The Electrochemical Method in Transport Phenomena

The Electrochemical Method in Transport Phenomena

.

T MIZUSHINA Department of Chemical Engineering. Kyoto University. Kyoto. Japan

I. Introduction . . . . . . . . . . . . . . . . . . . . . . I1. Electrochemical Reaction under the Diffusion-Controlling Condition

. . . . . . . . . . . . . . . . . . . . . . .

I11. Application to Mass Transfer Measurements . . . . . . . . A . Principle and Method of Measurements . . . . . . . . .

B. Free Convection . . . . . . . . . . . . . . . . . . . C . Forced Convection . . . . . . . . . . . . . . . . . . IV . Application to Shear Stress Measurements . . . . . . . . . A Principle and Method of Measurements . . . . . . . . . B. Shear Stress in a Well-Developed Flow . . . . . . . . . C . Shear Stress in the Boundary Layer . . . . . . . . . . . V Application to Fluid Velocity Measurements . . . . . . . . . A . Principle and Method of Measurements . . . . . . . . . B Time-Smoothed Velocity . . . . . . . . . . . . . . . C . Fluctuating Velocity . . . . . . . . . . . . . . . . . Symbols . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

.

.

.

87 89 94 94 98 103 136 136 140 142 144 144 147 153 159 160

.

I Introduction

The electrochemical method cannot be applied directly to heat transfer measurements . Nevertheless. many investigators of heat transfer have adopted the electrochemical method since the knowledge of the mass transfer obtained by this method can be correlated with heat transfer by the analogy between both transport phenomena . Especially. the chemical engineers who are mostly interested in operations involving liquids make use more frequently of the method of electrochemical reaction under the diffusion-controlling condition 87

88

T.MIZUSHINA

because this method enables them to measure not only the average value but also the local value of the mass transfer rate and shearing stress at a liquid-solid interface. From the limiting current density at the cathode, the transfer rate of mass or momentum can be calculated. Since the response of the cathode is very fast, we can obtain the fluctuation value as well as the time-smoothed value; further, a hot wire-type device to measure the velocity of liquid can be made from such cathodes of thin wire. Though the diffusion-controlling electrochemical reaction is a very strong weapon to attack transport phenomena in liquids, it should be noted that there are several limits in using this method. First, it is limited to liquids; accordingly, the data of mass transfer are limited for high Schmidt numbers. Second, only certain kinds of liquid mixtures can be used, i.e., those in which a diffusion-controlling electrolytic reaction occurs. Furthermore, we cannot use this method for a velocity larger than a critical flow rate at which the reaction resistance at the cathode becomes relatively significant compared with decreasing resistance of diffusion. For practical purposes, it is better to choose the condition in which the limiting current is reached before the potential becomes larger than the hydrogen overvoltage. Otherwise, the discharge current of hydrogen ions is added and the limiting current cannot be found. First, the electrochemists took up the problem of concentration polarization in analyzing the reaction in polarography (I)and electrolytic cells (2). Not very many years ago, the diffusion-controlled electrode reaction also began to be applied to the measurements of transport phenomena. Lin et al. (3) made a systematic study of the transfer rates of ions and other reacting species in electrochemical reactions in several kinds of mixtures and measured the mass transfer coefficients in laminar and turbulent flows by this method. Ranz (4) discussed the problems in applying this method to the measurement of the velocity of liquids and suggested that a long development and considerable study of chemical mechanisms are necessary in order to make practical instruments. Mitchell and Hanratty ( 5 ) developed the shear-stress meter and used this for a study of turbulence at a wall. In the following chapter, the method of the diffusion-controlling electrochemical reaction and its applications in the study of transport phenomena will be discussed. At the end of introduction, however, it seems necessary to mention the electroconductivity method, one of the electrochemical methods other than those described above. Nukiyama et al. (5a, b) developed the electroconductivity method to simulate complex problems in thermal con-

THEELECTROCHEMICAL METHOD

89

ductivity. Lamb et al. (5c) made an electrical conductivity probe to measure liquid-phase concentration fluctuations, and Manning and Wilhelm (5d) applied this method to measurement of turbulent characteristics in an agitated vessel.

II. Electrochemical Reaction under the Diffusion-Controlling Condition In electrode reactions, there exist concentration polarization and chemical polarization in series, i.e., the ions move from the bulk of the solution to the surface of the electrode where chemical and physical changes occur. I n measuring the rates of mass transfer by the use of electrochemical reactions, it is better and more usual to make the chemical polarization negligible because the mass transfer coefficients are most easily obtained from the limiting current when the concentration at the liquid-solid interface can be assumed to be zero. The ions are transferred from the bulk of the solution to the surface of the electrode principally by (a) migration due to the potential field, (b) diffusion due to the concentration gradient, and (c) convection by the flow. Assuming that the transfer is steady and unidirectional in the y-direction perpendicular to the surface of the electrode, the rate of transfer of a reacting species is expressed as NA

=

t9

+

€Y')

+

cAtntdRT)(ay/Ylay)- (9

cD>taCA/@)

+

OCA

tl)

and the current density at the electrode is expressed as i/An,F

=N A

(2)

The three terms on the right of Eq. (1) represent the contributions of migration, diffusion, and convection, respectively. The last term for convection vanishes in the redox processes because there is no net bulk flow in the y-direction. But it does not vanish in the processes of metal depositing on the electrode because there is a net bulk flow. However, its effect is very small at ordinary conditions and usually negligible. For example, Wilke et al. (6) showed that the error from neglecting this effect was never larger than 0.3 yofor the maximum flux of deposit in their experiments. Consequently, it may be assumed that the . zero. last term of Eq. (1) is Next, a simplification can be achieved concerning the migration term by adding a large excess of an unreactive electrolyte to the solution. If such electrolytes, which do not react at the electrode, exist in the solution in relatively high concentrations and have high conductivity

90

T. MIZUSHINA

compared with the reacting species of ions, there should be no sharp potential gradient near the electrode, i.e., a!P/ay may be assumed to be zero. The migration current then becomes negligible and almost all of the electrolysis current arises from the reaction of ions which reach the electrode surface by diffusion. I n this case the migration current was estimated to be of the order of 1 yo of the total current1 (8). Thus, the migration term in Eq. (1) can be neglected and only the diffusion term remains to give NA = 4 9 9)ac,/?Y (3)

+

I n the general case, the integration of Eq. (3) gives the following expression for the rate of mass transfer:

NA = k(Cb - C i )

(4)

Substituting Eq. (2) into Eq. (4), one obtains i/An,F

= k(c, - ci)

(5)

If the effect of ionic migration in the potential field is eliminated as mentioned above, the analogy between heat and mass transfer is valid as discussed by Agar (9). Thus the results of experiments by the electrochemical method may be correlated with the knowledge of heat transfer at a wall of constant temperature. At the limiting current, the concentration at the surface of the electrode, ci , becomes zero, and Eq. ( 5 ) is simplified to i/An,F = kcb

(6)

The limiting current can be seen easily from the potential-current curve as shown in Fig. 1. As the negative potential of the cathode is increased, the positive ions migrate, to a certain extent, from the bulk of the solution into the double layer. The accumulation of the ions at the surface of the cathode is equivalent to the charging current. T o keep a finite current between two electrodes in an electrolytic cell, the applied potential difference must exceed the equilibrium one by a finite voltage of concentration overpotential. As the applied potential is made higher, the current increases exponentially and thereafter approaches a constant value, i.e., a limiting current, asymptotically. Under the condition of a limiting current, the ions transferred to the electrode surface react very soon and the increasing potential does not result in an increase in the If the indifferent electrolyte is not added to the solution, the current due to the migration is of the order of 10% of the total current (8).

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91

rate of the desired reaction. Lin et al. (3) indicated that the limiting currents observed under the same flow condition were exactly proportional to the bulk concentration of the ions which reacted at the cathode. This fact proves Eq. (6) to be valid. However, it should be noted that this theoretical limiting current density cannot be perfectly achieved. T h e polarization developed at the cathode is approximately equal to the electromotive force of a concentration cell of two solutions of the reacting ion at concentrations cb and ci ,

E

= (RT/n,F)In yb/yi

(7)

(RT/n,F)In cb/ci

Therefore, the value of ci decreases exponentially toward zero with the increase of the cathode potential, but it never quite reaches zero.

-

0.6

.

0.4

-

0.2

-

#

N

E

u

\

2

Y z 9

t .-

v)

C

a

’0 +

C

E! 3

u

0

I

I

FIG. 1.

I

I

Limiting current.

I n Fig. 1, it is seen that a further increase of the potential over the limiting current region causes a steep increase in current density due to the discharge by a secondary reaction such as hydrogen evolution on the cathode. As the diffusion rate of ions is made to increase, e.g., by increasing the flow rate, under the same conditions of electrolysis, the value of the limiting current is raised and finally the flat portion of the polarization curves disappears above a certain upper limit of the flow rate as shown in Fig. 2. I n such situations the limiting current is no longer indicated since the reaction is too slow to remove all ions reaching the electrode surface. This upper limit of flow rate is called the “critical flow rate.” T h e higher

92

T. MIZUSHINA

the reaction rate is, the higher the critical flow rate obtained. Since the diffusivity is an approximately linear function of temperature while the reaction rate constants vary exponentially with the temperature, the critical flow rate is very sensitive to temperature. The data above the critical flow rate should not be used in correlations since the concentration at the electrode surface is not zero.

c

40-

N

5

\

a

E

30-

x

c

0, C

0 c

20-

g, =I

u

IOuc 0

0

-0.2

Ucrlllcol I

I

I

-04 -0.6 -0.8 Cathode potential ( V )

-

FIG. 2. Critical flow rate.

The electrolytic solutions to be used to measure the mass transfer rates are those of redox couples such as ferrocyanide and ferricyanide ions in NaOH as an unreactive electrolyte and those for reduction of metal ion such as Cu2+deposition on Cu cathode in H2S0, solution. The reactions of these systems are: (1)

+

Fe(CN)e3- e + Fe(CN),4Fe(CN)e4- + Fe(CN),3-

(2) Cu*+

+ 2e

Cu

+

-+

Cu Cus+

+ 2e

+e

at cathode at anode

at cathode at anode

I n the former solutions, the presence of dissolved oxygen influences the mass transfer measurement. It is better to remove oxygen from the test solution and to seal the surface of the solution with an inert gas. When the latter solutions are used, it should be noted that the conditions of the cathode surface are changing due to metal deposition during electrolysis.

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93

It is recommended that a metal which has as high a hydrogen overvoltage as possible and is inert in acidic solutions be selected as the cathode. In the case of CuSO, in H,SO, solution, H,SO, reacts with H,O to form oxonium ions, H,O+, and sulfate ions, SO,2-. It is desirable that the potential applied be sufficient for the reduction of Cu2+ ions to Cu metal at the cathode but not for hydrogen evolution by reduction of H,O+ ions. If the hydrogen overvoltage on that cathode is lower than the potential for the limiting current, the polarization curve obtained may not have a flat part just as that obtained when the flow rate is over the critical one. However, in this case the polarization curve can be corrected theoretically with the information on the hydrogen overvoltage on that cathode, and hence it is possible to estimate the “true” limiting current. Suppose that simultaneous discharges of Cu2+and H,O+ on a copper cathode surface are occurring and that the hydrogen discharge current is superimposed on the limiting current due to Cu2+ ion discharge. The overvoltage for the former process seems to be mainly caused by the charge transfer overvoltage while that for the latter is evidently by the diffusion overvoltage. Therefore, it may be assumed that these processes do not interfere with each other and that these current densities can be added. The hydrogen overvoltage is expressed as a function of the current density of hydrogen ions by Tafel’s experimental equation TH = a

+Pin 5~

(8)

The constants 01 and j3 in Eq. (8) are dependent on the combination of cathode and electrolyte. For example, for 2N-H2S0, on Cu cathode, f H = 10-7-10-3 A/cm2 at 20°C, 01 = 0.78 and fl = 0.043. On the other hand, the cathode potential, E , at the given current density is expressed as follows: E, = (RT/F)In YH - TH (9) From Eqs. (8) and (9), the current density by hydrogen discharge under the given potential can be calculated as

5br ,=exp[(l/@)((RT/F)In

- EC

-

(10)

Subtracting this calculated current density of hydrogen ions from the observed total current density at that potential, one can obtain the “true” limiting current for copper deposition as shown in Fig. 3. When this electrochemical method is applied to measurements of transport phenomena, in general an anode of larger surface area and a cathode of smaller surface area are coupled. This arrangement of

94

T. MIZUSHINA

Electrode potential ( V )

FIG.3.

