Negative fouling resistances: The effect of surface roughness

Chemrcal Engrneering Science. Vol. 43, No. 4. pp. 829-838. Printed in Great Britain.

NEGATIVE

School

1988.

0009~2509 88 f3.00 + 0.00 Pergamon Press p,c

FOULING RESISTANCES: THE SURFACE ROUGHNESS

B. D. CRITTENDEN+ of Chemical Engineering, University or Bath, Claverton

EFFECT

Down.

Bath BA2

OF

7AY.

U.K.

N. J. ALDERMAN Schlumberger Cambridge Research Ltd, Cambridge, U.K. (Received

19 January

1987; accepted

21 July

1987)

Abstract-The limiting diffusion current technique has been used to study the effect of deposit characteristics on mass transfer rates in a simple mixing cell. The presence of particles on the cathode has two effects on mass transfer rates. Firstly, the resistance to mass transfer through the deposit increases with the number of particle layers. Secondly, the presence of particles, large with respect to the original roughness of the unfouled surface, reduces the resistance to mass transfer through the fluid film at the fluid-deposit interface. At relatively high Reynolds numbers the latter effect can predominate to such an extent that the overall resistance to mass transfer can become less than that of the unfouled surface. The application of these findings to the analogous case of heat transfer across a fouled surface explains the existence of apparently negative fouling resistances which are often observed shortly after start-up from clean conditions. INTRODUCTION

The process analysis approach to fouling research calls, inter alia, for a better understanding of the influence of deposit characteristics such as thickness, mass, morphology, physical properties and chemical and microbiological analyses (Somers&es, 1981). The most important physical properties are density, thermal conductivity, porosity, roughness, adhesion and strength. Surface roughness can have a significant effect on momentum, mass and heat transfer under turbulent flow conditions. The magnitude of this roughness effect is dependent on the ratio of roughness height to hydraulic diameter or to the laminar sub-layer thickness (Perkins and McEligot, 1973). Whether transfer rates are increased or decreased depends largely on the nature of the roughness, that is on the number per unit surface area, size, shape, orientation and distribution of the roughness elements (Mahato and Shemilt, 1968). If a fouling uncommon

resistance is calculated

from eq. (l), it is not

to obtain

f+;_;

(1) c

D

deceptively low or even negative values of R,during the initial stages of the fouling process (Epstein, 1978; Bott and Gudmundsson, 1978; Crittenden and Khater, 1987). It has been suggested that such an event is caused by small amounts of deposit creating a rough surface in the early stages of fouling and thereby increasing the film heat transfer coefficient to an extent sufficient to counteract the additional thermal resistance due to the deposit itself (Epstein, 1978; Crittenden

+To whom

correspondence should be addressed.

and Khater, 1987). However, there is no experimental validation of this hypothesis. Variations in surface roughness occur as fouling proceeds. Roughness eradication, that is an increase in substrate surface roughness due to rapid growth on the peaks followed by a decrease in this roughness due to the filling up of the valleys, can take place rapidly when the constituents

of a fouling

fluid

have

dimensions

comparable to the roughness dimensions (Rankin and Adamson, 1973; Lund and Sandu, 1981). Even more complex effects can occur in biofilm formation (Bott, 1979). The present research was initiated in order to study the relative effects of roughness and deposit resistance in a model experimental system in which complete control of deposit characteristics could be assured. EXPERIMENTAL

APPARATUS

Use of the limiting (LDCT) for measuring

AND

PROCEDURE

diffusion current technique mass transfer rates, and, by

analogy, heat and momentum transfer rates, has been reviewed many times in the literature (e.g. Mizushina, 1971; Wragg, 1977; Selman and Tobias, 1978). At the limiting current, the electrochemical reaction proceeds at its maximum rate, and since the surface ion concentration is zero, the mass flux is given by eq. (2):

Hence the Sherwood equation (3):

number

may be obtained

Two previous papers (Galloway, Hartt, 1981) report the use of

from

1973; Wolfson and the electrochemical

B.D.

