FLUID MECHANICS IN CHEMICAL ENGINEERING
I
K. M. KISER' and H.
E.
HOELSCHER
Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Md.
Chemical Reactions in a Water Tunnel Better dynamic control is needed for investigating interfacial reactions during dissolution. In this water tunnel for studying liquid-solid systems, flow can be carefully regulated
H E T E R O G E N E O U S c H m c I c ~ L reactions comprise a large portion of kinetics. Those occurring at the interface between a liquid and a solid are among the most intriguing, but quantitative analyses of data from these have been only partially successful. Most work has been done in batch reactors using a centrally located agitator and one of the primary problems has been defining a parameter that adequately characterized the system. Rotational speed of the agitator is probably a n inadequate measure of turbulence intensity, and might not even be related to it. A Reynolds number defined in terms of revolutions per minute of the agitator probably does not describe the dynamical system. I n batch systems, baffles are commonly introduced, which further reduces the probability of establishing a suitable parameter based on rotational speed. Because of a n inadequate description of the transport properties of fluid field, kinetic studies in batch agitated systems have yielded no significant information on surface reaction rates or diffusion velocities. Nernst, the first to analyze relative magnitudes of these two mechanisms, assumed that chemical phenomena at the interface are much faster than transport processes, and that equilibrium is
Present address, Research Laboratories, General Electric Co., Schenectady, N. Y .
970
established almost instantaneously at this interface. Considerable data have been amassed since Nernst first proposed this theory. Bircumshaw and Riddiford ( 3 ) , Moelwyn-Hughes (72), and Amis ( 7 ) have reviewed these experiments thoroughly. Generally, Nernst's theory applied to most of the data. Exceptions are few but significant enough to question the theory as a universal description of such processes. There are four assumptions inherent in the Nernst theory: The chemical process at the surface always proceeds much faster than either of the transport processes; in well-stirred s!-stems, the concentration gradient is confined to a thin layer adhering to the solid surface; within this layer, concentration varies linearly with distance measured normal to the surface; and thickness of this layer, although a function of stirring rate and geometry of the system, is independent of the coefficient of diffusion of the solute, viscosity of the solution, and temperature. Thus, for a firstorder reaction,
Applying data to this relationship, a film thickness of approximately 3 X to 8 X 10-3 cm. is obtained for many reactions, provided the observed velocity constant is obtained from data taken duringviolent agitation (turbulent range). For reactions such as dissolution of benzoic acid in dilute sodium hydroxide
INDUSTRIAL AND ENGINEERING CHEMISTRY
solutions, zinc in acetic acid and hydrogen chloride, magnesium in dilute hydrogen chloride, and sodium in liquid ammonia, rates are dependent on agitation. Thus, as assumed by Nernst, they may be diffusion controlled, at least in part. O n the other hand, dissolution of glass in alkali, and possibly iron in dilute acetic acid solutions, above a certain agitation rate is independent of agitation rate; therefore, contrary to Nernst, dissolution rate may be controlled by surface phenomena. For this study, dissolution of zinc in dilute acetic acid was selected as a representative system. General observations made on this sj-stem apply equally ~7ellto catalytic reactions. A closed return water tunnel was constructed to provide a working section wherein the Reynolds number could be varied over a wide range, such parameters as temperature, concentration, turbulcnce intensity, and flow rate could be controlled independently, and the velocity profile was flat. With this arrangement it was felt that each contributing mechanism could be studied individually. Equipment and Experimental Procedure
Design and Construction. The closed return water tunnel contains the reacting system and at the same time provides controllable and reproducible operating conditions in the working section.