“True” limiting current.

electrodes makes the current density at the anode smaller than that at the cathode, and the process at the anode has no noticeable effect on the shape of the applied potential-current curve. However, to obtain more accurate data, it is recommended that a standard electrode be used to measure the liquid potential. The addition of an unreactive electrolyte to the solution for eliminating the migration contributions has an advantage in practical operations. It makes the ohmic resistance of the solution negligibly small.

III. Application to Mass Transfer Measurements

A. PRINCIPLE AND METHOD OF MEASUREMENTS Utilizing the electrochemical method, three kinds of mass transfer coefficients (i,e., a space-time-averaged value, a local time-smoothed value, and a local fluctuating value at a solid-liquid interface in many different flow systems) can be easily measured. If one chooses an electrolytic system in which only one kind of reaction occurs at the electrode to be used as a mass transfer surface, the limiting current is reached provided that the opposite electrode does not limit the rate of the reaction. These conditions are satisfied when the opposite electrode is

THEELECTROCHEMICAL METHOD

95

large enough and when the reaction occurring at the measuring electrode is sufficiently rapid. And then the limiting current is a direct measure of the mass flux of a specific species of ion at the measuring surface. If the concentration of this species of ion in the bulk of the solution, c, , is known, the mass transfer coefficient can be calculated from Eq. (6). The time-smoothed value of the limiting current on the whole electrode surface gives the space-time-averaged value of the mass transfer coefficient, K. On the other hand, point electrodes which are electrically isolated from the surrounding electrode surface as shown in Fig. 4 and

Polyvinyl chloride

Isolated cathode

Cothode

FIG. 4. Isolated point cathode.

held at the same potential as other parts of the electrode are used to obtain the distribution of local time-smoothed values, k, . Furthermore, the instantaneous fluctuating values of these point electrodes give the local fluctuating values, A,’, from which information concerning the turbulent behavior of diffusion processes is obtained. The electrochemical method to measure the local values of the mass transfer coefficients gives more accurate results than those obtained by the ordinary measurement of mass or heat since the electrical insulation of an isolated cathode is an easy method. Figure 5 shows the electrical circuit of an experimental apparatus to measure the mass transfer rate in an agitated vessel. A potential applied between the anode and cathode is adjusted with a rheostat and the current in the circuit is measured. After the flow conditions are set, current is passed through the cell and increased in small increments at intervals of some time until the limiting current is reached and usually increased further to the hydrogen evolution point. Approximately one minute is recommended as the time interval at each rheostat setting because this is long enough to make the potential and current reach a steady state. Thus, the polarization curve can be drawn.

96

T. MIZUSHINA

On the other hand, Wilke et al. (6) in their experiments on free convection mass transfer at a plate studied various ways of obtaining current-potential curves with changes in the time of duration of electrolysis and intervals for potential measurement following each current setting. They concluded that the limiting current density was independent of the initial shape of the current-potential curve of each different time program. T h e maximum deviation from the average value was only 3.2%.

FIG. 5. Agitated vessel to measure mass transfer coefficients.

For metal depositing processes, especially, it is desirable to repeat the limiting current determinations three or four times to ensure reproducibility. When fluctuating values of the mass transfer coefficient are measured, the instantaneous limiting current of point cathodes are recorded to obtain the turbulent characteristics, such as intensity of turbulence, from the fluctuation curve. As discussed in Section 11, the cathode potential is measured against the saturated calomel electrode through a capillary in the illustrated circuit. The effect of location of capillary junctions along the cathode surface proved to be negligible on the observed limiting current. The shape and dimensions of the cathode are designed to form the transfer surface to be investigated, and the test sections should be electrically insulated from the rest of the experimental apparatus, such as

THEELECTROCHEMICAL METHOD

97

the pumping system, by means of rubber gaskets. T h e whole surface of electrodes including point cathodes should be carefully machined and polished to eliminate any roughness or discontinuity on the surface. For ferri-ferro redox systems, platinum or nickel is generally used as the electrode material. However, it has been reported that a platinum electrode containing a small percentage of rhodium showed a downward drift in the limiting current. Before each run, the electrodes should be cleaned with CC1, and buffed with soft paper. It is recommended that the electrodes be cleaned cathodically in 5% NaOH solution to make sure of eliminating chemical polarization for redox system. After each run a check must be made for corrosion deposits formed on the electrode surface. For Cu2f reduction system, copper metal is usually used for electrodes. T o obtain reproducible quality after machining, the electrodes must be polished by emery paper No. 400, then washed and degreased and used immediately after preparation. It should be noted again that the redox system should be kept under nitrogen to avoid oxidation of the ferrocyanide and reaction of the dissolved oxygen at the electrode although this effect is very small. Furthermore, it is recommended that nitrogen be bubbled through the storage tank to remove any oxygen prior to the runs. In addition, it is better to keep the test solution away from light. T h e light decomposes potassium ferrocynaide slowly to hydrogen cyanide, which will poison the electrodes. In fact, it is better to use the solutions which have been prepared just before the experiment, using specially treated distilled water, since colloidal ferrihydroxide is formed and the color of the solution becomes brown after a few days. It is important to know the actual effect of dissolved oxygen in a ferro-ferricyanide redox couple when the presence of air is unavoidable, as in two-phase air-liquid flow systems. However, there is uncertainty concerning this effect since the effects of oxygen in contaminating the solution and the electrodes are complicated. The actual electrochemical reaction involving oxygen may not occur at such a relatively low voltage as applied in this method since oxygen activation polarization is generally high. But mixed potential and oxide films on electrodes might have some influence. Anyway, absorption of oxygen by the solution and contamination of electrodes depend upon time. Therefore, suitable measurements may be finished before such an effect of contamination becomes controlling. Sutey et al. (7) indicated that the mass transfer coefficients measured at oxygen saturations below 70 % were within 5 % error for an operating time of 275 min. Finally, precautions must be taken to keep the temperature of the test

T. MIZUSHINA

98

section constant since the physical variables and, accordingly, the mass transfer rate depend on the temperature. Lin et al. (3) studied the following two systems as well as redox systems in measuring mass transfer rate in annular flow. (1) Reduction of quinone on a silver cathode in a strongly buffered solution. C,H402

+ 2Hf + 2e

-f

C,H,(OH),

(2) Reduction of oxygen on a silver cathode in a NaOH solution. 0,

+ 4e + 2H,O

-f

40H-

I n these two systems the electrode reactions are rapid enough, and the electrode materials are inert to the solutions. In the case of reduction of quinone, hydrogen ion is also a reacting material. In the presence of a strong buffer, however, the concentration of hydrogen ion is essentially constant across the boundary layer because the buffer reaction rate is more rapid than the diffusion rate of quinone. With regard to the reduction of oxygen, the situation is not complex since water always exists in large excess. It was found that for reduction of quinone, the critical flow rates were comparatively low, whereas the limiting current was still obtained at Re = 17,500 with reduction of oxygen and up to Re = 29,800 with ferricyanide reduction. Consequently, the ferro-ferricyanide redox couple may be best since in alkaline solutions it is stable and the chemical polarization on the cathode is so small that the critical flow rate is very high. An additional advantage is that the bulk composition of the solution is constant since the same reaction (but in opposite directions) occurs at cathode and anode.

B. FREECONVECTION 1. Free Convection Mass Transfer at a Plate a. Free Convection Mass Transfer at a Vertical Plate. The free convection mass transfer at a vertical plate has been studied by Wagner (lo), Wilke et al. (6),(ZZ), and Ibl et al. (22, 13). Wagner measured the limiting current by the deposition of copper from an acidic solution of copper sulphate, Wilke et al. by the deposition of copper and silver, and Ibl et al. by the deposition of copper and the reduction of ferricyanide in a solution of K,Fe( CN)6-K,Fe( CN),NaOH, respectively. The apparatus used by Wilke et al. is made of an electrolytic cell containing a solution of CuSO, and equipment for measuring cathode potential relative to the bulk solution in the cell and total current.

THEELECTROCHEMICAL METHOD

99

Fifteen different solutions of CuSO, concentration ranging from 0.01 to 0.74 mole/liter and that of H,SO, ranging from 1.38 to 1.57 moles/liter were used. I n Fig. 6 the results of Wilke et al. are plotted in the range of Rayleigh numbers = 4 x lo6 6 x loll. T h e correlating equation of these results is N

Sh

= 0.671

(Gr S C ) O . ~ ~

(1 1)

T h e Sherwood number or the Nusselt number for diffusion is given by Sh

=

KxX,/B

(12)

where k is the averaged mass transfer coefficient over x, x is the vertical height of electrode surface, and Xr is the logarithmic mean volume fraction of nondiffusing species.

1o3 ul

102

Gr Sc Wilke FIG. 6. Free convection mass transfer coefficients on a vertical plate. Key: (0)

et al.; ( 0 ) Ibl et al.; ( X ) Wagner.

T h e Grashof number is given by the following equation since the free convection in the ion transfer is caused by density differences due to concentration change between bulk solution and electrode surface: Gr

= g(Pb - Pi) PX3/P2

(13)

where p b and pi are the fluid densities in bulk solution and at the electrode surface, respectively. Equation (1 1) is in close agreement with the equation which correlates the data of the experiment on solid dissolution. This is an encouraging indication of the general validity of the electrochemical method. However, these experimental results of the mass transfer disagree with the predic-

T.MIZUSHINA

100

tions of Eq. (14), which is obtained from the modification of the SchmidtPohlhausen-Beckmann theory for heat transfer to air. Sh

= 0.525

(Sc * Gr)l/4

(14)

The disagreement may be expected since the numerical coefficient of

Eq. (14) is applicable only to systems with Pr or Sc

0.7.

Wagner measured the mass transfer rate from three electrodes of heights 1,4, and 16 cm in a solution of 0.1 mole of CuSO, and 1.0 mole of H,SO, per liter. I n his experiment it was found that deviations from the vertical position of the electrode surface up to 15" had no appreciable effect. Sherwood numbers calculated from his results are plotted in Fig. 6. It is seen that they are in good agreement with the results of Wilke et al. Using a solution of 0.2-0.025 mole of ferricyanide, 0.1 mole of ferrocyanide, and 2 moles of NaOH per liter, Ibl et al. obtained the mass transfer data not only in the laminar range, i.e., 3 x los < Sc . Gr < 4 x lo8 but also in the transition range, i.e., 1 x 1012 < Sc Gr < 1 x 10l6, while Wilke et al. and Wagner covered only the laminar region. Their data in the laminar region are correlated by Eq. (11) when the effect of the existence of a diaphragm is negligible. As shown in Fig. 6, the data of Ibl et al. for the transition region fall on an extension of Eq. (11) and are correlated by Sh = 0.31 (Sc * Gr)0*28 (15) Wagner divided a cathode into five electrically isolated sections, the heights of which are, respectively, 1, 3, 1, 7,and 1 cm from the bottom, and measured the current density of each section separately. The results are plotted in Fig. 7, which indicates the distribution of local mass transfer coefficients. b. Free Convection Mass Transfer at a Horizontal Plate. Fench and Tobias (14) studied the free convection mass transfer at a horizontal plate using the cathodic reduction of Cua+ to Cu from CuSO, solutions. Making use of standard electrodes, they measured the concentration polarization directly instead of measuring the applied potential. The anode was separated from the cathode by a $in. thick ceramic diaphragm so that convection currents generated by the dissolving anode would not influence convection in the cathode compartment. I n the process of the experiment, an interesting problem was presented. It is desirable to make the electrolysis period as short as possible to avoid decreasing the bulk concentration notably, especially in such a case as 0.01 moIe/liter solutions, and roughening the cathode surface excessively. On the other hand, the duration of electrolysis must be long

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101

enough to allow natural convection to reach a steady state. As a result of the preliminary runs, 3-5 minutes were used for reaching the limiting current. The experimental data of Fench et al. are plotted in Fig. 8. The correlating equation is Sh

= 0.19

(Sc * Gr)”a

(16)

where the vertical distance between cathode and diaphragm was chosen as the characteristic dimension,

H

10.0

2.5

-0 k,

(cm/sec)

FIG. 7. Distribution of the local coefficients of free convection mass transfer on a vertical plate.

