640
INDUSTRIAL AND ENGINEERING CHEMISTRY
eddy viscosity to kinematic viscosity close t o the wall derived from Rannie’s expression varies with the second power of the distance parameter, y+. Rannie’s expression has undoubtedly the advantage, but the mass transfer equation derived with the use of Rannie’s eddy distribution relationship deviates markedly from the experimental data in liquid streams where the Schmidt group is extremely high. Acknowledgment The authors are indebted to B. H. Sage and W. G. Schlinger for numerous suggestions and criticisms of the manuscript. References (1) Barnet, W. I., and Kobe, K. A., IND.ENG.CHEM., 33,436 (1941). ( 2 ) Boelter, L. M. K., Wfartinelli, R. C., and Jonassen, F., Trans. Am. Soc. Mech. E ? L Q ~6s3. , 447 (1941). (3) Chilton, T . H., and Colburn, A. P., ISD.ENG.CHEM.,26, 1183
(1934). (4) Colburn, A. P., Ibid., 22, 967 (1930). (5) Colburn, A. P., and Coghian, C. A., Trans. Am. Soc. X e c h . Engrs., 63, 561 (1941). ( 6 ) Dunn, L. G., Powell, TT. B., an’d Seifert, H. S., “Heat Transfer Studies Relating t o Rocket Po%-er-Plant Development,” Third Angl0-~4mericanAeronautical Conference, published by RoyaI Aeronautical Society, England, 1951. (7) Eagle, A., and Ferguson, R. M., Pioc. Roy. Soc. London, A127, 540 (1930).
Vol. 45, No. 3
Fage, A , , and Townend, H. C. H., Ibid., A135, 656 (1932). Goldstein, S.,“Modern Development in Fluid Mechanics,” London, Oxford University Press, 1938. Jackson, M. L., and Ceaglske, N. H., IND. ENG.CHEM., 42, 1188 (1950). Johnstone, H. F., and Pigford, R. L., Trans. A m . Inst. Chenz. Engrs., 38, 25 (1942). KLrmLn, T . yon, Trans. A m . S O CMech. . Engrs., 61, 705 (1935). Levich, B., Acta Physicochim. U.R.S.S., 17, 256 (1942). Lin, C. S.,Denton, E. B., Gaskill, H. L., and Putnam, G. L., IND. ENG.CHEM., 4 3 , 2136 (1951). Lin, C. S., Moulton, R. W., and Putnam, G. L., deposited with the American Documentation Institute, Washington 25, D. C., Doc. 32845 (1952). Linton, W. H., Jr., and Sherwood, T. K., Chem. Eng. Progr., 46, 258 (1950). Martinelli, R. C., Trans. Am. SOC.Mech. Engrs., 6 9 , 547 (1947). Xikuradse, J., VDI-Forschungsheft. H356 (1932). Prandtl, L., P h y s i k . Z . , 2 9 , 487 (1928). Reichardt, R., Natl. Advisory Comm. Aeronaut., Tech. Mem. 1047 (1943). Reynolds, O., Proc. Manchester Literary and Phzlosophical Soc , 8 (1874). Rothfus, R. R., Moniad, C. C., and $enecal, V. E.. ISD. ETG. CHEX, 42, 2511 (1950). SherTTood, T. K.. IND. ENC.CHEM.,4 2 , 2077 (1950). Sherwood, T. K., Trans. Am. I n s t . Chem. Engrs., 36, 817 (1940). Taylor, G. I., Great Britzin Advisory Comm. Aeronaut., R e p t . Mem. 2272 (1916-17). RECEIVED for review July 7 , 1051.
ACCEPTEDOctober 23, 1992.