WORKING SECTION
FRONT
ELEVATION SOLI DI TY 64 tc 34.1 34.6 3 5.2 35.2
In this water tunnel, the velocity profile i s flat, Reynolds numbers can b e varied over a wide range, and parameters such as temperature, concentration, and turbulence can b e controlled
With this apparatus, it was possible to operate over a wide range of Reynolds numbers, which, although admittedly they provide no information concerning properties of fluid turbulence, were used as a convenient measure of the degree of agitation. In the working section, such factors as temperature, concentration, turbulence intensity, could be controlled by external means. Little information is available on design of water tunnels. Generally, basic wind tunnel designs are copied, using necessary adjustments in characteristic parameters. The St. Anthony Falls Hydraulic Laboratory at the University of Minnesota, the Hydrodynamic Laboratory of the California Institute of Technology, and David Taylor Model Basin, Department of the Navy, Carderock, Md., are a few of the water tunnels in existence throughout the country. N o such device has been employed to study interfacial chemical reactions. For this work, one major consideration is the reaction surface to fluid volume ratio, which in a water tunnel, is low. This presents difficult analysis problems
for catalytic reactions, because rate of reaction at the surface must be obtained from concentration changes in the fluid phase, O n the other hand, dissolution reactions can be studied easily despite the low surface to volume ratio; the rate may be followed by weighing the test piece after prescribed time intervals. Since large volumes are conducive to small changes in concentration, the tunnel may be considered as a differential reactor when operated over short periods of time, or as an integral reactor when operated over large time intervals. A square cross section was chosen for the tunnel because this was most economical and easiest to fabricate. Furthermore, the flat sides are more adaptable for inserting test specimens and various pieces of analytical equipment. No fillets were placed in the corners, as suggested by Pankhurst and Holder (73)) because the corners have no appreciable effect on velocity profile in the center section. The tunnel has a square cross section of 8 inches on a side, and for this work, a working section length of 19 inches.
Since the tunnel would be used to study catalytic reactions, fluids containing dissolved gases might be involved. According to the continuity equation and Bernoulli’s equation for steady motion, when a fluid passes through a contraction, its velocity must increase and pressure decrease. Since saturation concentration of gases in liquids is inversely proportional to pressure, any contraction could result in gas evolution. Thus, no contraction could be used in the tunnel, even though it would have provided distinct advantages; therefore, no diffuser was required. Besides being an energy-conserving device, contraction provides a uniform velocity profile in the working section. Such a velocity profile was considered essential to this study, and was obtained with a series of stainless steel wire mesh screens. Four screens with mesh sizes ranging from 4 to 30 and solidity ratios from 30 to 40% were placed in series just upstream of the working section. Mesh size decreased in the downstream direction. Spacing of these screens is such that each lies at least 60-mesh VOL. 49, NO. 6
JUNE 1957
971
/
H+/
1
4
/
TOP
“C
1.04
HOR I20N TA L
1
1.00
0.9a
\
FRONT
Figure 2.
4
1
I
I
0
0.2
0.4
I
I
0.6
0.e
Figure 1 ,
I
0 cl
X/L
@
lengths downstream of the screen immediately preceding it. Previous data in (2) indicate that 10 to 30 mesh lengths is sufficient to ensure a reasonably isotropic velocity field. Velocity profiles are shown in Figure 1 for two different mesh lengths downstream of the last screen. These were measured using a Pitot tube constructed from a standard hyperchrome hypodermic tubing. For a tunnel 15 cm. square (75),a boundary layer thickness of approximately 1 cm., 44 cm. downstream from a contraction, was to be expected. This reduced the length of the test piece usable in the tunnel. A series of turbulence intensities at the test piece was obtainable by changing the last screen before the working section or by working at various distances downstream from any given screen. According to Schubauer and Spangenburg (18, 79), transverse turbulence intensity attains the isotropic decay region a t a shorter distance downstream than does the axial value. Corner sections of the tunnel were designed according to Krober ( 9 ) . A 40-mesh screen (solidity of 647,) was placed just downstream of the turning vanes to stabilize the flow leaving the trailing edges. An AllisChalmers 8 X 6, Type SS-EB. centrifugal pump was used, capablk of delivering 1000 gallons per minute at a head of 7 feet of water when driven at 580 r.p.m. This pump was selected for its relatively low cost and small space features. Its only significant disadvantage was that the discharge is not free of pulsations, is highly turbulent, and is nonuniform. These effects, however, were taken care of in another way. The pump contributes energy to the fluid, some appearing as heat which gradually raises fluid temperature. To control temperature in the tunnel, a
972
INDUSTRIAL AND ENGINEERING
Concentration profiles
--+
Y*
Mean velocity profiles
YIM
UC
22.5 22.5 292.5 292.5
185
130 185
130
copper cooling coil was placed in the return leg and a 1000-watt heater with thermostatic control was placed in the bottom.
