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Theoretical description of the diffusion stage of reactant transport from turbulent flow to a macroscopically inhomogeneous metal surface
J. EieciroanaL Chem, 180 (1984) 437-444 Elsevter Sequoia S.A.. Lausanne - Prmted m The Netherlands
437
THEORETICAL DESCRIPTION OF THE DIFFUSION STAGE OF REACTANT TRANSPORT FROM TURBULENT FLOW TO A MACROSCOPICALLY INHOMOGENEOUS METAL SURFACE *
YS
KRYLOY
A N. Frumkm Insr;rwe of Elecirochemrstty of the Academy of ScrenceJ of the U SS R, Moscow (LI S S R.) (Recetved 27th May 1984)
ABSTRACT A new method of calculation of the local and dverage tome mass transfer coefficients IS developed for the case of turbulent electrolyte solution flow znslde a roughened-wal! metalbc tube. The method is based on a sophisticated hydrodynanuc model and free of empirical parameters found by mass transfer measurements m the system under constderatlon
Up to recent time, there were no adequate theoretical approaches to the quantitative description of convective mass transfer betweek turbulent liquid flow and a rough wall, particularly in cases where the elements of roughness are located far enough from one another. In these cases the flow near the wall can be divided into two types of region with an essentially different flow pattern: along the smooth parts of the surface liquid moves relatively quietly, whereas belAd the rough protuberances, due to the separative character of flow past such obstacles, stagnant zones urlth intensive circulation of liquid are formed. The i?echanisms of momentum transfer in these two types of region are completely different. Inside the stagnant zones, the flow resistance results from the difference between the static pressures at the front and the back walls of the protuberance. In the “outer” regions, i.e. the regions adjacent to the smooth-parts of the wall, the flow resistance is connected mainly with deceleration of flow by viscous friction. Therefore it is evident that the viscous sublayer thicknesses at the areas near the roughness elements md far from them must differ from each other. The principal importance of taking into account this difference in working out quantitative methods of description of transport processes in roughened wall channels was emphasized in the monograph of Reynolds [l]. Below we will present one of the possible means for taking into account * Dedxated to the memory of Professor Dr. Dr. h. c. Kurt Schwabe 0022-0728/84/$03
00
@ 1984 Else&x
,Szquoia S.A.
438
the effect of wall roughness on hydrodynamics and mass transfer in systems of interest from the electrochemical engineering point of view. In previous papers [2], the local characteristics of near-wall turbulence were analysed for the particular case of a crrculsr tube-with the elements of roughness having the shape of annular protuberances cf a rectangular profile with height h and width b located at the same distance p + i from one another (see Fig. 1): It was postulated that, at the back edge of the top of each protuberance, flow separation occurs and that the separation line is straight with an angle of slope 8. (This angle was supposed to be the same for all the protuberances and dependent neither on the vrscosity nor on the geometrical parameters h, b and p.) Furthermore, it was accepted that the vrscous subls.yer thickness 6, was constant along the whole area of the wall between the roughness elements_ In contrast with the latter postulate, we will now suppose that each of the hydrodynamically selected regions-the stagnant zone and the “outer” zone adjacent to the smooth part of the wall-has its own vrscous sublayer with the thickness Se, determined by the relation &, = YV/U*,
0)
where y 1s a constant, v is the kinematrc viscosity and vt, is the characterrstrc dynamic turbulent velocity of the z-th region (indexes I = 1,2 refer to the stagnant and the outer zones, respectively). The velocity u *, is connected with the shear stress, T,, on the z-th region of the wall by the relation d,u;,=(P+b)‘q
(2)
where p 1s the density of the liquid and Z, is the length of the region (I, = h cot 8, I,=p+b-1,).
