy viscosity, the latter being obtained from a velocity traverse. The eddy diffusivity may also be obtained from the flow rate, the friction data, and the duct dimensions, using the semitheoretical Equation 26. I n gas flow, under the conditions of the tests described, 28 to 57 per cent of the total resistance to diffusion is in the main turbulent core of the fluid stream, and the remainder is divided about equally between the two narrow regions adjoining the two walls of the duct.

Nomenclature

1

I

a0

c

C

0.4

D E f

0.3

g

IC I

R

NA p,* pw

0.2

p,*

Q

0 .I

0

R

RQ 0.2

0.4

0.6

CENTER I-

0.8

I.o

WALL

u

‘ FIGURE 8. CORRELATION OF DEVIATING VELOCITIES u’ AND vf

I n some ways this theory is borne out remarkably by the present data. Equation 8 is found to represent the velocity traverses; i. e., e is found to be constant over the main central portion of the duct. E is likewise constant in this region although Ep is approximately 1.6 times e and not equal to it. It is interesting to compare the theory with data of a particular run. Integration of Equation 7 on the assumption that e is proportional to y3 and equal to E from Figure 6 at y = yf leads to an expression for the velocity distribution in the film, from which the velocity a t y = yf may be calculated. This is equated to the velocity given by Equation 8, and y~ is obtained. For run A-19, at the low Reynolds number of 5700, this “film thickness” is calculated to be 1.18 cm. Values used for the purpose were: E = 0.00075, p = 0.00018, p = 0.00118, Urn = 75, UO= 63.5, and a. = 2.63. From this result it would be expected that the concentration would be linear in y over a region 2.63 - 1.18 = 1.45 cm. either side of the center. The curve shown in Figure 2 indicates this to be very nearly the case. The theory may be tested further by calculating A p through the “film” of 1.18 cm. This is done by integrating Equation 13, first substituting [ E ( y / ~ f ) ~ D] for the single constant E , taking E = 2.3 from Figure 7 and D as 0.25 sq. cm./second. The result is 7.6 mm. of mercury, which compares favorably with the value of 28.3 - 22.4 = 5.9 mm. read from Figure 2. Similarly encouraging results were not obtained when these calculations were repeated for runs at higher Reynolds numbers. Apparently the calculation of yf by the method described is quite sensitive and was found to be imaginary for run A-5 a t a Reynolds number of 25,800. It would appear that Murphree’s theory holds remarkably well a t low values of Re but breaks down a t the higher fluid velocities.

+

Conclusions The eddy diffusivity is found to be essentially constant over the main central portion of the gas stream in turbulent

1041

Re tl ta tl

t6

T

u

5’ u

uf

= distance from wall t o center of duct, em.

constant in Equation 8 solute concentration in gas stream, gram moles/cc. molecular diffusivity, sq. cm./sec. = eddy diffusivity, sq. cm./sec. = friction factor for turbulent flow, defined by Equation 3 = acceleration due to gravity, 981 cm./(sec.)z = a constant = mixing length, cm. = rate of transfer of solute gas, gram.moles/(sec.) (sq. cm.) = vapor pressure of water over solution of CaC12, mm. Hg = partial pressure of water vapor in air, mm. Hg = vapor pressure of water at temperature of water film, mm. Hg = liquid flow rate, cc./(sec.) (cm. film width) = correlation between deviating velocities u and u (Equation 2) = perfect-gas law constant, (cc.) (mm. Hg)/(gram mole) ( ” K.) = Reynolds number = [4ao(oUa uf)pI/p = inlet water temperature, C. = temperature of water leaving upper 0: test section, O C. = temperature of inlet CaClz solution, C. = temperature of outlet ‘+CL solution, C. = absolute temperature, K. = instantaneous deviating velocit,y in direction x, cm./sec. = root mean square value of u = cm./sec. = mean (with time) value of u, cm./sec. = downward surface velocity of liquid film on duct wall, cm./sec. = mean velocity in x-direction, cm./sec. = velocity at center of duct, cm./sec. = av. velocity across a traverse 22.2 cm. from one narrow end wall, cm./sec. = instantaneous deviating velocity in direction u, cm./sec. = root mean square value of v, cm./sec. = mean (with time) value of v, cm./sec. = instantaneous deviating velocity in direction z, cm./sec. = distance from wall of duct, em. = thickness of film as defined by Murphree, cm. = fluid density, grams/cc. = eddy viscosity, grams/(cm.) (see.) = fluid viscosity, grams/(cm.) (sec.) = viscosity of liquid layer flowing down duct wall, grams/ (cm.)(sec.) = shear stress on a lane parallel to direction of flow, grams/(cm.) (secJ2 = shear stress at wall of duct, grams/(cm.) (sec.)z = = =

+

O

40,

Literature Cited Bakhmeteff, B. A,, “Mechanics of Turbulent Flow,” Princeton Univ. Press, 1936. Dryden, H. L., IND.ENG.CREM.,31, 416 (1939). Durand, W. F., “Aerodynamic Theory,” Berlin, Julius Springer, 1935. Izakson, A., Tech. Phys. U.S. S. R., 4, No. 2, 155 (1937). Xarman, Th. yon, J. Aeronaut. Sci., 1, 1 (1934). Murphree, E. V., IND.ENQ.CHEM.,24, 727 (1932). Reichardt, H., 2. angew. Math. Mech., 13, 177 (1933). Rouse, H., Proc. Am. Soc. Civil Engrs., 62, 21 (1936). Taylor, G. I., Proc. Roy. Soc. (London), A159, 496 (1937). Towle, W. L.,Sherwood, T . K., and Seder, L. A., IND.ENQ. CHEV.,31, 457-63 (1939). Townend, H. G. H., Proc. Roy. SOC. (London), A145, 180 (1934). Wattendorf, F. L., J. Aeronaut. Sei., 3, 200 (1936).

PRESENTED before the meeting of the American Institute of Chemical Engineers, Akron, Ohio.