(Mass Transfer between Solid Wall and Fluid Streams)
Interferometric Measurements of Concentration Profiles in Turbulent and Streamline HOW
A
LTHOUGH the rate of mass or heat transfer between solid walls and turbulent fluid streams has been the subject of considerable study, the fundamental mechanism of the transferring process in the vicinity of the wall is still inconclusive. Progress made in the development of the analogies between mass or heat and momentum transfer by employing concepts of laminar films existing adjacent t o the wall has not been proved because of lack of experimental evidence ( 7 ) ,and because of failure to explain the experimental results of mass transfer rates in turbulent liquid streams (10, 13) where the diffusivities of materials are very low in comparison with those in gaseous streams. Instead of using the sublaminar layer for pure molecular diffusion, the authors, in the preceding paper ( 1 1 ) , have developed a new eddy distribution relationship in the vicinity of the wall. The equation derived for calculating the mass transfer coefficients in turbulent fluid streams agrees well with the experimental results of mass transfer rates covering the whole range of Schmidt groups available in the literature. hIeasurements of mass and thermal eddy diffusion in turbulent fluid streams have been contributed by several investigators (14, 16), but in order to reveal the existence of the small eddies adjacent to the wall, accurate experimental concentration or temperature distribution data are very desirable. Moreover, the eddy diffusion of momentum near the wall is extremely low in comparison with the molecular diffusion of momentum, and i t seems impossible to obtain any quantitative information about these small eddies from the velocity profile measurements. Temperature or concentration distribution measurements encounter similar difficulties when the molecular diffusion of heat or material has the same order of magnitude as the molecular diffusion of momentum-i.e., the Prandtl or Schmidt group is near 1 or less than 1. It is believed, however, that measurements of concentration profiles in a liquid stream very close to the a a l l
will reveal the nature of these small eddies. Since in the liquid stream the molecular diffusion of material is very low in comparison with the molecular diffusion of momentum, the quantitative effect of these small eddies becomes appreciable and can be detected in the concentration distribution measurements. For streamline flow in a tube, the material is transferred by molecular diffusion with parabolic velocity distribution (10, 15) and hence the theoretical calculation of the concentration distribution is possible (5). For a flat plate, mass transfer coefficients and concentration profiles can be predicted with the use of boundary layer concepts (1). The purposes of this work were to measure quantitatively the concentration distributions near the surfaces of concentrationpolarized electrodes in turbulent fluid streams and t o verify the eddy distribution relationship introduced by the authors for the analogy between mass transfer and momentum transfer ( 1 1 ) Since the electrochemical reaction on the so-called concentrationpolarized electrode is diffusion-controlled, the rate of deposition or reactions of materials or ions depends on the rate of supply of materials or ions from the main body of the fluid to the surface by diffusion and convection (10). The concentration distributions in streamline flow were also measured, in order to demonstrate the laminar diffusion boundary layer on a flat surface. Experimental The concentration distribution near the surface of the electrode was measured by means of light interference. Local changes of concentration in the fluid, with resultant changes in the refractive index, cause the displacement of interference fringes. The measurement of the displacement of the local fringes gives an accurate quantitative value of local concentration a t any point in the fluid. The optical method is superior t o the ordinary eani-
March 1953
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
phng method, since i t eliminates the disturbance of flow patterns by sampling probes and, in addition, concentration distributions even a t distances of the order of 0.01 nun.from the electrode surface can be measured by magnifying the object either photographically or microscopically. A Mach-Zehnder type interferometer was chosen for the present work. A clean and smooth electrode surface, near which the concentration distribution is measured, was essential to the success of the experiment. Through preliminary experiments i t was found t h a t electrodeposition of cadmium metal from cadmium sulfate solution onto a thin layer of fresh mercury gave satisfactory results; t h e mercury layer waB supported on a smooth silver plate. COMPENSATING CHAMBER
64 1
axis parallel t o the picture by H . The screw, Vs,was supplementary coarse adjustment t o V . The adjustment was so delicate that t h e minimum displacement of each plate was several wave lengths. For changing the path length in the two beams of the instrument, the bases of plates 2 and 3 can be moved by the traversing screws, T,sliding through grooves. I n general, the instrument has various independent adjusting devices for focusing the interference fringes. Adjusting the Instrument. Several schemes have been suggested (5, 15, 18) for adjusting the instrument. Most of these methods are laborious and time-consuming. The following is a simple and reliable method as described by Winckler (18), with slight modifications.
1. The four plates should be adjusted approximately parallel t o one another, and the path length should be equalized ~ t 9close as possible on the optical bench with the aid of auxiliary oDtical apparatus. 2. Locate cross hair a just before plate 1(Figure 1). Another cross hair, b, is secured t o the front of t h e collimating lens, Ll. With the use of a telescope placed a certain distance away from plate 4, the images of both cross hairs can be found. If the interferometer is out of adjustment, each cross hair appears double because of the action of the beam splitters. Plate 4 is tilted while observing a until two images coincide, and the procedure is duplicated using plate 1 and b. a is readjusted, and so on until both a and b appear single, simultaneously. Since the two plates are nearly independent for this adjustment, the process converges very rapidly, and fringes should soon be seen. 3. The paths of the two beams are then equalized by turning the traversing screws with filter removed t o cover half of the field. When the path equality is a p roached, colored fringes of white light should be seen. When tge path length is equalized, the contrast of fringes can be increased t o maximum. 4. The fringes can be localized in any desired plane either in the front of plate 4 or behind it by turning the adjustment screws on plates 1 and 4. The telescope is focused on a cross hair placed in this plane, and the interferometer is then adjusted t o the maximum visibility of fringes at the same plane. I
Figure 1.