diameter and upstream velocities, from 5000 to 120,000. The over-all chemical reaction, stoichiometrically, is
Reaction Studies
Zn(s)
Chemical reactions classifiable as interfacial are numerous. Among these are a number of catalytic (chemical) reactions as well as many dissolution (physical) reactions. Many of these reactions were not considered possible for this study for a number of reasons-they are gaseous or vapor phase systems, require high temperatures or pressures, or could not be followed by simple analytical techniques of sufficient accuracy. After surveying possible systems (a), dissolution of zinc in dilute acetic acid was selected for the rate studies. Molten zinc was cast into cylinders 8 inches long and of varying diameters by pouring into borosilicate glass tubing. After cooling, the glass was cracked away, and the cylinders, turned down on a lathe to prescribed diameters, were polished with No. 0 emery paper. When a metal such as zinc dissolves in acid, hydrogen is liberated at the surface. To prevent surface activity from varying with time, a depolarizer (oxidizing agent) was added. If concentration of depolarizer is large enough, hydrogen is reoxidized as fast as it is produced, maintaining a constant surface activity. I n this work, potassium nitrate was used as a depolarizer in concentrations such that the observed rates were not functions of depolarizer concentrations. Reaction rates were measured by weighing the cylinder after arbitrarily selected time intervals. Runs of 0.5and 1-hour duration were made. Rates were measured for four different acid concentrations, three potassium nitrate concentrations, two temperatures, and Reynolds numbers, based on cylinder
At low initial acid concentrations, concentration of acetate ions iernains constant and this ion does not enter the reaction directly. Hydrogen produced at the metal surface is oxidized there by the depolarizer as rapidly as it is formed, thus preventing its release and resulting decrease in dissolution rate with time. Acetate ion concentration is constant from the wall to the bulk of the fluid. Hydrogen ion concentration must diminish as the surface is approached; it is replaced with zinc ions so that the solution remains elcctrically neutral at every point (Figure 2). A study of these profiles suggests that diffusion of zinc ions from the surface into the bulk of fluid might be the controlling diffusion phenomenon in this process. Diffusivity of hydrogen ions as measured by mobility, is between six and seven times that of zinc ions; therefore, any diffusional contribution to the over-all process must arise from resistance to diffusion of the zinc ions. Thus, data on diffusivity of zinc ions are probably necessary for analytical description of the system. Since experimental values are not available in the literature, they must be calculated. Equations first developed by Onsager (7) permit this calculation, using a value for mobility of zinc ions in solution, available, for example, from Taylor and Taylor (20). Thus, values for diffusivity of zinc ions at 18’, 27’, and 34’ C. were calculated and used throughout subsequent calculations. They are roughly 2.5 times the value for diffusivity of zinc acetate in aqueous solution.
CHEMISTRY
+ 2H(Ci”02) Zn(CzHaOz)(soln.)
f Hz(g)
(2)