Under the condition h -=z R, where R is the tube radius, the turbulence structure inside the core of the flow IS insensitive to the details of the wall profile and is homogeneous along the tube axis. For the core the logarithmrc drag law is vahd [3]: ii/v,=xln(R/k)+B
(3)
Here v* = m, +ris the average shear stress on the wall, U is the average axial velocity of the turbulent flow, x and B are universal constants, and k 1s the effective
3
A
h
0
I+
&-
Fig 1. Schematic representation of the roughness
tu;bulent
flow structure near a wall with rectangular
elements of
439
roughness height, dependent in general on both the geometrical parameters of the roughness and the molecular viscosity. There exist semi-empirical approaches to the analysis of the quantity k for particular types of roughness [1,4]. Below we suggest a new approach that eliminates the use of adjustable empirical parameters. As was shown in ref. 2, at high enough Reynolds numbers the quantity k can be identified with the average volume of the stagnant zones per unit square of the channel surface. The effect of viscosity on the volume of zone 1, V,, can be described by taking into account the correction for the viscous sublayer volume: V, = +a cot e( h - 6,,)’
(4)
where u is the approximate average perimeter of zone 1. (In further developments the same quantrty a will be accepted as the characteristic width of all the roughness elements.) In order to take into account the contribution of the viscous friction forces to the flow resistance of the “outer” regton 2, we will first consider some particularities of turbulent flow in a smooth wall channel. The hydrauhc drag in thus case is described by the well-known Prandtl-Nikuradze law [l]: C/u* = 2.5 ln( U*R/V)
+ 3.75
(5)
One can observe that within the accuracy of a constant coefficient tl-ns equatton can be obtained from eqn. (3) if, m the latter, the characteristic roughness height k is replaced by the value X = c&_ where S,,, is the thickness of the viscous sublayer determined by eqn. (1). In other words, to perform the transition from eqn. (5) to eqn. (3), one should introduce such a mechanism of interaction between the viscous sublayer and the turbulent core at which the value h would play the role of the height of some roughness_ The nature and physrcal meanmg of this meshamsm may be illustrated by a simple model of two-layer turbulent flow with two regions-the turbulent core and the laminar sublayer inside of which the turbulent veloctty fluctuations do not penetrate. The turbulent and the lammar regions within the framework of such a model exist Independently of each other and do not interact with one another. As a matter of fact, the laminar sublayer IS unstable and turbulent velocity fluctuations cause significant perturbations \ inside it. Hence, confining oneself to the two-layer model, one can suggest the following way for involving the interaction effects m the flow pattern. Namely, it is possible to suppose that the viscous sublayer behaves like an elastic film. Perturbations caused by turbulence can be approxtmately considered as a system of waves propagating along the film surface. For the outer zone 2, the waves would play the role of roughness, and due to the existence of roughness there would be separation of flow and formation of stagnant zones analogous to zone 1. The volume of the stagnant zones per urnt square of region 2 may be identifted with the value X mentioned above. Thus, the accepted model of interaction between the turbulent core and the laminar sublayer enables an expression for the effective roughness height A for the outer region 2 to be obtained. The volume of the stagnant zones in this region, V,, is
Vz = crS,,l,a
(6)
440
Because the stagnant zones do not interact directly, the total expression for the effective roughness height k can be written as the sum of the volumes of the stagnant zones in r-egrons 1 and 2: k=
(7)
(v, + Y,)/++b)
Relative to the turbulent core, the stagnant zones can be approximated on average as a slab of constant thtckness k. Due to hquid circulation in the stagnant zones of region 1 and due to liquid motron inside the viscous sublayer in region 2, the boundary between the effective near-wall layer (of thickness k) and the turbulent core must move relative to the wall with some linear velocity u,. Let us estimate this velocity. The velocity at the outer boundary of the i-th zone is proportional to u*,. If the shear stress 5 is constant over the length I,, the outer velocity I(,, can be written as US,= c,U*,r,(p+5)-’
(8)
where Ci is a constant. Because the partial herght It, = v:/( p + b)a is an additive variable, the average value (over all zones) af the outer velocity is 24\ = (u,th, + u*,hz)/(
Iz, + 122)
(9)
In deriving eqn (3), we supposed that the velocity U was determined relattve to an immovable wall. In the model considered, the role of the wall IS played by the surface of the stagnant zones, and the velocity of motnon of this surface depends on te type of roughness and the Reynolds number. Therefore, in order that the quantity B in eqn. (3) should be a universal constant, the veloctty U must be replaced by Z - I(,. In this way, we arrive at the universal drag law, whtch is valid both for smooth and rough-walled channels: u/u,=xln(R/k)+uJu,+B
00)‘
Let us derive the distributton of the mcmentum flux over regions 1 and 2. The quantity r,, hke the quantity h,, is an additive variable and consequently must be proportronal to h,. If a roughness element moves with the velocity u,,, the general expression for T, should be r, = K~,(U us,)‘, where the coef,“lcient K IS the same for all zones. In our case, ~(,r = 0, tlS1= yu+ *_ Therefore by dividing 5 by 7 = or + 72 we obtain the following dtstribution: r,/r = Iz,/( h, + PAZ)
rJ7
= &Ah,
+ I&)
01)
where p = (1 - yv * JE>‘_ Usin g the eqns. (3)-(11) one can calculate u*, and S,,. By introducing the dimensionless values h, = hu_/v and X, = 8,/h, we can obtain the followmg expressions: x, = (y/lz+)(2G/‘(l
-x1))“’
x2 = ( y/h+)2’3\:G/ap)1’3
02)
where cot e(1 -x,)z+cxpxz
p+b h
- cot B(1 - X,)
Ii
441
At high Reynolds numbers it is possible to evaluate roughness parameters without solving the system (12).