Diagrammatic Sketch of Interferometer and Associated Optical Apparatus
Interferometer and Associated Optical Apparatus. The theory of formation of interference fringes in the Mach-Zehnder interferometer has been described in the literature (9,6, 16). The instrument has been applied t o gas dynamics (18) and t o heat convection in air ( 4 ) . A small size instrument of this type was built for the investigations described in this paper.
_
Figure 1 is a diagrammatic sketch of the interferometer and associated optical apparatus. The light from source H, passing through a 5-mm. aperture, S, is made parallel by lens h. The parallel beam is then split into two beams by beam splitter plate 1; the transmitted beam, A , proceeds t o plane mirror 2 and the reflective beam, B, t o mirror 3. The two beams are recombined by plate 4, which is identical with plate 1. (Glass plates and plane mirrors were purchased from Gaertner Scientific Go.) These beams are coherent and produce interference fringes which are focused on the photographic screen, P, by t h e camera lens, Lz. Plates 1 and 4 were rectangular, 15 X 25 mm. Their flatness and parallelism were corrected t o 1/10 wave length. Aluminum coatings on t h e plates providing for half transmission and half reflection were made by evaporating aluminum metal in vacuum. Plane mirrors were circular, 19 mm. in diameter, with their flatness corrected t o wave length. The light source was a medium pressure, 400 watt, Hanovia DH-1 type mercury arc lamp with the protecting glass envelope removed t o prevent the spreading of the mercury spectrum by excessive heating. Both a Bausch-Lomb 5461 A. line interference filter and a Kodak Wratten filter No. 77A were used t o provide monochromatic light for producing contrast fringes. r*
Interference filters gave anarrower band spectrum than the Wratten filter. However, since the lamp itself does not produce a widely spread spectrum, a single Wratten filter gave light sufficiently monochromatic to provide contrast of the fringes, and also gave a higher light intensity for photography than did the interference filter. The photograph in Figure 2 shows the interferometer and its mountings. Plates 1 and 3 and plates 2 and 4 were held, re spectively, perpendicular to each of the vertical aluminum supports, the latter mounted on separated riders on a lathe bed-type optical bench. The base of each plate and each aluminum support could be tilted by adjusting screws. Each plate was held on a rectangular frame, Ft, which was connected t o another rectangular frame, Fz, by means of a thin piece of bronze metal having springlike flexibility. The frame, F1, could be rotated by the lever mechanism, V , about the axis perpendicular t o t h e picture. Frame Fz was connected to the base through another piece of flexible bronze metal and could be rotated about the
Figure
2. Interferometer and Mountings
Electrolytic Cell and Flow System. The electrolytic cell was rectangular; inside dimensions were 0.75 inch high, 1.25 inches wide, and 5.5 inches long, as shown in Figure 3. The cathode a t the bottom, fastened by screws against a stainless steel plate, was made of '/Finch thick silver plate coated with a thin layer of fresh mercury. The mercury was introduced and removed through a capillary hole drilled through the silver plate. A glass capillary probe was located downstream near the corner of the silver plate t o measure the potential of the cathode, near the surface of which the concentration gradients were measured.
INDUSTRIAL AND ENGINEERING CHEMISTRY
642
T h e anode a t the top was made of '/&nch thick cadmium late fastened by screws against the other stainless steel plate, insurated with rubber gaskets and bakelite. The inner side of the upper stainless steel plate, supporting the anode, was painted with Glyptal insulation t o prevent the leakage of current, since the cathode was grounded t o the exterior metal parts of the cell. BRASS FLANGE
BAKELITE INSULATION
411
\CADMIUM
I
ANODE)
'PARALLEL GL.ASS WINDOW
Figure 3.
Cell Construction
The windows were of good quality optical glass, uniform in thickness and flatness, thereby avoiding distortion of the interference fringes. Selected front aluminized mirror glasses purchased from the Edmund Scientific Go., the aluminum removed by dilute nitric acid, were found satisfactory for this purpose. The glasses were carefully cut to proper size and mounted parallel t o each other in accurately machined grooves on the stainless steel plates, with rubber gasket and putty as the sealing mediums. The cell was placed in path B of the interferometer, as shown in Figure 1. In order t o equalize the beam path of the instrument, a compensating chamber made of methyl methacrylate plastic, of identical nidth and with glass windows similar t o those of the cell, was placed in path A . Two rectangular ducts, one 4 feet long and the other ll/z feet long, were prepared of methyl methacrylate plastic. The ducts had the same inside dimensions as the cell and were connected smoothly t o the cell, upstream and downstream, respectively, to make the calming sections (Figure 4). A pair of electrodes 3 inches long were built in the upstream calming duct immediately in front of the cell, and an additional pair of electrodes 6 inches long placed in front of the 3-inch electrodes. The two pairs of electrodes were of the same thickness, width, and material as those of the optical cell, and were fastened tightly by means of screws into grooves cut into the methyl methacrylate plastic and fitted continuously and smoothly with the inner surface of the duct. Capillary holes %ere provided for introducing or removing the mercury on the silver suri'ace. Each pair of electrodes, including the pair in the optical cell, was insulated and separated by a distance of '/*inch from each other. Two vertical square ducts of methyl methacrylate plastic, 4 X 4 inches in cross section, were connected to the inlet and outlet of the horizontal rectangular duct in order t o provide a constant head for uniform flow. The solution was circulated by means of a stainless steel pump through a 15-gallon glass storage tank, in which a thermoregulator was provided to keep the temperature constant within 0.02" C. I n order to prevent impurities from contaminating the circulating electrolyte, stainless steel valves and pipes and rubber tubing were used throughout the system (Figure 4). The flow rates were measured with calibrated sharp edge stainless steel orifices.