FLUID M E C H A N I C S 1001
Diffusivity for Zinc Ions Temp., (en++), O c. Sq. Cm./Seo. 18 27 34
0
= 0.610*
0.020 EN.
0
o
0.61 X 10-6 0.78 X lo-' 0.97 X 1 s '
-
0 Z7*
1.6 OC.
x s4* 1.0
1.4801f; o.oao CN. IY/
0-
o c
2.40C
Results $
Experimental Measurements. Effects of Reynolds number on observed rate of reaction for a temperature of 27 O C. and an 0.08M acid concentration, shown in Figure 3, are typical of those for other temperatures and acid concentrations (8). Effect of acid concentration for Reynolds numbers less than 40,000 is shown in Figure 4. The observed rate is linear with acid concentration over the range investigated. This is expected if the process is diffusion controlled, since Fick's law is linear in concentration. Nernst observed this behavior for dissolution reactions; in fact, it is the basis of his diffusion film theory. However, Figure 4 alone is not proof of a diffusion-controlled mechanism. If the surface reaction is first order compared with concentration of one reactant, then the linear relationship between acid concentration and rate will still exist. At 27' and 34' C. (Figure 3) the measured rate was increased approximately 10% by increasing the operating temperature 7' C. As a 10' rise in temperature should have little effect on diffusion rate, it appears that the observed rate of reaction for dissolution of zinc in dilute acetate solution is diffusion-controlled. Over-all energy of activation calculated from the data, was constant at 4100 f 100 calories per gram-mole from Reynolds numbers of 10,000 through 100,000. This again indicates that the process is essentially diffusion-controlled, even a t high flow rates. Turbulence intensity was altered by operating at different distances downstream from a 30-mesh screen of 35y0 solidi-ty-Le., between 22.5 and 292.5 mesh lengths from the screen. Turbulence intensity in the tunnel was O.8Y0, and had little or no effect on the observed rate of reaction. This is not surprising; Maisel and Sherwood ( 7 7) found that turbulence intensities of less than 4% did not cause measurable variations in vaporization rate from solid or liquid surfaces into a turbulent air stream. Calculated Results. The Chilton and Colburn JD-factor relationship for mass transfer is J D = NstNs," (3) where n is generally assigned the value of 2/3. I n Figure 6, JD is plotted us.
P 0
I
IO
20
30
I
I
I
I
I
I
I
40
50
80
70
00
SO
100
RE x
Figure 3.
IO-'
Dissolution rates for zinc in acetic acid
Acetic acid 0.08M; potassium nitrate, 0.05M
*
'Of
TEMP. a ZT 1.5 OC 0 s= 1.50fOJO CY.
CONC. HAC
Figure 4.
MOLARITY
Dissolution rates for zinc in acetic acid
30t Figure 5. Nusselt correlation
I
10 4
REYNOLDS NUMBER (RLX IO-")
VOL. 49, NO. 6
.
JUNE 1957
973
culated for a typical example, experimental run 1000 (8), using boundary layer theory.
‘Or
ZL 2
I
6
4
8
IO
Figure 6.
J H
=
e 2
than about 50,000) the simple linear relationship apparently no longer applies. Following the suggestion of Levich ( I O ) , N u ( P \ ~ s J - ~was / ~ plotted as a function of Reynolds number (Figure 5). The data follow closely the predicted linear relationship between Nu(NBc)- l I 3 and Re,”. For the low Reynolds number region (less than 40,000), the data can be fitted with good accuracy into the equation Re,0.57
(4)
The exponent, 0.57, agrees with the value of 0.50 usually predicted for this region. For higher Reynolds numbers, considerable deviation is again noted from this simple power relationship. Frank-Kamentskiy (6) and Von KArm&n (27) have pointed out that for the limiting example of high turbulence, k,, should be independent of diffusivity and viscosity for a diffusion-controlled system. The equation k , = Ng, L‘,
*
300
Equation 6 results in Nsc ~*km
==
Nscys,
c‘,
+{ (skin friction) coefficient (7)
-
( 4 ) . At high Reynolds numbers (greater
r \ ’ ~ ( N s ~ ) - ~=I 30.211
loo
PI0
( R E m X 10-3)
Jn correlation
the Reynolds number for Schmidt numbers based upon the diffusivity of the zinc ion. The other two curves are from Maisel and Sherwood ( 7 7 ) for vaporization of water and the Chilton and Colburn analogy, J n =
60
40
20
R E Y N O L D S NUMBER
wherein the temperatui e dependency has been removed. This is Equation 3 forn = 1. For many reactions occurring a t room temperature, the Stanton number approaches 5 X 10-5. In this work, the experimental value is approximately 3 X 10-5. For the dissolution of zinc cylinders in aqueous iodine, &/Urn* nearly attains the asymptotic value under conditions far removed from high turbulence (74). From this viewpoint, the observed rate might not be entering surface rate-control region at all. At present, high Reynolds number data are insufficient to predict whether or not a surface rate control is being approached. Assuming that diffusion of zinc ions from the surface of the cylinder to the bulk of fluid is the controlling mechanism, local values of __ were cal-