S,-,, as a function of the Because in this case the
condition x, -=K1 is valid, we obtain: x2 = ( ~/Jz,)~‘~( G/c@)“~
x1 = (Y/&)(~G)“’
G-hcot6/2(p+b)
(13)
It follows from eqn. (13) that the ratio 6,,/&
= a(
ay/h+)1’3G1’6
(14)
is practically msensitive to the roughness parameters and decreases with increasing Reynolds number proportionally to Re-‘i3. The model considered above contains six universal constants: cot 8, B, C,, C,, y and (Y. Analysis of the literature data given m ref. 2 enabled three of them to be determined: cot 19= 7.54, B = - 3.34; C, = 1.15. The remaining constants are interconnected by the condition that eqn. (3) must be equivalent to the empirical drag law (eqn. 5) in the case of turbulent flow in a channel with hydraulically smooth walls when k = ayv/u,
u, = cgJ*
(B+C,)x-‘-ln(ay)-0X2=0
From this condition it is possible to calculate the parameter y if the constants C, and cy are specified. We determined these constants by minimizing the sum of the squares of deviations of the theoretical values of the function u,=ti/u,+2.5ln(h/R)t3.75 from the corresponding experimental values obtained in ref. 5 for turbulent flow in a tube with five types of rough surfaces at the following combinauons of the values
I
2
3
-h+
Fig. 2. Dependence of u, on h, for Mferent profdes of roughness elements; figures on the curves and the expenmental data sets correspond to the figures used m the text to denote the types of roughness element.
442
(p + b)//z and h/R: (1) 10 and 0.08; (2) 10 and 0.04; (3) 20 and 0.03; (4) 40 and 0.04; and (5) 10 and 0.02. The range of Reynolds numbers used was from 4 x lo3 to 105. The final calculated values were: C, = 6.0, (Y = 0.2, and y = 6.5. In Fig. 2 the curves U, = f( h +) are presented. The quantity U,U* is the velocity at theheight of the protuberance, h. The dependence of the friction coefficient f = 2(u,/E)’ on the Reynolds number is shown in Fig_ 3. From these two figures it can be concluded that the similanty flow regime is achieved approximately at h, = 40 (Re = 104) for all the types of roughness except for the one having the smallest height (nutiber 5 for which h, = 50, Re = 2.5 X 104). It was supposed that at deviations off and zl, from hmiting values less than 5%, the similarity regime occurs. Knowledge of the hydrodynamic structure of liquid flow near a rough wall enables a method of calculating the ionic mass transfer rate to be developed_ In analysing the mass transfer problem we will confine ourselves to considering processes which occur at the metal-solution interface in the presence of an excess of an indifferent electrolyte. In this case one can neglect the contribution of the migration mechanism of ionic mass transport. Let us denote by J the total mass flux to the surface ABCD which is the characterisiic element of local inhomogeneity of the surface. Let AC be the difference between the ionic concentrations at the surface and m the bulk of the solution. Wlthin the framework of the hydrodynamic model considered above, the expresslon for J can be written as
where D is the diffusion
coefficient,
I (=p
-p’7--
-I- 6 + 2h)
is the total length of the
o-7
0.6
A
-H-~Y~-A-
-2
a-3
Y”
+++=-*--a-”
Y
o-
4
v -
5
Y’
log Re Fxg. 3 The frxtlon coefficient as a function of the Reynolds number The meamng of the figures is the same as m FI$ 2
443