2. Free Convection Mass Transfer at a Cylinder Free convection mass transfer at a horizontal cylinder was studied by Schutz (15). In his experiments, the limiting currents were measured by the electrolysis of CuSO, in H,SO, as the supporting electrolyte. The experimental results of the space-time-averaged mass transfer coefficients are shown in Fig. 9. All the data for Gr Sc < loQ come together on a line of the gradient of 0.25. The correlation is expressed as

-

Sh

= 0.53 (Gr

-

(17)

T. MIZUSHINA

102

10’ L

Y,

lo2

.-

in‘

10’

10’

do

I 013

10”

Gr- Sc

FIG.8. Free convection mass transfer coefficients on a horizontal plate. Key:( 0 ) without glycerol; ( 0 ) with glycerol. Sh = 0.190 (Gr *

loj

cn

lo2

10 lo7

108 Gr-Sc

1o9

100

FIG.9. Free convection mass transfer coefficients on a cylinder.

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103

This equation is in good agreement with the ordinary equation obtained for free convection heat transfer from horizontal cylinders. The data for Gr Sc > lo0 seems to belong to the transition region. By rotating the cylinder, the local mass transfer coefficients are measured by an isolated cathode prepared on the cylinder surface. T h e results of the local mass transfer measurements are shown in Fig. 10. For Gr . Sc > 3 x los, all the correlating curves have minimum points at the separation points in the range between 130 and 180". Figure 11 indicates that the limiting current to the isolated cathode, i.e., the local transfer coefficient, is fluctuating in the turbulent region but not in the laminar region.

3. Free Convection Mass Transfer at a Sphere Free convection mass transfer at a sphere was also measured by Schutz (15). T h e experimental technique was the same as that for the cylinder. T h e experimental results of the space-time-averaged and local mass transfer coefficients are shown in Figs. 12 and 13, respectively. The correlating equation of the space-time-averaged mass transfer coefficients is Sh

=2

+ 0.59 (Gr - S C ) O ~ ~

(18)

C. FORCED CONVECTION

1. Forced Convection Mass Transfer in Tube Flow a. Fully Developed Mass Transfer in Turbulent Flow. Several equations have been proposed for predicting the mass transfer coefficients between a pipe wall and turbulent flow in the region of a fully developed concentration profile. T h e equations differ from each other in the effect of the Schmidt number on the mass transfer coefficients. I n the empirical correlations of the heat transfer coefficients by Chilton and Colburn (16) and Friend and Metzner (27) the exponent for the Prandtl number is #. The semitheoretical equation of Lin et al. (28) predicts that the mass transfer coefficients are proportional to Sc2I3 for high Schmidt numbers, and a similar treatment of Deissler (19) leads to an exponent of $. Since the concentration gradient in the direction of flow (x-coordinate) is much smaller than that in the direction perpendicular to the wall (y-coordinate), the mass flux in the fully developed mass transfer of turbulent flow along a wall is usually expressed by Eq. (3).

I 200

k 100 2.60 .

Id

3.99 ' 10'

-

0

0

90

45

135

180

e (deg) FIG. 10. Local coefficients of free convection mass transfer on a cylinder.

180'

0.20/&

-

0.15 -

oo

1 20°

-

c

3' 0

aE 0 . 1 0 - w 1 5 5 °

v

.-

0.05

-1

50'

-

u 20 40

O O

t

FIG. 11. a cylinder.

(sec 1

Fluctuation of the local coefficients of free convection mass transfer on

THEELECTROCHEMICAL METHOD

FIG. 12. Free convection mass transfer coefficients on a sphere.

400

I 7.68.10'

2.94.109

5.18. lo8

2.d.108

- 0

0

45

90

135

180

0 (deg)

FIG. 13. Local coefficients of free convection mass transfer on a sphere.

105

T. MIZUSHINA

106

Assuming that diffusion in the direction of flow is ignored, and in addition that the concentration boundary layer is so thin that the wall curvature is negligible, one obtains the following equation for mass balance on the diffusing species: u+-ac+ = _ a [(sc-l+ ax+ ay+

g]

+)

The boundary conditions are c+ = 0

at

1 c+ = 1

at

c+ =

at

x+

> 0, y+ = 0 y+ = a3

x+

(20)

<0

The concentration profile becomes fully developed for very large values of x+, and ac+/ax+, at any value of y+, becomes constant. In such a region a mass balance gives -ac+/ax+ g -ac,+/axt

= 4Km+/Re

(21)

where K,+ = k,/u,. This term is very small because Km+is of the order of lo-*, while Re is of the order of 104. Taking account of the smallness of ac+/ax+, one integrates Eq. (19) as

Accordingly,

The fully developed mass transfer rate K,+ depends on the distribution of the eddy diffusivity, cD/u, near the wall, especially at high Schmidt numbers. The function describing eD/u near the wall may be expanded in a Taylor series, and usually only the leading term in the series €&

=

C(y+)"

(24)

is used. Substituting Eq. (24) into Eq. (23), one obtains K,+

= ( n / r )C1/lasin(r/n)(Sc)-(n-l)'n

(25)

I n general, the coefficient C and the exponent n in Eq. (24) could be a function of the Schmidt number and n may change slightly with y+.

THEELECTROCHEMICAL METHOD

107

Arguments can be presented to show that the exponent n should be three or greater. As described in Section 111, A, use of electrochemical methods offers a number of important advantages, the most significant of which is that surface roughness can be kept very small compared with the dissolving wall method. This is particularly important at high Schmidt numbers because the surface roughness influences the mass transfer rate significantly in this case. Typical experimental apparatus is given schematically in Fig. 14.

v Recorder

Potentiostat

Reference electrode -Luggin

-

capillary

I

Anode

Pump

U

Electrolyte reservoi r

Manometer

FIG. 14. Experimental apparatus for forced convection mass transfer in a tube flow.

T h e mass transfer coefficient is calculated by Eq. ( 6 ) from the experimental results of the electrochemical method provided that the necessary conditions are satisfied. From Eq. (19) the mass transfer rate is a function of Sc and x+, K,+

=

K,+(SC,

Xf)

(26)

At very large values of xf, K,+ = K,f. T h e average mass transfer rate over the length of the mass transfer section L f is expressed as K+

=

(l/L+)

's:

K,+ dxf

T. MIZUSHINA

108

If the length of the mass transfer section is very large, the entrance effects are negligible, i.e., L++m lim

K+

= K,+

(28)

Therefore, it is necessary to check that the measured value of the spaceaveraged mass transfer coefficient is a true asymptotic one to use as the fully developed mass transfer coefficient. Rather than such a method to measure K,+,however, use of an isolated cathode may be recommended at a downstream position where the concentration profile has been fully developed to measure the local value of K,+ directly. Lin et al. (3) measured the mass transfer rate between flow in the annulus of a double tube and the wall surface of the inner tube over a range of Schmidt numbers from 300 to 3000 and Reynolds numbers from 260 to 30,000, using four different systems of electrochemical reactions as mentioned in Section 11. Results for laminar flow are presented in Section 111, C, 1, d. Shaw and Hanratty (20) measured the fully developed mass transfer rate at Sc = 2400 and Re = 8000-50,OOO and found that K,+ is independent of Reynolds number. The maximum deviation of twenty nine measurements from the average was 4%. The process chosen was the reduction of ferricyanide ions at a nickel cathode in the presence of a large excess of sodium hydroxide (0.001 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 2 moles of NaOH per liter). Hubbard and Lightfoot (21) used the diffusion-controlled reduction of potassium ferricyanide in excess caustic (0.005 mole of K,Fe( CN), and K,Fe(CN), , respectively, and 1 4 moles of NaOH/liter) in their measurement. Experiments were made at Sc = 1700-30,000 and Re = 7000-60,OOO in a rectangular channel. The data of Lin et al. (3), Shaw and Hanratty (20), Hubbard and Lightfoot (21) and unpublished data of Mizushina et at., along with the results of Harriott and Hamilton (22) which were obtained by the dissolving-wall method, are summarized in Fig. 15 and compared with predictions calculated from equations of Son and Hanratty (23), Lin et al. ( l a ) ,and Mizushina et al. (24) for eddy diffusivity. The broken line in Fig. 15 is given by Son et al. (23) and expressed as

K,f

SC-9'4

(29)

Re ( f / 2 ) l I 2Sc1I4

(30)

= 0.121

This equation is equivalent to Sh,

= 0.121

THEELECTROCHEMICAL METHOD

109

On the other hand, Hubbard and Lightfoot (21) and Harriott and Hamilton (22) have obtained Sh,

=

K

Ref

(31)

Sc1f3

Figure 15 indicates that the result of Son and Hanratty is not really conclusive as to the Schmidt number dependence of K,+. An equation containing S r 2 I 3rather than Sc3I4 seems to represent the data better. Accordingly, Eq. (31) seems to be a better correlation than Eq. (30).

I

lo2

I

I

I I l l 1 1

I

I

I

I

I I I I I

1 o3

104

1

I

I

I I l l l l

I

lo5

sc

FIG. 15. Variation of Km+with Sc. Key: (0)Hubbard et al.; ( A ) Lin et al.; (V) Harriott et al.; ( 0 ) Shaw et al.; (0) Mizushina et al.; (-) Mizushina et al.; (---) Son et al.; (--.--.-) Lin et al.

By comparing Eq. (29) with Eq. (25) and by assuming that the coefficient C is independent of Schmidt number, Son and Hanratty (23) obtained the following relation: ED/V =

0.00032(~+)~

(32)

However, the calculated curves based on Eq. (23) and the following equations for eddy diffusivity distributions represent the data better than Eq. (32).

T. MIZUSHINA

110 Lin et al. (18): Mizushina et al. (24):

F{ 1 - exp(-1IF)}

= (E&)

E&

F= EM/v

1

0.0344(~M/~)~/' SC

+ 0.240(~M/~)'/* Scl"

= 4.16

x 10-4(y+)9

(34) (344 (34b)

b. Turbulent Flow Mass Transfer in the Mass Transfer Entry Region. When a fluid in fully developed turbulent flow enters a mass transfer section, the local value of the mass transfer coefficient decreases from infinity at the inlet to a minimum value downstream. If the transfer section is long enough, that minimum will be the fully developed value. Accordingly, the space-averaged mass transfer coefficients decrease with an increase of the mass transfer section. Though the heat transfer measurements in such an entry region have been reported by several investigators, such studies are complicated because thermal isolation of a small transfer section from the remaining part of the pipe is very difficult. In this respect, the mass transfer experiments by electrochemical methods are superior for entry region investigations because electrically isolated cathodes are easily made. Shaw et al. (25) used the reduction of ferricyanide in determining the effect of the length of mass transfer section on the average rate of transfer, i.e., the relation between K+ and L+. Ten lengths of the mass transfer section, from 0.0177 to 4.31 diameters were used. T h e Schmidt number was constant at about 2400, while the Reynolds number was varied from 1000 to 75,000. The electrolyte concentration was 2 moles of NaOH and 0.01-0.0001 mole of K,Fe(CN), and K,Fe(CN), per liter. In Fig. 16 the results are plotted and compared with curves of predictions calculated numerically by substituting Eqs. (32), (33), and (34), respectively, into Eq. (19). Schutz (26) measured the local transfer rate K,+ in the mass transfer entry region. The Reynolds number was varied from 20,000 to 50,000 and the Schmidt number was 2170. T h e used solution contains 0.025 mole of K,Fe(CN), , 0.025 mole of K,Fe(CN), and 2 moles of NaOH per liter. I n Fig. 17 the results are plotted and compared with calculated curves based on the eddy diffusivities which are predicted by the equations of Lin et al. (18), Mizushina et al. (24), and Son and Hanratty (23).

THEELECTROCHEMICAL METHOD

111

+

Y

lo-&

1 o2

L’

(-

FIG. 16. Variation of K+ with L+. Key: ( 0 ) Son et al.; (-) Son at al.; (--.--.-) Lin et ul. Sc = 2400.

- -)

Mizushina et al.;

* Yx

X*

(-

FIG. 17. Variation of K.+ with x+. Key: (0)Schutz; (-) - -) Son et ul.; (--.--.-) Lin et al. Sc = 2170.