Vol. 45, No. 3
The cell, the compensating chamber, and the rectangular duct were provided with clamps and adjusting screws, and were carefully leveled by the use of a carpenter's level. The whole apparatus was supported by rigid steel frames which were fastened to the concrete floor t o avoid any mechanical vibration, The parallel light beams in the two paths of the interferometer were adjusted perpendicular, respectively, t o the windows of the cell and t o the windows of the compensating chamber at a position 3 inches from the upstream end of the cell (see Figure 4) with the aid of the optical bench and auxiliary mirrors. Procedure. I n measuring the concentration gradients, the interferometer was first adjusted with the aid of a telescope, according to the methods described, until fringes of proper width, perpendicular t o the surface of the electrode, appeared. The fringes were localized in a vertical plane just beyond the glass window a t the side of plate 4 of the interferometer. (The concentration distribution measured is the average distribution over the entire width of the liquid path, independent of the position a t which the fringes are localized.) The telescope was then replaced by a camera with a Tessar lens 3 inches in focal length. The camera was adjusted so that the fringes were magnified to approximately 10 diameters. Kodak ortho-contrast process films with 3 to 5 seconds' exposure were found satisfactory for this use. The magnification factor was determined by measuring the length of the photographic negative image of a small thin wire placed where the fringes were localized, while the actual length of the object was measured by a microscope micrometer. The negatives were again magnified on photographic paper, and the same procedure was then used to determine the magnification of the negatives. The oyer-all magnification was approximately 100 diameters.
I1
COMPENSATING CHAMBER
iiL
h
1
ORIFICE M E T E R
1 I !'\
RESERVOIR
PUMP
Figure 4.
Diagram of Flow System
The concentration of cadmium sulfate solution used was 0.01 M ; C.P. cadmium sulfate and distilled water were used t o make the solution. The temperature of the solution was kept at 23.8" =k 0.05" C. (less than 0.5" C. higher than room temperature). Several photographs were taken when the solution was circulated through the cell and the compensating chamber without electrolysis. These photographs indicated that a temperature difference of less than 0.5" C. between the solution and the surrounding air in the room did not distort the fringes to any observable extent. I n each test, photographs were taken '/z to 1 minute after the electric circuit was closed, in order to permit equilibrium conditions to be established on the electrode surface. At low flow rates, pictures were usually taken only after several minutes of electrodeposition. a f t e r each run, the mercury surface was washed with dilute nitric acid, followed by thorough washing with water. The mercury surface was always kept clean and
March 1953
Figure 5.
INDUSTRIAL AND ENGINEERING CHEMISTRY
Interference Fringe Displacement at Re = 7200 Test69
T a b l e 1.
643
passing through two identical liquid paths, one of which ie adjacent to the electrode surface. If the concentration were uniform throughout, the center line of fringe A instead of the center line of fringe B would pass through point P. The shift of one fringe at point P indicates that there is one less wave light in the path through P than there would b e i f the concentration at P were the same as at &. The number of fringe shift from one path t o the other is proportional t o the difference of refractive indices between two paths. I n the dilute solution, a linear relationship between concentration and refractive index map be assumed ( l 7 ) , and consequently the fringe shift from one path to the other will be proportional to the concentration difference between two paths. The following equation was derived for calculating the concentration distribution from interference fringe displacements ( l a ) :
'
Conditions for Concentration P r o f i l e Measurements in Turbulent Flow Region
Temp. = 23.8' C. p = 0.00918 poise D = 1.02 X 10-5 sq. cm./sec. = 900 P U d = 2 . 3 8 cm. Length of velocity approaching sect. = 127.3 om. 1 5 . 2 om. Length of concn. approaching sect. Test Teat Test Test Test Test 67 55 69 56 87 62 U ,om./sec. 15.1 21.0 27.1 31.6 36.6 46.7 dUP 4000 5560 7200 8400 9700 12400 &rent densitya. amp./sq. cm. X 108 0 52 0 78 1 04 1 02 1 06 1 40 C, , g. moles/cc. 0 01 0 01 0 01 0 01 0 01 0 01 A Wmax 3 23 2 94 3 24 2 93 2 65 2 94 iw, om. X 102 4 14 3 18 2 58 2 29 2 10 1 63 Current densities on the cathode of the optical cell were measured. The circuit of the 3-in. approaching electrodes were closed; the currents on the 3-in. electrode surface were not measured.