dRc
Analysis of Results Assuming that diffusion of zinc ions from the surface of the cylinder to the bulk of the fluid is the controlling mechanism, particularly a t low Reynolds numbers, standard boundary layer analysis should permit calculation of local hTusselt numbers. Applying the boundary layer approximations of Prandtl, Pohlhausen, and particularly Eckert and Livingood (5), approximate solutions have been obtained which are valid for the limiting case of diffusion control. I n the coordinate system shown in Figure 8, the ?c* axis lies along the surface of the cylinder in the direction of fluid motion, the ,y* axis lies normal to the surface, and the z* axis lies along the length of the cylindrical normal to the mean fluid motion. Differential equations which describe the motion of fluid are the well-known h’avier-Stolies and continuity equations. They constitute a set of differential equations which express the condition of equilibrium that exists between the forces acting on each particle of fluid. The boundary layer theory requires that it be assumed that there exists a very thin layer in the immediate neighborhood of the body in which the velocity gradient normal to the wall is very large-this being the boundary layer. An analogous argument permits the establishment of a “mass transfer boundary layer concept.” Diffusive terms in the generalized mass transfer equation become the same order of magnitude as the convective terms only if the concentration gradient is large-Le., only if there is a concentration boundary layer a t the wall within which lies the entire concentration gradient. With these assumptions, the continuity equation
(5)
0
T
=
27OC
X
T
=
S4’6
constitutes a definition of Nst, the Stanton (Margoulis) number. As the limiting case is approached
k , = N~~ u,
--t
constant
(6)
This constant is approached asymptotically with increasing Reynolds numbers and turbulence. The experimental data behave in this manner (Figure 7). The asymptotic value is a function of temperature, but this also is expected, since k , is temperature dependent. Equation 6 may be derived from the Reynolds analogy. If this is done, the constant contains the Schmidt number. Introducing the Schmidt number into
974
INDUSTRIAL AND ENGINEERING CHEMISTRY
I
0
1
I
20
I
I
40
60
REYNOLDS
Figure 7.
I
1
NUMBER
I
100
BO
( Re,
x
i
I20
d
Effect of Reynolds number on Stanton number
FLUID MECHANICS
L"*
written for two dimensions, the NavierStokes equations simplified for the boundary layer, and the generalized mass transfer equation reduce to the following set of three simultaneous differential equations (8, 76)
e
u, (9)
I n solving these equations, the procedure is similar to that of Eckert and Livingood ( 5 ) using Von K6rmBn's integrated momentum equation and an analogous integrated mass-transfer equation. Incompressible flow with constant property values throughout the system is assumed.
-
STAONATlON POINT
c9 0
CON~CNTRATlON. BOUNDARY LAYlCR
Substituting Figure 8.
The cylinder showing notation .
U =
.\/Re,
negligible. After evaluating the coefficients of this expansion from the boundary conditions a t the wall and putting the velocity profile into dimensionless form, Equation 21 transforms to
Equations 8, 9, and 10 transform to the dimensionless forms
au
au
au
u - + u - = u - + ay ax ax
ac
u - + v E = ax ay
azU
ayz
nl, ($)
(12)
u = o
c = c w u-u
y = o
y = o
y = o
y-6
(2 Re,-')u(?)
= Aq
- B$
(22)
where
(13)
where the following boundary conditions now apply u = o
where 0, the dimensionless concentration, is given by
With these definitions, Equations 14 and 15 transform to
c-+o
The form of the concentration profile can be taken as a power series in q :
y-+&
e
Integrating Equation 12 over y from y = 0 toy = 6 leads to
By a similar technique, Equation 13 becomes
For simplification, a displacement thickness, &, can be defined for the momentum boundary layer
C = 1 -
[Q
C W
T o obtain an approximate solution from these two equations, it is necessary only to choose appropriate expressions for the velocity and concentration profiles within the boundary layers. As present interest is in solving the mass transfer problem only, it is sufficient that the expression for the velocity profile be accurate only within the concentration boundary layer. Detailed calculations for the following solution are available (8). As an approximation, the form of the velocity profile is taken as a Taylor series expansion about y* = 0. Thus, u*cy*) = u * ( O )
Similarly, a convection thickness, 6,,, for the concentration boundary layer, and a momentum thickness, &, for the momentum boundary layer may be defined thus :
=
+ ;u*'(0jy* + u*"(O)y*2
+ ...