Mizushina et al.;

112

T. MIZUSHINA

c. Fluctuations of Mass Transfer Rates. The small eddies in the region of the viscous sublayer of a turbulent flow are known to play an important role in determining the transfer rates to the wall, especially in the case of high Schmidt or Prandtl numbers. Since the flow of these small eddies is unsteady, the local mass transfer rates to the wall are fluctuating. For studying the unsteady flow, optical methods are not useful because the quantitative data are difficult to obtain and hot-wire anemometer measurements are limited in that the size of the probe is so big relative to the region being studied that the flow may be disturbed by the probe. To overcome these difficulties an electrochemical method to measure the fluctuations in the local rate of mass transfer to the wall has been developed by Shaw and Hanratty (20). The fluctuating values of the local mass transfer rate can be calculated directly from the fluctuations of the electrical current to the isolated cathodes surrounded by the active cathode surface with the following equation: k,’ = i,‘/(A,n,Fc,) (35) The root-mean-square, the frequency spectra, and the scale of fluctuations in the local mass transfer coefficients are calculated from the measured values of fluctuation. Shaw and Hanratty measured the instantaneous rate of transfer in a fully developed boundary layer at Reynolds numbers ranging from 10,OOO to 60,000 and a Schmidt number of about 2400. As shown in Fig. 18, the size of the isolated cathode affects the measurements of the mass transfer fluctuations remarkably because the fluctuations of the small circumferential scale tend to be averaged over the cathode surface and larger cathodes give smaller signals. However, the correction by assuming a circumferential scale of As+ = 2.1 and an - all data into exponential form for the correlation coefficient brought agreement and indicated that the true local value of [(km’)a]1/2/ikm 0.48. Maximum deviation of the data from this value was about 10%. The axial correlation coefficients obtained with two isolated cathodes of 0.65-mm diameter which are located at a distance of 5 to 30 mm (center to center) in the axial direction are shown in Fig. 19. By integrating the curve in Fig. 19 graphically, longitudinal integral scales of the fluctuation are determined as Az+ E 350, which is about lo2 times as large as the circumferential scale. The frequency spectra are plotted against the dimensionless frequency ++ = + Y / ( v * ) ~ in Fig. 20.

d. Mass Transfer of Laminar Flow. A theoretical equation for the heat transfer in fully developed laminar flow with constant wall tempera-

1 o4

1o5

Re

FIG. 18. Cathode size effects on fluctuation measurements. Electrode diameter: (1) 0.0157 in.; (2) 0.0259 in.; (3) 0.0640 in.; (4) 0.1250 in.

0.0

I

0

I

2

I

1

4

6 (x;

1

8

- x; 1

Y

I

4

10

12

16'

FIG. 19. Axial correlation coefficients.

I

14

16

114

T. MIZUSHINA

ture was given by Graetz. For short tube lengths, the Graetz solution reduces asymptotically to LCvCque's equation, which is modified for mass transfer as Sh = 1.62(Re Sc * D/L)Il3 (36) This is equivalent to the following equation for a circular pipe:

(37)

K+ = 0.81(L+)-'/S ( S C ) - ~ ' ~

The mass transfer rates measured by Lin et al. (3) in laminar flow in an annulus are plotted in Fig. 21, and compared with Eq. (36). Their data are correlated well with Eq. (36). 1.0 I

I

0.9

-

0.8

-

0.7

-

0.6

-

0.b

-

0.3

-

0.2

-

0.1

-

0.5

0.0

I

0

I

2

I

1

4

I

I

6

I

I

8 @*B

1

I

10

I

I

12

I

I

14

I

I

I

16

18

104

FIG.20. Frequency spectra of the mass transfer fluctuations. Key: ( 0 ) Re = 8700;

( A ) Re = 23,150; ( 0 ) Re = 50,700.

Son and Hanratty (23) measured the mass transfer rate of laminar flow in a circular pipe. Their data, along with Eq. (37), were used to calculate the Schmidt numbers. The results for lengths ranging from 0.0174 to 1.91 pipe diameters and Reynolds numbers from 335 to 2200 are plotted in Fig. 22. The agreement of their results with Eq. (37) means

THEELECTROCHEMICAL METHOD I

r Ill

115

I

1

-

lo2

ReScDIL

FIG. 21. Mass transfer coefficients in laminar flow.System: (0)oxygen; ( 0 )quinone;

(0) ferricyanide ion; ( A ) ferrocyanideion.

sc = 2,LOO

FIG.22. Variations of K + with Lf in laminar flow.

T. MIZUSHINA

116

that the exponent for L+ in Eq. (37) and thus that for LCvCque’s equation are confirmed experimentally.

e. Dynamic Response of the Mass Transfer Coe@cient. I n Kyoto University the dynamic response of the mass transfer coefficient itself is being studied by Mizushina et al. using the flow of solution of ferriferrocyanide redox system in a circular tube and a rectangular duct. In Fig. 23, an experimental result of a step response of the space-averaged mass transfer coefficient is compared with a theoretical calculation.

Shear stress at the wall

1.0

0.5 t

1.5

2.0

(secl

FIG.23. Step response of the mass transfer coefficient.

2. Forced Convection Mass Transfer from Cross Flow Many investigators have studied the transfer of heat from a cylinder to fluids in cross flow. Only a few corresponding studies of the mass transfer have been made, however. A knowledge of the diffusional transport of mass in cross flow is of more than academic interest. For example, it may be of some practical importance in heterogeneous catalytic processes. An electrochemical method based on a diffusion-controlled electrode reaction seems suitable for studying mass transfer to a cylinder in a liquid. This enables us to obtain a lot of valuable data which are very difficult to get by any other method. Dobry and Finn (27) measured the space-time-averaged mass transfer coefficients to very fine wires at Reynolds numbers ranging from 0.08 to 10. Grassmann et al. (28) made an experiment at Reynolds numbers from 120 to 12,000 and a Schmidt number of 2780. They measured the local and the average mass transfer coefficients to cylinders and confirmed the analogy between heat and mass transfer. Vogtlander and Bakker (29) also obtained mass transfer data which agree with heat transfer results in the Reynolds number range between 5 and 100.

THEELECTROCHEMICAL METHOD

117

Dimopoulos and Hanratty (30) used electrochemical techniques in studies of flow around cylinders to measure the velocity gradient in the boundary layer. They also measured local mass transfer rates to cylinders and confirmed good agreement with calculated values from measured velocity gradients for Re = 60-360. These investigators used reduction of ferricyanide on the test cylinders in the system composed of ferricyanide, ferrocyanide, and sodium hydroxide as supporting electrolyte. Their experiments were carried out in different Reynolds number ranges and their correlating curves cannot be linked in a single curve, probably due to the effect of different turbulent intensities in the approaching flow. In construction of the experimental apparatus, it is necessary to satisfy the following requirements: (1) The ratio between the diameter of the test cylinder and that of the channel in which they are mounted, i.e., the blockage ratio, is small enough. (2) The approaching flow has a flat velocity profile and its fluctuations are as small as possible. For this purpose, a head tank or a water tunnel which has a convergent nozzle and calming section containing honeycombs and screens is used. I n addition, it is necessary to measure the turbulence intensity in the approaching flow. An experimental apparatus which is being used in Kyoto University is shown schematically in Fig. 24. A test cylinder of 1-cm diameter is 16'

JU

'4

screen

'61

i:Test cylinder ( l c m * cathode1

30' 30' I

k

!

Heat Electric heater exchanger NI Orifice

Storage tank

Frc. 24.

Experimental apparatus for mass transfer in cross flow.

118

T. MIZUSHINA

mounted horizontally in a channel of 8 x 16 cm cross section and has a platinum cathode of 0.5-mm diameter, which is embedded in, but isolated electrically, from the main cathode covering around the middle part of the cylinder surface in width of 2 cm. The cylinder can be rotated during the experiments so as to make the position of the isolated cathode relative to the front stagnation point vary. The anode, which is a large nickel plate of 20 x 10 cm, is located downstream. The circulating solution contains 0.01 mole of K,Fe(CN), and K,Fe(CN), ,respectively, and 2 moles of KOH per liter, and is kept at 30°C. T h e oxygen in the solution is purged with nitrogen before each experimental run. In order to make the velocity profile as flat as possible, five screens are installed in series in the duct before the test cylinder. The turbulent intensities are varied from 0.8 to 2.3 yo by changing the screens located 21 cm upstream from the test cylinder. Reynolds numbers are calculated from the velocity based on the corrected free cross section and vary from 3900 to 10,400.

a. Space-Averaged Mass Transfev Coe$&ents. In Fig. 25 the results of Dobry and Finn (27), Grassmann et al. (28), and Vogtlander and Bakker (29) are plotted together. Compiling - the results of many investigators, mainly on heat transfer, van der Hegge Zijnen (31) obtained the following correlation:

+

Sh = 0.38 Sc0u2 (0.56 Re0.5 10'

+ 0.001 Re) S C O . ~ ~

.

100

10.'

1o2

t

I

lo-'

1

I oo

I

I

10'

10'

1o3

I'0

lo5

Re

FIG.25. Mass transfer coefficients in cross flow. Key: ( 0 ) Dobry et al.; (0)Vogtliinder et al.; ( A , V, 0 ) Mizushina et al., 100(i/i)l/a/ub = 0.8, 1.2, and 2.1, respectively.

THEELECTROCHEMICAL METHOD

119

This equation, which is represented for Re = 1-500 and Sc = 1000 in Fig. 25, correlates the data by Vogtlander and Bakker for the middle range of Reynolds numbers quite well while it predicts j factor values which are somewhat higher than the data by Dobry and Finn for low Reynolds numbers. On the other hand, the results of Grassmann et al. for Reynolds numbers which are greater than 160 appear to be rather larger than predictions by Eq. (38). Perhaps this can be explained by two effects, i.e., the effect of blockage and turbulence in approaching flow. Since the ratio between the diameter of the test cylinders and the duct in which they were mounted was not very small in the experiments of Grassmann et al., the liquid might be accelerated near the cylinders. When Re is based on the mean velocity in the smallest cross section instead of in approaching flow, the results in Fig. 25 go down about 10yo. T h e unpublished results of Mizushina and Ueda of Kyoto University are also plotted in Fig. 25. T h e experiments were carried out with various levels of turbulent intensity in the approaching flow, namely 0.8, 1.2, and 2.1 %. T h e data for turbulent intensities of 1.2 and 2.1 yo are in good agreement with the heat transfer results by Comings et al. (32) and the corrected values of Grassmann et al., and the data for the intensity of turbulence (0.8%) agree with Hilpert’s heat transfer data for the intensity (0.3 %) (33). T h e effect of turbulence was studied by Kestin and Maeder (34) in heat transfer. Their results together with the results of Mizushina and Ueda are shown in Fig. 26. From this, the analogy between heat and mass transfer is confirmed even in the effect of turbulence. I

I

I

1.0

I

2.0

100J;r;/ur

FIG. 26. Effect of turbulent intensity on the mass transfer coefficients in cross flow.

120

T.MIZUSHINA

b. Local Mass Transfer Coeficients. T h e distribution of the local mass transfer coefficients obtained by Grassmann et al. (28) are shown for various Reynolds numbers in Fig. 27. It is shown that the concentration boundary layer starts at the front stagnation point and develops, as it proceeds around the cylinder, and reaches the separation region.

'*0°

k

800

i lJY

.40°

t 30

60

90

120

150

100

8 (deg)

FIG.27. Distribution of the local mass transfer coefficients in cross flow. Key: ( 0 ) Re = 10,380; (0)Re = 5210; (0)Re = 7240; ( A ) Re = 3870. 100(u'a)l/a/ub = 2.3.

Dimopoulos and Hanratty (30) measured the local mass transfer rates and velocity gradients on the wall in the range of Reynolds numbers from 115 to 356 and compared their mass transfer data with the prediction based on the measured velocity gradients. Except in the case of Re = 115, the predicted mass transfer rates are in good agreement with the measured values as shown in Fig. 28. For the higher Reynolds number region the local mass transfer coefficients are also sensitive to turbulence in the approaching flow. As seen in Fig. 27, there is a minimum at the front stagnation point. This may be due to the effect of turbulence. The fluctuations of the local mass transfer coefficients near the front stagnation point are plotted in Fig. 29 from which it is seen that the fluctuation is negligible at the 2.5", and decreases again to front stagnation point, increases to 0 9 = 5". Such a behavior of fluctuation corresponds to the distribution of the mass transfer coefficients near the front stagnation point in Fig. 27.

THEELECTROCHEMICAL METHOD I

I

I

I

1

1

I

121 I

Calculated line

1.0 -

-

0.8-

A A ”

-

A A A

-

0.2 0

A

I

I

I

I 10

I

1

I

I

1 30

I

20

I

1 40

I

L

50

t (sec) FIG. 29. Fluctuations of the local mass transfer in cross flow at various angles.