830

CRITTENDEN

method for monitoring the growth of calcareous deposits on cathodic steel surfaces in sea water. In this study the expeximental apparatus comprised a perspex cylinder, 76 mm ID and 150 mm high which was secured between top and bottom plates (Fig. 1). The anode was a nickel 200 plate, of 58 mm diameter and 3.5 mm thickness, perforated with 91 holes of 1.7 mm diameter in order to ensure that the anode area exceeded that of the cathode which was a nickel 200 plate of 90 mm diameter and 3.5 mm thickness (Mizushina, 197 1). The electrolytic solution was 0.005 M potassium ferricyanide, 0.005 M potassium ferrocyanide in 0.5 M sodium hydroxide as the supporting electrolyte; this solution is commonly used for mass transfer studies (Tagg et al., 1979). The solution was agitated by a 50 mm diameter stainless steel, six flat-bladed impeller placed centrally through the anode, using a nylon bearing, at a distance of 30 mm from the cathode. Wall mounted baffles were not required as vortexing was eliminated by the presence of the perforated anode. To minimize degradation of the solution, the walls of the vessel were covered with orange celluloid film and nitrogen was passed slowly

and N.J. ALDERMAN

over the free liquid surface. The contents of the vessel were replaced frequently with fresh solution, the concentration of which was checked by titrimetric analysis (Alderman, 1986). The fouling deposit was simulated by glass spheres, of various sizes and number of layers, resting on the cathode. The electrodes were polished with progressively finer grades of emery paper, washed with detergent, rinsed with distilled water and finally degreased with carbon tetrachloride. The electrodes were then activated in 2 M sodium hydroxide solution for five minutes. The mean peak to valley height of the prepared cathode was found by a Talysurf machine to be 1.27 pm. The potential difference across the cell was provided by a lA/30V variable output stabilized power supply (Weir Instrumentation Ltd, Model 400), and measured by a Solartron digital multimeter. The current flowing through the cell was monitored on a chart recorder via the potential difference across a standard 10 R resistor. For several different impeller speeds in the range 0 to 350 rpm the current-potential curve was obtained by deriving the average cell current from the current-time

nitrogen J top

top

plate

plate

cou\er

__)

gasket!

-

f electrolytic solution

nylon

--

bearing

--

I anodes

tie

-

rod-

I

base

I

Cathode

II

otate

Fig. 1. Electrolyticcell.

831

Negative fouling resistances traces which were recorded for applied cell potentials in the range 0 to 2 V. Similar curves were also obtained with glass spheres, of various sizes and number of layers, resting on the cathode. Example plots are shown in Fig. 2. It was concluded that for all mass transfer experiments the limiting diffusion current could be obtained with an applied cell potential of 0.9 v. RESULTS

Forced convection mass transfer at a smooth surface is expressed in the form of eq. (4) (Gilliland and Sherwood, 1934): Sh = a RebScC

(4)

Re = d:N Y SC = ;

PD = (2.50 f 0.17) x lo-to T

with p in poise, D in cm’ s- ’ and T in K. The mass transfer experiments were carried out with solution temperatures in the range 14 to 24 “C. The physical property values which correspond to these temperatures are as follows: v from 1.24 x 1O-6 to 9.96 x lo-’

mZ s- 1

D from 5.64 x IO- to to 7.30 x lo-

lo mz s- r

SC from 2199 to 1364. Mass

transfer

at a smooth

surface

For experiments with no solids resting on the cathode, the derived data are plotted in Fig. 3. Assuming the index on the Schmidt number to be l/3 and using linear least squares regression, the 210 data points were correlated by eq. (9): Sh = 1.033 Re”-‘6

c = 0.333.

(7)

The density and kinematic viscosity of the electrolytic solution were determined as a function of temperature by means of a hydrometer and a U-tube viscometer, respectively. Several methods have been used to determine the diffusivity of the ferricyanide ion in an equimolar solution of potassium ferricyanide and potassium ferrocyanide in excess sodium hydroxide (Alderman, 1986). For this study, the effective diffusivity was obtained from equation (8) (Bazan and Arvia, 1965).