r
-
-
In using Equation 1, it should be noticed that the glass windows and the liquid paths of the cell and the compensating chamber are identical. The air path of the two light beams are compensated by each other. The validity of this equation under present experimental condition has been demonstrated ( 1 2 ) . I n evaluating the concentration profiles from the interference fringe displacements, parallel lines passing through the center of each fringe were drawn, as illustrated in Figures 5 and 6. The maximum displacement of each fringe at the surface of the electrode was located. Since the fringes are equally spaced, a measurement of the ratio of the distance displaced a t any point to the maximum distance displaced will give directly the concentration distribution.
(1
fresh. It was found necessary t o adjust the thickness of mercury layer on the silver plate so that the mercury covered the whole silver surface evenly, yet was not so thick t h a t ripples were induced b y the flow of solution. Concenfration Profile Measurements in Turbulent Flow Region Photographs of the interference fringe displacements near the electrode surfaces a t various flow rates during electrolysis were taken. A sample of the photographs at a flow rate corresponding t o a Reynolds number equal t o 7200 is shown in Figure 5. Interference fringe displacement pictures at other flow rates are available (9). I n each test the circuit of the 3-inch electrodes (in parallel with those in the cell) was closed, and the concentration gradients were measured at a distance of 6 diameters (equivalent diameter) of the duct downstream from the position where the concentration gradients started to build. As indicated by mass transfer coefficient measurements on the electrode surfaces in the same apparatus, the mass transfer rates reached the steady state condition a t the position where the concentration gradients were measured (12). Linton and Sherwood ( l a ) also reported that in measuring the solution rates of cast tubes of solid materials the entrance effect becomes negligible when the ratio of the tube length t o the tube diameter is larger than 6, and therefore the concentration gradients at a distance of 6 diameters of the pipe, downstream, will be uniform. The concentration distributions a t any point near the electrode surface were evaluated from the photographs of the interference fringe displacements. As illustrated in Figure 6, A and B are two adjacent interference fringes formed by monochromatic light
Figure 6.
Interference Fringe Displacement at Re = 2000 Tort 71
The concentration distributions evaluated from each photograph are presented in Table 11. Table I shows the experimental conditions for each test, The flow rates vary from R e = 4000 to 12,000. The equivalent diameter (d =
di + di'
where
dl = height of the cell and dz = width of the cell) was used to calculate the Reynolds numbers. The concentration profile for test 87 is plotted in Figure 7 for illustration. The solid line represents the experimental result calculated from the interference fringe displacement picture. To verify the concentration distribution expression derived on the basis of eddy relationship near the wall (presented in the preceding paper) with the use of the experimental concentration profile data, i t is necessary t o know the diffusivity of cadmium ion during electrolysis. I n electrolyzing a pure electrolyte, the ions discharged on the cathode, in addition to diffusion and convection, are supplied by ionic migration from the bulk of the solution. Levich (8) has shown t h a t for the discharge of cations from a pure electrolyte, the ionic migration term
INDUSTRIAL AND ENGINEERING CHEMISTRY
644
Vol. 45, No. 3
can be included in the diffusion term. The over-all rate coefficient is called effective diffusivity, which may be expressed as follows:
X
The authors have made a brief derivation of the above equation ( l a ) . With the diffusivity values of cadmium ion, D1 = 0.72 X 1 0 - 6 square centimeters per second, and sulfate ion, D z = 1.08 X 10-6 square centimeters per second at 25" C., calculated from the conductivities of ions a t infinite dilution (6),the value of effective diffusivity computed from Equation 2 should be 0.863 X 10-5 square centimeters per second.
EXPERJMENTA L
RE 4000 5650
0
7200
A
8400
9100 12400
X
THEORETICAL
-
-
EXPERIMENTAL
- - -CALCULATED
d, = 0.210m m 0.0 0
O.2
I
2
I 3
4
I
5
I7
6
I
8
Yt Figure 8.
0.0 0.0
I
0.2
0.1
0.3
Generalized Concentration Distribution in Turbulent Flow Region
MM
Figure 7.