122

T. MIZUSHINA

In Kyoto University, Mizushina and Ueda are studying the effect of turbulence on the local mass transfer coefficients. The experiments are limited to the laminar boundary layers at the front half of the cylinder for the subcritical Reynolds number region from 7240 to 10,380. Since data for zero turbulent intensity are not available, those values were predicted by Dienemann's approximate method based on the static pressure distribution around the cylinder. In Fig. 30, Sh,/( Rell2 Sc1/3) are plotted against 8, the angle from the front stagnation point, and different curves were obtained for different turbulent intensity. From these curves the ratios of measured Sherwood number to those for zero intensity and then (Sh/Sh, - 1) are calculated and plotted against 0 on a semilog scale as Fig. 3 1. Different straight lines were obtained for each turbulent intensity. With increase of the angle 8, this ratio increases, i.e., the effect of turbulence increases. Since it becomes very large near the separation point, it is expected that the turbulence influences the separation of the boundary layer. With respect to the measurement of the separation point, two electrochemical methods, i.e., the shear-stress meter and the measurement of the local mass transfer fluctuation, are used. The former will be described in Section IV. I n the experiments of Mizushina and Ueda the latter was used. Since large vortices are shed at the separation point, the local mass transfer fluctuations have a sharp maximum there. T h e separation points at Re = 10,OOO were determined experimentally to be at 6 = 81", 82", and 84" in the flow, the turbulent intensities there were 2.3, 1.2, and 0.8 yo,respectively.

3. Forced Convection Mass Transfer in an Agitated Vessel The space-time-averaged and the local time-smoothed values and the local fluctuation of the mass transfer coefficients at the wall of an agitated vessel with paddle-type impellers were measured by Mizushina et al. (35) using an aqueous solution of 0.001 mole of CuSO, and 2 moles of H2S04per liter. As shown in Fig. 5, the cylindrical wall of the vessel is the cathode for measuring the average transfer coefficients. Nine isolated cathodes of 1.7-mm diameter for measuring the local transfer coefficients and of 0.3-mm diameter for measuring the fluctuation intensity are mounted at 1-cm intervals in the vertical direction on the cylindrical wall while the bottom plate is the anode. Both electrodes are made of copper. Since the limiting current was not found distinctly, owing to the discharge current of hydrogen ions in this case, it was determined by the calculation described in Section 11.

1

i

i

40

20

60

i

80

0 (deg 1

FIG.30. Turbulence effects on the local mass transfer coefficients in cross flow.

1.0

0.8 0.6 c

I

0.2

(

0.1

1

0

FIG.31. Turbulence

20

1

40

8 ldeg)

1

60

effects on the local mass transfer coefficients in cross flow.

T. MIZUSHINA

124

From the results of measurements of the mass transfer coefficients,

j factors are calculated by the following equation:

where V Bis the fluid velocity outside the laminar film and calculated from Vo = 1.52Rd (40) Equation (40) was obtained by measuring the time lag of the fluctuations at two horizontally separated electrodes. This equation was examined and found to be valid in the range of z / H = 0.1-0.9 for b/H = 0.7 and a / H = 0 . 3 4 . 7 for b/H = 0.2. Hence, it is assumed that Eq. (40) is valid in the whole range of z.

a. Space-Averaged Transfer Factors. T h e values of j , are plotted against Re, in Fig. 32. On the other hand, the friction factors in the to-‘

-k(

.t o 2

._

0

10

1o2

lo3

I 0‘

Red

FIG. 32. j factors of the mass transfer and the friction factors at the wall of an agitated vessel. Key: ( 0 ) mass, (0)momentum, b/H = 0.2; ( A ) mass, ( A ) momentum, b / H = 0.4; ( W) mass, (0) momentum, b / H = 0.5; (+) mass, (0) momentum, b/H = 0.7. j o = f / 2 = 0.15

tangential direction were measured in another agitated vessel which was designed for this purpose, and the measured values of f/2 are also plotted in Fig. 32. Both data are correlated with the same equation as follows:

THEELECTROCHEMICAL METHOD

125

Chilton et al. (36) obtained the following equation for heat transfer in agitated vessels with paddle-type impellers: Nu

= 0.36

Re:/3 Pr1’3(p/pt)0*14

(42)

Neglecting the correction term for viscosity, substituting Eq. (40) into Eq. (42) and taking into account that d / D = 0.66 in this experiment gives

jH

= 0.156

Re&”3

(43)

Thus, it is recognized that the average factors of transfer of momentum, mass, and heat are correlated with similar equations. This implies that the heat and mass transfer at the wall are caused mainly by tangential motion of the fluid and that those by vertical motion of fluid are very small.

b. Local Transfer Coe8cients. The vertical distributions of the mass transfer coefficients and local tangential friction factors which are obtained in a specially designed experimental apparatus are shown in Figs. 33 and 34, respectively. It may be recognized again that both transfer phenomena are quite similar. The distribution curves of fz/f and K,/k have peaks at the height of the impeller. The peak becomes lower with an increase of the width of the impeller and separates into two at a certain value of this width. The curves become flatter also with an increase of Reynolds number. The peak seems to be caused by the jet flow issuing from the impeller. Thus the two peaks seem to be made by the two separate jet flows issuing from the upper and lower edges of the impeller. On the other hand, the flat parts of the curves are attributed to the rotating motion of the fluid in the vessel. In Figs. 35 and 36 the local values of fz/2 and jDz at z / H = 0.5 are plotted against Re for b/H = 0.7 and 0.2, respectively. For the large value of b/H, the momentum and mass transfer factors are well correlated by a single curve. However, the plot for the small value of b/H shows that (jD)z,N=0.5 is a little bit larger than (f )z,a=o.5/2.This means that there is a vertical flow at this point by the jet flow from the impeller, and this flow increases the mass transfer coefficients to some extent. c. Local Mass Transfer Fluctuation Intensity. The electrical currents accompanying the fluctuations which were caused by the turbulent flow were measured with an AC amplifier and a square circuit. The vertical distribution of the fluctuation intensity is shown in Fig. 37. For the small value of b / H , this distribution curve has a peak

0.5

Red=16500

b 1

l 5 0

Y

0.5

y

N

I j

1

1

0

1

1

-

FIG. 33. Distribution of the local mass transfer coefficients at the wall of an agitated vessel.

1

. I -

x

3

kIk

0

0

1

-

0

0

1

i--.

1-

0 1

I

F

6

-

I

FIG.34. Distribution of the local friction factors at the wall of an agitated vessel.

T. MIZUSHINA

0 1.0

126

N

I

\

1 1 0

m=0.7

I B

- 3I

n 2

5

HlZ

I

1

Red =7OOO

p-++---

I

2

-i b/H=O/.

- OO

L: l':

1.0

127

THEELECTROCHEMICAL METHOD

Red

FIG.35. Variations of

jo, and fJ2 at zlH = 0.5 with Red for b / H = 0.7. Key:

( 0 ) mass; ( 0 ) momentum.

FIG.36. Variations of j D z and fJ2 at z l H

( 0 ) mass; ( 0 ) momentum.

=

0.5 with Rea for blH

=

0.2. Key:

1 -1 i

T. MIZUSHINA

128 1c

I

; 0.5

~/H=0.7

blH =0.5

b/H =OX

b/H=02

I n

"0

L

01

Q20

01

0

I

01

0

01

FIG.37. Distributions of the local mass transfer fluctuation intensities at the wall of an agitated vessel.

at the height of the impeller whereas for the large value of b/H, it has no such peak as seen in the local mass transfer coefficients distribution.

4.Forced Convection Mass Transfer from Rotating or Vibrating Bodies a. Mass Transfer in the Annulus of Concentric Rotating Cylinders. Many investigators have carried out theoretical analyses of instability and experimental analyses of transport phenomena of tangential flow in the annulus of concentric rotating cylinders. But special difficulties are met in measuring the local values because the wake behind any measuring probe, such as a thermocouple or pitot tube, which is inserted into the annulus between the cylinders is carried round with fluid flow. Mizushina et al. (37) have successfully employed the diffusion-controlled electrolytic mass transfer technique in studies of transport phenomena in fluid flowing tangentially between concentric rotating cylinders. T h e mass transfer rate to the inside surface of a stationary outer cylinder was measured by a depositing reaction of Cu2+ions. Figure 38 shows their experimental apparatus. Two kinds of inner rotors (44 and 58 mm in diameter) and one stationary outer cylinder (94 mm in diameter) are made of copper. The outer cylinder was used as a cathode for measuring the space-time-averaged mass transfer rate, and the inner one was used as the anode, the electrical circuit to which was connected by means of a mercury well. T h e electrolytic

THEELECTROCHEMICAL METHOD Inside surface of outer cylinder

129

I ~

Point cathodes detail

/ couple ' Copper anode

FIG.38. Apparatus for the mass transfer of the flow in an annulus of concentric rotating cylinders.

solutions contained 0.0005-0.0075 mole of CuSO, and 2 moles of H,SO, per liter. To change the viscosity of the solution, 0-55 weight per cent of glycerin was added. The experimental conditions were: Number of revolutions = 13-425 rpm, Taylor number = 59-19,500, and Schmidt number = 3 x 103-7.7x lo5. Figure 39 shows the experimental results of the space-time-averaged mass transfer coefficients together with correlations of the heat transfer coefficients presented by other investigators (38,39). It is seen that there exists an analogy between the heat and mass transfer in this case. In order to measure the local rates of mass transfer, the inside surface of the outer cylinder was equipped with 71 point cathodes which were embedded axially at intervals of 4 mm and angularly at intervals of 45", respectively, and insulated electrically from the surrounding main cathode (See Fig. 40). I n tangential flow between concentric rotating cylinders, a wide range of transition from laminar to turbulent flow occurs in the form of toroidal vortices, i.e., Taylor vortices spaced regularly along the axis. In Fig. 41, an example of an axial distribution of Sherwood numbers and a corresponding pattern of Taylor vortices are indicated. T h e timesmoothed mass transfer rates do not change significantly in the transition

T. MIZUSHINA

130

range, but the fluctuation of the instantaneous local mass transfer rates is very sensitive to the change of flow pattern. The oscillograms of the fluctuating limiting currents to embedded point cathodes and the axial distributions of the time-smoothed, local mass transfer coefficients are shown in Fig. 42. The former distinguishes the steps of transition from laminar to turbulent flow more precisely than the latter.

t

U '

3

m

FIG. 39. Transfer coefficients of the flow in an annulus of concentric rotating cylinders. Key: (1) Nu = 0.88 Talls Pro*s (Aoki et ul.); (2) N u = 0.84 Tall* Pr1/' (Tachibana et ul.); (3) Sh = 0.74 Tal/aS C ' / ~(Mizushina et d.).

Eisenberg et al. (40) obtained the space-time-averaged mass transfer rate to the inner rotating cylinder of an experimental apparatus similar to that shown in Fig. 38 using the redox system of ferrocyanide and ferricyanide. Their correlation is j D = 0.0791

for lo00

(44)

< Re, < 100,OOO.

b. Mass Transfer to Other Rotating or Vibrating Bodies. Noordsij and Rotte (41) applied an electrolytic redox reaction method to a study of the average mass transfer coefficients on a rotating and a vibrating sphere. I n their experiments a nickel sphere (25.3 mm in diameter), to which mass transferred, was used as the cathode.

THEELECTROCHEMICAL METHOD

FIG. 40.

131

Location of the isolated point cathodes in the experimental apparatus

M: 5 points at 4-mm intervals. L: 36 points at 4-mm intervals.

FIG. 41. Taylor vortices and the distribution of Sherwood numbers at the wall of the outer cylinder.

Their results can be summarized as follows: Rotating sphere:

k D / 9 = 10

+0 . 4 3 ( ~ D ~ / v ) ~ / ~

(v/.9)ll3

for 0.8 x lo3 < wD2/v < 27

(45)

Vibrating sphere:

+

k D / B = 2 0.24(+Da/~)l/~ for lo2 < +Da/v < 16 x lo2, 0.03

x lo3

< a / D < 0.063

(46)

T. MIZUSHINA

132 JJA

sh/s50.41

t

1 ' 1

0

1

2

la)

1

3 +

min

FIG.42. Transition from laminar to turbulent flow in an annulus of concentric rotating cylinders. (a) laminar flow, Ta < 41.2; (b) laminar vortex flow, 41.2 < Ta < 800; ( c ) transition flow, Ta g 1000; (d) turbulent vortex flow, 2000 < Ta < 10,000-15,000; (e) turbulent flow, Ta > 15,000.

Okada et al. (42) studied average rates of mass transfer on a rotating disk by means of electrolytic redox reactions. In a cylindrical cell, the distance between a rotating circular disk (60 mm in diameter) and a stationary disk (61 mm in diameter) was varied in the range of 0.2 to 2 cm. Their result is: k D / 9 = 0 . 6 1 ( ~ D ~ / v( v) /~9') ~l 1 3

for 4 x lo3 < wD2/v < 2 x lo* (47)

5 . Forced Convection Mass Transfer from Falling Film, Flow in a Packed Red, Jet Flow, etc.

a. Mass Transfer from Falling Liquid Film. Iribarne et al. (43) reported the results of a theoretical and experimental study on mass transfer between a falling liquid film and a wall, employing a diffusioncontrolled electrolytic reaction in aqueous solutions of 0.005 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 0.5-6 moles of NaOH per liter, which are flowing on the outer surface of a vertical cylinder.