inhomogeneity element, I, is the length of the characteristic hydrodynamic region, and 8, is the corresponding thickness of the laminar diffusron layer. To calculate the diffusion layer thicknesses 6, and S,, one can use the known 161 relation sr, = a,(Re)Sc(‘-““”
06)
Here St, = OfiS, is the Stanton number, which is in fact a drmensionless mass transfer coefficient; SC = v/D is the Schmidt number; a, is the numerical coefficient depending on the Reynolds number; and the parameter n characterizes the degree of dampling of turbulent velocity fluctuations on approaching a solid wall. Equation (16) can be rewritten in the form s, = CYo,flS?2
07)
where f, = 2(vl,/iQ2 is the partial (corresponding to the z-th region) friction coefficient and a,-,, is the viscous sublayer thickness determined by eqn. (1). Using the relations (13) obtain: 6, = (~~~/u*~)(Gfi)‘~Sc-“~
W
8, = (x2y/u*2)f~‘2(Ghu*.Jy)“3Sc-1’2
09)
where x, and xZ are numerical parameters. The corresponding Stanton numbers are: St, = (&m)
--‘SC” -nyn
Sr, = (~~~)-‘(Ghu*Jv)-~‘~Sc(~--~)/~
(20) (21)
Fig 4 Dependence of the ratio Sf,/S~, on the Reynolds number for region 2 Experunental points are given for four values of the parameter (+ b)/h- 10 (a), 7 (b). 5 (c) and 3 (d)
444
It follows from eqns. (20) and (21) that the Stanton number is independent of the Reynolds number in region 1 and changes proportionally to Re-‘i3 in region 2. To check these conclusions we used the experimental data for the local electrochemical mass transfer coefficients obtained in ref. 7. In this paper the local mass transfer rares were measured for a wide range of geometrical sizes corresponding to rectangular profiles of roughness. In Fig. 4, the experimental data for region 2 are presented in the form of the dependence of the ratio S~,/SZ, on the Reynolds number (St, is the corresponding Stanton number for a smooth-walled tube). All the expenmental points correspond to n = 4, SC -= 1500 and different values of the parameter (p + b)/h. The sohd line is the theoretical curve calculated from eqn. (21) using only one ad-lusting parameter: x1 = 2.7. As is very evidently seen, the agreement between the theoretical predictions and the experimental results is quite good. As for the comparison of the theory and the experimental data for region 1, we should not that according to the data [7] the structure of the so-called stagnant zone (region 1) is in fact much more comphcated than supposed in our theory. The main feature of the real structure which was discarded in the theory consists in the existence of two clearly distinguished subzones m region 1: m one of them, the outher one, liquid is involved m very intensive turbulent motion; the mner subzone is hydrodynamically “passive” and there is very slow liquid motion inside it. This complex structure of course cannot be described more or less adequately by such a simple equation as eqn. (20). The proper description demands a more sophisticated theory which is being developed now. REFERENCES 1 A J Reynolds, Turbulent Flows tn Engtneertng Wdey, London, New York, Sydney, 2 VP. Vorotthn, V S Krylov, A.D Davydov and L I. Kheifetz. Elektrokhnniya, 19 (1984) 214 3 A S Monin and A M. Yaglom, Statisttcal Hydromechanics, Part 1. Nauka, Moscow, 4 B A Kader, Tear. Osn. Khirn. Teknol . 11 (1977) 393 5 R L. Webb, E R.G Eckert and R J Goldstem, Int J. Heat Mass Transfer 14 (1971) 6 D A Frank-Kamenetakn, Diffusion and Heat Transfer rn Chemtcal Kmetics, Nauka, 237. 7 F P. Berger. K -F , F-L. Hau, Int J Heat Mass Transfer, 22 (1979) 1645.