Calculated and Experimental Concentration Profile at Re = 97,200
able when the diffusivity is very low or the Schmidt group is very high.
Test 87
laminar Diffusion Boundary Layer in Streamline Flow Region The effective diffusivity was also evaluated from mass transfer coefficient data obtained from limiting current density measurements for cadmium ion deposition from a pure cadmium sulfate solution ( l a ) . The limiting current densities were measured in the same apparatus. The temperature of the solution was kept a t 23.8" & 0.05" C., while the concentration of the cadmium sulfate solution was 0.0125 M . It has been found t h a t the mass transfer data can be best correlated with the authors' mass transfer equation (11) in the turbulent flow region and with Piusselt's equation in the streamline region for a rectangular duct ( 3 ) when the effective diffusivity is 1.02 X square centimeters per second, corresponding to the Schmidt group equal t o 900. Since the diffusion of ions and salts is rather complicated, deviations from nonideality for the solution used would be expected. The experimental value presented in this paper is used for calculating theoretical concentration profiles with the authors' concentration distribution expression (11). P
The calculated concentration profile a t - = 900, test 87, is PD depicted in Figure 7 by the dotted line. Since the concentration gradient is not extended appreciably to the main body of the fluid at high Schmidt group, CaV. can be replaced by C,,. Experimental data from Table I1 are plotted in the generalized coordinates in Figure 8. The solid line represents the theoretical generalized concentration distribution. The experimental results agree well with the concentration distribution relationship derived on the basis of the assumed eddy values existing a t the vicinity of the wall (11). The results prove definitely t h a t the laminar type of film for molecular diffusion does not exist near the wall. These small amounts of eddies become very appreci-
Concentration profiles in the streamline flow region a t Re = 2000 with various electrode lengths, evaluated from the photographs of interference fringe displacements, are given in Table IV. Table I11 shows the experimental conditione for each test. Figure 6 shows a sample of the photographs. Other photographs are available ( 9 ) . The theoretical calculation of unsteady d a t e
d
Figure 9 , Flow Boundary and Diffusion Boundary Layers on a Flat Plate
diffusion with parabolic velocity distribution in a rectangular duct is difficult. The electrode in the rectangular duct may be considered as a flat plate along which the fluid flows. As shown in Figure 9, the hydrodynamic boundary layer begins at the entrance of the rectangular duct ( 1 ) . At the end of 50,the diffusion boundary layer starts t o build at the edge of the electrode surface. Both increase their thickness, 6 and a,, in the direction of flow. The thickness of the hydrodynamic boundary layer is (1):
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953 Table II. Teat 67
3
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
om. 102 0.41 0.83 1.24 1.66 2.07 2.48 2.90 3.32 3.72 4.14
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
0.23 0.46 0.69 0.92 1.15 1.37 1.60 1.83 2.06 2.29
y+
0.5 1 .o
The calculated concentration profiles from Equations 4 and 5 are shown in the second colTest 69 c c. umn of Table IV. Figure 10 Y Y em. 2 102 2 loa - C G 7 Z C 8 is a sample of the concentra0.26 0.28 tion distribution curve in the 0.52 0.52 laminar diffusion boundary 0.66 0.77 0.78 1.03 layer. T h e solid line repre1.29 0.85 1.55 0.90 sents the experimental value, 1.81 0.94 while the dotted line is cal2.06 0.96 2.32 0.97 culated from Equations 4 and 0.98 2.58 5. The boundary layer thickTest 62 ness increases with the elec0.16 0.30 trode length and is demon0.53 0.33 0.49 0.64 strated clearly by the present 0.73 0.65 experiment. Since the diffu0.82 0.80 0.98 0.84 sion boundary layer is thin, 1.14 0.89 1.30 0.92 the effect of the restriction 1.47 0.94 at the edges of the electrode 1.63 0.96 by vertical walls may be small. I n general, no serious errors have been made in the assumptions for the calculation of concentration profiles in the streamline region, as shown by comparing the calculated values with experimental results.
Concentration Profiles in Turbulent Flow Region Test 55
c - c.
f x lo2- c,,. 0.174 0.348 0.522 0.696 0.870 1.04
1.22 1.39 1.57 1.74 Test 56
-
C, 0.35 0.58 0.73 0.82 0.88 0.92 0.94 0.96 0.97 0.98
em.
10s 0.32 0.64 0.95 1.27
%'
- c;
c
loa emu,
- C.