THEELECTROCHEMICAL METHOD

133

The nickel electrode on the surface of the cylinder was divided into fifteen different lengths electrically insulated from each other. These sections were used as cathodes and anodes. It was taken into account that the anode was larger than the cathode. In addition, since the ohmic drop in potential which occurs in a thin film is a serious problem, the distance between cathode and anode should be as short as possible. Their experimental results of the space-time-averaged mass transfer rates for Re < 700 are in good agreement with the predictions by a LkvvCque-type solution, and those for Re > lo00 agree with the solutions of the equation of turbulent flow using the total-viscosity formula of van Driest (43a). Wragg et al. (44) carried out a similar experiment with a flat vertical surface. Their experimental values of the mass transfer coefficients are a little smaller than the predictions of LCv&que'stheoretical equation. Vibrator and velocity transducer

1

voltage power

0-100 kn

FIG.43. Apparatus of Goren and Mani for the mass transfer from an artificially waved liquid layer.

b. Mass Transferfrom an Artificially Waved Liquid Layer. Goren and Mani (45) studied the effect of artificial standing waves of controlled amplitude and frequency on the steady state rate of mass transfer in thin horizontal liquid layers. They measured oxygen transferring through aqueous potassium hydroxide solution to a horizontal silver cathode at the bottom of liquid layer in the diffusion-controlled condition. A schematic view of the apparatus is shown in Fig. 43. The silver cathode in a Plexiglas trough has a porosity of 60-70 yo by volume. The trough was connected to another Plexiglas box which houses a nickel anode. The two electrodes should be separated because the oxygen liberated at the anode by the reverse reaction must be kept away from the cathode. They found that vibrations increased the transfer rate up to

T. MIZUSHINA

134

more than one order of magnitude. Their data at low frequencies are correlated with the following equation:

(i - i0)/(4Fcb/h)= &!UC$~'~T'''&~''

(48)

where ( i - i,) is the increase of electrical current to the cathode by making a wave, A is the surface area of the cathode, a is the amplitude, q5 is the frequency of the wave motion, r is the thickness of the liquid layer on the cathode, and B is the distance between blades of the wave generator.

c. Mass Transfer in Packed Beds. Jolls and Hanratty (46)studied the transition from laminar to turbulent flow in liquid flow through a packed bed of spheres of I-in. diameter. T h e transition was detected by oscillograms of the instantaneous local mass transfer rate to one of the packings. T h e test sphere was plated with nickel and fourteen isolated nickel cathodes were embedded in its surface so as to make a meridian circle around the sphere. T h e solution contained 0.01 mole of K,Fe(CN), and K,Fe(CN), , respectively, and 1 mole of NaOH per liter. Its Schmidt number was 1700. Keeping the main cathodes on the test sphere inactive, they measured the fluctuating mass transfer rates to the isolated cathodes and found that the transition occurred at Re = 110-150 where the Reynolds numbers were based on the diameter of the sphere and the fluid velocity in an empty column. They (46a) also correlated the space-time-averaged mass transfer coefficients to a single sphere in a packed bed by the following equations: Sh = 1.59 Re0eS6Sc1I3

for Re > 140

Sh = 1.44

for

Sc1I3

35

< Re < 140

(49)

(50)

These correlations are in good agreement with the experimental results of the heat transfer coefficients between air and a single sphere in a packed bed but they predict coefficients significantly larger than the experimental values of the mass transfer coefficients in a packed bed in which all of the packing are active in the mass transfer. This difference is easily explained by the fact that the thickness of the concentration boundary layer of a single active sphere is much less than that of a bed of many active spheres. Their measurements of the local mass transfer coefficients varied more around a meridian than on an equator owing to the difference of the variations of the boundary layer thickness. Ito et al. in Osaka University are going to study the mass transfer between packing solids and flowing liquid film in a packed column in

THEELECTROCHEMICAL METHOD

135

which nitrogen gas flows up. This experiment is of practical importance as a model of gas-liquid reaction in a catalyst-packed column.

d. Mass Transfer from a Jet Flow. T h e local mass transfer rates in a wall jet have been studied mainly by means of solid dissolution. But the electrolytic technique has a much larger advantage than the solid dissolution method in measuring the fluctuations of the local mass transfer rates. Kataoka et al. of Kobe University are studying the local mass transfer in a two-dimensional wall jet with a deposition reaction of Cuz+ ions. T h e experimental values of the Sherwood numbers are approximately constant in the impingement region and then decrease in proportion to the dimensionless distance from the stagnation point. Mizushina et al. of Kyoto University studied mass transfer from two-dimensional multiple impinging and sucking jets as shown in Fig. 44. This study was done for a simulation of mass transfer from a Taylor vortex flow which was described in Section 111, C, 4,a. T h e space-time-averaged mass transfer coefficients are correlated as follows: Sh = 0.24 Sc1l3 for lo2 < Re < lo4 (51) where Re

=

2 ~ , ( 7 D / S )B~/ vl ~

u, is the fluid velocity in nozzle,

D is the gap of nozzle, S is the distance between nozzle exit and wall, and B is the distance between impinging and sucking nozzles. T h e distribution of the local time-smoothed mass transfer coefficients

0'

0

5

I

1

I

10

15

20

Z(cm) Embedded ~ o i n tcathodes

FIG.44.

Multiple impinging and sucking jets.

T. MIZUSHINA

136

in laminar vortex flow is shown in Fig. 44.T h e flow pattern was detected by oscillograms of local fluctuating limiting currents to be laminar for Re < 200, laminar vortex for 200 < Re < 500, transition for 500 < Re < 1050 and turbulent for Re > 1050. Weder (47) measured the mass transfer rates from liquid to a horizontal plate around a single circular gas nozzle in the center of the plate. This study was carried out as a model of boiling heat transfer around a bubble evolved on a heated plate. Eleven concentric nickel ring-shaped cathodes are embedded around the gas nozzle to measure the local time-smoothed mass transfer coefficients as a function of the distance from the nozzle.

IV. Application to Shear Stress Measurements

A. PRINCIPLE AND METHOD OF MEASUREMENTS

It is very difficult to measure directly the velocity gradient close to a surface because the boundary layer thickness is so small that any measuring instrument disturbs the flow. However, a diffusion-controlled electrochemical reaction at small cathodes embedded in the wall can be applied to obtain the velocity gradient at the wall. If the test electrode is made quite small in length, the concentration boundary layer is very thin owing to a large value of the Schmidt number. Therefore, the curvature of the surface may be neglected, and it can be assumed that the velocity gradient in the concentration boundary layer is linear. Thus, the electrode is analogous to a constant-temperature hot-wire anemometer with the characteristics that the surface concentration is constant and the electrical current in the circuit depends on the surface shear stress. Furthermore, several limitations on both measurements are also similar. High frequency velocity fluctuations cannot be measured by either method owing to the thermal inertia of the wire of the anemometer and the capacitance effect of the concentration boundary layer over the electrode. Nonlinear response is caused in both systems by large turbulent intensities. I n addition, if there is nonuniform flow over the wire length or the electrode width, it may result in some error. Consider a rectangular cathode embedded in the surface with its short side parallel to the direction of flow. T h e length L of the cathode is much smaller than the width so as to make the concentration boundary layer two-dimensional. Assuming use of a redox system of ferro-ferricyanide, the mass balance for the ferricyanide ion gives aclat

+

ac/ax +

atlay

= 9 a2c/ap

(52)

THEELECTROCHEMICAL METHOD

137

where x and y are the coordinates in the direction of the flow and perpendicular to the surface, respectively. Boundary condition: c=O

at y = O ,

c =cb

at

y =

c=cb

at

x
O
(53)

a,

T h e diffusion in the flow direction has been neglected in Eq. (52) since the condition of s L z / 9 > 5000 is satisfied in the ordinary case owing to the very small values of the diffusion coefficients of the ions. But it must be taken into account in the flow near the separation point. T h e velocity and the concentration are expressed as the sum of timesmoothed and fluctuating components. u=ii+u‘,

Z)=V+Z)’,

c=c+c’

(54)

Since the velocity gradient can be assumed to be linear in the very thin concentration boundary layer, ii is given by

u = sy

(55)

From the continuity equation, E is calculated as

where s is the time-smoothed velocity gradient at the wall and is a function of x. In this region, the fluctuating velocity can be represented by

where s‘ is the fluctuation in the velocity gradient at the wall and is a function of t and x. Using the above-mentioned equations of velocity distribution and neglecting the second-order terms in fluctuating equations, give the equations of time-smoothed and fluctuating concentration fields as follows: For time smoothed concentration,

138

T. MIZUSHINA

Boundary condition: E=O

at

y=O,

C=c, E=c,

at

y=00 x=O

at

O
(60)

For fluctuating concentration,

Boundary condition: c' = 0

at

y

c'=O

at

x=O

=

00

and y = 0,

(62)

These equations can be simplified in each case and solved to relate the mass transfer coefficient to the velocity gradient at the wall.

1 . Shear Stress in a Fully Developed Flow a . Time-Smoothed Velocity Gradient. For a fully developed flow, 0. Therefore, from Eq. (56), the second term of Eq. (59) becomes zero. Hence, the equation of time-smoothed concentration is simplified as follows: sy aqax = 9 a z E / a p (63) Boundary condition: =

t=cb

at

E=c,

at y = c o

s=O

at y = O ,

x=O

O
T h e solution of Eqs. (63) and (64) is

where

-77 = y(s/99x)1/3

T h e average mass transfer coefficient over the electrode surface is

THEELECTROCHEMICAL METHOD therefore s =

139

1.90 k 3 L / 9

When circular electrodes are used instead of rectangular ones, the effective length in the direction of flow is calculated by L,

(69)

= 0.81360

b. Velocity Gradient Fluctuation. as follows:

In pipe flow, Eq. (61) is simplified

Boundary condition: c'=O

at y =

c' = 0

at x = 0, at y = O ,

c'=O

00,

(71) O
Analyzing the fluctuations of the mass transfer coefficient and the velocity gradient into harmonic oscillations, one solves Eq. (70) for the oscillation of frequency, 4, to give &'/s = 3fl

+ O . O ~ ( ~ T C # J L ~)/ }~ / W&'/k~ S ~ / ~ 1'2

(72)

where f,' and L,' are the amplitudes of oscillations of frequency, 4,in the velocity gradient and the mass transfer coefficient. When the time constant or L2/9sa is very small as a result of making the electrode size and the Schmidt number small, the capacitance effect is negligible except for very large values of 4. T h e velocity gradient fluctuation intensity is calculated from the mass transfer coefficient fluctuation intensity and the spectral distribution function of the mass transfer fluctuations, W, by the following equation: )

((S')2)ll2/s

[SrnW4(l + 0.06(2m$L2/3/91/3~2/3)2f 41"' (73)

3[((K')")1/2/k]

+

0

Since W, for large values of is much smaller than that for small values of 4,Eq. (73)practically becomes ((s')2)'/2/S

=

3(0")'/"k

(74)

Since Eqs. ( 5 5 ) and (57) are assumed near the wall, ((u')2)'/2/u = ((s')2)1/2/s

(75)

140

T. MIZUSHINA

Problems which are similar to those for a hot-wire anemometer are encountered in using this method, i.e., the time response, the nonuniformity of the flow over the cathode, and the effect of nonlinearities. T h e time response has been already taken into account in the calculations. Since nonuniformities in the structure of turbulence are much greater in the circumferential direction than in the direction of flow, the measurements must be corrected, if necessary, by the use of measured values of the circumferential correlation coefficient. T h e assumption of the linearities is not a serious problem since it was found to give an error of less than 3 yo.

2. Shear Stress in the Boundary Layer Solving Eq. (59) by using a similarity transformation gives the concentration gradient at a position on the electrode surface.