60

Current ,

mA

SC’.~~.

(9)

Equation (9) is compared with correlations established by other workers in Fig. 4. Reasonable agreement exists, any differences probably being due to different geometrical configurations for the electrolytic cells. Mass transfer at a rough surface Up to seven layers of spherical glass particles of various diameters, Table 1, were placed on the cathode to simulate a rough deposit. Limiting diffusion currents were obtained as a function of impeller speed, but only up to the speed at which the particles began to

_

o-5

1.0

1-5

2-O Potential,

Fig.

2. Current-potentialdata for

1 layer of 3 mm glass spheres

resting

volts

on the cathode.

832

and N. J. ALDERMAN

B. D. CRITTENDEN

Sh

Sh=

12-7 Re0.56

L

1

10'

I

I ,,I,,

lo*

I

transfer

lo4

data with no solids

move. Data were correlated by eq. (lo), and the results are summarized in Table 1. An example plot is shown in Fig. 5. Sh = A Reb.

(10)

The value of K calculated from eq. (2) is the overall mass transfer coefficient for the combined resistances due to the fluid film and to the deposit thickness. Table 1 shows that for particles of a fixed diameter, d,, the exponent b is reasonably independent of the number of layers in the deposit, n. The dependence of 6 including the upper and lower bounds of b, is in Fig. 6. Using least squares analysis, the data

are fitted by eq. (11) which h = 0.56 +

is plotted

I

I ,,I,,

10” Fig. 3. Mass

on d,, shown

I

resting

Ill111 10”

Re on the cathode. DISCUSSION

The transport of reacting ions from the bulk ution to the cathode involves two steps:

The overall rate of transport can be described by the model proposed by Galloway (1973) for the growth of calcareous deposits on a cathodic steel surface in sea water:

=,

_I_&=-

3.15 d,)).

1

A =

12.7exp[-7.78~1’.~~ - exp ( - 4.0d,)

(exp(-0.27dr) )-1.

(12)

(13)

x

z+r

(11) The coefficient A decreases non-linearly with increasing numbers of layers, n. An example plot is shown in Fig. 7. The dependence of A on d, is more complex. An example plot is shown in Fig. 8. As dp increases, the coefficient A decreases to a minimum at d, = 0.8 mm. For d, > 0.8 mm, the coefficient A tends to the value obtained for d, = 0, that is the value for the smooth cathode. This trend for d, z 0.8 mm is to be expected as large diameter spheres create very little effective roughness. For each value of n, the minimum value of A occurs at d, ‘v 0.8 mm. Using non-linear least squares analysis, the data are fitted by eq. (12), which is plotted on Figs 7 and 8

sol-

(1) convection of ions from the bulk solution to the deposit-solution interface {2) diffusion of ions through the deposit to the cathode.

on Fig. 6.

1.Ol {exp (- 0.29 dp) - exp (-

I

F

in which D, for a simulated

fouling

deposit

is given by:

DFZ$

(14)

The tortuosity of the simulated fouling deposit, fined as the ratio of the path length travelled by reacting species in the deposit to the thickness of deposit is independent of d, and is calculated from

dethe the eq.

(15) (Alderman, r =

1986). 1.023~1 - 0.023 ~~ 0.933n + 0.067

for n 3 1.

(15)

Since the deposit voidage E is independent of d,, then D, is also predicted to be independent of the particle diameter. Thus the first term in the denominator of eq.

Negative fouling resistances lo4

I

I

I ,‘“I,

I

I

It11111

833

t

t

I1111

ISh

/

32’3

-

Fig. 4. Comparison of mass transfer correlations with no solids resting on the cathode. curve 1