Toronto, 1974 (1983) 1657, 20 1965. 601. Moscow.
1967, p
437
THEORETICAL DESCRIPTION OF THE DIFFUSION STAGE OF REACTANT TRANSPORT FROM TURBULENT FLOW TO A MACROSCOPICALLY INHOMOGENEOUS METAL SURFACE *
YS
KRYLOY
A N. Frumkm Insr;rwe of Elecirochemrstty of the Academy of ScrenceJ of the U SS R, Moscow (LI S S R.) (Recetved 27th May 1984)
ABSTRACT A new method of calculation of the local and dverage tome mass transfer coefficients IS developed for the case of turbulent electrolyte solution flow znslde a roughened-wal! metalbc tube. The method is based on a sophisticated hydrodynanuc model and free of empirical parameters found by mass transfer measurements m the system under constderatlon
Up to recent time, there were no adequate theoretical approaches to the quantitative description of convective mass transfer betweek turbulent liquid flow and a rough wall, particularly in cases where the elements of roughness are located far enough from one another. In these cases the flow near the wall can be divided into two types of region with an essentially different flow pattern: along the smooth parts of the surface liquid moves relatively quietly, whereas belAd the rough protuberances, due to the separative character of flow past such obstacles, stagnant zones urlth intensive circulation of liquid are formed. The i?echanisms of momentum transfer in these two types of region are completely different. Inside the stagnant zones, the flow resistance results from the difference between the static pressures at the front and the back walls of the protuberance. In the “outer” regions, i.e. the regions adjacent to the smooth-parts of the wall, the flow resistance is connected mainly with deceleration of flow by viscous friction. Therefore it is evident that the viscous sublayer thicknesses at the areas near the roughness elements md far from them must differ from each other. The principal importance of taking into account this difference in working out quantitative methods of description of transport processes in roughened wall channels was emphasized in the monograph of Reynolds [l]. Below we will present one of the possible means for taking into account * Dedxated to the memory of Professor Dr. Dr. h. c. Kurt Schwabe 0022-0728/84/$03
00
@ 1984 Else&x
,Szquoia S.A.
438
the effect of wall roughness on hydrodynamics and mass transfer in systems of interest from the electrochemical engineering point of view. In previous papers [2], the local characteristics of near-wall turbulence were analysed for the particular case of a crrculsr tube-with the elements of roughness having the shape of annular protuberances cf a rectangular profile with height h and width b located at the same distance p + i from one another (see Fig. 1): It was postulated that, at the back edge of the top of each protuberance, flow separation occurs and that the separation line is straight with an angle of slope 8. (This angle was supposed to be the same for all the protuberances and dependent neither on the vrscosity nor on the geometrical parameters h, b and p.) Furthermore, it was accepted that the vrscous subls.yer thickness 6, was constant along the whole area of the wall between the roughness elements_ In contrast with the latter postulate, we will now suppose that each of the hydrodynamically selected regions-the stagnant zone and the “outer” zone adjacent to the smooth part of the wall-has its own vrscous sublayer with the thickness Se, determined by the relation &, = YV/U*,
0)
where y 1s a constant, v is the kinematrc viscosity and vt, is the characterrstrc dynamic turbulent velocity of the z-th region (indexes I = 1,2 refer to the stagnant and the outer zones, respectively). The velocity u *, is connected with the shear stress, T,, on the z-th region of the wall by the relation d,u;,=(P+b)‘q
(2)
where p 1s the density of the liquid and Z, is the length of the region (I, = h cot 8, I,=p+b-1,).