0.33 0.59 0.72 0.83 0.87 0.91 0.92 0.94 0.95 0.96
1.58 1.91 2.22 2.54 2.86 3.18
Test 87
0.0965 0.193 0.299 0.386 0.481 0.578 0.674 0.770 0.866 0.962
0.32 0.56 0.71 0.79 0.86 0.91 0.93 0.95 0.97 0.98
Table 111. Conditions for Concentration Profile Measurements in Laminar Diffusion Boundary Layers in Streamline Flow Region
-
Temp. 23.8' C. M = 0.00918poise D = 1.02 X 1 0 - 6 sq. cm./sec. = 900 PD d = 2.38cm. d U p = 2000
Acknowledgment
U = 7.52om./sec. Test
Test Test 71 83 84 Av. current density", 0.44 amp./sq. em. X 108 0.38 0.32 0.01 0.01 Cma=.,g. moles/cc. 0.01 3.23 3.17 3.11 WUW. 7.6 L,om. 15.2 30.4 127.3 127.3 127.3 2 , om. 119.7 112.1 96.9 20, om. 3.51 &, om. X 102 4.39 5.07 a Average current density on the total electrode surfaces, including the approaching electrode surfaces.
-6 =
a
645
I
The authors are indebted to K. C. Clark for valuable suggestions in preparing the optical apparatus and to B. H. Sage, W. G. Schlinger, and W. D. Rannie for numerous suggestions and criticisms of the manuscript. They appreciate the assistance of J. L. Sundling and W. H. Antonius, whose skill and ingenuity provided various essential parts of the equipment.
I
I
I
I
I
I
I
I
-I
I
EXPERIMENTAL
4.64
4 7
_ - - CALCULATED
(3)
bc a 0.507rnm x = 50.1 inch*, X. = 41.1 inchar
with u8 equal to the velocity a t the center of the duct and x I I I I I I I .I I I 0.0 equal to the distance from the entrance of the duct to the point oa 0.1 0.2 0.3 0.4 0.5 MM where the concentration gradients were measured. 6 computed from Equation 3 is 1.27 om., corresponding to the flow rate Re = Figure I O . Calculated and Experimental Concentration 2000. Actually, the maximum boundary layer is extended to the Distribution in Laminar Diffusion Boundary Layer center of the duct and is equal to 0.952 cm., the distance from the electrode surface to the center of the duct. This value is used Test a4 at present for the calculation of diffusion boundary layer. The diffusion boundary - layer thickness, b,, can be computed Table IV. Concentration Profiles in Laminar Diffusion Boundary Layers in Streamline Flow Region from:
3= 6
(-E)
-
(yyl/a (4)
and the concentration profile in the diffusion boundary layer is expressed as ( 1 ) :
c
- c,
crux.
- c,
2
(Calcd. from Eq. 5)
0.1 0.2 0.3
0.15 0.30
6.
Test 71
Test 83
c c . Y, cm.*. - c. om. X 101 (exptl.) om. X 102 ?I!
0.35 0.70 1.05 1.41 1.75 2.11 2.46 2.81 3.16 3.51
0.44 0.88 1.32
- c. Cmax. - c, (exptl.)
Test 84
c
0.19 0.37 0.53
tl,
om. X 102
cm,,.
- c, - c, (exptl.) c
INDUSTRIAL AND ENGINEERING CHEMISTRY
646
Nomenclature
= fluid density, grams/cc.
p
= viscosity, grams/(cm.)(sec.) = kinematic viscosity, s cm./sec. T = shearing stress, dynesjsq. em. T O = shearing stress at the wall, dynes/sq. cm. n- = 3 . 1 4 1 6 c p = Reynolds number, dimensionless
p
C C’
= mean concentration a t any position, gram moles/cc. = fluctuation concentration, gram moles/cc.
= concentration at the surface, gram moles/cc. CaV.= average concentration of the main fluid, gram moles/
cc. C,,,. = concentration at center of the conduit, gram moles/cc. C , = specific heat, gram cal./(gm.)( C). D = d s u s i v i t y of solute or effective diffusivity of electrolyte, sq. cm./sec. D 1 = diffusivity of ion species 1,sq. cm./sec. Dz = diffusivity of ion species 2, sq. cm./sec. d = diameter or equivalent diameter of the conduit, em. E = mass eddy diffusivity, sq. cm./sec. EH = heat eddy diffusivity, sq. cm./sec. f = friction factor, dimensionless G = mass velocity, grams/(sq. cm.)(sec.) h = heat transfer coefficient, gram cal./(sq. cm.)(sec.) ( ” C.) k = thermal conductivity, gram cal.,/(cm.)(sec.) k , = mass transfer coefficient, cm./sec. 1 = mixing length, em. AT = mass transfer rate, gram moles/(sq. cm.)(sec.) n1 = valence of ion species 1 n2 = valence of ion species 2 U = average velocity of the main fluid, cm./sec. u = mean velocity a t any position, cm./sec. u’ = fluctuation velocity parallel t o the direction of the flowing fluid, cm. /see. U. = velocity of fluid beyond the floiv boundary layer, cm./sec. u* = friction velocity, cm./sec. u u+ = -, dimensionless u* v‘ = fluctuation velocity normal to the direction of main flowing fluid, cm./sec. v’ = root mean square value of L ‘, cm./sec. TV = number of light waves AJV = number of light waves displaced a t any position ~ b r= ~number ~ ~ of. light waves displaced a t the surface z = distance downstream from the leading edge of plate, em. zo = length of plate preceding the leading edge of electrode, cm. 21 = distance from the surface, em. O
+ ;
=
6 6, -da
= = = = = =
6.’ 6, E
(:)
.