2 3 4 5 6 7 8

d, 0.076 m 0.10 m 0.28 m 0.28 m 0.28 m 0.28 m 0.28 m 0.28 m

dildv 0.66 0.66 0.22 0.333 0.475 0.22 0.333 0.475

(13) is dependent on particle diameter but not on number of layers, whilst the second term is dependent on both the number of layers and particle diameter. Galloway evaluated the mass transfer coefficient for the unfouled surface, k. As fouling proceeded, he assumed that the mass transfer coefficient for the fluid-deposit interface remained equal to k. This assumption is only valid for layers comprising extremely fine deposits which do not alter the surface roughness. In this study, the particle sizes were large, and therefore they have an important effect on the film mass transfer coefficient. Large particles tend to reduce the resistance to transfer across the film, but on the other hand, an increasing number of layers of particles increases the resistance to transfer through the deposit itself. Prediction of k for this study is therefore precluded and the rate of transport of ions has been expressed instead in terms of an overall mass transfer coefficient, K, in eq. (2). The Sherwood number was based on this overall coefficient and correlated with Re by eq. (10). CES

13:*--6

this study Mizushina et al. (1969) Lelan and Angelino (1975) 7, 3, 7. 3, 7. 3, 1. 3. .. *.

In this study it has not been possible to evaluate the exponent on SC. Whilst the value of l/3 has been considered to be equally suitable for transport across a film and through a packed bed (Selman and Tobias, 1978; Karabelas et al., 1971), Davies (1972) reasoned that the value should be f. Experimental work (Davies, 1983) on the deposition of small aerosol particles (SC - 10”)onto very rough surfaces confirmed, at least for such a case, that an exponent of + was more appropriate. The similarity between the term {exp ( - 0.29 dp) -exp(-3_15d,)) in eq. (11) for b and the term fexp (-0.27d,)-exp (-4.OOd,)) in eq. (12) for A suggests that these parameters are related in a broadly similar manner to the variables which describe the structure of the simulated fouling deposit, namely, porosity, tortuosity and roughness. Values of the exponent b in excess of unity for a certain range of d, are interesting (Fig. 6), and it is worthwhile noting that Cornet et al. (1969) predicted that the exponent on Re for the portion of a roughened centrifugal disc in

B. D. CRKTENDEN

834

Table

Particles n

d,(mm)

0

0

1 2 3 1 2 3 4 5 1 2 3 4 5 6 1 2 3 4 1 2 3 4 5 6 7 1 2 3 4 5 6 1 2 3 4 5 6 7 1 2 3 4 1 2 3

0.368 ( + 0.058)

(k

0.695 0.055)

0.805 (_t 0.130)

1.29 (-+ 0.110)

(f

1.34 0.175)

1.55 (kO.15)

3.0 (k

0.5)

(2

5.0 0.5)

7.0

( f 0.5) (+

10.0 0.5) 12.0

: 1 2

(-t 0.5)

No. of data points 210 7 5 4 40 38 35 24 21 35 21 29 26 27 21 18 11 7 19 11 10 12 12 10 8 9 48 z!: 43 48 24 68 54 54 35 18 14 10 19 19 15 18 30 30 26 21 26 30 22

1. Experimental

A

Correlation Sh=AReb

12.7 0.35 0.065 0.045 0.072 0.005 0.11 0.029 0.0066 0.03 1 0.005 0.0022 0.0022 0.0@07 0.0034 0.032 0.018 0.020 0.028 0.20 0.023 0.022 0.01 0.025 0.029 0.028 0.091 0.028 0.053 0.104 0.065 0.11 0.49 0.58 0.34 0.40 0.45 0.63 0.3 1 1.18 1.47 0.88 0.94 1.65 1.04 1.25 2.68 1.46 5.30 3.75

turbulent flow should be equal to unity. The range of d, found in typical particulate fouling deposits is about 1 to 50pm (Gudmundsson, 1981; Newson et al., 1983; Watkinson and Epstein, 1970; Hopkins and Epstein, 1974), which is well within the range 0 to 12000 pm used in this study. Taking d, for an average particulate fouling deposit to be 25 pm, it may be predicted from eqs (4), (7), (11) and (12), that Sh = 1.033 exp { - 0.69n0~“} For the moderate

SC (-

and N. J. ALDERMAN

I~c”.~~SC”.“.