(21)
I t is assumed further that all terms beyond the third in this expression are
+ bq + + d q 3 ] cg2
(23)
where a, b, c, and d are constants independent of q. These four constants can be evaluated from the following four boundary conditions : 6 = 1
g = O
e'=O
q = o
q = 1
7 = 1
g"-O
7 = 0
The resulting equation for the velocity profile is
e
=
1
-
3 1 - q + 2 2
73
(24)
Choosing this simple power series representation for the concentration profile in the boundary layer implies that the profiles are similar over all values of %*-that is, shape of the concentration profile is unaffected by the pressure gradient in the direction of flow. Although this may not be true for the present problem, it is as an approximation. VOL. 49, NO. 6
JUNE 1957
975
0.24
0
1
LOCAL
14
PROW
[12A - S B ]
(25)
U p to this point, no assumption as to how C, varies along the interface has been made. For this work, it is assumed independent of x* (hence, a diffusioncontrolled system). Then, substituting for the concentration gradient at the wall from Equation 24, Equation 20 reduces to
For any given system, it is possible to solve simultaneously Equations 25 and 26 for 6,) once the initial value ( x = 0) has been found. This initial value can be obtained from Equation 26 by substituting Equations 25 into 26 and performing the indicated differentiations :
where, at x = 0 Ng,
>,,(
E = O
L*
U,*R
G = ,
20R
STAGNATION
0457
POINT
1
I
0522
0.587
7
0 699-
(X)
where the overbar denotes average values. Defining the fractional contribution by
Growth of concentration boundary layer thickness
With the simple approximate form chosen for the velocity profile, it is unnecessary to solve the momentum equation for the momentum boundary layer thickness, 6. In fact, the chosen velocity profile will not satisfy this equation since it does not satisfy all of the boundary conditions. The group, U6,,, is readily determined from Equation 20 by substituting Equations 22 and 24 for O(q) and u(q). After integration and simplification, the equation can be put into the form m e , = -7&j-
I
0 327
DlUENSiONLESS DISTANCE
Figure 9.
NUSSELT NUMBER
1
I
0.196
0 065
quadrant of the cylinder. Because of symmetry. however, it is the same for equivalent values of x bet\veen the front stagnation point and the separation point on both faces of the cylinder. At present, no theory has been developed to handle the back of the cylinder. Some data from heat transfer experiments (77), however, indicate the percentage contribution to the total heat transfer from the back of the cylinder. An average, or over-all, transfer coefficient can be defined on the basis of these measurements. Over the whole cylinder, the total transfer is given by
iw
= 0
With the property values given for any particular system, G and F may be calculated and hence, ( 6 c ) x 3 ~ determined. With 6, calculated at x = 0,
33,
and
d dx
from Equation 26. The value of U6,, is zero at x = 0, because at this point U = 0 and a,, is not infinite. With this value as a starting point, 6, can be calculated for all values of x up to the separation point by the method of isoclines. Once the concentration boundary layer thickness, 6,: has been determined, the local mass transfer coefficient can be obtained. This local coefficient, k L , can be defined by the relationship, W ( L * )= kLC,
(29)
Substituting the wall boundary condition,
the average Nusselt number over the whole cylinder is
The boundary layer thickness 6, was calculated as a function of x* from x = 0 to the separation point (Figure 9) Knowing the boundary layer thickness Nu a t every point, values of .\/Rem were computed over this distance using Equation 32. These values are also plotted in Figure 9. From this latter Nu
curve, an average value of __
.\/Re, was
Multiplying through by L*/cBC,,
computed for the front portion of the cylinder-Le., from the forward stagnation point to the separation point. From measurements taken on the cylinder (the separation point was visible)