Under the condition h -=z R, where R is the tube radius, the turbulence structure inside the core of the flow IS insensitive to the details of the wall profile and is homogeneous along the tube axis. For the core the logarithmrc drag law is vahd [3]: ii/v,=xln(R/k)+B
(3)
Here v* = m, +ris the average shear stress on the wall, U is the average axial velocity of the turbulent flow, x and B are universal constants, and k 1s the effective
3
A
h
0
I+
&-
Fig 1. Schematic representation of the roughness
tu;bulent
flow structure near a wall with rectangular
elements of
439
roughness height, dependent in general on both the geometrical parameters of the roughness and the molecular viscosity. There exist semi-empirical approaches to the analysis of the quantity k for particular types of roughness [1,4]. Below we suggest a new approach that eliminates the use of adjustable empirical parameters. As was shown in ref. 2, at high enough Reynolds numbers the quantity k can be identified with the average volume of the stagnant zones per unit square of the channel surface. The effect of viscosity on the volume of zone 1, V,, can be described by taking into account the correction for the viscous sublayer volume: V, = +a cot e( h - 6,,)’
(4)
where u is the approximate average perimeter of zone 1. (In further developments the same quantrty a will be accepted as the characteristic width of all the roughness elements.) In order to take into account the contribution of the viscous friction forces to the flow resistance of the “outer” regton 2, we will first consider some particularities of turbulent flow in a smooth wall channel. The hydrauhc drag in thus case is described by the well-known Prandtl-Nikuradze law [l]: C/u* = 2.5 ln( U*R/V)
+ 3.75
(5)
One can observe that within the accuracy of a constant coefficient tl-ns equatton can be obtained from eqn. (3) if, m the latter, the characteristic roughness height k is replaced by the value X = c&_ where S,,, is the thickness of the viscous sublayer determined by eqn. (1). In other words, to perform the transition from eqn. (5) to eqn. (3), one should introduce such a mechanism of interaction between the viscous sublayer and the turbulent core at which the value h would play the role of the height of some roughness_ The nature and physrcal meanmg of this meshamsm may be illustrated by a simple model of two-layer turbulent flow with two regions-the turbulent core and the laminar sublayer inside of which the turbulent veloctty fluctuations do not penetrate. The turbulent and the lammar regions within the framework of such a model exist Independently of each other and do not interact with one another. As a matter of fact, the laminar sublayer IS unstable and turbulent velocity fluctuations cause significant perturbations \ inside it. Hence, confining oneself to the two-layer model, one can suggest the following way for involving the interaction effects m the flow pattern. Namely, it is possible to suppose that the viscous sublayer behaves like an elastic film. Perturbations caused by turbulence can be approxtmately considered as a system of waves propagating along the film surface. For the outer zone 2, the waves would play the role of roughness, and due to the existence of roughness there would be separation of flow and formation of stagnant zones analogous to zone 1. The volume of the stagnant zones per urnt square of region 2 may be identifted with the value X mentioned above. Thus, the accepted model of interaction between the turbulent core and the laminar sublayer enables an expression for the effective roughness height A for the outer region 2 to be obtained. The volume of the stagnant zones in this region, V,, is
Vz = crS,,l,a
(6)
440
Because the stagnant zones do not interact directly, the total expression for the effective roughness height k can be written as the sum of the volumes of the stagnant zones in r-egrons 1 and 2: k=
(7)
(v, + Y,)/++b)
Relative to the turbulent core, the stagnant zones can be approximated on average as a slab of constant thtckness k. Due to hquid circulation in the stagnant zones of region 1 and due to liquid motron inside the viscous sublayer in region 2, the boundary between the effective near-wall layer (of thickness k) and the turbulent core must move relative to the wall with some linear velocity u,. Let us estimate this velocity. The velocity at the outer boundary of the i-th zone is proportional to u*,. If the shear stress 5 is constant over the length I,, the outer velocity I(,, can be written as US,= c,U*,r,(p+5)-’
(8)
where Ci is a constant. Because the partial herght It, = v:/( p + b)a is an additive variable, the average value (over all zones) af the outer velocity is 24\ = (u,th, + u*,hz)/(
Iz, + 122)
(9)
In deriving eqn (3), we supposed that the velocity U was determined relattve to an immovable wall. In the model considered, the role of the wall IS played by the surface of the stagnant zones, and the velocity of motnon of this surface depends on te type of roughness and the Reynolds number. Therefore, in order that the quantity B in eqn. (3) should be a universal constant, the veloctty U must be replaced by Z - I(,. In this way, we arrive at the universal drag law, whtch is valid both for smooth and rough-walled channels: u/u,=xln(R/k)+uJu,+B
00)‘
Let us derive the distributton of the mcmentum flux over regions 1 and 2. The quantity r,, hke the quantity h,, is an additive variable and consequently must be proportronal to h,. If a roughness element moves with the velocity u,,, the general expression for T, should be r, = K~,(U us,)‘, where the coef,“lcient K IS the same for all zones. In our case, ~(,r = 0, tlS1= yu+ *_ Therefore by dividing 5 by 7 = or + 72 we obtain the following dtstribution: r,/r = Iz,/( h, + PAZ)
rJ7
= &Ah,
+ I&)
01)
where p = (1 - yv * JE>‘_ Usin g the eqns. (3)-(11) one can calculate u*, and S,,. By introducing the dimensionless values h, = hu_/v and X, = 8,/h, we can obtain the followmg expressions: x, = (y/lz+)(2G/‘(l
-x1))“’
x2 = ( y/h+)2’3\:G/ap)1’3
02)
where cot e(1 -x,)z+cxpxz
p+b h
- cot B(1 - X,)
Ii
441
At high Reynolds numbers it is possible to evaluate roughness parameters without solving the system (12).