1800) used in this study,

(16) the

data

b 0.56 0.92 1.08 1.11 1.12 1.44 1.02 1.18 1.35 1.25 1.45 1.50 1.48 1.62 1.42 1.25 1.29 1.27 1.22 1.03 1.27 1.26 1.33 1.22 1.18 1.18 1.14 1.27 1.18 1.10 1.15 1.07 0.95 0.92 0.95 0.92 0.91 0.86 0.93 0.82 0.78 0.82 0.80 0.79 0.82 0.78 0.73 0.79 0.64 0.65

Sh at Re=5000 1500 885 640 575 1000 1060 650 670 650 1300 1150 780 655 690

610 1350 1065 loo0 910 1290 1150 1010 830 815 670 650 1500 1400 I230 1220 1170 loo0 1600 1470 1110 1010 1045 955 855 1275 1130 950 855 1380 1120 960 1345 1220 1235 950

Correlation coefficient 0.976 0.999 0.998 0.994 0.995 0.971 0.970 0.985 0.985 0.98 1 0.995 0.99 1 0.997 0.988 0.994 0.994 0.993 0.995 0.9x3 0.996 0.996 0.997 0.993 0.996 0.997 0.993 0.985 0.993 0.995 0.989 0.996 0.994 0.980 0.982 0.978 0.976 0.997 0.997 0.994 0.996 0.997 0.998 0.998 0.992 0.994 0.996 0.986 0.996 0.990 0.984

exponent on SC of l/3 has been assumed for the fouled surface as well as the smooth surface. Figure 9 shows the comparison between eq. (16) for the cathode fouled by 1 to 5 layers of 25 pm particles and eq. (9) for the unfouled cathode. At relatively high Reynolds numbers, it is clear that the Sherwood number can become greater than that for an unfouled surface. At relatively low Reynolds numbers, the deposit thickness has a much greater influence than the additional surface roughness, causing the Sherwood number to remain less than that for the unfouled surface. The sphere

Negative fouling resistances

Sh

=

0.0022

835

Re1-5

103,

J

7

P

/

102=_

.

l

/. lo'&

I

I

lllll

I

I

lo2

I

I

I

lllll

lo3

10”

I

I

lllll

10”

Re

Fig. 5. Mass transfer data with 3 layers of 0.80 mm spheres resting on the cathode.

I

z

0

I

I

,

I

I

I

I

I

1

=

equation

I

(11 )

1

I

I

I

, 15

10

5

d p’

mm

Fig. 6. Dependence of exponent 6 on particle diameter.

diameter

which is equivalent to the roughness of the - 2.5 pm, is small compared with unfouled cathode, the average particulate fouling deposit size, - 25 pm. Thus for the above analysis it is safe to assume that the cathode was a smooth surface. It is reasonable to assume that for the analogous case of heat transfer, results could be expressed in the form of eq. (17), if care is taken to re-evaluate Nu = a RebPr0.333

(17)

the coefficient a (Berger and Ziai, 1984). The schematic representation of heat transfer from a surface through a foulant layer to the bulk fluid shown in Fig. 10(a) demonstrates the importance of Re in whether or not apparently negative fouling resistances will be observed. At Re = Rel , the overall Nu is always less than that for the unfouled surface and gives rise to a steady increase in apparent fouling resistance with time [Fig. 10(b)]. At Re = Re2, the overall Nu after some short period of time exceeds the initial value, but then

B. D.

836

lzi2 0

CRITTENDEN

and N. J. ALDERMAN

I

I

I

I

I

I

I

1

2

3

4

5

6

7

8

n

Fig. 7. Dependence of coefficient A on number of layers of particles of diameter 3 mm

equation

(12)

5

0

10

15

d p,mm

Fig. 8. Dependence of coefficient A on particle diameter for 1 layer of particles.

1000

Sh/5c”-33

100

10

-1

increasing

-

equation

(9)

-----

equation

(161 1s~

5

n

1 100

1000

10000

100000 Re

Fig. 9. Comparison

of mass transfer data for 1 to 5 layers of 25 pm particles with data for the unfouled cathode.