Except in the region close to the front stagnation point (0 < 5") in the case of cross flow, the variation of s over the small length of cathode is negligible. Assuming that s is constant, one integrates Eq. (76) from 0 to L to give the space-averaged value of the mass transfer coefficient, k, over the cathode k L / 9 = (l/cb)

&/8y 0

I

dx = 0 . 8 0 7 ( ~ L ~ / 9 ) ~ / ~

1/-0

(77)

therefore s = 1.90(ksL/92)

B. SHEARSTRESSIN

A

WELL-DEVELOPED FLOW

Mitchell and Hanratty (48) measured the shear stress and the velocity fluctuations in a fully developed turbulent flow using measuring cathodes embedded in the wall of I-in. i.d. Lucite pipe. A fore-flow section of 180 diameters in length preceded the test section. T h e electrolyte solution contained 0.01 mole of K,Fe(CN), , 0.01 mole of K,Fe(CN),, and 2 moles of NaOH per liter. T h e test electrodes consisted of several nickel sheets which has a length of 0.003-0.021 in. and a width of 0.020 to 0.062 in. On the other hand, Reiss and Hanratty (49) used three sizes of nickel wire of 0.0398-0.1636 cm in diameter as cathodes which were arranged so as to measure longitudinal and circumferential correlations.

THEELECTROCHEMICAL METHOD

141

Different configurations of the rectangular cathodes such as shown in Fig, 45(a)-(d) were used for different purposes, i.e., (a) for the longitudinal velocity intensity, (b) for the circumferential correlation coefficient, (c) for the longitudinal correlation coefficient, and (d) for the circumferential velocity intensity. T o check the usefulness of this technique, the following experiments were carried out.

FIG.45.

Configurations of rectangular cathodes for various measurements.

A momentum balance in fully developed turbulent flow in a circular tube gives s = Um2f/2V

(79)

From Eqs. (67) and (79)

f

=

3.80(k/u,,J3 ( S C ) (L/D)(Re) ~

(80)

Thus the values of the friction factor can be calculated from the measurement of the mass transfer coefficients. On the other hand, the friction factors are calculated from Blasius' equation f

= 0.079

(81)

I n Fig. 46, the values of the friction factors calculated from Eqs. (80) and (81) are compared. T h e data obtained in Kyoto University in a circular tube are plotted also. T h e agreements of both predictions indicate that this technique is useful. Mitchell and Hanratty measured longitudinal and circumferential correlation coefficients also, and found that the circumferential integral scale was only about one-thirtieth of the longitudinal one.

T. MIZUSHINA

142

Using Eqs. (74) and (75), they calculated the velocity fluctuation intensity near the wall and obtained its value as 0.32, independent of Reynolds number. This result agrees with the measurements by other investigators (50)-(52) with the hot-wire anemometer.

f x lo3 (Blasius’ equation)

FIG.46. Comparison of the friction factors calculated from measurements and LID = 0.003; ( 0 )LID = 0.005; ( 0 ) LID = 0.007;( A ) Blasius’ equation. Key: (0) L/D = 0.0021 (Hanrattyet d.); ( 0 ) LID = 0.0424, ( r ) L / D= 0.0410 (Mizushinaet d).

C. SHEARSTRESSIN

THE

BOUNDARYLAYER

Dimopoulos and Hanratty (30) studied flow crossing cylinders and measured the velocity gradient in the boundary layer as described in Section 111, C, 2. Direct application of Eq. (77) to this case causes an error because diffusion in the x-direction and natural convection by density difference in the concentration boundary layer are neglected in deriving Eq. (52). Assuming that the interaction of the effects of natural convection and diffusion in the flow direction are negligible because both effects are small, one adds the correction terms for each effect to Eq. (77) as in Eq. (82). kL/B = 0.807(~L~/B)~’~ 0.19(~L~/9)-~’~

+

f 0.253(Gr,/Re,)(S~/sL~/B)~’~ sin B

(82)

THEELECTROCHEMICAL METHOD

143

where the plus sign is for aiding Aows and the minus sign is for opposing flows. These correction terms, however, are important only at very low Reynolds numbers and near the separation point. The test cylinder of 1-in. diameter was mounted in a 1 x 1 ft duct as the axis of the cylinder was perpendicular to the flow direction. T h e test section was connected to a gravity flow tunnel. Two kinds of test cylinders were used, one for velocity gradient measurements and another for mass transfer studies. The former was equipped with a platinum cathode of 0.020 x 0.500 in., and the latter with an isolated round platinum cathode of 0.020-in. diameter which was insulated electrically from the main cathode covering the whole cylinder surface. Both cylinders were rotated so as to change the position of the test cathode with respect to the front stagnation point. The mass transfer measurements have been described in Section 111, C, 2. A result of the measurements of the velocity gradient at the wall for Re = 151 is plotted in Fig. 47, and compared with the boundary layer calculation for Re = 174. T h e measurments agree with the calculations fairly well.

0 deg FIG. 41. Comparison between the velocity gradient measurements and the boundary Re = 174 layer calculation in cross flow. Key: (0)Re = 151 (measurements); (-) (boundary layer calculations).

T. MIZUSHINA

144

T h e separation angles determined by finding the position where the velocity gradient became zero are plotted against Reynolds number in Fig. 48 and are in reasonable agreement with the results of other investigators (53). I60

- 140 --

-

I

-

I

I

-

-

0

-

U

120-

-

O D

0 0 0

O g

rooI

I

al

Q

-

I

Re

FIG. 48. Measurements of the separation angles from the front stagnation point of the cylinders. Key: (0) 4-in. splitter plate; (0) without splitter plate.

V. Application to Fluid Velocity Measurements

A.

PRINCIPLE AND

METHODOF MEASUREMENTS

For measuring the velocity of air flow and its fluctuation, there are many devices. For liquid flow, however, convenient and reliable measuring techniques have not been developed. T h e pitot tube has a defect in response to the velocity fluctuations, and the hot-wire anemometer needs compensations for phase shift and amplitude and has limits in mechanical strength and operating temperature. Furthermore, it has a limit in the length of wire because the shorter wire causes the larger error due to the cooling effect of the supports. A diffusion-controlled electrochemical process on an extremely small cathode was found applicable to these purposes, and a device and technique for this process have been developed. T h e compensations for phase shift and amplitude attenuation of a fluctuating signal is not a serious problem because the measuring probe has practically no capacity for the transferred quantity. I n addition, this probe is extremely sensitive to low velocities and can measure the value of a few mmlsec. T h e maximum velocity which can be measured by this method was discussed in Section 11. O n the other hand, a capacitance effect of the concentration

THEELECTROCHEMICAL METHOD

145

boundary layer over the cathode limits the frequency range of the velocity fluctuations to be measured. Ranz (4) first suggested the usefulness of this method in measuring the liquid flow velocity and measured it in several kinds of electrolytic solutions. From the current-voltage curves obtained, he discussed the chemical reaction mechanism. The measuring probe which he used was a cylindrical type as shown in Fig. 49(a). Several types of probe other than this have been developed. Those are also shown in Fig. 49. A spherical type probe (b) was devised by Ito and Urushiyama (54) to determine the flow direction. A blunt-nose type (c) and hot-wire type (d) probes are used by Mizushina et al. in Kyoto University. The mass transfer coefficients calculated from the limiting currents to those electrodes are related to the liquid velocity. Since the electrodes in this case are so small, the boundary layers on these electrode surfaces are always laminar. The relations between Sh and Re for cylindrical electrodes was described in Section 111, C, 2. When the electrode is mounted at the front stagnation point of a sphere, the relations given experimentally by Brown et al. (55) are Sh

=

s

1.29 Re0.5S C O * ~ ~

(83)

,Stainless steel support

Glass

surface coating (a )

,Glass fusing

tfBare lcl

platinum surface

Glass fusing support platinum surface (b)

f t 1

Plat1 num wire

(d)

FIG.49. Various types of measuring probes. (a) cylindrical type, (b) spherical type, (c) blunt-nose type, (d) hot-wire type.

T. MIZUSHINA

146

and analytically by Sibulkin (56) are Sh

=

1.32 Re0.5Sco**

(84)

I n general, the mass transfer rate to a laminar boundary layer is proportional to Re1/2, so the limiting current is proportional to u1/2,i.e.,

where 0: is the natural convection term and /3 is the laminar convection term. For practical use, LY. and /3 are determined by experimental calibrations. As described in Section IV, A, this method is analogous to the hot-wire anemometer, not only in its characteristics but also in the limits of application. As a result, considerations similar to those for hot-wire anemometers must be given to the application of the electrochemical method. Lighthill (57) solved the linearized problem of the response of skin friction and heat transfer to the fluctuation in the velocity of incompressible laminar two-dimensional flow. He calculated the amplitude ratio relative to the quasi-steady amplitude and the phase lag in two-dimensional stagnation flow for Pr = 0.7. This calculation procedure was applied to axisymmetrical flows. I n this calculation it is assumed that the velocity at the outer edge of the boundary layer is equal to K X , where K is a constant. The results of the amplitude ratio for Pr or Sc = 0.7 and 2431 are represented in Fig. 50. It is seen that the amplitude ratio decreases more rapidly in the case of a higher Schmidt number and that the response of the mass transfer to liquid velocity fluctuation is worse than that of the heat transfer to air. For the comparison of the response between hot-wire anemometers and the electrochemical method, the critical frequencies, +crit , at which the amplitude ratios become 0.9 when the velocity measuring probes are placed in the stream of u = 100 cm/sec, are given as follows: Pr

= 0.7,

SC = 2431,

wire (D = 0.001 cm),

$crit =

wire ( D = 0.001 cm),

$crtt = 6.50

sphere (D = 0.001 cm),

$crlt

=

4.59 x lo4 Hz

x 10s Hz

5.16 x lo3 Hz

In the use of the electrochemical method in measuring the fluctuating velocity, this limitation in the ability of response must be always taken into account.

147

THEELECTROCHEMICAL METHOD 1.0

0.8

0.2

0. L

0.6

0.8

1.0

W I P

FIG. 50. Amplitude ratio in response to the fluctuations.

B. TIME-SMOOTHED VELOCITY 1. Time-Smoothed Velocity in a Tube Flow

To approach a wall in measuring the velocity of fluid flow, it is necessary to make the probe as small as possible. However, a small pitot tube is inadequate for measuring the fluctuating velocity because it needs a very long response time, especially in liquid flow. For this purpose the electrochemical method is very useful because the probe can be made very small to measure the velocity of bulk flow from the center to the vicinity of the wall and another probe embedded in the wall can measure the velocity gradient in the laminar sublayer at the wall as described in Section IV, and, as a result, the velocity profile in the whole cross section of flow can be obtained easily. I n addition, this method is conveniently used to determine the direction of flow. A hot-wire type probe designed for this purpose can be constructed smaller than a hotwire anemometer, which has a lower limit in wire length owing to the

148

T. MIZUSHINA

cooling effect of the supports. A blunt-nose type probe has an appreciable advantage. Flowing dirt particles hardly cling to the probe during an experiment, and the output signal is most reliable. These two types of probes, which have been developed and used in Kyoto University, are shown in Fig. 49. The hot-wire type probe has a platinum wire cathode of 0.003-cm diameter at the tips of the prongs, which are coated with glass. The blunt-nose type probe has a platinum wire cathode of 0.01-cm diameter which is embedded in the blunt glass nose and smoothed flush with nose surface so that only the tip of the wire is exposed. Before the experimental run, the probe must be calibrated with the electrolytic solution which will be used in the experiment. I n Kyoto University, a redox system containing 0.01 mole of K,Fe(CN)6 , 0.01 mole of K,Fe(CN), , and 2.0 moles of KOH/l. is used. The apparatus for calibration in Kyoto University consists of an arm of brass which is fixed perpendicular to a rotating shaft and a plastic basin which is of doughnut shape and placed around the shaft. The probe attached at the end of the arm is dipped in the electrolytic solution contained in the basin. By changing the rotating speed of the shaft, the traveling speed of the probe can be changed from 5 cm/sec to 5 m/sec. T h e probe must be attached to its place very carefully to meet the direction of flow. I n the basin a narrow passage for the probe is made with two parallel screens which depress the wake flow of the probe. In addition, the diameter of the circle of the passage is 90 cm, which is large enough to decay the wake within the period of one cycle. A typical plot of velocity versus limiting current for the calibration is shown in Fig. 51. In the calibration of a hot-wire type probe, its angular property is investigated by directing the axis of wire at various angles to its moving direction. The rate of mass transfer is determined mainly by the velocity component perpendicular to the wire axis. The effect of the velocity component parallel to the wire becomes noticeable only when the normal velocity component is very small. Thus, if the wire makes an angle t,b with the direction of flow, the mass transfer is accomplished essentially by the component usin#, and onIy at small values of $ does the component u cos t,b also assume importance. Consequently, the effective value of velocity is obtained approximately from the relation

where the factor K has a value between 0.1 and 0.4 and increases with decreasing velocity. A typical calibration curve for angular property is shown in Fig. 52, from which K is determined to be 0.39. However, for

THEELECTROCHEMICAL METHOD

149

the range of practical use, i.e., 30” < i+h 5 go”, it seems enough to consider the effective value of velocity to be u sinn 4. Thus, the following equation is obtained instead of Eq. (85):

i

=

01

+ p(sinn

t,/~ * ~ 4 ) ~ ’ ~

where a,8, and n are determined by calibration.

fi

(cm”2/sec1’2 )

FIG. 51. Calibration of the measuring probe.