Y
C.
(y) d;,
dimensionless flow boundary layer thickness, cm. diffusion boundarv laver thickness, cni. sublaminar layer thickness, cm. diffusion sublayer thickness, cni. wall layer thickness, cm. eddy viscosity, sq. cm./sec.
Vol. 45, No. 3
P
-k = Schmidt group, dimensionless PD = Prandtl group, dimensionless
a IC
Literature Cited (1) Eckert. E., “Introduction to the Transfer of Heat and Mass,”
New York, hlcGraw-Hill Book Co., 1950. Hansen, G., 2.tech. Physik, 1 2 , 4 3 6 (1931). Jakob, &I., “Heat Transfer,” 5’01. 1, pp. 45-64, Sew York, John Wiley & Sons, 1949. (4) Kennard. R. B., J. Research .VatZ. Bur. Standards, 8, 787-805 (2) (3)
(1932).
( 5 ) Kinder, TV., Optik, 1 , 413 (1946). ( 6 ) Kolthoff, I. M., and Lingane, J. J., “Polarography,” New York,
lnterscience Publishers. 1946.
(7) RlcAdams, \I7.H., Chem. Eng. Progr., 4 6 , 121 (1950). (8) Levich, B., Acta Physicochim. U.R.S.S., 17, 256 (1942). (9) Lin, C . S., Ph.D. thesis in chemical engineering, University of (10)
Washington. 1952. Lin, C. S., Denton, E. E., Gaskill, H. 8.. and Putnam, G. L., IND.E m . CHEM.,4 3 , 2 1 3 6 (1951).
( 1 1 ) Lin, C. S., hloulton, R. TT‘., and Putnam, G. L., Ibid., 45, 636 (1953).
Lin, C. S., Moulton, R. W.,and Putnam, G. L., deposited with the American Documentation Institute, Xashington 25, D. C , Doc. 3845 (1952). (13) Linton, W . H., Jr., and Sherwood, T. K., Chem. E n g , Progr., (12)
46, 258 (1950).
Page, F., Jr., Corcoran, TV. H., Schlinger, TT‘. G., and Sage, B. H., IND.ENG.CHEX.,4 4 , 4 1 0 , 419 (1952). (15) Schardin, H., 2. Instrumentenk., 53; 3 9 6 , 4 2 4 (1933). (16) Sherwood, T. K., and Woetz, B. B., Trans. A m . Inst. Chem. (14)
Engrs., 35, 517-40 (1939).
Weissberg, A., et al., “Physical Methods of Organic Chemistry,” 2nd ed., Vol. 1, Part 1, pp. 1150, New York, Interscience Publishers, 1949. (18) Winckler, J., Rev.Sci. Instr., 1 9 , 3 0 7 - 2 2 (1948). (17)
RECEIVED for review December 4, 1951. ACCEPTEDOctober 23, 1952. For material supplementary t o this article order Document 3845 from American Documentation Institute, c/o Library of Congress, Washington 25, D. C . , remitting $1.75 for microfilm (images 1 inch high on standard 35-mm. motion picture film) or $2.50 for photostats readable without optical aid.
0 0 0
0 0
By Fusion with agnesium and Potassium Sulfates G. 1. BRIDGER
AND
D. R. BOYLAN
Deporfmenf of Chemical and Mining Engineering, lowu State College, Ames, lawa
OST of the phosphorus in phosphate rock is present aa fluorapatite, CaloFz(P04)a,a compound so stable that the phosphorus is not readily available as a murce of plant food. To make the phosphorus available, it is necessary t o destroy the fluorapatite structure of the phosphate rock and form compounds that are soluble in soil solutions. This may be accomplished by acid treatment or by thermal treatment.
Treatment of phosphate rock with acids such as sulfuric or phosphoric acid is widely used for production of normal or triple superphosphates. However, in view of the current shortage of sulfur, used for production of both these acids, interest in processes that do not require sulfur has been accentuated. Such a process is the electric furnace fusion of phosphate rock with magnesium silicates-for example, olivine and serpentine.