Negative fouling resistances

837

Fig. 10. (a) (Top) Typical plot of mass or heat transfer data as fouling proceeds with time. (b) (Bottom) Typical fouling curves showing, or not showing, negative fouling resistances.

declines steadily as the deposit thickness continues to increase. This latter scenario describes how fouling resistances can sometimes appear to be very small or even negative in the early stages of deposit growth on a tube surface (Epstein, 1978; Bott and Gudmundsson, 1978; Crittenden and Khater, 1987).

CONCLUSIONS

The limiting diffusion current technique has been used to study the relative effects of particle diameter, number of particle layers and Reynolds number on the overall mass transfer coefficient in a small stirred vessel. Whether or not the coefficient initially increases or decreases as the number of particle layers is increased depends critically on particle size and Reynolds number. As the number of layers is increased at relatively high Reynolds numbers, the overall coefficient can increase above the initial value, subsequently decreasing to below the initial value.

The analogy between heat transfer and electrochemical mass transfer can be used to explain why very small or even negative fouling resistances appear to occur in the early stages of fouling of a heat transfer surface. Acknowledgement-The authors gratefully acknowledge the financial support of the Science and Engineering Research Council. NOTATION

coefficient Zl b

L d

D

in eqs (4) and (17)

coefficient in eq. (10) exponent in eqs (4) and (10) exponent in eq. (4) concentration of reacting species in bulk fluid, kgmm3 cathode diameter, m diffusivity of reacting species in bulk fluid, m’s_r

B.D.

838 D,

diffusivity m2 s-l

d,

impeller diameter, m particle diameter, m stirred vessel diameter, m Faraday’s constant limiting current, A mass flux of reacting species, kg m -’ sfilm mass transfer coefficient, m s-l overall mass transfer coefficient, m s-l number of particle layers impeller speed, Hz number of baffles Nusselt number Prandtl number Reynolds number fouling resistance, (kw m- * K- ’ ) ’

d, d, F II_ J k K n N "B

NU Pr Re R, s SC

Sh t T u, u, x z zi

of reacting

CRITTENDEN

species in fouling

deposit,

’

electrode surface area, m2 Schmidt number Sherwood number tortuosity temperature, K clean overall heat m2 K-’

transfer

coefficient,

kw

dirty overall heat transfer coefficient, m --2 K-1 deposit thickness, m charge number of reacting species height of impeller from base of vessel

kw

Greek

letters

& I9

porosity of fouling deposit time, s viscosity of electrolytic solution, Ns mP2 kinematic viscosity of electrolytic solution, m2 s-1

P V

REFERENCES Alderman, N. J., 1986, Electrochemical studies of mass and heat transfer in a simulated fouling deposit. PhD Thesis, University of Bath. Bazan, J. C. and Arvia, A. J., 1965, The diffusion of ferro- and ferricyanide ions in aqueous solutions of sodium hydroxide. Elecrrochim. Acta 10, 1025-1032. Berger, F. P. and Ziai, A., 1984, Problems of electrochemical modelling of heat transfer process. Proc. 11th Annual Research Meetina on Heat Transfer. Catalvsis and Catalvtic Reactors, IChe&E, Rugby, pp. -129-l 34. ’ Bott, T. R., 1979, Biological fouling of heat transfer surface. Effluent Wate; Treat-J. 19, 453261. Bott, T. R. and Gudmundsson, J. S., 1981. Rippled silica deposits in heat exchanger tubes. Proc. 6th Int. Conj: on Heat Transfer, Vol. 4, pp. 373-378. Hemisphere, Washington. Cornet. I., Lewis. W. N. and KaDnesser. R.. 1969. The effect of surface roughness on mass- &nsf& to a rotating disk. Trans. Insl. them. Engrs 47, T222-T226. Crittenden, B. D. and Khater, E. M. H., 1987. Fouling from

and N.J.