S,-,, as a function of the Because in this case the
condition x, -=K1 is valid, we obtain: x2 = ( ~/Jz,)~‘~( G/c@)“~
x1 = (Y/&)(~G)“’
G-hcot6/2(p+b)
(13)
It follows from eqn. (13) that the ratio 6,,/&
= a(
ay/h+)1’3G1’6
(14)
is practically msensitive to the roughness parameters and decreases with increasing Reynolds number proportionally to Re-‘i3. The model considered above contains six universal constants: cot 8, B, C,, C,, y and (Y. Analysis of the literature data given m ref. 2 enabled three of them to be determined: cot 19= 7.54, B = - 3.34; C, = 1.15. The remaining constants are interconnected by the condition that eqn. (3) must be equivalent to the empirical drag law (eqn. 5) in the case of turbulent flow in a channel with hydraulically smooth walls when k = ayv/u,
u, = cgJ*
(B+C,)x-‘-ln(ay)-0X2=0
From this condition it is possible to calculate the parameter y if the constants C, and cy are specified. We determined these constants by minimizing the sum of the squares of deviations of the theoretical values of the function u,=ti/u,+2.5ln(h/R)t3.75 from the corresponding experimental values obtained in ref. 5 for turbulent flow in a tube with five types of rough surfaces at the following combinauons of the values
I
2
3
-h+
Fig. 2. Dependence of u, on h, for Mferent profdes of roughness elements; figures on the curves and the expenmental data sets correspond to the figures used m the text to denote the types of roughness element.
442
(p + b)//z and h/R: (1) 10 and 0.08; (2) 10 and 0.04; (3) 20 and 0.03; (4) 40 and 0.04; and (5) 10 and 0.02. The range of Reynolds numbers used was from 4 x lo3 to 105. The final calculated values were: C, = 6.0, (Y = 0.2, and y = 6.5. In Fig. 2 the curves U, = f( h +) are presented. The quantity U,U* is the velocity at theheight of the protuberance, h. The dependence of the friction coefficient f = 2(u,/E)’ on the Reynolds number is shown in Fig_ 3. From these two figures it can be concluded that the similanty flow regime is achieved approximately at h, = 40 (Re = 104) for all the types of roughness except for the one having the smallest height (nutiber 5 for which h, = 50, Re = 2.5 X 104). It was supposed that at deviations off and zl, from hmiting values less than 5%, the similarity regime occurs. Knowledge of the hydrodynamic structure of liquid flow near a rough wall enables a method of calculating the ionic mass transfer rate to be developed_ In analysing the mass transfer problem we will confine ourselves to considering processes which occur at the metal-solution interface in the presence of an excess of an indifferent electrolyte. In this case one can neglect the contribution of the migration mechanism of ionic mass transport. Let us denote by J the total mass flux to the surface ABCD which is the characterisiic element of local inhomogeneity of the surface. Let AC be the difference between the ionic concentrations at the surface and m the bulk of the solution. Wlthin the framework of the hydrodynamic model considered above, the expresslon for J can be written as
where D is the diffusion
coefficient,
I (=p
-p’7--
-I- 6 + 2h)
is the total length of the
o-7
0.6
A
-H-~Y~-A-
-2
a-3
Y”
+++=-*--a-”
Y
o-
4
v -
5
Y’
log Re Fxg. 3 The frxtlon coefficient as a function of the Reynolds number The meamng of the figures is the same as m FI$ 2
443
inhomogeneity element, I, is the length of the characteristic hydrodynamic region, and 8, is the corresponding thickness of the laminar diffusron layer. To calculate the diffusion layer thicknesses 6, and S,, one can use the known 161 relation sr, = a,(Re)Sc(‘-““”