4 AT and substituting for the concentration gradient at the wall,
Except for the constant, 3/2, this result is identical with that obtained by Eckert and Livingood ( 5 ) . The value, 3/2, is a consequence of the form of the equation chosen to represent the concentration profile in the boundary layer. Eckert and Livingood chose a trigonometric representation which gave 53-
976
Equation
the value of - (U6,,) can be obtained
a constant of 2. I t follows that
substituting into rearranging
Actually, Equation 32 has been calculated in such a manner as to give the local Kusselt number for only one
INDUSTRIAL AND ENGINEERING CHEMISTRY
N
0.46
From the data of Schmidt and Wenner (77) f - 0 42 Thus, using the result shown as Equation 34, the calculated mean value of Nu/ Z/=, around the cylinder was 6.8 and the experimental value was 7.4. Agreement between these two results is good. These calculations are shown in detail (8). This solution to the problem is not necessarily the only solution. Many other paths are available-for example. the calculations may be simplified somewhat by assuming that the boundary layer thickness at the forward
FLUID M E C H A N I C S stagnation point is zero. This is similar to Prandtl’s solution for the flat plate. Also, by assuming that both the concentration and velocity profiles in the boundary layer are linear, a solution in a closed form may be obtained. However, such solutions cannot yield results as accurate as that shown. Hence, although examined from curiosity, they are not presented here. Conclusions
I n the past, practically all studies of reactions occurring a t the interface between a solid and liquid phase have been made in systems with poorly defined operating conditions. Different investigators have often disagreed concerning the relative importance of diffusion and surface reaction. The major contribution from this work is thought to be the development of an apparatus which provides well-defined boundary conditions and a well-defined fluid dynamical atmosphere. The use of this apparatus on a relatively simple chemical reaction, dissolution of zinc in dilute acetic acid solutions, has been demonstrated. More specifically, the results and conclusions are :
1. The water tunnel described here is a better device for studying liquid-solid interfacial reactions over a wide range of Reynolds numbers, or any other criterion employed to describe the degree of agitation, than any apparatus heretofore used. 2. For all but the slowest chemical reactions, a set of conditions exists, under which the over-all reaction is controlled by diffusion through a thin boundary layer of fluid a t the solid surface. 3. Boundary layer analysis following classical lines is applicable to this process. 4. The over-all process should be controlled by surface reaction at sufficiently high Reynolds numbers, turbulence intensities, or reactions whose surface rate occurs with high energies of activation. None of these conditions was found in this study. Nomenclature
AF = frontal surface area AB
AT C
= back surface area = total surface area = concentration of component in
fluid diffusion coefficient of component in fluid, sq. cm. per second D = diameter of zinc cylinder .f = fractional transfer contribution from back of cylinder B = friction factor G = effective length of zinc cylinder H = grams of zinc dissolved in time, t J D = Chilton-Colburn mass transfer number k = chemical reaction rate constant ~ L F= frontal transfer coefficient a,
=
~ L B=
ka k, kL
L M n
P r
R T t u
back transfer coefficient
= total transfer coefficient = observed (over-all) reaction rate