ALDERMAN

vaporizing kerosine. Trans ASME. J. Heat Transfer 109, 583-589. Davies, J. T.. 1972, Turbulence Phenomena, pp. 144-147. Academic Press, New York. Davies, J. T., 1983, A new theory of aerosol deposition from turbulent fluids. Chem. Engng Sci. 38, 135-139. Epstein, N., 1981, Fouling of heat exchangers. Proc. 6th Int. Conf. on Heat Transfer, Vol. 6, pp. 235-253. Hemisphere, Washington. Galloway, T. R., 1973, Heat transfer fouling through growth ofcal&eous film deposits. Int. .J. Heat kass T&nsfer 16, 443-460. Gilliland, E. R. and Sherwood, T. K., 1934, Diffusion of vaoours into air streams. Ind. Enana - - Chem. 26. 516-523. Gudmundsson, J. R., 1981, Particulate fouling, in Fouling of Heat Transfer Equipment {Edited by Somerscales, E. F. C. 357-387. Hemisphere, and Knubsen, J. G.), pp. Washington. Hopkins, R. M. and Epstein, N., 1974, Fouling of heated stainless steel tubes by a flowing suspension of ferric oxide in water. Proc. 5th Inr. Heat Transfer ConJ Vol. 5, pp. 18s-184. Japan Society of Mechanical Engineers, Tokyo. Karabelas, A. J., Wegner, T. H. and Hanratty, 7‘. J., 1971, Use of asymptotic relations to correlate mass transfer data in packed beds. Chem. Engng Sci. 26, 1581-1589. LeLan, A. and Angelino, H., 1975a, Transferts de matiere a la paroi d’une cuve mechaniquement agitee. Chem. Engng Sci. 29, 1557-1565. LeLan, A. and Angelino, H., 1975b, Study of the analogy between heat and mass transfer to the wall of a stirred tank. Int. .I. Heat Mass Transfer 18, 163-167. Lund, D. and Sandu, C., 1981, Chemical reaction fouling due to foodstuffs, in Fouling of Heat Transfer Equipment (Edited by Somerscales, E. F. C. and Knudsen, J. G.), pp. 437476. Hemisphere, Washington. Mahato, B. K. and Shemilt, L. W., 1968, Effect of surface roughness on mass transfer. Chetn. Engng Sci. 23,183-185. Mizushina, T., Ito, R., Hiraoka, S., Jbusuki, A. and Sakaguchi, I., 1969, Transport phenomena at the wall of agitated vessels. J. them. Engng Japan 2, 89-94. Mizushina, T., 1971, The electrochemical method in transport phenomena. Adv. Heat Transfer 7, 87-161. Newson, 1. H., Bott, T. R. and Hussain, C. I., 1983, Studies of magnetite deposition from a flowing suspension. Chem Engng Commun. 20, 335-353. Perkins, K. R. and McEligot, D. N., 1973, Roughness of heat transfer surfaces. Inc. J. Hear Mass Transfer 16, 679-681. Rankin, B. H. and Adamson, W. L., 1973, Scale formation as related to evaporator surface conditions. Desalination 13, 63-87. Selman, J. R. and Tobias, C. W., 1978, Mass transfer measurements by the limiting current technique. Ado. them. Engng

IO,

2 1 l-3

18.

Somerscales, E. F. C., 1981, Introduction and summary, in Transfer Equipment (Edited by Fouling oj’ Heat Somerscales. E. F. C. and Knudsen. J. G.).,, __ DD. l-27. Hemisphere: Washington. Taga, -- D. J., Patrick, M. A. and Wragg, -- A. A., 1979, Heat and mass transfer downstream of abrupt nozzle expansions in turbulent flow. Trans. Inst. them. Engrs 57, 176181. Watkinson, A. P. and Epstein, N., 1970, Particulate fouling of sensible heat exchangers. Proc. 4th Int. Heat Transfer Conf., Vol. 1, Paper HE1.6. Elsevier, Amsterdam. Wolfson, S. L. and Hartt, W. H., 1981, An initial investigation of calcerous deposits upon cathodic steel surfaces in sea water. Corros., Houston 37, 7C-76. Wragg, A. A., 1977, Applications of the limiting diffusion current technique in chemical engineering. The Chemical Engineer, No. 316, 3949.