I

90

I

70

1

I

1

50

30

JI ( d e g 1

I

FIG. 52. Calibration of the angular property of the hot-wire type probe.

10

150

T. MIZUSHINA

T h e essential part of the experimental apparatus in Kyoto University to measure the time-smoothed velocity profile in tube flow is a polyvinyl chloride plastic tube of 2-in. diameter, through which an oxygen-free redox system is circulated at controlled rates of flow. T h e test tube has a sharp leading edge with a trip wire from which the turbulent boundary layer develops. At various cross sections in the fully developed turbulent region and in “the entrance region” from the leading edge to the cross section where the boundary layer thickness becomes equal to the radius of tube, measurements of velocity profiles were made with a blunt-nose type probe of platinum cathode of 0.01-cm diameter. T h e anode is located at the tube wall downstream from the measuring cross section. T h e measurements of velocity were also carried out by a total pressure tube of 0.6-mm 0.d. T o achieve a flat velocity profile and a small turbulence level, a Hori type (58) water tunnel was constructed. T h e liquid from the convergent nozzle is passed into the leading edge of the test tube where the flow close to the wall of the nozzle is cut off to the bypass. Thus, a fairly flat velocity profile at the leading edge was obtained. On the other hand, the velocity gradients near the wall were measured by the electrochemical method described in Section IV. By combining the results of both experiments, the velocity distributions in fully developed tube flow were obtained. T h e result is represented by an ordinary equation of universal profile as shown in Fig. 53.

FIG. 53. Velocity distribution in a tube flow. Key: (- - -) fully developed region; entrance region; (o)x/D= 5.92; ( 0 ) x/D= 12.30;(0) x / D = 15.47;(A) x / D = 28.83; ( 0 )x/D= 73.40. (-)

151

THEELECTROCHEMICAL METHOD 2. Time-Smoothed Velocity in the Boundary Layer

T h e velocity distributions in the boundary layer in the entrance region of the same experimental apparatus were plotted in Fig. 53. As seen from Fig. 53, the similarity in velocity distributions at various cross sections may be assumed to be expressed in the form Uf =

C(y+)l’m

(88)

The value of m is a characteristic value in the entrance region and a little smaller than that in the fully developed region as follows. m

= 6.6,

C

=

7.6

at

Re,,, = 5 x lo4

m

=

6.7,

C

=

7.9

at

Re,,,

=

lo5

Using Eq. (88) and these experimental values, one can calculate the displacement thickness Sl and the momentum thickness 6, in the entrance region of the tube from the following equations:

jD’’

= (2/O)

(u/ub)(l

-r)

- u/ub)(D/2

dy

(90)

The shape parameters defined as P = 6J6, are calculated and plotted in Fig. 54.This indicates that the turbulent boundary layer develops from the leading edge owing to the existence of a trip wire. The macroscopic mass balance for the circular tube flow leads =

0

10

(D/4)(1

(91)

- um/ub)

20

30

40

X I D

FIG. 54. Shape parameter in the entrance region of a tube flow. Key: (0)Re, 5 x lo4; (0)Re, = lo6.

=

152

T. MIZUSHINA

Substituting Eq. (89) into this equation, one can obtain the experimental values of u&,. On the other hand, the momentum equation written in terms of P and 6, is given as

o'oal 0.06

xlD

FIG.55. Development of the momentum thickness in the entrance region of a tube theory, Re, = 5 x lo4; (---) Re, = lo6.

flow. Key: (-*-) Re, = 5 x 10';

theory, Re, = 10'; (0)exp.,

( 0 ) exp., 1

1

I

/'

/-

1.20-

/'

1

1

1.25-

0

/'

/

/'

1

0

.HO

,/'

-

/94

1.15-

'6

-

/',)

/;/'

1.101.05-

<;.'

4

&

-

0

/

-

p'

1.00

1

-

B

)I(

1

I

I

I

I

1

THEELECTROCHEMICAL METHOD

153

Solving this equation by the increment method using an electric computer gives the values of 6, , 6, , and u&, as functions of x. The experimental and predicted values of 26,/D and ub/umare compared in Fig. 55 and Fig. 56, respectively.

C. FLUCTUATING VELOCITY 1. Fluctuating Velocity in a Tube Flow The experimental apparatus is the same as described in Section V, B. The electric circuit used to measure the velocity fluctuation is shown in Fig. 57. The electric potential of the probe was held at -0.200V relative to the mercury oxide reference electrode with a potentiostat. Thus Cathode

Flow

-

DC component

- DC amp.

T I I \ r n n o

-Potentio- d e stat -

rms of fluctuation

c-

Power spectrum a nalvr er

FIG. 57. Electrical circuit for measuring the fluctuating velocities.

limiting currents through the circuit were ensured. The fluctuation of current was amplified 106 times, and then root-mean-squares and power spectrum densities of output signals were measured. The measured values of turbulent intensity in flow direction, ((u')2)1/2/u, at x / D = 4.638, 12.05, and 29.77 downstream from the leading edge are shown in Fig. 58. It may be noticed that the similarity in the distribution of turbulent intensity has been established even in the entrance region of tube flow. The relative turbulent intensity of free stream was roughly 0.7%. They are also compared with Klebanoff's results (59) measured in flow along a flat plate with no pressure gradient. It is shown that these results agree. The frequency spectrum densities at various positions in a cross section in the entrance region are shown in Fig. 59. The measurements

"i \

0 6

Flow along flat p l a t e

4

2

0.5

1.0

1.5

2.0

YJb

FIG.58. Distribution of turbulent intensities in the boundary layer in the entrance region of a tube flow. Key: ( A ) x/D = 4.638;(0) x / D = 12.05; (0)x / D = 29.77.

FIG. 59. Power spectrum density of u' in the entrance region of a tube flow. Key:

( A ) y / s = 1.086; ( 0 ) y/6 = 0.136; Mizushinaetal. (x/D = 12.05); (---) (-) y/S = 0.20 Klebanoff (with zero-pressure gradient).

y/8

=

1.00,

THEELECTROCHEMICAL METHOD

155

by Klebanoff (59) in a boundary layer on a flat plate with zero pressure gradient are also represented. There is a remarkable similarity in these measurements. T h e discrepancy between these measurements at very low frequency seems due to the difference in the mechanism of intermittent flow in the region of the outer side of the boundary layer. As described in Section V, A, the electrochemical method has a limit in the frequency which can be measured. This limit corresponds to cr 20 in Fig. 59. Accordingly, the measurements up to u = 20 are reliable. T h e correlation of the plots has two extensive regions. One is the inertial subrange where the spectrum varies in proportion to c r 5 I 3 . T h e other is the largest wave number region where viscosity plays an important role and the spectrum is proportional to 0-'. At the end of this section it should be mentioned that there was no difference in spectra obtained with different applied electrode potentials. This indicates that the reaction kinetics at the electrode surface does not influence the high frequency components of the fluctuations of mass transfer measurements and that the concentration of the diffusing ion is always regarded as zero at the surface of electrodes.

2. Fluctuating Velocity in an Agitated Vessel I n Kyoto University, time-smoothed and fluctuating velocities in an agitated vessel are measured with a hot-wire type electrode. T h e experiments have been carried out in cross sections where the vertical velocity component does not have a significant efiect on the measurements of the tangential and the radial velocity component. T h e experimental results are shown in Figs. 60-63. Figure 60 shows the equiturbulent intensity curves from which we may guess the movement of the turbulent eddies. I n Fig. 61, the radial distributions of the time-smoothed tangential and radial velocities, turbulent intensity in tangential and radial directions, and Reynolds stress and longitudinal spatial micro scale in the cross section at the center of impeller blades are plotted. All measured values have been reduced to a nondimensional form by dividing by a characteristic length d or a velocity Od where d is the impeller diameter and Q is the rotation speed, respectively. T h e Reynolds stress was calculated from the measured values of turbulent intensities in the direction of *45" to the tangential by the following equation:

- ((-)1/2 is the turbulent intensity in $45" where ( ( ~ - ' ) z ) l / ~is the turbulent intensity in -45" direction.

direction and

T. MIZUSHINA

156

The spatial micro scale in the flow direction, A,, was obtained from the time correlation curve of the fluctuating velocity and the Taylor hypothesis. Figure 62 shows the energy balance in the jet flow issuing from the impeller. From this, the following conclusions may be obtained: (1) The energy gain by convection balances the work by pressure, and the work by shear stress balances the conversion to turbulent energy. (2) The energy of the time-smoothed velocity first converts into turbulent energy which then dissipates into heat but does not directly convert into heat.

3.0 cm/sec

zt

0.5

0

0

0.5

I .o

r+

FIG.60. Equiturbulent intensity curves in an agitated vessel.

Figure 63 shows the turbulent energy balance in the same jet flow. In this diagram the loss by dissipation was calculated by assuming the isotropic turbulence theory in this flow. The gain by production is the same as the loss by conversion to turbulent energy in Fig. 62. From the total turbulent energy balance, the loss by turbulent diffusion was calculated and represented by a dotted curve. This result indicates that the local production is not equal to the local dissipation because of advection and diffusion effects and that the turbulent energy which is produced near the tip of the impeller is transferred toward the wall of the vessel.

THEELECTROCHEMICAL METHOD

157

0.10 '

0.05

0-

I .o

0.5

rt G.

61.

Distribution of the velocities and turbulent characters in an agitated vessel.

24

1.0 16 .-C

C

.-Q

rd

13

8

0

0

111 I 0

0.5

-8

UI

0"

4

-1 6

-0.5

1.o

0.5

r*

r+ FIG. 62. Energy balance in an agitated vessel. convection by mean motion; ( 0 ) work by Key: (0) pressure; ( x ) work by shear stress; (+) conversion to turbulent energy.

1.0

0.5 FIG. 63.

Turbulent energy balance in an agitated vessel. Key: (0)production; ( 0 ) dissipation; ( x ) advection, (- - -) turbulent diffusion.

THEELECTROCHEMICAL METHOD

159

ACKNOWLEDGMENTS This survey has been prepared in cooperation with Professor R. Ito of Osaka University, Assistant Professor F. Ogino, Messrs H. Ueda and Y. Hirata of Kyoto University, Mr. S. Hiraoka of Nagoya Institute of Technology, and Mr. K. Kataoka of Kobe University. The author acknowledges their great help. He also wishes to express his thanks to Professor P. Grassmann of E. T. H., Zurich, who first stimulated the interest of the author and his co-workers in the electrochemical method.

SYMBOLS A A, a

6

C

C

D

9 d E

F

f

g Gr

H 1

iD

in

K K+

k L m

N

no n

P

Pr

R

R18

Re

sc Sh

surface area of mass transfer surface area of isolated cathode at x amplitude width of agitating impeller experimental constant concentration diameter moiecular diffusivity diameter of agitating impeller electromotive force Faraday constant friction factor gravitational acceleration Grashof number depth of liquid in agitated vessel electric current j factor of mass transfer J’ factor of heat transfer experimental constant nondimensional mass transfer rate = k/o* mass transfer coefficient or its space-time-averaged value length of mass transfer section exponent mass transfer rate valence charge of an ion exponent shape parameter = &/A, Prandtl number gas constant correlation coefficient of mass transfer fluctuations Reynolds number Schmidt number Sherwood number or Nusselt number of mass transfer

velocity gradient near wall absolute temperature Ta Taylor number t time u velocity in x- or tangential direction w velocity in y- or radial direction w* friction velocity W spectral distribution function w velocity in z-direction x, y , z coordinates s

T

experimental constant experimental constant activity displacement thickness of boundary layer momentum thickness of boundary layer eddy diffusivity of mass eddy diffusivity of momentum angle from front stagnation point experimental constant integral scale of fluctuation spatial microscale in fiow direction viscosity kinematic viscosity current density by hydrogen ion hydrogen overvoltage density number of wave frequency electrostatic potential angle of wire with the direction of flow number of rotations per unit time angular velocity

T. MIZUSHINA SUPERSCRIPTS

+

fluctuating nondimensional

-

b

i

m

OVERLINES

,.

SU~SCRIPTS

amplitude time-smoothed

x

z co

bulk interface mean local value at x local value at z fully developed region

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161

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