constant, cm. per second diffusion rate constant = characteristic length; equals diameter = mesh size of screen = a constant = fluid pressure = rate = radius of zinc rod = fluid temperature = time of run = x component of velocity inside boundary layer x component of potential velocity field = velocity of fluid far upstream = y component of velocity in boundary layer milligrams of zinc dissolved per hour per sq. cm. of surface = distance along cylinder in flow direction ‘ = distance normal to cylinder surface = distance along cylinder normal to flow direction = fluid density = fluid viscosity coefficient = fluid kinematic viscosity coefficient = momentum boundary layer thickness = displacement thickness = momentum thickness = thermal boundary layer thickness = concentration boundary layer thickness = convection thickness = diffusion layer thickness, centimeters = dimensionless distance defined by Equation 22 = Reynolds number based on cylinder diameter and upstream velocity = Nusselt number = Schmidt number = Stanton number = Prandtl number = energy of activation =
u = u, v
w = x
Y 2
P f i
v
6 6d
6i 61
6, 6cc
6*
v Re Nu NsO
N.,
Ni; AI3
Superscripts
*
= derivative = dimensional quantity
Subscripts A = on concentration C, denotes HAC Ac- = on concentration, C,denotes ace-
B 9
k
K
= = = =
tate ion concentration in bulk difftxiion chemical reaction on concentration, C, denotes
KNO, m w m
= observed = concentration at wall = value far upstream
Acknowledgment
The authors gratefully acknowledge the support of this work by a grant from
the National Science Foundation. They are also indebted to Stanley Corrsin for his many suggestions relative to the construction of the tunnel and the boundary layer solutions, to Robert Sparks and William H. Schwarz for their continued interest and assistance, and to Bernard Baker for his substantial contribution in the construction of the tunnel. Literature Cited
(1) Amis, E. S., “Kinetics of Chemical Change in Solution,” p. 309, Macmillan, New York, 1949. (2) Baines, W. D., Peterson, E. G., Trans. Am. Soc. Mech. Engrs. 73,467 (1951 ). (3) Bircumshaw, L. L., Riddiford, A. C., Quart. Revs. Chem. SOC. 6, 157 (1952). (4) Chilton, T. H., Colburn, A. P., IND. ENC.CHEM.26, 1183 (1934). (5) Eckert, E. R. G., Livingood, J. N. B., Natl. Advisory Comm. Aeronaut. Rept. and Tech. Mem. E51F22, 1951. (6) Frank-KamanetskiI, D. A., “Diffusion and Heat Exchange in Chemical Kinetics,” p. 166, Princeton Univ. Press, 1955. (7) Glasstone, S., Laidler, K. J., Eyring, H., “Theory of Rate Processes,” p. 555, McGraw-Hill, New York, 1941. (8) Kiser, K. M., “Chemical Reactions in a Water Tunnel,” Dr. Eng. dissertation, Johns Hopkins University, 1956. (9) Krober, G., Natl. Advisory Comm. Aeronaut.. Tech. Mem. 722, 1933. (10) Levich, M., .Discussions Faraday‘Soc. 1, 37 (1947). (11) Maisel, D. S., Sherwood, T. K., Chem. Eng. Progr. 46, 131 (1950). (12) Moelwyn-Hughes, E. A,, “Kinetics of Reactions in Solution,” 2nd ed., p. 365, Clarendon Press, 1947. (13) Pankhurst, R. C., Holder, D. W., “Wind Tunnel Technique,” p. 36, Putnam, London, 1952. (14) Riddiford, A. C., Bircumshaw, L. L., J . Chem. SOC. (London) 1, 701 (1952). (15) Ripken, J. F., Ser. B, Tech. Paper 9, St. Anthony Falls Hydraulic Laboratory, Univ. of Minnesota, Minneapolis, Minn., 1951. (16) Schlichting, H., “Boundary Layer Theorv,” P. 94, McGraw-Hill, New York, ‘1955. . (17) Schmidt, E., Wenner, K., Natl. Advisorv Comm. Aeronaut.. Tech. Mem. 1050, 1943. (18) Schubauer, G. B., Natl. Advisory Comm. Aeronaut., Tech. Rept. 524, 1935. (19) Schubauer, G. B., Spangenburg, W. G., Natl. Advisory Comm. Aeronaut., Tech. Note 2001, 1950. (20) Taylor, H. S., Taylor, A. A., “Elementary Physical Chemistry,” p. 334, Van Nostrand, New York, 1942. (21) Von KArmh, T., Trans. Am. SOC. Mech. Engrs. 61, 705 (1939).
RECEIVED for review January 2, 1957 ACCEPTED March 16, 1957 Division of Industrial and Engineering Chemistry, ACS, Symposium on Fluid Mechanics in Chemical Engineering, Lafayette, Ind., December 1956. VOL. 49, NO. 6
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