06)
Here St, = OfiS, is the Stanton number, which is in fact a drmensionless mass transfer coefficient; SC = v/D is the Schmidt number; a, is the numerical coefficient depending on the Reynolds number; and the parameter n characterizes the degree of dampling of turbulent velocity fluctuations on approaching a solid wall. Equation (16) can be rewritten in the form s, = CYo,flS?2
07)
where f, = 2(vl,/iQ2 is the partial (corresponding to the z-th region) friction coefficient and a,-,, is the viscous sublayer thickness determined by eqn. (1). Using the relations (13) obtain: 6, = (~~~/u*~)(Gfi)‘~Sc-“~
W
8, = (x2y/u*2)f~‘2(Ghu*.Jy)“3Sc-1’2
09)
where x, and xZ are numerical parameters. The corresponding Stanton numbers are: St, = (&m)
--‘SC” -nyn
Sr, = (~~~)-‘(Ghu*Jv)-~‘~Sc(~--~)/~
(20) (21)
Fig 4 Dependence of the ratio Sf,/S~, on the Reynolds number for region 2 Experunental points are given for four values of the parameter (+ b)/h- 10 (a), 7 (b). 5 (c) and 3 (d)
444
It follows from eqns. (20) and (21) that the Stanton number is independent of the Reynolds number in region 1 and changes proportionally to Re-‘i3 in region 2. To check these conclusions we used the experimental data for the local electrochemical mass transfer coefficients obtained in ref. 7. In this paper the local mass transfer rares were measured for a wide range of geometrical sizes corresponding to rectangular profiles of roughness. In Fig. 4, the experimental data for region 2 are presented in the form of the dependence of the ratio S~,/SZ, on the Reynolds number (St, is the corresponding Stanton number for a smooth-walled tube). All the expenmental points correspond to n = 4, SC -= 1500 and different values of the parameter (p + b)/h. The sohd line is the theoretical curve calculated from eqn. (21) using only one ad-lusting parameter: x1 = 2.7. As is very evidently seen, the agreement between the theoretical predictions and the experimental results is quite good. As for the comparison of the theory and the experimental data for region 1, we should not that according to the data [7] the structure of the so-called stagnant zone (region 1) is in fact much more comphcated than supposed in our theory. The main feature of the real structure which was discarded in the theory consists in the existence of two clearly distinguished subzones m region 1: m one of them, the outher one, liquid is involved m very intensive turbulent motion; the mner subzone is hydrodynamically “passive” and there is very slow liquid motion inside it. This complex structure of course cannot be described more or less adequately by such a simple equation as eqn. (20). The proper description demands a more sophisticated theory which is being developed now. REFERENCES 1 A J Reynolds, Turbulent Flows tn Engtneertng Wdey, London, New York, Sydney, 2 VP. Vorotthn, V S Krylov, A.D Davydov and L I. Kheifetz. Elektrokhnniya, 19 (1984) 214 3 A S Monin and A M. Yaglom, Statisttcal Hydromechanics, Part 1. Nauka, Moscow, 4 B A Kader, Tear. Osn. Khirn. Teknol . 11 (1977) 393 5 R L. Webb, E R.G Eckert and R J Goldstem, Int J. Heat Mass Transfer 14 (1971) 6 D A Frank-Kamenetakn, Diffusion and Heat Transfer rn Chemtcal Kmetics, Nauka, 237. 7 F P. Berger. K -F , F-L. Hau, Int J Heat Mass Transfer, 22 (1979) 1645.
Toronto, 1974 (1983) 1657, 20 1965. 601. Moscow.
1967, p